arab journal of mathematical sciences number 2 arab

Quarto trim size: 174mm x 240mm

Arab

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27 Issu

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e-ISSN 2588-9214p-ISSN 1319-5166

Volume 27 Issue 2 2021

Arab Journal of Mathematical

Sciences

Emerald publishing services

www.emeraldinsight.com/loi/ajms

Volume 27 Issue 2 2021

Number 2

129 Editorial advisory board

130 Further study on the Brück conjecture and some non-linear complex di� erential equations Dilip Chandra Pramanik and Kapil Roy

139 On classifi cation of ( n + 6)-dimensional nilpotent n -Lie algebras of class 2 with n ≥ 4 Mehdi Jamshidi , Farshid Saeedi and Hamid Darabi

151 Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics Tarikul Islam and Armina Akter

171 Generalized cyclic contractions and coincidence points involving a control function on partial metric spaces Sushanta Kumar Mohanta

189 Multistep-type construction of fi xed point for multivalued ρ -quasi-contractive-like maps in modular function spaces Hudson Akewe and Hallowed Olaoluwa

214 Fixed point theorems for Geraghty-type mappings applied to solving nonlinear Volterra-Fredholm integral equations in modular G -metric spaces Godwin Amechi Okeke and Daniel Francis

235 Existence of positive solutions for p-Laplacian systems involving left and right fractional derivatives Samira Ramdane and Assia Guezane-Lakoud

249 A direct computation of a certain family of integrals Lorenzo Fornari , Enrico Laeng and Vittorino Pata

Arab Journal of Mathematical Sciences


The Arab Journal of Mathematical Sciences is the official science journal of the Saudi Association for Mathematical Sciences. It is dedicated to the publication of original and expository papers in pure and applied mathematics, and is reviewed and edited by an international group of scholars.The Arab Journal of Mathematical Sciences will accept submissions in the mainstream areas of pure and applied mathematics, including algebra, analysis, geometry, diff erential equations, and discrete mathematics.

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EDITORIAL BOARD

Dr Rafik AguechKing Saud University, Saudi [email protected]

Dr Fawzi Al-ThukairKing Saud University, Saudi [email protected]

Professor Mongi BlelKing Saud University, Saudi [email protected]

Professor Youssef BoudabbousUniversity of La R�eunion, [email protected]

Professor Souhail ChebbiKing Saud University, Saudi [email protected]

Professor Christian DelhommeUniversity of La R�eunion, [email protected]

Professor Sharief DeshmukhKing Saud University, Saudi [email protected]

Professor Alberto FerreroUniversit�a del Piedmonte Oriental, [email protected]

Professor Filippo GazzolaPolitecnico di Milano, [email protected]

Professor Mohammed GuediriKing Saud University, Saudi [email protected]

Professor Mourad IsmailUniversity of Central Florida, [email protected]

Dr Wissem JedidiKing Saud University, Saudi [email protected]

Professor Abdellatif LaradjiKing Saud University, Saudi [email protected]

Professor Elisabetta MalutaPolitecnico di Milano, [email protected]

Professor Nabil OurimiKing Saud University, Saudi [email protected]

Professor Vicentiu RadulescuUniversity of Craiova, [email protected]

Professor Bassem SametKing Saud University, Saudi [email protected]

EDITORIAL ADVISORY BOARD

Professor A.M. AbouammohKing Saud University, Saudi [email protected]

Professor M. Al-GwaizKing Saud University, Saudi Arabia

Professor V. AnandamInstitute of Mathematical Science, [email protected]

Professor Vieri BenciUniversity of Pisa, [email protected]

Professor Christian BergUniversity of Copenhagen, [email protected]

Professor Bang-Yen ChenMichigan State University, [email protected]

Professor S. HedayatUniversity of Illinois, [email protected]

Professor S. KabbajKing Fahad University, Saudi [email protected]

Professor Mokhtar KiraneUniversite de la Rochelle, [email protected]

Professor Zuhair NashedUniversity of Central Florida, [email protected]

Professor Maurice PouzetUniversite Claude-Bernard, [email protected]

Professor David YostFederation University, [email protected]

Editorialboards

129

Arab Journal of MathematicalSciences

Vol. 27 No. 2, 2021p. 129

Emerald Publishing Limitede-ISSN: 2588-9214p-ISSN: 1319-5166


Further study on the Br€uckconjecture and some non-linearcomplex differential equations

Dilip Chandra Pramanik and Kapil RoyDepartment of Mathematics, University of North Bengal, Siliguri, India

Abstract

Purpose –The purpose of this current paper is to deal with the study of non-constant entire solutions of somenon-linear complex differential equations in connection to Br€uck conjecture, by using the theory of complexdifferential equation. The results generalize the results due to Pramanik et al.Design/methodology/approach – 39B32, 30D35.Findings – In the current paper, wemainly study the Br€uck conjecture and the variousworks that confirm thisconjecture. In our study we find that the conjecture can be generalized for differential monomials under someadditional conditions and it generalizes some works related to the conjecture. Also we can take the complexnumber a in the conjecture to be a small function. More precisely, we obtain a result which can be restate in thefollowing way: Let f be a non-constant entire function such that σ2ðf Þ < ∞, σ2ðf Þ is not a positive integer andδð0; f Þ > 0. Let M ½f � be a differential monomial of f of degree γM and αðzÞ; βðzÞ∈Sðf Þ be such thatmaxfσðαÞ; σðβÞg < σðf Þ. If M ½f � þ β and f γM − α share the value 0 CM, then

M ½f � þ β

f γM � α¼ c;

where c≠ 0 is a constant.Originality/value – This is an original work of the authors.

Keywords Entire function, Br€uck conjecture, Small function, Differential monomial

Paper type Research paper

1. Introduction and main resultsIn this paper, by meromorphic function we shall always mean a meromorphic function in thecomplex plane. We adopt the standard notations in the Nevanlinna theory of meromorphicfunctions as explained in [1–4]. It will be convenient to let E denote any set of positive realnumbers of finite linear measure, not necessarily the same at each occurrence.

For any non-constant meromorphic function f ðzÞ, we denote by Sðr; f Þ any quantitysatisfying Sðr; f Þ ¼ oðTðr; f ÞÞ as r→∞; r∉E, where Tðr; f Þ is the Nevanlinnacharacteristic function of f. A meromorphic function α is said to be small with respect tof ðzÞ ifTðr; αÞ ¼ Sðr; f Þ. We denote by Sðf Þ the collection of all small functions with respectto f. Clearly ℂ ∪ f∞g⊂ Sðf Þ and Sðf Þ is a field over the set of complex numbers.

AJMS27,2

130

JEL Classification — 39B32, 30D35.© Dilip Chandra Pramanik and Kapil Roy. Published in Arab Journal of Mathematical Sciences.

Published by Emerald Publishing Limited. This article is published under the Creative CommonsAttribution(CCBY4.0) licence. Anyonemay reproduce, distribute, translate and create derivativeworks of this article (forboth commercial and non-commercial purposes), subject to full attribution to the original publication andauthors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode

This researchwork is supportedby theCouncil of Scientific and IndustrialResearch,ExtraMuralResearchDivision, CSIR Complex, Library Avenue, Pusa, New Delhi-110012, India, Under the sanctioned file no. 09/285(0069)/2016-EMR-I.

Authors would like to thank referees for their valuable comments and suggestions.

The current issue and full text archive of this journal is available on Emerald Insight at:

https://www.emerald.com/insight/1319-5166.htm

Received 27 August 2020Revised 28 August 2020Accepted 28 August 2020

Arab Journal of MathematicalSciencesVol. 27 No. 2, 2021pp. 130-138Emerald Publishing Limitede-ISSN: 2588-9214p-ISSN: 1319-5166DOI 10.1108/AJMS-08-2020-0047

For any two non-constant meromorphic functions f and g, and α∈ Sðf Þ ∩ SðgÞ, we saythat f and g share α IM(CM) provided that f − α and g − α have the same zerosignoring(counting) multiplicities.

For any complex number a, the quantity defined by

δða; f Þ ¼ lim infr→∞

mðr; 1f�a

ÞTðr; f Þ ¼ 1� lim sup

r→∞

Nðr; 1f�a

ÞTðr; f Þ

is called the deficiency of a with respect to the function f ðzÞ.We also need the following definitions:

Definition 1.1. Let f ðzÞ be a non-constant entire function, then the order σðf Þ of f ðzÞ isdefined by

σðf Þ ¼ lim supr→þ∞

logTðr; f Þlog r

¼ lim supr→þ∞

log logMðr; f Þlog r

and the lower order μðf Þ of f ðzÞ is defined by

μðf Þ ¼ lim infr→þ∞

logTðr; f Þlog r

¼ lim infr→þ∞

log log Mðr; f Þlog r

:

The type τðf Þ of an entire function f ðzÞwith 0 < σðf Þ ¼ σ < þ∞ is defined by

τðf Þ ¼ lim supr→þ∞

logMðr; f Þrσ

;

where and in the sequel

Mðr; f Þ ¼ maxjzj¼r

jf ðzÞj:

Definition 1.2. Let f be a non-constant meromorphic function. Then the hyper-orderσ2ðf Þ of f ðzÞ is defined as follows:

σ2ðf Þ ¼ lim supr→þ∞

log logTðr; f Þlog r

:

Definition 1.3. Let f be a non-constant meromorphic function. A differential monomialof f is an expression of the form

M ½f � ¼ a0ðzÞf n0�f ð1Þ�n1�

f ð2Þ�n2

. . .�f ðkÞ�nk

; (1)

where n0; n1; n2; . . . ; nk are non-negative integers and a0ðzÞ∈Sðf Þ. The degree of thedifferential monomial is given by γM ¼ n0 þ n1 þ n2 þ . . .þ nk.

Rubel andYang [5] proved that if a non-constant entire function f and its derivative f 0 share twodistinct finite complex numbers CM, then f ≡ f 0. What will be the relation between f and f 0, if anentire function f and its derivative f 0 share one finite complex number CM? Br€uck [6] made aconjecture that if f is a non-constant entire function satisfying σ2ðf Þ< ∞, where σ2ðf Þ is not apositive integer and if f and f 0 share one finite complex number aCM, then f 0 − a ¼ cðf − aÞ forsome finite complex number c≠ 0. Br€uck [6] himself proved the conjecture for a ¼ 0. Br€uck alsoproved that the conjecture is true for a≠ 0 provided that f satisfies the additional assumptionNðr; 1

f 0Þ ¼ Sðr; f Þ and in this case the order restriction on f can be omitted. After that manyresearchers [7–10] have proved the conjecture under different conditions.

Further studyon differential

equations

131

In 2017, Pramanik et al. [11] investigated on the non-constant entire solution of some non-linear complex differential equations related to Br€uck conjecture and proved the followingtheorems:

Theorem 1.1. Let f ðzÞ and αðzÞ be two non-constant entire functions and satisfy0 < σðαÞ ¼ σðf Þ < þ∞ and τðf Þ > τðαÞ. Also, let PðzÞ be a polynomial. If f is anon-constant entire solution of the following differential equation

M ½f � � α ¼ ðf γM � αÞePðzÞ;then PðzÞ is a constant.

Theorem 1.2. Let f ðzÞ and αðzÞ be two non-constant entire functions and satisfy0 < σðαÞ ¼ σðf Þ < þ∞and τðf Þ > τðαÞ.Also, let PðzÞbe a polynomial. If f is a non-constantentire solution of the following differential equation

M ½f � þ βðzÞ � αðzÞ ¼ ðf γM � αðzÞÞePðzÞ;where βðzÞ is an entire function satisfying 0 < σðβÞ ¼ σðf Þ < þ∞ and τðf Þ > τðβÞ, thenPðzÞ is a constant.

Theorem 1.3. Let f ðzÞ and αðzÞ be two non-constant entire functions satisfyingσðαÞ < μðf Þ and PðzÞ be a polynomial. If f is a non-constant entire solution of thefollowing differential equation

M ½f � þ βðzÞ � αðzÞ ¼ ðf γM � αðzÞÞePðzÞ;where βðzÞ is an entire function satisfying σðβÞ < μðf Þ. Then σ2ðf Þ ¼ degP.

Regarding Theorems 1.1–1.3, one can ask the the following

(1) What will happen if PðzÞ is an entire function?

In this paper we answer the question by proving the following theorems:

Theorem 1.4. Let f ðzÞbe a non-constant entire function such that σ2ðf Þ < ∞, σ2ðf Þ is nota positive integer and δð0; f Þ > 0. Let M ½f � be a differential monomial of f of degree γM asdefined in (1), fðzÞ be an entire function and αðzÞ∈ Sðf Þ be such that σðαÞ < σðf Þ. If f is asolution of the following differential equation

M ½f � � αðzÞ ¼ ðf γM � αðzÞÞefðzÞ; (2)

then M ½f �− αðzÞf γM − αðzÞ ¼ c, where c≠ 0 is a constant.

Theorem 1.5. Let f be a non-constant entire function such that σ2ðf Þ < ∞, σ2ðf Þ is not apositive integer and δð0; f Þ > 0. Let M ½f � be a differential monomial of f of degree γM asdefined in (1), fðzÞ be an entire function and αðzÞ; βðzÞ∈ Sðf Þ be such that σðαÞ < σðf Þ andσðβÞ < σðf Þ. If f is a solution of the following differential equation

M ½f � þ βðzÞ ¼ ðf γM � αðzÞÞefðzÞ; (3)

then M ½f �þβðzÞf γM − αðzÞ ¼ c, where c≠ 0 is a constant.

2. Preparatory lemmasIn this section we state some lemmas needed to prove the theorems.

Lemma 2.1. [2] Let f ðzÞ be a transcendental entire function, νðr; f Þ be the central index off ðzÞ. Then there exists a set E ⊂ ð1; þ∞Þ with finite logarithmic measure such that

AJMS27,2

132

r∉ ½0; 1� ∪ E, consider z with jzj ¼ r and jf ðzÞj ¼ Mðr; f Þ, we getf ðjÞðzÞf ðzÞ ¼

�νðr; f Þ

z

�j

ð1þ oð1ÞÞ; for j ∈ N :

Lemma 2.2. [12] Let f ðzÞ be an entire function of finite order σðf Þ ¼ σ < þ∞, and letνðr; f Þ be the central index of f. Then

lim supr→þ∞

log νðr; f Þlog r

¼ σðf Þ:

And if f is a transcendental entire function of hyper order σ2ðf Þ, then

lim supr→þ∞

log log νðr; f Þlog r

¼ σ2ðf Þ:

Lemma 2.3. [13] Let f ðzÞ be a transcendental entire function and let E ⊂ ½1;þ∞Þ be a sethaving finite logarithmic measure. Then there exists fzn ¼ rne

iθng such thatjf ðznÞj ¼ Mðrn; f Þ; θn ∈ ½0; 2πÞ; lim

n→þ∞θn ¼ θ0 ∈ ½0; 2π�; rn ∉E and if 0 < σðf Þ < þ∞, then

for any given ε > 0 and sufficiently large rn,

rσðf Þ−εn < νðrn; f Þ < rσðf Þþεn :

If σðf Þ ¼ þ∞, then for any given large K > 0 and sufficiently large rn,

νðrn; f Þ > rKn :

Lemma 2.4. [2] Let PðzÞ ¼ bn zn þ bn−1 zn−1 þ . . .þ b0 with bn ≠ 0 be a polynomial.

Then for every ε > 0, there exists r0 > 0 such that for all r ¼ jzj > r0 the inequalities

ð1� εÞjbnjrn ≤ jPðzÞj ≤ ð1þ εÞjbnjrn

hold.

Lemma 2.5. [14] Let f ðzÞ and AðzÞ be two entire functions with 0 < σðf Þ ¼ σðAÞ ¼ σ <þ∞; 0 < τðAÞ ¼ τðf Þ < þ∞, then there exists a set E ⊂ ½1; þ∞Þ that has infinitelogarithmic measure such that for all r∈E and a positive number κ > 0, we have

Mðr; AÞMðr; f Þ < expf−κrσg:

Lemma 2.6. [14] Let g : ð0; ∞Þ→ℝ; h : ð0; ∞Þ→ℝ be monotone increasing functionssuch that gðrÞ ≤ hðrÞ outside an exceptional set Ewith finite linear measure, or gðrÞ≤ hðrÞ,r∉ H ∪ ð0; 1�, where H ⊂ ð1; ∞Þ is a set of finite logarithmic measure. Then for any α > 1,there exists r0 such that gðrÞ≤ hðαrÞ for all r≥ r0.

3. Proof of main theoremsIn this section we present the proofs of the main results of the paper.

3.1 Proof of Theorem 1.4We will consider the following two cases:


equations

133

Case I: Let αðzÞ≡ 0. Then

M ½f �f γM

¼ efðzÞ: (4)

Now,

M ½f �f γM

¼a0ðzÞf n0

�f ð1Þ�n1

. . .�f ðkÞ�nk

f n0þn1þ...þnk

¼ a0ðzÞ f ð1Þ

f

!n1 f ð2Þ

f

!n2

. . .

f ðkÞ

f

!nk

:

(5)

From (4) and (5), it follows that

Tðr; efÞ ¼ mðr; efÞ ¼ m

�r;

M ½f �f γM

≤Xki¼1

nim

r;

f ðiÞ

f

!þmðr; a0Þ

¼ OðlogðrTðr; f ÞÞÞ;outside an exceptional set E0 of finite linear measure.

Thus there exists a constant K such that

Tðr; efÞ≤K logðrTðr; f ÞÞ for r∉E0:

By Lemma 2.6 there exists r0 > 0 such that for r≥ r0, we have

Tðr; efÞ≤K logðηrTðηr; f ÞÞ for η > 1: (6)

From (6), we can deduce that σðefÞ≤ σ2ðf Þ < ∞ and hence fðzÞ is a polynomial.Proceeding similarly as in [11], Theorem 3, we obtain that σ2ðf Þ ¼ degf, which is a

contradiction to our assumption that σ2ðf Þ is not a positive integer. Hence fðzÞ is only aconstant.

Case II: Let αðzÞu0 and d ¼ γM . Taking the logarithmic derivative of (2), we get

f0ðzÞ ¼ M 0½f � � α0ðzÞM ½f � � αðzÞ � df d−1f 0 � α0ðzÞ

f d � αðzÞ : (7)

Subcase I: Let f0ðzÞ≡ 0. Then fðzÞ ¼ c1; c1 is a constant.Subcase II: Let f0ðzÞu0. Then it follows from (7) that

mðr; f0Þ ¼ Sðr; f Þ: (8)

We can rewrite (7) in the following form:

f0 ¼ f d

"M ½f �f d

:1

M ½f �:M 0½f � � α0ðzÞM ½f � � αðzÞ � 1

f ddf d−1f 0 � α0ðzÞ

f d � αðzÞ

#

¼ f d

αðzÞ

"M ½f �f d

:M 0½f � � α0ðzÞM ½f � � αðzÞ �M 0½f �

f d� df d−1f 0 � α0ðzÞ

f d � αðzÞ þ df 0

f

#:

(9)

AJMS27,2

134

We set

ψ ¼ M ½f �f d

:M 0½f � � α0ðzÞM ½f � � αðzÞ �M 0½f �

f d� df d−1f 0 � α0ðzÞ

f d � αðzÞ þ df 0

f: (10)

Then we have

mðr; ψÞ ¼ Sðr; f Þ:Therefore it follows from (9) and (10) that

αðzÞf d

¼ ψðzÞf0ðzÞ : (11)

Since f is an entire function, then we have

m

�r;

1

f d

≤ m

�r;

αðzÞf d

þm

�r;

1

αðzÞ

≤ m�r;

ψðzÞf0ðzÞ

�þ Sðr; f Þ

≤ mðr; ψðzÞÞ þm

�r;

1

f0ðzÞ

þ Sðr; f Þ

¼ Tðr; f0ðzÞÞ þ Sðr; f Þ¼ mðr; f0Þ þ Sðr; f Þ¼ Sðr; f Þ

0m

�r;

1

f

¼ Sðr; f Þ: (12)

It follows from (12) that

δð0; f Þ ¼ lim infr→∞

mðr; 1fÞ

Tðr; f Þ ¼ 0;

which contradicts our hypothesis.Thus the proof is completed.

3.2 Proof of Theorem 1.5We will consider the following two cases:

Case I: Let αðzÞ≡ 0. Then from (3) it follows that

M ½f � þ βðzÞ ¼ f γM efðzÞ

0efðzÞ ¼ M ½f � þ βðzÞf γM

:

Proceeding similarly as in Case I of Theorem 1.4, we can prove that fðzÞ is a constant.Case II: Let αðzÞu 0 and d ¼ γM . Eliminating ef from (3) and its derivative, we get

f0 ¼ M 0½f � þ β0ðzÞM ½f � þ βðzÞ � df d−1f 0 � α0ðzÞ

f d � αðzÞ : (13)

Subcase I: Let f0ðzÞ≡ 0: Then fðzÞ ¼ c2; c2 is a constant.


equations

135

Subcase II: Let f0ðzÞu 0. Then it follows from (13) that

mðr; f0Þ ¼ Sðr; f Þ: (14)

Now,

M 0½f � þ β0ðzÞM ½f � þ βðzÞ ¼ f d

βðzÞM ½f �f d

1

M ½f � �1

M ½f � þ βðzÞ�ðM 0½f � þ β0ðzÞÞ

�

¼ f d

βðzÞM 0½f � þ β0ðzÞ

f d�M ½f �

f d:M 0½f � þ β0ðzÞM ½f � þ βðzÞ

�;

(15)

and

df d−1f 0 � α0ðzÞf d � αðzÞ ¼ f d

αðzÞ

1

f d � αðzÞ �1

f d

��df d−1f 0 � α0ðzÞ

�

¼ f d

αðzÞ

"df d−1f 0 � α0ðzÞ

f d � αðzÞ � df 0

fþ α0ðzÞ

f d

#:

(16)

Therefore from (13), (15) and (16) we have

f0 ¼ f d

βðzÞM 0½f �f d

�M ½f �f d

:M 0½f � þ β0ðzÞM ½f � þ βðzÞ þ β0ðzÞ

f d

�

� f d

αðzÞ



fþ α0ðzÞ

f d

#

¼ f d


�M ½f �f d

:M 0½f � þ β0ðzÞM ½f � þ βðzÞ

�� f d

αðzÞ



f

#

þ β0ðzÞβðzÞ �

α0ðzÞαðzÞ :

0f0 � β0ðzÞβðzÞ þ

α0ðzÞαðzÞ ¼ f d


�M ½f �f d


�

� f d

αðzÞ



f

#:

(17)

Let

ψ 1 ¼M 0½f �f d

�M ½f �f d


and

ψ 2 ¼df d−1f 0 � α0ðzÞ


f:

Then we have mðr; ψ1Þ ¼ Sðr; f Þ and mðr; ψ2Þ ¼ Sðr; f Þ.

AJMS27,2

136

Thus it follows from (17) that

f0 � β0ðzÞβðzÞ þ

α0ðzÞαðzÞ ¼ f d

ψ 1

βðzÞ �ψ 2

αðzÞ �

01

f d¼

ψ 1

βðzÞ �ψ 2

αðzÞ �


α0ðzÞαðzÞ

:

(18)

Since f is an entire function, from (18) we have

m

�r;

1

f d

≤m

�r;

ψ 1

βðzÞ �ψ2

αðzÞ�þm r;

1


α0ðzÞαðzÞ

0BB@

1CCA

≤ mðr; ψ 1Þ þmðr; ψ 2Þ þ Tðr; f0Þ þ Sðr; f Þ¼ Sðr; f Þ

0m

�r;

1

f

¼ Sðr; f Þ: (19)

It follows from (19) that

δð0; f Þ ¼ lim infr→∞

mðr; 1fÞ

Tðr; f Þ ¼ 0;

which is a contradiction.Hence the proof is completed.

Corollary 3.1. Let f ðzÞbe a non-constant entire function such that σ2ðf Þ < ∞, σ2ðf Þ is nota positive integer and δð0; f Þ > 0. Let M ½f � be a differential monomial of f of degree γM asdefined in (1), fðzÞ be an entire function and αðzÞ∈ Sðf Þ be such that σðαÞ < μðf Þ. If f is asolution of the following differential equation

M ½f � � αðzÞ ¼ ðf γM � αðzÞÞefðzÞ;then M ½f �− αðzÞ

f γM − αðzÞ ¼ c, where c≠ 0 is a constant.

Corollary 3.2. Let f be a non-constant entire function such that σ2ðf Þ < ∞, σ2ðf Þ is not apositive integer and δð0; f Þ > 0. Let M ½f � be a differential monomial of f of degree γM asdefined in (1), fðzÞ be an entire function and αðzÞ; βðzÞ∈ Sðf Þ be such that σðαÞ < μðf Þ andσðβÞ < μðf Þ. If f is a solution of the following differential equation

M ½f � þ βðzÞ ¼ ðf γM � αðzÞÞefðzÞ;then M ½f �þβðzÞ

f γM − αðzÞ ¼ c, where c≠ 0 is a constant.

References

[1] Hayman WK, Meromorphic function. Oxford: Clarendon Press; 1964.

[2] Laine I. Nevanlinna theory and complex differential equations. Berlin: Walter de Gruyter; 1993.

[3] Yang L. Value distributions theory. Berlin: Springer-Verlag; 1993.


equations

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[4] Yi HX, Yang CC. Uniqueness theory of meromorphic functions. (in Chinese). Beijing: SciencePress; 1995.

[5] Rubel L, Yang CC. Values shared by an entire function and its derivative. Lecture Notes Math.1977: 599; 101-3.

[6] Br€uck R. On entire functions which share one value CM with their first derivative. Results Math.1996: 30; 21-24.

[7] Al-Khaladi A. On meromorphic functions that share one value with their derivatives. Analysis.2005: 25; 131-140.

[8] Chang JM, Zhu YZ. Entire functions that share a small function with their derivatives. J. Math.Anal. Appl. 2009: 351; 491-96.

[9] Gundersen GG, Yang LZ. Entire functions that share one value with one or two of theirderivatives. J. Math. Anal. Appl. 1998: 223; 85-95.

[10] Li XM, Cao CC. Entire functions sharing one polynomial with their derivatives. Proc. Indian Acad.Sci. Math. Sci. 2008; 118: 13-26.

[11] Pramanik DC, Biswas M, Mandal R. On the study of Br€uck conjecture and some non-linearcomplex differential equations. Arab J. Math. Sci. 2017: 23; 196-204.

[12] He YZ, Xiao XZ. Algebroid functions and ordinary differential equations. Beijing: Sciencepress; 1998.

[13] Mao ZQ. Uniqueness theorems on entire functions and their linear differential polynomials.Results Math. 2009; 55: 447-56.

[14] Wang H, Yang L-Z, Xu H-Y. On some complex differential and difference equations concerningsharing function. Adv. Differ. Equ. 2014: 274; 1-10.

Corresponding authorDilip Chandra Pramanik can be contacted at: [email protected]

For instructions on how to order reprints of this article, please visit our website:www.emeraldgrouppublishing.com/licensing/reprints.htmOr contact us for further details: [email protected]

AJMS27,2

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On classification of(n+ 6)-dimensional nilpotent n-Lie

algebras of class 2 with n ≥ 4Mehdi Jamshidi and Farshid Saeedi

Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad,Iran, and

Hamid DarabiEsfarayen University of Technology, Esfarayen, Iran

Abstract

Purpose – The purpose of this paper is to determine the structure of nilpotent ðnþ 6Þ-dimensional n-Liealgebras of class 2 when n≥ 4.Design/methodology/approach – By dividing a nilpotent ðnþ 6Þ-dimensional n-Lie algebra of class 2 by acentral element, the authors arrive to a nilpotent ðnþ 5Þ dimensional n-Lie algebra of class 2. Given that theauthors have the structure of nilpotent ðnþ 5Þ-dimensional n-Lie algebras of class 2, the authors have access tothe structure of the desired algebras.Findings – In this paper, for each n≥ 4, the authors have found 24 nilpotent ðnþ 6Þ dimensional n-Liealgebras of class 2. Of these, 15 are non-split algebras and the nine remaining algebras are written as directadditions of n-Lie algebras of low-dimension and abelian n-Lie algebras.Originality/value –This classification of n-Lie algebras provides a complete understanding of these algebrasthat are used in algebraic studies.

Keywords Nilpotent n-lie algebra, Classification, Nilpotent n-Lie algebra of class 2


1. IntroductionIn 1985, Filippov [1] introduced the concept of n-Lie (Filippov) algebras, as an n-arymultilinearand skew-symmetric operation ½x1; . . . ; xn�, which satisfies the following generalized Jacobiidentity

½½x1; . . . ; xn�; y2; . . . ; yn� ¼Xn

i¼1

½x1; . . . ; ½xi; y2; . . . ; yn�; . . . ; xn��:

Clearly, such an algebra becomes an ordinary Lie algebra when n ¼ 2. Beside presentingmany examples of n-Lie algebras, he also extended the notions of simplicity and nilpotencyand determined all ðnþ 1Þ-dimensional n-Lie algebras over an algebraically closed field ofcharacteristic zero.

The study of n-Lie algebras is important, since it is related to geometry and physics.Among other results, n-Lie algebras are classified in some cases. For example, Bai et al. [2]classified all n-Lie algebras of dimension nþ 1 over a field of characteristic 2. Also, theyshowed that there is no simple n-Lie algebra of dimension nþ 2. Then, Bai et al. [3] classified

Classificationof nilpotent

n-Lie algebras

139

JEL Classification — Primary 17B05, 17B30, Secondary 17D99© Mehdi Jamshidi, Farshid Saeedi and Hamid Darabi. Published in Arab Journal of Mathematical

Sciences. Published by Emerald Publishing Limited. This article is published under the CreativeCommons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and createderivative works of this article (for both commercial and non-commercial purposes), subject to fullattribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode



Received 24 September 2020Revised 26 September 2020Accepted 26 September 2020


Vol. 27 No. 2, 2021pp. 139-150


DOI 10.1108/AJMS-09-2020-0075

n-Lie algebras of dimension nþ 2 on the algebraically closed fields with characteristic zero.(see [4–7] for more information on the Filippov algebras).

In 1986, Kasymov [8] studied some properties of nilpotent and solvable n-Lie algebras. Ann-Lie algebra A is nilpotent if As ¼ 0 for some nonnegative integer s, where Ai is definedinductively byA1 ¼ AandAiþ1 ¼ ½Ai; A; . . . ; A�. The n-Lie algebraA is nilpotent of class c,if Acþ1 ¼ 0 and Ai ¼ 0 for each i≤ c. The ideal A2 ¼ ½A; . . . ; A� is called the derivedsubalgebra of A. The center of A is defined by

ZðAÞ ¼ fx∈A : ½x; A; . . . ; A� ¼ 0g:Let Z0ðAÞ ¼ h0i. Then the ith center of A is defined inductively by

ZiðAÞZi−1ðAÞ ¼ Z

�A

Zi−1ðAÞ�

for all i≥ 1. Clearly, Z1ðAÞ ¼ ZðAÞ.The nilpotent theories of many algebras attract more and more attention. For example, in

[9,10], and [11], the authors studied nilpotent Leibniz n-algebras, nilpotent Lie and Leibnizalgebras, and nilpotent n-Lie algebras, respectively.

The ðnþ 3Þ-dimensional nilpotent n-Lie algebras and ðnþ 4Þ-dimensional nilpotent n-Liealgebras of class 2 were classified in [12]. Hoseini et al. [13] classified ðnþ 5Þ-dimensionalnilpotent n-Lie algebras of class 2.

In this paper, we have interest for algebras of class 2 (the minimal class for nonabeliancase). The concept of filiform n-Lie algebras (maximal class) has been studied in some papers.For example, see [14].

The rest of our paper is organized as follows: Section 2 includes the results that are usedfrequently in the last section. In Section 3, we classify ðnþ 6Þ-dimensional n-Lie algebras ofclass 2 when n≥ 4. For the case n ¼ 2, this problem is dealt with by Yan et al. [15]. Also, thecase n ¼ 3 stated in [16].

2. PreliminariesIn this section, we introduce some known and necessary results. We denote d-dimensionalabelian n-Lie algebra byFðdÞ. An important category of n-Lie algebras of class 2, which playsan essential role in classification of nilpotent n-Lie algebras, are algebras whose derived andcenter are equal. We call an n-Lie algebraA, a generalized Heisenberg of rank k, ifA2 ¼ ZðAÞand dimA2 ¼ k. The particular case k ¼ 1, is called the special Heisenberg n-Lie algebras.The structure of this algebras defined as follows.

Theorem2.1. [17] Every special Heisenberg n-Lie algebra has dimensionmnþ 1 for somenatural number m, and it is isomorphic to

Hðn; mÞ ¼ hx; x1; . . . ; xnm : ½xnði−1Þþ1; xnði−1Þþ2; . . . ; xni� ¼ x; i ¼ 1; . . . ; mi:

Theorem 2.2. [18] Let A be a d-dimensional nilpotent n-Lie algebra, and let dimA2 ¼ 1.Then, for some m≥ 1, it follows that

A≅Hðn; mÞ⊕Fðd �mn� 1Þ:

Theorem 2.3. [18] Let A be a nonabelian nilpotent n-Lie algebra of dimension d≤ nþ 2.Then A is isomorphic to Hðn; 1Þ; Hðn; 1Þ⊕Fð1Þ or An;nþ2;1, whereAn;nþ2;1 ¼ he1; . . . ; enþ2 : ½e1; . . . ; en� ¼ enþ1; ½e2; . . . ; enþ1� ¼ enþ2i.

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For unification of notation in what follows, the tth d-dimensional n-Lie algebra is denotedby An;d;t.

Theorem 2.4. [12] The ðnþ 3Þ-dimensional nonabelian nilpotent n-Lie algebras for n > 2over an arbitrary field are An;nþ3;ið2≤ i≤ 8Þ: Moreover nilpotent classes of An;nþ3;2 andAn;nþ3;5 is two, nilpotent classes ofAn;nþ3;3; An;nþ3;4 andAn;nþ3;8 is three and finally, nilpotentclasses of An;nþ3;6 and An;nþ3;7 is four (maximal class).

Theorem 2.5. [12] The only ðnþ 4Þ-dimensional nilpotent n-Lie algebras of class 2 areHðn; 1Þ⊕Fð3Þ; An;nþ4;1; An;nþ4;2; An;nþ4;3; Hð2; 2Þ⊕Fð1Þ; Hð3; 2Þ; L6;22ðεÞ; and L2

6;7ðηÞ:Theorem 2.6. [13] The ðnþ 5Þ-dimensional nilpotent n-Lie algebras of class 2 forn > 2 over an arbitrary field are Hðn; 1Þ⊕Fð4Þ; An;nþ5;ið1≤ i≤ 7Þ; Hð3; 2Þ⊕Fð1Þand Hð4; 2Þ:Theorem 2.7. [19] Let A be a nilpotent n-Lie algebra of class 2. Then, there exista generalized Heisenberg n-Lie algebra H and an abelian n-Lie algebra F suchthat A ¼ H ⊕F.

3. Main resultsIn this section, we classify ðnþ 6Þ-dimensional nilpotent n-Lie algebras of class 2. If n-Liealgebra A is nilpotent of class 2, then A is nonabelian and A2 ⊆ ZðAÞ. The nilpotent n-Liealgebra of class 2 plays an essential role in some geometry problems such as the commutativeRiemannian manifold. Additionally, the classification of nilpotent Lie algebras of class 2 isone of the most important issues in Lie algebras.

The following theorems define the structure of generalized Heisenberg n-Lie algebras ofrank 2 with dimension at most 2nþ 3.

Theorem 3.1. [18] Let A be a nilpotent n-Lie algebra of dimension d ¼ nþ k for3≤ k≤ nþ 1 such that A2 ¼ ZðAÞ and dimA2 ¼ 2. Then

A≅ he1; . . . ; enþk : ½ek−1; . . . ; enþk−2� ¼ enþk; ½e1; . . . ; en� ¼ enþk−1i:Remark. In the above theorem for n ¼ 2 and k ¼ 3, we obtain

A≅ he1; e2; e3; e4; e5 : ½e1; e2� ¼ e4; ½e2; e3� ¼ e5i:This algebra appears many times in differential geometry in the study of Pfaffian systems. Itwas developed by P. Libermann and introduced in [20].

Theorem 3.2. [19] Let A be a generalized Heisenberg n-Lie algebra of rank 2 withdimension 2nþ 2. Then

A≅An;2nþ2;1 ¼ he1; . . . ; e2nþ2 : ½e1; . . . ; en� ¼ e2nþ1; ½enþ1; . . . ; e2n� ¼ e2nþ2i:

Theorem 3.3. [19] Let A be a generalized Heisenberg n-Lie algebras of rank 2 withdimension 2nþ 3. Then, A is isomorphic to one of the following n-Lie algebras:

An;2nþ3;1 ¼ he1; . . . ; e2nþ3 : ½e1; . . . ; en� ¼ e2nþ3; ½e2; . . . ; enþ1� ¼ ½enþ2; . . . ; e2nþ1� ¼ e2nþ2i:

An;2nþ3;2 ¼ he1; . . . ; e2nþ3 : ½e1; . . . ; en� ¼ ½enþ1; . . . ; e2n� ¼ e2nþ3; ½e2; . . . ; enþ1�¼ ½enþ2; . . . ; e2nþ1� ¼ e2nþ2i:

For n ¼ 2, we obtain also a Lie algebra of the previous type.


n-Lie algebras

141

Now we are going to classify ðnþ 6Þ-dimensional nilpotent n-Lie algebras of class 2.According to Theorem 2.7, we can writeA ¼ H ⊕F, whereH is a generalized Heisenberg

n-Lie algebra of rank 2 and F is abelian. Therefore, first we classify the generalizedHeisenberg n-Lie algebra of rank 2.

By the classification of nilpotent n-Lie algebras of class 2, we have the followingtheorem. All the algebras defined in theorem 3.4 and follow are in Table 1 at the end ofthe paper.

Theorem 3.4.

(1) The only ðnþ 4Þ-dimensional generalized Heisenberg n-Lie algebra of rank 3 isAn;nþ4;3.

(2) The only ðnþ 5Þ-dimensional generalized Heisenberg n-Lie algebras of rank 3 areAn;nþ5;4 and An;nþ5;5.

(3) The only ðnþ 5Þ-dimensional generalized Heisenberg n-Lie algebra of rank 4 isAn;nþ5;6.

The following lemma defines the structure of ðnþ 6Þ-dimensional generalized Heisenberg n-Lie algebras of rank 2.

Theorem 3.5. Let A be a generalized Heisenberg n-Lie algebra of rank 2 with dimensionnþ 6. Then

A≅An;nþ6;1 ¼ he1; . . . ; enþ6 : ½e1; . . . ; en� ¼ enþ5; ½e5; . . . ; enþ4� ¼ enþ6i:

Proof. For n ¼ 4, we have nþ 6 ¼ 2nþ 2. Thus by Theorem 3.2, if n≥ 5, thennþ 3 < nþ 6≤ 2nþ 1. Applying Theorem 3.1 completes the proof. ▪

Theorem 3.6. The only ðnþ 6Þ-dimensional generalized Heisenberg n-Lie algebras ofrank 3 are

An;nþ6;2; An;nþ6;3; An;nþ6;4; An;nþ6;5; and An;nþ6;6ðεÞ:

Nilpotent n-Lie algebras of class 2 Nonzero multiplications

An;nþ4;1 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4

An;nþ4;2 ½e1; . . . ; en� ¼ enþ3; ½e3; . . . ; enþ2� ¼ enþ4 ðn≥ 3ÞAn;nþ4;3 ½e1; . . . ; en� ¼ enþ1; ½e2; . . . ; en; enþ2� ¼ enþ3;

½e1; e3; . . . ; en; enþ2� ¼ enþ4


An;nþ5;2 ½e1; . . . ; en� ¼ enþ4; ½e3; . . . ; enþ2� ¼ enþ5 ðn≥ 3ÞAn;nþ5;3 ½e1; . . . ; en� ¼ enþ5; ½e4; . . . ; enþ3� ¼ enþ4 ðn≥ 3ÞAn;nþ5;4 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;

½e1; e3; . . . ; enþ1� ¼ enþ5

An;nþ5;5 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e2; . . . ; en; enþ2� ¼ enþ5

An;nþ5;6 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e3; . . . ; enþ2� ¼ enþ5

An;nþ5;7 ½e1; . . . ; en� ¼ enþ1; ½e1; e2; e4; . . . ; en; enþ2� ¼ enþ5;½e1; e3; . . . ; en; enþ2� ¼ enþ4; ½e2; . . . ; en; enþ2� ¼ enþ3

Table 1.

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Proof. Suppose thatA is an ðnþ 6Þ-dimensional generalized Heisenberg n-Lie algebra of rank3 with basis fe1; . . . ; enþ6g, which n≥ 4. Also, suppose that A2 ¼ henþ4; enþ5; enþ6i. In thiscase, A=henþ6i is an ðnþ 5Þ-dimensional nilpotent n-Lie algebra of class 2 with derivedalgebra of dimension 2. By Theorem 2.6, we have three possibilities for A=henþ6i: Case 1: LetA=henþ6i≅An;nþ5;1. Then the brackets in A can be written as8>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>:

½e1; . . . ; en� ¼ enþ4 þ αenþ6;

½e2; . . . ; enþ1� ¼ enþ5 þ βenþ6;

½e1; . . . ; bei; . . . ; en; enþ1� ¼ αienþ6; 2≤ i≤ n;

½e1; . . . ; bei; . . . ; en; enþ2� ¼ βienþ6; 1≤ i≤ n;

½e1; . . . ; bei; . . . ; en; enþ3� ¼ γienþ6; 1≤ i≤ n;

½e1; . . . ; bei; . . . ; bej; . . . ; en; enþ1; enþ2� ¼ χ ijenþ6; 1≤ i < j≤ n;

½e1; . . . ; bei; . . . ; bej; . . . ; en; enþ1; enþ3� ¼ δijenþ6; 1≤ i < j≤ n;

½e1; . . . ; bei; . . . ; bej; . . . ; en; enþ2; enþ3� ¼ λijenþ6; 1≤ i < j≤ n;

½e1; . . . ; bei; . . . ; bej; . . . ; bek; . . . ; en; enþ1; enþ2; enþ3� ¼ fijkenþ6; 1≤ i < j < k≤ n:

Regarding a suitable change of basis, one can assume that α ¼ β ¼ 0.Since dimðA=henþ4; enþ5iÞ2 ¼ 1, we haveA=henþ4; enþ5i≅Hðn; 1Þ⊕Fð3Þ. According to

the structure of n-Lie algebras, we conclude that one of the coefficients

λijð1≤ i < j≤ nÞ; fijkð1≤ i < j < k≤ nÞ

is equal to one, and the others are zero. We have four possibilities:

(1) λ12 ¼ 1; λij ¼ 0ð1≤ i < j≤ n; ði; jÞ≠ ð1; 2ÞÞ; and fijk ¼ 0ð1≤ i < j < k≤ nÞ. In thiscase, the brackets in A can be written as

½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e3; . . . ; en; enþ2; enþ3� ¼ enþ6;

which we denote it by An;nþ6;2.

(2) λ23 ¼ 1; λij ¼ 0ð1≤ i < j≤ n; ði; jÞ≠ ð2; 3ÞÞ; and fijk ¼ 0ð1≤ i < j < k≤ nÞ. In thiscase, the brackets in A can be written as

½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e1; e4; . . . ; en; enþ2; enþ3� ¼ enþ6;


(3) λij ¼ 0ð1≤ i < j≤ nÞ; f123 ¼ 1; andfijk ¼ 0ð1≤ i < j < k≤ n; ði; j; kÞ≠ ð1; 2; 3ÞÞ.In this case, the brackets in A can be written as

½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e4; . . . ; enþ3� ¼ enþ6:

One can easily see that this algebra is isomorphic to Anþ6;3.

(4) λij ¼ 0ð1≤ i < j≤ nÞ;f234 ¼ 1;fijk ¼ 0ð1≤ i < j < k≤ nÞ; ði; j; kÞ≠ ð2; 3; 4ÞÞ. In thiscase, the brackets in A can be written as

½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e1; e5; . . . ; enþ3� ¼ enþ6;

which we denote it by An;nþ6;4:


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143

Case 2. Let A=henþ6i≅An;nþ5;2. Then the brackets in A can be written as8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

½e1; . . . ; en� ¼ enþ4 þ αenþ6;½e2; . . . ; enþ1� ¼ enþ5 þ βenþ6;½e1; . . . ;bei; . . . ; en; enþ1� ¼ αienþ6; 1≤ i≤ n;½e1; . . . ;bei; . . . ; en; enþ2� ¼ βienþ6; 1≤ i≤ n;½e1; . . . ;bei; . . . ; en; enþ3� ¼ γienþ6; 1≤ i≤ n;½e1; . . . ;bei; . . . ;bej; . . . ; en; enþ1; enþ2� ¼ χ ijenþ6; 1≤ i < j≤ n;

ði; jÞ≠ ð1; 2Þ;½e1; . . . ;bei; . . . ;bej; . . . ; en; enþ1; enþ3� ¼ δijenþ6; 1≤ i < j≤ n;½e1; . . . ;bei; . . . ;bej; . . . ; en; enþ2; enþ3� ¼ λijenþ6; 1≤ i < j≤ n;½e1; . . . ;bei; . . . ;bej; . . . ;bek; . . . ; en; enþ1; enþ2; enþ3� ¼ fijkenþ6; 1≤ i < j < k≤ n:

Regarding a suitable change of basis, one can assume that α ¼ β ¼ 0.Since dimðA=henþ4; enþ5iÞ2 ¼ 1, we have A=henþ4; enþ5i≅Hðn; 1Þ⊕Fð3Þ. According to

the structure of n-Lie algebras and ZðAÞ ¼ henþ4; enþ5; enþ6i, we conclude that one of thecoefficients

γið1≤ i≤ nÞ; δijð1≤ i < j≤ nÞ;λijð1≤ i < j≤ nÞ; fijkð1≤ i < j < k≤ nÞ

is equal to one, and the others are zero. Similar to case 1, up to isomorphism, we have thefollowing algebras:

½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e2; . . . ; en; enþ3� ¼ enþ6;½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e1; e2; e4; . . . ; en; enþ3� ¼ enþ6;½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e2; e4; . . . ; enþ1; enþ3� ¼ enþ6;½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e1; e2; e5; . . . ; enþ1; enþ3� ¼ enþ6:

One can easily see that the first and second algebras are isomorphic to An;nþ6;2 and An;nþ6;3,respectively. The third and fourth algebras are denoted byAn;nþ6;5 andAn;nþ6;6, respectively,that is,

An;nþ6;5 ¼ he1; . . . ; enþ6 : ½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e2; e4; . . . ; enþ1; enþ3� ¼ enþ6i;An;nþ6;6 ¼ he1; . . . ; enþ6 : ½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5; ½e1; e2; e5; . . . ; enþ1; enþ3� ¼ enþ6i;

Case 3. Let A=henþ6i≅An;nþ5;3. Then the brackets in A can be written as8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

½e1; . . . ; en� ¼ enþ4 þ αenþ6;½e2; . . . ; enþ1� ¼ enþ5 þ βenþ6;½e1; . . . ;bei; . . . ; en; enþ1� ¼ αienþ6; 1≤ i≤ n;½e1; . . . ;bei; . . . ; en; enþ2� ¼ βienþ6; 1≤ i≤ n;½e1; . . . ;bei; . . . ; en; enþ3� ¼ γienþ6; 1≤ i≤ n;½e1; . . . ;bei; . . . ;bej; . . . ; en; enþ1; enþ2� ¼ χ ijenþ6; 1≤ i < j≤ n;½e1; . . . ;bei; . . . ;bej; . . . ; en; enþ1; enþ3� ¼ δijenþ6; 1≤ i < j≤ n;½e1; . . . ;bei; . . . ;bej; . . . ; en; enþ2; enþ3� ¼ λijenþ6; 1≤ i < j≤ n;½e1; . . . ;bei; . . . ;bej; . . . ;bek; . . . ; en; enþ1; enþ2; enþ3� ¼ fijkenþ6; 1≤ i < j < k≤ n;

ði; j; kÞ≠ ð1; 2; 3Þ:Regarding a suitable change of basis, one can assume that α ¼ β ¼ 0.

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Since dimðA=henþ4; enþ5iÞ2 ¼ 1, we haveA=henþ4; enþ5i≅Hðn; 1Þ⊕Fð3Þ. According tothe structure of n-Lie algebra, we conclude that one of the coefficients

αi; βi; γið1≤ i≤ nÞ; χ ij; δij; λijð1≤ i < j≤ nÞ;fijkð1≤ i < j < k≤ n; ði; j; kÞ≠ ð1; 2; 3ÞÞ

is equal to one, and the others are zero. Similar to case 1, up to isomorphism, we have thefollowing algebras:

½e1; . . . ; en� ¼ enþ5; ½e4; . . . ; enþ3� ¼ enþ4; ½e2; e3; . . . ; enþ1� ¼ enþ6;½e1; . . . ; en� ¼ enþ5; ½e4; . . . ; enþ3� ¼ enþ4; ½e1; e2; e3; e5; . . . ; enþ1� ¼ enþ6;½e1; . . . ; en� ¼ enþ5; ½e4; . . . ; enþ3� ¼ enþ4; ½e2; e3; e5; . . . ; enþ2� ¼ enþ6:

One can easily see that these algebras are isomorphic to An;nþ6;3; An;nþ6;4 and An;nþ6;6,respectively. Therefore, there is no new algebra in this case. ▪

Theorem 3.7. The only ðnþ 6Þ-dimensional generalized Heisenberg n-Lie algebras ofrank 4 are

An;nþ6;7; An;nþ6;8; An;nþ6;9; An;nþ6;10; An;nþ6;11; An;nþ6;12 and An;nþ6;13:

Proof. Suppose thatA is an ðnþ 6Þ-dimensional generalized Heisenberg n-Lie algebra of rank4 with basis fe1; . . . ; enþ6g, which n≥ 4. Also, suppose that A2 ¼ henþ3; enþ4; enþ5; enþ6i. Inthis case, A=henþ6i is an ðnþ 5Þ-dimensional nilpotent n-Lie algebra of class 2 with derivedalgebra of dimension 3. By Theorem LABEL:?, we have three possibilities for A=henþ6i:Case 4. Let A=henþ6i≅An;nþ5;4. Then the brackets in A can be written as8>>>>>><

>>>>>>:

½e1; . . . ; en� ¼ enþ3 þ αenþ6;½e2; . . . ; enþ1� ¼ enþ4 þ βenþ6;½e1; e3; . . . ; enþ1� ¼ enþ5 þ γenþ6;½e1; . . . ;bei; . . . ; en; enþ1� ¼ αienþ6; 3≤ i≤ n;½e1; . . . ;bei; . . . ; en; enþ2� ¼ βienþ6; 1≤ i≤ n;½e1; . . . ;bei; . . . ;bej; . . . ; en; enþ1; enþ2� ¼ χ ijenþ6; 1≤ i < j≤ n:

Regarding a suitable change of basis, one can assume that α ¼ β ¼ γ ¼ 0.Since dimðA=henþ3; enþ4; enþ5iÞ2 ¼ 1, we have A=henþ3; enþ4; enþ5i≅Hðn; 1Þ⊕Fð2Þ.

According to the structure of n-Lie algebras, we conclude that one of the coefficients

αi ð3≤ i≤ nÞ; βi ð1≤ i≤ nÞ; χ ij ð1≤ i < j≤ nÞ

is equal to one, and the others are zero. According to ZðAÞ ¼ fenþ3; enþ4; enþ5; enþ6g, thecoefficients αið3≤ i≤ nÞ cannot be equal to one. We have three possibilities:

(1) β1 ¼ 1; βi ¼ 0 ð2≤ i≤ nÞ; αi ¼ 0 ð3≤ i≤ nÞ; χij ¼ 0 ð1≤ i < j≤ nÞ.

In this case, the brackets in A can be written as

½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e1; e3; . . . ; enþ1� ¼ enþ5; ½e2; . . . ; en; enþ2� ¼ enþ6;


(2) β3 ¼ 1; βi ¼ 0 ð1≤ i≤ n; n≠ 3Þ; αi ¼ 0 ð3≤ i≤ nÞ; χij ¼ 0 ð1≤ i < j≤ nÞ.


n-Lie algebras

145

In this case, the brackets in A can be written as

½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1�enþ4;½e1; e3; . . . ; enþ1� ¼ enþ5; ½e1; e2; e4; . . . ; en; enþ2� ¼ enþ6;


(3) Only one of χ ijsð1≤ i < j≤ nÞ is equal to one and the others are zero. Up toisomorphism, we have the following algebras:� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;

½e1; e3; . . . ; enþ1� ¼ enþ5; ½e3; . . . ; enþ2� ¼ enþ6;� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e1; e3; . . . ; enþ1� ¼ enþ5; ½e2; e4; . . . ; enþ2� ¼ enþ6;� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e1; e3; . . . ; enþ1� ¼ enþ5; ½e1; e2; e5; . . . ; enþ2� ¼ enþ6:

One can easily see that the first and second algebras are isomorphic to An;nþ6;7 and An;nþ6;8,respectively. The third algebras is denoted by An;nþ6;9.

Case 5. Let A=henþ6i≅An;nþ5;5. Then the brackets in A can be written as8>>>>>><>>>>>>:

½e1; . . . ; en� ¼ enþ3 þ αenþ6;½e2; . . . ; enþ1� ¼ enþ4 þ βenþ6;½e2; . . . ; en; enþ2� ¼ enþ5 þ γenþ6;½e1; . . . ;bei; . . . ; en; enþ1� ¼ αienþ6; 2≤ i≤ n;½e1; . . . ;bei; . . . ; en; enþ2� ¼ βienþ6; 2≤ i≤ n;½e1; . . . ;bei; . . . ;bej; . . . ; en; enþ1; enþ2� ¼ χ ijenþ6; 1≤ i < j≤ n:



αi ð3≤ i≤ nÞ; βi ð1≤ i≤ nÞ; χ ij ð1≤ i < j≤ nÞ

is equal to one, and the others are zero. We have two possibilities:

(1) Only one of αið3≤ i≤ nÞ and βið1≤ i≤ nÞ is equal to one and the others are zero.Without loss of generality, we assume α2 ¼ 1. Thus, the brackets inA can bewritten as

½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e2; . . . ; en; enþ2� ¼ enþ5; ½e1; e3; . . . ; enþ1� ¼ enþ6:

One can easily see that this algebra is isomorphic to An;nþ6;7.

(2) Only one of χ ijsð1≤ i < j≤ nÞ is equal to one and the others are zero. Up toisomorphism, we have the following algebras:� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;

½e2; . . . ; en; enþ2� ¼ enþ5; ½e3; . . . ; enþ2� ¼ enþ6;� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e2; . . . ; en; enþ2� ¼ enþ5; ½e1; e4; . . . ; enþ2� ¼ enþ6:

One can easily see that the first algebra is isomorphic to An;nþ6;7. The second algebra isdenoted by An;nþ6;10.

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Case 6. Let A=henþ6i≅An;nþ5;6. Then the brackets in A can be written as8>>>>>><>>>>>>:

½e1; . . . ; en� ¼ enþ3 þ αenþ6;½e2; . . . ; enþ1� ¼ enþ4 þ βenþ6;½e3; . . . ; enþ2� ¼ enþ5 þ γenþ6;½e1; . . . ;bei; . . . ; en; enþ1� ¼ αienþ6; 2≤ i≤ n;½e1; . . . ;bei; . . . ; en; enþ2� ¼ βienþ6; 1≤ i≤ n;½e1; . . . ;bei; . . . ;bej; . . . ; en; enþ1; enþ2� ¼ χ ijenþ6; 1≤ i < j≤ n; ði; jÞ≠ ð1; 2Þ:



αi ð2≤ i≤ nÞ; βi ð1≤ i≤ nÞ; χ ij ð1≤ i < j≤ n; ði; jÞ≠ ð1; 2ÞÞ

is equal to one, and the others are zero. Up to isomorphism, we have the following algebras:� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e3; . . . ; enþ2� ¼ enþ5; ½e1; e3; . . . ; enþ1� ¼ enþ6;� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e3; . . . ; enþ2� ¼ enþ5; ½e1; e2; e4; . . . ; enþ1� ¼ enþ6;� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e3; . . . ; enþ2� ¼ enþ5; ½e1; e3; . . . ; en; enþ2� ¼ enþ6;� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e3; . . . ; enþ2� ¼ enþ5; ½e1; e2; e4; . . . ; en; enþ2� ¼ enþ6;� ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e3; . . . ; enþ2� ¼ enþ5; ½e1; e2; e5; . . . ; enþ2� ¼ enþ6:

One can easily see that the first and second algebras are isomorphic to An;nþ6;7 and An;nþ6;8,respectively. The third, fourth and fifth algebras are denoted by An;nþ6;11, An;nþ6;12 andAn;nþ6;13, respectively. ∎

Theorem3.8. The only ðnþ 6Þ-dimensional nilpotent n-Lie algebras of class 2where n≥ 4are 8<

:Hðn; 1Þ⊕Fð5Þðn≥ 4Þ; Hð4; 2Þ⊕Fð1Þ; Hð5; 2Þ;An;nþ5;1 ⊕Fð1Þ; An;nþ4;1 ⊕Fð2Þ; An;nþ3;1 ⊕Fð3Þ; An;nþ5;4 ⊕Fð1ÞAn;nþ5;5 ⊕Fð1Þ; An;nþ4;3 ⊕Fð2Þ; An;nþ5;6 ⊕Fð1Þ; and An;nþ6;ið1≤ i≤ 14Þ:

Proof.Assume thatA is an ðnþ 6Þ-dimensional nilpotent n-Lie algebra of class 2, where n≥ 4and A ¼ he1; . . . ; enþ6i. If dimA2 ¼ 1, then by Theorem 2.2, A is isomorphic to one of thefollowing algebras:

Hðn; 1Þ⊕Fð5Þðn≥ 4Þ; Hð4; 2Þ⊕Fð1Þ; Hð5; 2Þ:Now, assume that dimA2 ≥ 2 and that henþ5; enþ6i⊂A2. Therefore, A=henþ6i is anðnþ 5Þ-dimensional nilpotent n-Lie algebra of class 2. It follows from Theorem 2.5 thatA=henþ6i is one of the following forms:

Hðn; 1Þ⊕Fð4Þ; Hð4; 2Þ; An;nþ5;ið1≤ i≤ 7Þ:If A=henþ6i is isomorphic to Hðn; 1Þ⊕Fð4Þ or Hð4; 2Þ, then dimA2 ¼ 2. According toLemma 2.7, we can write A ¼ H ⊕F, where H is a generalized Heisenberg n-Lie algebra of


n-Lie algebras

147

rank 2 and F is abelian. The center ofA has a dimension at most 5; thus the possible cases ofAareH0; H1 ⊕Fð1Þ; H2 ⊕Fð2Þ; H3 ⊕Fð3Þ, whereH0; H1; H2; H3 are generalized Heisenbergn-Lie algebras of rank 2 with dimensions nþ 6; nþ 5; nþ 4; nþ 3; respectively. Thesealgebras read as follows:

An;nþ6;1; An;nþ5;1 ⊕Fð1Þ; An;nþ4;1 ⊕Fð2Þ; An;nþ3;1 ⊕Fð3Þ:If A=henþ6i is isomorphic to An;nþ5;1; An;nþ5;2 or An;nþ5;3, then dimA2 ¼ 3. According toLemma 2.7, we can write A ¼ H ⊕F, where H is a generalized Heisenberg n-Lie algebra ofrank 3 and F is abelian. According to ?, these algebras read as follows:

An;nþ6;2; An;nþ6;3; An;nþ6;4; An;nþ6;5; An;nþ6;6;An;nþ5;4 ⊕Fð1Þ; An;nþ5;5 ⊕Fð1Þ; An;nþ4;3 ⊕Fð2Þ:

Also, IfA=henþ6i is isomorphic toAn;nþ5;4; An;nþ5;5 orAn;nþ5;6, then dimA2 ¼ 4. According toLemma 2.7, we can write A ¼ H ⊕F, where H is a generalized Heisenberg n-Lie algebra ofrank 4 and F is abelian. According to ?, these algebras read as follows:

An;nþ6;7; An;nþ6;8; An;nþ6;9; An;nþ6;10;An;nþ6;11; An;nþ6;12; An;nþ6;13; An;nþ5;6 ⊕Fð1Þ:

Finally, If A=henþ6i≅An;nþ5;7, then A2 ¼ ZðAÞ ¼ henþ1; enþ3; enþ4; enþ5; enþ6i. Thebrackets in A can be written as8>>>><

>>>>:

½e1; . . . ; en� ¼ enþ1 þ αenþ6;½e2; . . . ; en; enþ2� ¼ enþ3 þ βenþ6;½e1; e3; . . . ; en; enþ2� ¼ enþ4 þ γenþ6;½e1; e2; e4; . . . ; en; enþ2� ¼ enþ5 þ fenþ6;½e1; . . . ; bei; . . . ; en; enþ2� ¼ βienþ6; 4≤ i≤ n:

With a suitable change of basis, one can assume that α ¼ β ¼ γ ¼ f ¼ 0. Thus, the bracketsin A are 8>>>><

>>>>:

½e1; . . . ; en� ¼ enþ1;½e2; . . . ; en; enþ2� ¼ enþ3;½e1; e3; . . . ; en; enþ2� ¼ enþ4;½e1; e2; e4; . . . ; en; enþ2� ¼ enþ5;½e1; . . . ; bei; . . . ; en; enþ2� ¼ βienþ6; 4≤ i≤ n:

By dim ZðAÞ, we must have βi ≠ 0 for some 4≤ i≤ n. Without loss of generality, assume thatβ4 ≠ 0. By applying the transformations

e04 ¼ e4 þ

Xn

j¼5

ð−1Þj βiβ4ej; e

0i ¼ ei ð1≤ i≤ nþ 5; i≠ 4Þ; e

0nþ6 ¼ β4enþ6;

we conclude that

A ¼ he1; . . . ; enþ6 : ½e1; . . . ; en� ¼ enþ1; ½e2; . . . ; en; enþ2� ¼ enþ3;

¼ enþ4; ½e1; e2; e4; . . . ; en; enþ2� ¼ enþ5;¼ enþ6i;which we denote it by An;nþ6;14. ∎

In Table 1, we show all ðnþ 4Þ-dimensional and ðnþ 5Þ-dimensional nilpotent n-Liealgebras of class 2.

In Table 2, we show all n-Lie algebras obtained in this paper.

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References

1. Filippov VT. n-Lie algebras, Sib. Math. J. 1985; 26: 879-91.

2. Bai R, Wang XL, Xiao WY, An HW. Structure of low dimensional n-Lie algebras over a field ofcharacteristic 2, Linear Algebra Appl. 2008; 428(8-9): 1912-20.

3. Bai R, Song G, Zhang Y. On classification of n-Lie algebras, Front. Math. China. 2011; 6(4):581-606.

4. de Azc�arraga J, Izquierdo JM, Pic�on M. Contractions of filippov algebras. J. Math. Phys. 2011;52(1): 24, 013516.

5. Darabi H and Saeedi F. On the Schur multiplier of n-Lie algebras, J. Lie Theory. 2017; 27: 271-81.

6. Furuuchi K, Tomino D, Supersymmetric reduced models with a symmetry based on filippovalgebra, J. High Energy Phys. 2009; 5(70): 17.

7. Grabowski J, Marmo G. On Filippov algebroids and multiplicative Nambu-Poisson structures.Differential Geom. Appl. 2000; 12: 35-50.

8. Kasymov SM. Theory of n-Lie algebras, Algebra Logic. 1987; 26: 155-66.

9. Camacho LM, Casas JM, Gomez JR, Ladra M and Omirov BA. On nilpotent Leibniz n-algebras, J.Algebra Appl. 11(3) 2012: 1250062(17).

10. Ray CB, Combs A, Gin N, Hedges A, Hird JT, Zack L. Nilpotent Lie and Leibniz algebras, Comm.Algebra. 2014; 42: 2404-10.

11. Williams MP. Nilpotent n-Lie algebras, Comm. Algebra. 2009; 37(6): 1843-49.

Nilpotent n-Lie algebras of class 2 Nonzero multiplications


An;nþ6;2 ½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5;½e3; . . . ; en; enþ2; enþ3� ¼ enþ6

An;nþ6;3 ½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5;½e1; e4; . . . ; en; enþ2; enþ3� ¼ enþ6

An;nþ6;4 ½e1; . . . ; en� ¼ enþ4; ½e2; . . . ; enþ1� ¼ enþ5;½e1; e5; . . . ; enþ3� ¼ enþ6

An;nþ6;5 ½e1; . . . ; en� ¼ enþ4; ½e3; . . . ; enþ2� ¼ enþ5;½e2; e4; . . . ; enþ1; enþ3� ¼ enþ6

An;nþ6;6 ½e1; . . . ; en� ¼ enþ4; ½e3; . . . ; enþ2� ¼ enþ5;½e1; e2; e5; . . . ; enþ1; enþ3� ¼ enþ6

An;nþ6;7 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e1; e3; . . . ; enþ1� ¼ enþ5; ½e3; . . . ; enþ2� ¼ enþ6

An;nþ6;8 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e1; e3; . . . ; enþ1� ¼ enþ5; ½e2; e4; . . . ; enþ2� ¼ enþ6

An;nþ6;9 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e1; e3; . . . ; enþ1� ¼ enþ5; ½e1; e2; e5; . . . ; enþ2� ¼ enþ6

An;nþ6;10 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e2; . . . ; en; enþ2� ¼ enþ5; ½e1; e4; . . . ; enþ2� ¼ enþ6

An;nþ6;11 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e3; . . . ; enþ2� ¼ enþ5; ½e1; e3; . . . ; en; enþ2� ¼ enþ6

An;nþ6;12 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e3; . . . ; enþ2� ¼ enþ5; ½e1; e2; e4; . . . ; en; enþ2� ¼ enþ6

An;nþ6;13 ½e1; . . . ; en� ¼ enþ3; ½e2; . . . ; enþ1� ¼ enþ4;½e3; . . . ; enþ2� ¼ enþ5; ½e1; e2; e5; . . . ; enþ2� ¼ enþ6

An;nþ6;14 ½e1; . . . ; en� ¼ enþ1; ½e2; . . . ; en; enþ2� ¼ enþ3;½e1; e3; . . . ; en; enþ2� ¼ enþ4; ½e3; . . . ; enþ2� ¼ enþ5;

½e1; e3; . . . ; en; enþ2� ¼ enþ6

Table 2.


n-Lie algebras

149

12. Eshrati M, Saeedi F, Darabi H. Low dimensional nilpotent n-Lie algebras, arXiv:1810.03782.

13. Hoseini Z, Saeedi F, Darabi H. On classification of (nþ5)-dimensional nilpotent n-Lie algebras ofclass two, Bull. Iranian Math. Soc. 2019; 45: 939-49.

14. Goze M, Goze N, Remm E. n-Lie algebras. Afr. J. Math. Phys. 2010; 8: 17-28.

15. Yan Z, Deng S. The classification of two step nilpotent complex Lie algebras of dimension 8,Czechoslovak Math. J. 2013; 63(138): 847-63.

16. Darabi H, Imanparast M. On classification of 9-dimensional nilpotent 3-ary algebras of class two.Bull. Iranian Math. Soc. 2020: 1-9.

17. Eshrati M, Saeedi F, Darabi H. On the multiplier of nilpotent n-Lie algebras, J. Algebra. 2016;450: 162-72.

18. Darabi H, Saeedi F, Eshrati M. A characterization of finite dimensional nilpotent Filippovalgebras, J. Geom. Phys. 2016; 101: 100-7.

19. Hoseini Z, Saeedi F, Darabi H. Characterization of capable nilpotent n-Lie algebras of class two bytheir Schur multipliers. Rend. Circ. Mat. Palermo. 2019; 68(2): 541–55.

20. Goze M, Haraguchi Y. Sur les r-systemes de contact. CR Acad. Sci. Paris. 1982; 294: 95-97.

Corresponding authorHamid Darabi can be contacted at: [email protected]


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Further fresh and general travelingwave solutions to some fractional

order nonlinear evolutionequations in mathematical physics

Tarikul Islam and Armina AkterDepartment of Mathematics,

HajeeMohammadDanesh Science andTechnology University, Dinajpur, Bangladesh

Abstract

Purpose – Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractionalderivative are considered to be revealed for well-furnished analytic solutions due to their importance in thenature of real world. In this article, the autors suggest a productive technique, called the rational fractionalðDα

ξG=GÞ-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili(PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinearspace-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformationtechnique is used to convert the considered equations into the fractional order ordinary differential equation.Then the method is employed to make available their solutions. The constructed solutions in terms oftrigonometric function, hyperbolic function and rational function are claimed to be fresh and further general inclosed form. These solutions might play important roles to depict the complex physical phenomena arise inphysics, mathematical physics and engineering.Design/methodology/approach – The rational fractional ðDα

ξG=GÞ-expansion method shows highperformance and might be used as a strong tool to unravel any other FNLEEs. This method is of the

form UðξÞ ¼Pni¼0aiðDα

ξG=GÞi=Pn

i¼0biðDαξG=GÞi.

Findings – Achieved fresh and further abundant closed form traveling wave solutions to analyze the innermechanisms of complex phenomenon in nature world which will bear a significant role in the of research andwill be recorded in the literature.Originality/value – The rational fractional ðDα

ξG=GÞ-expansion method shows high performance and mightbe used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

Keywords The rational fractionalðDαξG=GÞ -expansion method, Complex fractional transformation,

Conformable fractional derivative, Closed form solution, Fractional order nonlinear evolution equation


1. IntroductionFractional calculus originating from some speculations of Leibniz and L’Hospital in 1695 hasgradually gained profound attention of many researchers for its extensive appearance invarious fields of real world. Exact traveling wave solutions to fractional order nonlinearevolution equations (FNLEEs) are of fundamental and important in applied science because oftheir wide use to depict the nonlinear fractional phenomena and dynamical processes of natureworld. The FNLEEs and their solutions in closed form play fundamental role in describing,modeling and predicting the underlying mechanisms related to the biology, bio-genetics,

Solutions toFNLEEs in

mathematicalphysics

151

JEL Classification — 34A08, 35R11© Tarikul Islam and Armina Akter. Published in Arab Journal of Mathematical Sciences. Published

by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CCBY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article(for both commercial and non-commercial purposes), subject to full attribution to the original publicationand authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode



Received 29 September 2020Revised 13 November 2020Accepted 19 November 2020


Vol. 27 No. 2, 2021pp. 151-170


DOI 10.1108/AJMS-09-2020-0078

physics, solid state physics, condensed matter physics, plasma physics, optical fibers,meteorology, oceanic phenomena, chemistry, chemical kinematics, electromagnetic, electricalcircuits, quantum mechanics, polymeric materials, neutron point kinetic model, control andvibration, image and signal processing, system identifications, the finance, acoustics and fluiddynamics [1–3]. The closed form wave solutions of these equations [4–6] are greatly helpful torealize the mechanisms of the complicated nonlinear physical phenomena as well as theirfurther applications in practical life. Some attractive powerful approaches take into account inthe recent research area related to fractional derivative associated problems [7–9]. Therefore, ithas become the core aim in the research area of fractional related problems that how to developa stable approach for investigating the solutions to FNLEEs in analytical or numerical form.Many researchers have offered different approaches to construct analytic and numericalsolutions to FNLEEs as well as integer order and put them forward for searching travelingwave solutions, such as the He-Laplace method [10], the exponential decay law [11], thereproducing kernel method [12], the Jacobi elliptic function method [13], the

�G

0=G�-expansion

method and its various modifications [14–18], the exp-function method [19], the sub-equationmethod [20, 21], the first integral method [22], the functional variable method [23], the modifiedtrial equation method [24], the simplest equation method [25], the Lie group analysis method[26], the fractional characteristic method [27], the auxiliary equation method [28, 29], the finiteelement method [30], the differential transform method [31], the Adomian decompositionmethod [32, 33], the variational iteration method [34], the finite difference method [35], thehomotopy perturbation method [36] and the He’s variational principle [37], the new extendeddirect algebraic method [38, 39], the Jacobi elliptic function expansion method [40], theconformable double Laplace transform [41] etc. But eachmethod does not bear high acceptancefor the lacking of productivity to construct the closed form solutions to all kind of FNLEEs.That is why; it is very much indispensable to establish new techniques.

In this study, we offer a newly established technique, called the rational fractionalðDα

ξG=GÞ-expansion method [42], to investigate closed form analytic wave solutions to some

FNLEEs in the sense of conformable fractional derivative [43]. This effectual and reliableproductive method shows its high performance through providing abundant fresh andgeneral solutions to the suggested equations. The obtained solutions might bring up theirimportance through the contribution to analyze the inner mechanisms of physical complexphenomena of real world and make an acceptable record in the literature.

2. Preliminaries and methodology2.1 Conformable fractional derivativeAnew and simple definition of derivative for fractional order introduced byKhalil et al. [43] iscalled conformable fractional derivative. This definition is analogous to the ordinaryderivative

dψdx

¼ limε→0

ψðxþ εÞ � ψðxÞε

;

where ψðxÞ : ½0; ∞�→R and x > 0. According to this classical definition, dðxnÞdx

¼ nxn−1.According to this perception, Khalil has introduced α order fractional derivative of ψ as

TαψðxÞ ¼ limε→0

ψðxþ ε x1−αÞ � ψðxÞε

; 0 < α≤ 1;

If the function ψ is α differentiable in ð0; rÞ for r > 0 and limx→0þ

TαψðxÞ exists, then the

conformable derivative at x ¼ 0 is defined as Tαψð0Þ ¼ limx→0þ

TαψðxÞ. The conformable

integral of ψ is

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152

I rαψðxÞ ¼Z x

r

ψðtÞt1−α

dt; r≥ 0; 0 < α≤ 1:

This integral represents usual Riemann improper integral.The conformable fractional derivative satisfies the following useful properties [43]:If the functions uðxÞ and vðxÞ are α -differentiable at any point x > 0, for α∈ ð0; 1�, then(1) Tαðauþ bvÞ ¼ aTαðuÞ þ bTαðvÞ ∀ a; b∈R:

(2) TαðxnÞ ¼ n xn−α ∀ n∈R:

(3) TαðcÞ ¼ 0, where c is any constant.

(4) TαðuvÞ ¼ uTαðvÞ þ vTαðuÞ:(5) Tαðu=vÞ ¼ vTαðuÞ− uTαðvÞ

v2:

(6) if u is differentiable, then TαðuÞðxÞ ¼ x1−αdudxðxÞ.

Many researchers used this new derivative of fractional order in physical applications due toits convenience, simplicity and usefulness [44–46].

2.2 MethodologyIn this subsection, we discuss the main steps of the rational fractional ðDα

ξG=GÞ-expansionmethod to examine exact traveling wave solutions to FNLEEs. A fractional partial

differential equation in the independent variables t; x1; x2; . . . ; xn is supposed to be asfollows:

F�u1; . . . uk; D

αt u1; . . . ; D

αt uk; D

βx1u1; . . . ; D

βx1uk; . . .D

βxnu1; . . .D

βxnuk; . . .

�¼ 0 (2.2.1)

where 0 < α; β≤ 1; ui ¼ uiðt; x1; x2 ; . . . ; xnÞ, i ¼ 1; 2; 3; . . . ; k are unknown functions,F is a polynomial in ui and it’s various partial derivatives of fractional order. Maintain thefollowing steps to unravel Eqn (2.2.1) by the rational fractional ðDα

ξG=GÞ-expansion technique.Let us consider the nonlinear fractional composite transformation

ui ¼ uiðt; x1; x2; . . . ; xnÞ ¼ UiðξÞ; ξ ¼ ξðt; x1; x2; . . . ; xnÞ; (2.2.2)

which reduces Eqn (2.2.1) to the following ordinary differential equation of fractional orderwith respect to the variable ξ:

Q�U1; . . . ; Uk; D

αξ U1; . . . ; Dα

ξ Uk; Dβξ U1; . . . ;D

βξ Uk; . . .

� ¼ 0: (2.2.3)

We might take anti-derivative of Eqn (2.2.3) term by term as many times as possible andintegral constant can be set to zero as soliton solutions are sought.

Step 1: Suppose the traveling wave solution of Eqn (2.2.1) can be expressed as follows:

UðξÞ ¼Pni¼0

ai�Dα

ξG=G�i

Pni¼0

bi�Dα

ξG=G�i ; (2.2.4)

where a0i s and bi; s are unknown constants to be determined later and G ¼ GðξÞ satisfies

the following auxiliary nonlinear ordinary differential equation of fractional order:

D2αξ GðξÞ þ λDα

ξ GðξÞ þ μGðξÞ ¼ 0; (2.2.5)


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153

where λ, μ are arbitrary constants and DαξGðξÞ denotes the conformable fractional

derivative of order α for GðξÞwith respect to ξ.The nonlinear fractional complex transformation GðξÞ ¼ HðηÞ, η ¼ ξα=Γð1þ αÞ reducesEqn (2.2.5) into the following second order ordinary differential equation:

H00 ðηÞ þ λH

0 ðηÞ þ μHðηÞ ¼ 0; (2.2.6)

whose solutions are well-known. Since Dαξ GðξÞ ¼ Dα

ξ HðηÞ ¼ H0 ðηÞDα

ξ η ¼ H0 ðηÞ, with the

aid of the solutions of Eqn (2.2.6), we can obtain the solutions of Eqn (2.2.5) as follows:

�Dα

ξG=G�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2�4μ

p2

3

C1 sinh

ffiffiffiffiffiffiffiffiffiλ2�4μ

pξα

2Γð1þαÞ

!þC2 cosh


pξα

2Γð1þαÞ

!

C1 cosh


pξα

2Γð1þαÞ

!þC2 sinh


pξα

2Γð1þαÞ

!�λ

2; λ2�4μ>0 (2.2.7)

�Dα

ξG=G�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi4μ�λ2

p2

3

−C1 sin

ffiffiffiffiffiffiffiffiffi4μ�λ2

pξα

2Γð1þαÞ

!þC2 cos


pξα

2Γð1þαÞ

!

C1 cos


pξα

2Γð1þαÞ

!þC2 sin


pξα

2Γð1þαÞ

! �λ

2; λ2�4μ<0 (2.2.8)

�Dα

ξG=G�¼ C2Γð1þαÞ

C1Γð1þαÞþC2ξα�

λ

2; λ2�4μ¼0 (2.2.9)

where C1 and C2 are arbitrary constants.

Step 2: The positive constant ncan be determined by taking homogenous balance betweenthe highest order linear and nonlinear terms appearing in Eqn (2.2.3).

Step 3: Substitute (2.2.4) and (2.2.5) into Eqn (2.2.3) with the value of nobtained in step 2, weobtain a polynomial in ðDα

ξG=GÞ. Setting each coefficient of the resulted polynomial to zero

gives a set of algebraic equations for a0i s and bi; s by means of the symbolic computation

software, such as Maple, provides the values of constants.

Step 4: Inserting the values of a0i s and bi; s into (2.2.4) along with (2.2.7)–(2.2.9), the closed

form traveling wave solutions to the nonlinear evolution Eqn (2.2.1) are obtained.

3. Formulation of the solutionsIn this section, the exact analytic traveling wave solutions to the nonlinear space-timefractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-timefractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractionalKolmogorov–Petrovskii–Piskunov (KPP) equation are constructed.

3.1 The nonlinear space-time fractional PKP equationThis well-known equation is given as

1

4D4α

x uþ 3

2Dα

x uD2αx uþ 3

4D2α

y uþ Dαt

�Dα

x u� ¼ 0: (3.1.1)

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154

With the aid of the fractional compound transformation

uðx; y; tÞ ¼ UðξÞ; ξ ¼ xþ yþ c1=α t (3.1.2)

Eqn (3.1.1) is turned into the following ordinary differential equations of fractional order dueto the variable ξ:

1

4D4α

ξ U þ 3

2Dα

ξ U D2αξ U þ 3

4D2α

ξ U þ cD2αξ U ¼ 0 (3.1.3)

Taking anti-derivative of (3.1.3) yields

D3αξ U þ 3

�Dα

ξ U�2 þ ð3þ 4cÞDα

ξ U ¼ 0 (3.1.4)

Considering the homogenous balance to Eqn (3.1.4), the solution (2.2.4) becomes

UðξÞ ¼ a0 þ a1DαξG=G

b0 þ b1DαξG=G

(3.1.5)

Eqn (3.1.4) together with (3.1.5) and (2.2.5) becomes a polynomial in ðDαξG=GÞ equating whose

coefficients to zero and solving provides the following outcomes:

set 1: $a0 ¼ 1

b1

�a1 b0 � 2b21μþ 2b0 b1λ� 2b20

�; c ¼ 1

4

�4μ� λ2 � 3

�; (3.1.6)

where a1; b0; b1; λ and μ are free parameters.

set 2: a1 ¼ 2b0; b1 ¼ 0; c ¼ 1

4

�4μ� λ2 � 3

�; (3.1.7)

where a0; b0; λ and μ are free parameters.Insert the values appeared in (3.1.6) and (3.1.7) in the solution (3.1.5) provide the following

expressions for exact analytic solutions:

U1ðξÞ ¼�a1 b0 � 2b21μþ 2b0 b1λ� 2b20

�þ a1DαξG=G

b1�b0 þ b1 D

αξG=G

� ; (3.1.8)

U2ðξÞ ¼ a0

b0þ 2Dα

ξG=G; (3.1.9)

where ξ ¼ xþ yþ fð4μ− λ2 − 3Þ=4g1=αt.The expressions (3.1.8) and (3.1.9) along with (2.2.7)–(2.2.9) make available the following

closed form traveling wave solutions in terms of hyperbolic function, trigonometric functionand rational function:

3.1.1 Solution 1. When λ2 − 4μ > 0,

U 11 ðξÞ¼

�a1b0�2b21μþ2b0b1λ�2b20

�þa1

0BB@


p2

3

C1sin h

� ffiffiffiffiffiffiffiλ2�4μ

pξα

2Γð1þαÞ

�þC2cos h


pξα

2Γð1þαÞ

�

C1cos h


pξα

2Γð1þαÞ

�þC2sin h


pξα

2Γð1þαÞ

�� λ2

1CCA

b1

0BB@b0þb1

0BB@


p2

3

C1sin h


pξα

2Γð1þαÞ

�þC2cos h


pξα

2Γð1þαÞ

�

C1cos h


pξα

2Γð1þαÞ

�þC2sin h


pξα

2Γð1þαÞ

�� λ2

1CCA1CCA

(3.1.10)


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155

Choose c1 ≠ 0; c2 ¼ 0, then (3.1.10) becomes

U 11 ðξÞ ¼

�a1b0 � 2b21μþ 2b0b1λ� 2b20

�þ a1

� ffiffiffiffiffiffiffiffiffiλ2�4μ

p2

3tan h


pξα

2Γð1þαÞ

�� λ

2

�

b1

�b0 þ b1


p2

3tan h


pξα

2Γð1þαÞ

�� λ

2

�� ; (3.1.11)

where ξ ¼ xþ yþ fð4μ− λ2 − 3Þ=4g1=αt.When λ2 − 4μ < 0,

U 21 ðξÞ¼

�a1b0�2b21μþ2b0b1λ�2b20

�þa10BB@


p2

3

−C1sin

� ffiffiffiffiffiffiffi4μ�λ2

pξα

2Γð1þαÞ

�þC2cos


pξα

2Γð1þαÞ

�

C1cos


pξα

2Γð1þαÞ

�þC2sin


pξα

2Γð1þαÞ

��λ2

1CCA

b1

0BB@b0þb1

0BB@


p2

3

−C1sin


pξα

2Γð1þαÞ

�þC2cos


pξα

2Γð1þαÞ

�

C1cos


pξα

2Γð1þαÞ

�þC2sin


pξα

2Γð1þαÞ

��λ2

1CCA1CCA

(3.1.12)

The choice of c1 ≠ 0; c2 ¼ 0 gives way

U 21 ðξÞ ¼

�a1b0 � 2b21μþ 2b0b1λ� 2b20

�� a1

� ffiffiffiffiffiffiffiffiffi4μ�λ2

p2

3tan


pξα

2Γð1þαÞ

�þ λ

2

�

b1

�b0 � b1


p2

3tan


pξα

2Γð1þαÞ

�þ λ

2

�� ; (3.1.13)

where ξ ¼ xþ yþ fð4μ− λ2 − 3Þ=4g1=αt.When λ2 − 4μ ¼ 0;

U 31 ðξÞ ¼

�a1b0 � 2b21μþ 2b0b1λ� 2b20

�þ a1

�C2Γð1þαÞ

C1Γð1þαÞþC2ξα � λ

2

�

b1

�b0 þ b1

�C2Γð1þαÞ


2

�� (3.1.14)

Choosing c1 ¼ 0; c2 ≠ 0 yields

U 31 ðξÞ ¼

�a1b0 � 2b21μþ 2b0b1λ� 2b20

�� a1

�Γð1þαÞ

ξα � λ2

�

b1

�b0 � b1

�Γð1þαÞ

ξα � λ2

�� ; (3.1.15)

where ξ ¼ xþ yþ fð−3Þ=4g1=αt:

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156

3.1.2 Solution 2. When λ2 − 4μ > 0;

U 12 ðξÞ¼

a0

b0þ2

0BB@

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2�4μ

p2

3

C1 sinh


pξα

2Γð1þαÞ

�þC2 cosh


pξα

2Γð1þαÞ

�

C1 cosh


pξα

2Γð1þαÞ

�þC2 sinh


pξα

2Γð1þαÞ

��λ

2

1CCA; (3.1.16)

Assigning c1 ≠ 0; c2 ¼ 0 provides

U 12 ðξÞ ¼

a0

b0þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 � 4μ

p2

3 tan h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 � 4μ

pξα

2Γð1þ αÞ

!� λ

2

!; (3.1.17)

where ξ ¼ xþ yþ fð4μ− λ2 − 3Þ=4g1=αt.When λ2 − 4μ < 0;

U 22 ðξÞ ¼

a0

b0þ 2

0BB@

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4μ� λ2

p2

3

−C1 sin


pξα

2Γð1þαÞ

�þ C2 cos


pξα

2Γð1þαÞ

�

C1 cos


pξα

2Γð1þαÞ

�þ C2 sin


pξα

2Γð1þαÞ

� � λ

2

1CCA; (3.1.18)

Conveying c1 ≠ 0; c2 ¼ 0 offers

U 22 ðξÞ ¼

a0

b0� 2


p2

3 tan


pξα

2Γð1þ αÞ

!þ λ

2

!; (3.1.19)

where ξ ¼ xþ yþ fð4μ− λ2 − 3Þ=4g1=αt:When λ2 − 4μ ¼ 0,

U 32 ðξÞ ¼

a0

b0þ 2

�C2Γð1þ αÞ

C1Γð1þ αÞ þ C2ξα �

λ

2

�; (3.1.20)

The transmission c1 ¼ 0; c2 ≠ 0 puts forward

U 32 ðξÞ ¼

a0

b0þ 2

Γð1þ αÞξα

� λ; (3.1.21)

where ξ ¼ xþ yþ fð−3Þ=4g1=αt.

3.2 The nonlinear space-time fractional STO equationConsider the nonlinear space-time fractional STO equation

Dαt uþ 3β

�Dα

xu�2 þ 3βu2Dα

xuþ 3βuD2αx uþ βD3α

x u ¼ 0 (3.2.1)

Using the complex fractional transformation

uðx; tÞ ¼ UðξÞ; ξ ¼ k1=αxþ c1=αt; (3.2.2)

Eqn (3.2.1) reduces to the following fractional order ordinary differential equation withrespect to the variable ξ:

cDαξU þ 3k2β

�Dα

ξU�2 þ 3kβU 2Dα

ξU þ 3k2βUD2αξ U þ k3βD3α

ξ U ¼ 0; (3.2.3)


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157

Taking anti-derivative of Eqn (3.2.3) yields

cU þ 3k2βUDαξU þ kβU 3 þ k3βD2α

ξ U ¼ 0 (3.2.4)

Applying the homogeneous balance method to Eqn (3.2.4) the solution (2.2.4) takes theform (3.1.5).

Eqn (3.2.4) under the use of solution (3.1.5) and Eqn (2.2.5) creates a polynomial inðDα

ξG=GÞwhose coefficients assigning to zero and solving yields the outcomes:

Set 1: a0 ¼b0

nðb1λ� 2b0Þk

ffiffiffiffiffiffiffiffiffiffi−kβc

p73b1c

o±ðb1λ� 2b0Þk2β þ 3b1


p ; a1 ¼ ±b1

ffiffiffiffiffiffi−c

kβ

r; (3.2.5)

where b0; b1; k; c; β and λ are all arbitrary constants.

Set 2: a0 ¼ ±b0

ffiffiffiffiffiffi−c

kβ

r; a1 ¼ ±

2b0kffiffiffiffiffiffiffiffiffiffi−kβc

pk2βλ±3


p ; b1 ¼ 2b0k2β

k2βλ±3ffiffiffiffiffiffiffiffiffiffi−kβc

p ; (3.2.6)

where b0; k; c; β and λ are all unknown parameters.Utilizing the values available in (3.2.5) and (3.2.6) in (3.1.5) provide the following

expressions for analytic solutions:

U1ðξÞ ¼b0fðb1λ�2b0Þk

ffiffiffiffiffiffiffi−kβc

p73b1cg

±ðb1λ�2b0Þk2βþ3b1ffiffiffiffiffiffiffi−kβc

p ±b1ffiffiffiffi−ckβ

q �Dα

ξGG�

b0 þ b1�Dα

ξGG� ; (3.2.7)

U2ðξÞ ¼ ±b0

ffiffiffiffi−ckβ

qþ 2b0k


pk2βλ±3


p �Dα

ξGG�

b0 þ 2b0k2β

k2βλ±3ffiffiffiffiffiffiffi−kβc

p �Dα

ξGG� ; (3.2.8)

where ξ ¼ k1=αxþ c1=αt.The expressions (3.2.7) and (3.2.8) along with (2.2.7)–(2.2.9) make available the following

closed form traveling wave solutions in terms of hyperbolic function, trigonometric functionand rational function:


U 11 ðξÞ ¼

b0fðb1λ�2b0Þkffiffiffiffiffiffiffi−kβc

p73b1cg

±ðb1λ�2b0Þk2βþ3b1


p ± b1ffiffiffiffi−ckβ

q0BB@


p2

3

C1sin h


pξα

2Γð1þαÞ

�þC2cos h


pξα

2Γð1þαÞ

�

C1cos h


pξα

2Γð1þαÞ

�þC2sin h


pξα

2Γð1þαÞ

�� λ2

1CCA

b0 þ b1

0BB@


p2

3

C1sin h


pξα

2Γð1þαÞ

�þC2cos h


pξα

2Γð1þαÞ

�

C1cos h


pξα

2Γð1þαÞ

�þC2sin h


pξα

2Γð1þαÞ

�� λ2

1CCA

(3.2.9)

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158

Fixing c1 ≠ 0; c2 ¼ 0 serves

U 11 ðξÞ ¼


p73b1cg



q � ffiffiffiffiffiffiffiffiffiλ2�4μ

p2

3 tan h


pξα

2Γð1þαÞ

�� λ

2

�

b0 þ b1


p2

3 tan h


pξα

2Γð1þαÞ

�� λ

2

� (3.2.10)

where ξ ¼ k1=αxþ c1=αt.When λ2 − 4μ < 0;

U 21 ðξÞ ¼


p73b1cg



q0BB@


p2

3

−C1sin


pξα

2Γð1þαÞ

�þC2cos


pξα

2Γð1þαÞ

�

C1cos


pξα

2Γð1þαÞ

�þC2sin


pξα

2Γð1þαÞ

� � λ2

1CCA

b0 þ b1

0BB@


p2

3

−C1sin


pξα

2Γð1þαÞ

�þC2cos


pξα

2Γð1þαÞ

�

C1cos


pξα

2Γð1þαÞ

�þC2sin


pξα

2Γð1þαÞ

� � λ2

1CCA

(3.2.11)

Setting up c1 ≠ 0; c2 ¼ 0 provides

U 21 ðξÞ ¼


p73b1cg



p 7b1ffiffiffiffi−ckβ

q � ffiffiffiffiffiffiffiffiffi4μ�λ2

p2

3 tan


pξα

2Γð1þαÞ

�þ λ

2

�

b0 � b1


p2

3 tan


pξα

2Γð1þαÞ

�þ λ

2

� (3.2.12)

where ξ ¼ k1=αxþ c1=αt.When λ2 − 4μ ¼ 0,

U 31 ðξÞ ¼


p73b1cg



q �C2Γð1þαÞ


2

�

b0 þ b1

�C2Γð1þαÞ


2

� (3.2.13)

Putting c1 ¼ 0; c2 ≠ 0 gives out

U 31 ðξÞ ¼


p73b1cg



p 7 b1ffiffiffiffi−ckβ

q �Γð1þαÞ

ξα � λ2

�

b0 � b1

�Γð1þαÞ

ξα � λ2

� (3.2.14)

where ξ ¼ k1=αxþ c1=αt.


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159


U 12 ðξÞ ¼ ±

b0ffiffiffiffi−ckβ

qþ 2b0k


pk2βλ±3


p

0BB@


p2

3

C1sin h


pξα

2Γð1þαÞ

�þC2cos h


pξα

2Γð1þαÞ

�

C1cos h


pξα

2Γð1þαÞ

�þC2sin h


pξα

2Γð1þαÞ

�� λ2

1CCA

b0 þ 2b0k2β


p

0BB@


p2

3

C1sin h


pξα

2Γð1þαÞ

�þC2cos h


pξα

2Γð1þαÞ

�

C1cos h


pξα

2Γð1þαÞ

�þC2sin h


pξα

2Γð1þαÞ

�� λ2

1CCA

(3.2.15)

Selecting c1 ≠ 0; c2 ¼ 0 yields

U 12 ðξÞ ¼ ±


qþ 2b0k


pk2βλ±3


p� ffiffiffiffiffiffiffiffiffi

λ2�4μp

23tan h


pξα

2Γð1þαÞ

�� λ

2

�

b0 þ 2b0k2β



λ2�4μp

23tan h


pξα

2Γð1þαÞ

�� λ

2

� (3.2.16)

where ξ ¼ k1=αxþ c1=αt.When λ2 − 4μ < 0;

U 22 ðξÞ ¼ ±


qþ 2b0k


pk2βλ±3


p

0BB@


p2

3

−C1sin


pξα

2Γð1þαÞ

�þC2cos


pξα

2Γð1þαÞ

�

C1cos


pξα

2Γð1þαÞ

�þC2sin


pξα

2Γð1þαÞ

� � λ2

1CCA

b0 þ 2b0k2β


p

0BB@


p2

3

−C1sin


pξα

2Γð1þαÞ

�þC2cos


pξα

2Γð1þαÞ

�

C1cos


pξα

2Γð1þαÞ

�þC2sin


pξα

2Γð1þαÞ

� � λ2

1CCA

(3.2.17)

Assigning c1 ≠ 0; c2 ¼ 0 reduces

U 22 ðξÞ ¼ ±


q� 2b0k


pk2βλ±3



4μ�λ2p

23tan


pξα

2Γð1þαÞ

�þ λ

2

�

b0 � 2b0k2β



4μ�λ2p

23tan


pξα

2Γð1þαÞ

�þ λ

2

� ; (3.2.18)


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When λ2 − 4μ ¼ 0;

U 32 ðξÞ ¼ ±


qþ 2b0k


pk2βλ±3


p�

C2Γð1þαÞC1Γð1þαÞþC2ξ

α � λ2

�

b0 þ 2b0k2β


p�


α � λ2

� (3.2.19)

Using c1 ¼ 0; c2 ≠ 0, we obtain

U 32 ðξÞ ¼ ±


qþ 2b0k


pk2βλ±3


p�

Γð1þαÞξα � λ

2

�

b0 þ 2b0k2β


p�

Γð1þαÞξα � λ

2

� ; (3.2.20)


3.3 The nonlinear space-time fractional KPP equationThe nonlinear space-time fractional KPP equation is

Dαt u� D2α

x uþ auþ bu2 þ cu3 ¼ 0 (3.3.1)

The fractional complex transformation

uðx; tÞ ¼ UðξÞ; ξ ¼ k1=αxþ w1=αt (3.3.2)

reduces Eqn (3.3.1) to

wDαξU � k2D2α

ξ U þ aU þ bU 2 þ cU 3 ¼ 0 (3.3.3)

Applying the homogeneous balance method to Eqn (3.3.3) the solution (2.2.4) takes theform (3.1.5).

Using Eqn (3.1.5) and Eqn (2.2.5), Eqn (3.3.3) forms a polynomial in ðDαξG=GÞ whose

coefficients assigning to zero and solving gives up the following outcomes:

a0 ¼ 1; a1 ¼ab1

n�−b±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

p ��wþ λk2

�� 4ab1k2μo

�−b±


p ��2ab1k2μþ bwþ bλk2

�þ 2ac�wþ λk2

�;b0 ¼ −b±


p

2a

(3.3.4)

where b1; k; w; λ and μ are all unknown parameters.Inserting the values from (3.3.4) in (3.1.5) provides the following expressions for exact

wave analytic solutions:

UðξÞ ¼1þ ab1

�−b±

ffiffiffiffiffiffiffiffiffiffib2�4ac

p �ðwþλk2Þ�4ab1k

2μ�ðDα

ξG=GÞ�−b±


p �ð2ab1k2μþbwþbλk2Þþ2acðwþλk2Þ

−b±ffiffiffiffiffiffiffiffiffiffib2�4ac

p2a

þ b1�Dα

ξGG� ; (3.3.5)

where ξ ¼ k1=αxþ w1=αt.Eqn (3.3.5) together with (2.2.7)–(2.2.9) presents the following exact traveling wave

solutions:


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161

When λ2 − 4μ > 0;

U1;2ðξÞ ¼1þ

ab1

�−b±



2μ�0BB@

ffiffiffiffiffiffiffiλ2�4μ

p2

3

C1sin h


pξα

2Γð1þαÞ

�þC2cos h


pξα

2Γð1þαÞ

�

C1cos h


pξα

2Γð1þαÞ

�þC2sin h


pξα

2Γð1þαÞ

��λ2

1CCA

�−b±




p2a

þ b1

0BB@


p2

3

C1sin h


pξα

2Γð1þαÞ

�þC2cos h


pξα

2Γð1þαÞ

�

C1cos h


pξα

2Γð1þαÞ

�þC2sin h


pξα

2Γð1þαÞ

�� λ2

1CCA

(3.3.6)

Applying c1 ≠ 0; c2 ¼ 0 gives

U1;2ðξÞ ¼1þ

ab1

�−b±



2μ�� ffiffiffiffiffiffiffi

λ2�4μp

23tan h


pξα

2Γð1þαÞ

��λ

2

��−b±




p2a

þ b1


p2

3tan h


pξα

2Γð1þαÞ

�� λ

2

� (3.3.7)

where ξ ¼ k1=αxþ w1=αt.When λ2 − 4μ < 0;,

U3;4ðξÞ ¼1þ

ab1

�−b±



2μ�0BB@

ffiffiffiffiffiffiffi4μ�λ2

p2

3

−C1sin


pξα

2Γð1þαÞ

�þC2cos


pξα

2Γð1þαÞ

�

C1cos


pξα

2Γð1þαÞ

�þC2sin


pξα

2Γð1þαÞ

� �λ2

1CCA

�−b±




p2a

þ b1

0BB@


p2

3

−C1sin


pξα

2Γð1þαÞ

�þC2cos


pξα

2Γð1þαÞ

�

C1cos


pξα

2Γð1þαÞ

�þC2sin


pξα

2Γð1þαÞ

� � λ2

1CCA

(3.3.8)Using c1 ≠ 0; c2 ¼ 0 yields

U3; 4ðξÞ ¼1�

ab1

�−b±



2μ�� ffiffiffiffiffiffiffi

4μ�λ2p

23tan


pξα

2Γð1þαÞ

�þλ2

��−b±




p2a

� b1


p2

3tan


pξα

2Γð1þαÞ

�þ λ

2

� ; (3.3.9)

where ξ ¼ k1=αxþ w1=αt.

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When λ2 − 4μ ¼ 0;

U5; 6ðξÞ ¼1þ

ab1

�−b±



2μ��


α�λ2

��−b±




p2a

þ b1

�C2Γð1þαÞ


2

� (3.3.10)

Fixing c1 ¼ 0; c2 ≠ 0 gives way

U5; 6ðξÞ ¼1þ

ab1

�−b±



2μ��

Γð1þαÞξα �λ

2

��−b±




p2a

þ b1

�Γð1þαÞ

ξα � λ2

� ; (3.3.11)

where ξ ¼ k1=αxþ w1=αt.

4. Graphical representationsSome of the furnished solutions in this paper are depicted graphically for their physicalappearance which stands for different shapes of soliton, like, kink-type soliton, singular kink-type soliton, periodic soliton, singular periodic soliton etc. The solution (3.1.11) represents theshape of kink-type soliton for λ ¼ 4; μ ¼ b1 ¼ 3; b0 ¼ 2:9; a1 ¼ 1:9; α ¼ 1and y ¼ 0within−10≤ x; t ≤ 10 shown in Figure 1. Eqn (3.1.13) stands for the singular periodic soliton forλ ¼ 2; α ¼ μ ¼ 1; b0 ¼ b1 ¼ 2; a1 ¼ 5 and x ¼ 0 within −10≤ y; t ≤ 10, Eqn (3.1.15) takesthe form of singular kink shape soliton for λ ¼ 2; μ ¼ 1; b0 ¼ 4; b1 ¼ 3; a1 ¼ 1:5; α ¼ 0:5and y ¼ 0 in the range −10≤ x; t ≤ 10 exposed in Figure 2. Eqn (3.1.17) represents kink-typesoliton for λ ¼ 4; μ ¼ 3; α ¼ b0 ¼ 1 and a0 ¼ 0:5 within −10≤ x; t ≤ 10, Eqn (3.1.19) gives

10

5

0

–5

–10–7

–6–5

–4–3

–2–1

0–10

–50

510

x

t

Figure 1.Kink-type soliton ofsolution (3.1.11) forλ ¼ 4; μ ¼ b1 ¼ 3;b0 ¼ 2:9; a1 ¼ 1:9;α ¼ 1 and y ¼ 0in −10≤ x; t ≤ 10


mathematicalphysics

163

the shape of periodic soliton for λ ¼ 3; μ ¼ 2:5; b0 ¼ 0:5; a0 ¼ 1; α ¼ 1 and y ¼ 0 in theinterval −10≤ x; t ≤ 10 given away in Figure 3. Eqn (3.1.21) stands for the singular periodicsoliton for α ¼ λ ¼ a0 ¼ 1; b0 ¼ 0:5and y ¼ 0within the range−10≤ x; t ≤ 10. The solution(3.2.10) represents the kink-type soliton for λ ¼ 4; μ ¼ b1 ¼ 3; β ¼ b0 ¼ 2; α ¼ c ¼ 1 andk ¼ −1 within −10≤ x; t ≤ 10. Eqn (3.2.12) stands for periodic soliton with λ ¼ 2; μ ¼ 5;b0 ¼ 2; b1 ¼ 3; α ¼ β ¼ 1; k ¼ −1 and c ¼ 2 in the interval −10≤ x; t ≤ 10 shown inFigure 4. Eqn (3.2.14) presents singular kink soliton for λ ¼ 2; μ ¼ 5; b0 ¼ 0:2;b1 ¼ 0:3; α ¼ k ¼ c ¼ 1 and β ¼ −2 within the range −10≤ x; t ≤ 10 revealed in Figure 5.

2.5

2

1.5

1

0.5

0

–10

–5

0

5

1010

50

–5–10

x

t

10

5

0

–5

–10

–10–5

05

10 5 0 –5 –10t

x

Figure 2.Shape of solution(3.1.15) for λ ¼ 2;μ ¼ 1; b0 ¼ 4;b1 ¼ 3; a1 ¼ 1:5;α ¼ 0:5 and y ¼ 0 intherange −10≤ x; t ≤ 10

Figure 3.Periodic plot ofsolution (3.1.19)forλ ¼ 3;μ ¼ 2:5; b0 ¼ 0:5;a0 ¼ 1; α ¼ 1 andy ¼ 0within −10≤ x; t ≤ 10

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Eqn (3.2.16) takes the form of kink-type soliton for λ ¼ 4; μ ¼ 3; α ¼ k ¼ 1; c ¼ 2;b0 ¼ 0:5; b1 ¼ 1:5 and β ¼ −1 with −10≤ x; t ≤ 10. Eqn (3.2.18) gives the shape ofperiodic soliton for λ ¼ b0 ¼ 2; μ ¼ 5; α ¼ k ¼ c ¼ 1; b1 ¼ 3 and β ¼ −2 in the interval−10≤ x; t ≤ 10. Eqn (3.2.20) represents singular kink-type soliton for λ ¼ 2; μ ¼ k ¼ c ¼ 1;b0 ¼ 0:2; b1 ¼ 0:3; α ¼ 0:5 and β ¼ −2 within −10≤ x; t ≤ 10 shown in Figure 6. Thesolution (3.3.7) represents the kink-type soliton for λ ¼ 4; μ ¼ 3; a1 ¼ b0 ¼ 0:5;b1 ¼ 1:5; α ¼ k ¼ w ¼ p ¼ r ¼ 1 and q ¼ 2 in the range −10≤ x; t ≤ 10 made known inFigure 7. Eqn (3.3.9) stands for periodic soliton for λ ¼ 2; μ ¼ 5; b0 ¼ 0:2; α ¼ k ¼ w¼ p ¼ r ¼ 1; a1 ¼ 0:5; b1 ¼ 0:2 and q ¼ 2:5within the interval −10≤ x; t ≤ 10 given away

20

10

0

–10

–20

–30–10 –5 0 5 10

50

–5–10

0.60.40.2

0–0.2–0.4–0.6–0.81.0

–1.210

5

0

–5

–10 10

50

–5

–10

tx

Figure 4.Physical appearance of

solution (3.2.12) forλ ¼ 2; μ ¼ 5;

b0 ¼ 2; b1 ¼ 3;α ¼ β ¼ 1; k ¼ −1

and c ¼ 2in −10≤ x; t ≤ 10

Figure 5.Singular kink-typesoliton of solution

(3.2.14) forλ ¼ 2; μ ¼ 5;

b0 ¼ 0:2; b1 ¼ 0:3;α ¼ k ¼ c ¼ 1 and

β ¼ −2 in therange −10≤ x; t ≤ 10


mathematicalphysics

165

in Figure 8. Eqn (3.3.11) takes the form of singular kink-type soliton for λ ¼ 2; α ¼ μ ¼ w¼ k ¼ r ¼ 1, q ¼ 2; b0 ¼ 0:4; b1 ¼ 0:2 and p ¼ 0:5 in the range −10≤ x; t ≤ 10 exposed inFigure 9.

The physical appearance of solutions to FNLEEs bears great importance to depictdifferent phenomena arisen in various fields of nature in real world. This paper consists ofsome fresh and general solutions among which few are graphically brought up.

0.2

0.1

0

–0.1

–0.2

–0.3

–0.4

–0.5

–0.610 5 0 –5 –10

105

0–5

–10

t

x

0.6

0.5

0.4

0.3

0.210 5

0 –5–10

50

–5–10

tx

Figure 6.Plot of solution (3.2.20)for λ ¼ 2;μ ¼ k ¼ c ¼ 1;b0 ¼ 0:2; b1 ¼ 0:3;α ¼ 0:5; andβ ¼ −2 within−10≤ x; t ≤ 10

Figure 7.Physical appearance ofsolution (3.3.7) forλ ¼ 4; μ ¼ 3;a1 ¼ b0 ¼ 0:5;b1 ¼ 1:5; α ¼ k ¼w ¼ p ¼ r ¼ 1 andq ¼ 2 in theinterval−10≤ x; t ≤ 10

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5. ConclusionThe core aim of this study is to make available further general and fresh closed form analyticwave solutions to the nonlinear space-time fractional PKP equation, the nonlinear space-timefractional STO equation and the nonlinear space-time fractional KPP equation through thesuggested rational fractional ðDα

ξG=GÞ-expansion method. The offered method hassuccessfully presented attractive solutions to the considered equations and shown its highperformance. So far we know the achieved solutions are not available in the literature and

30

20

10

0

–10

–20

–30

–40

–50

–60

–10–5

05

10–5 0

5 10

tx

1.51

0.50

–0.5–1

–1.5

105

0–5

10

5

0

–5

–10

t

x

Figure 8.Periodic shape ofsolution (3.3.9) for

λ ¼ 2; μ ¼ 5;b0 ¼ 0:2; α ¼ k ¼ w

¼ p ¼ r ¼ 1;a1 ¼ 0:5; b1 ¼ 0:2

and q ¼ 2:5 within therange −10≤ x; t ≤ 10

Figure 9.Plot of solution (3.3.11)

for λ ¼ 2; α ¼ μ ¼w ¼ k ¼ r ¼ 1, q ¼ 2;b0 ¼ 0:4; b1 ¼ 0:2 and

p ¼ 0:5 within theinterval−10≤ x; t ≤ 10


mathematicalphysics

167

might create amilestone in research area to analyze the physical structure and behavior of thereal life events that correspond to the fractional related models. Therefore, it may be claimedthat the rational fractional ðDα

ξG=GÞ-expansion method in deriving the closed form analyticalsolutions is simple, straightforward and productive. Thismethodmight be taken into accountfor further implementation to investigate any FNLEEs arising in various fields of appliedmathematics and mathematical physics. The obtained solutions in terms of trigonometricfunction, hyperbolic function and rational function containing many free parameters areclaimed to be fresh and further general which will take place in the literature.

References

1. Oldham KB, Spanier J. The fractional calculus. NewYork, NY: Academic Press; 1974.

2. Samko G, Kilbas AA, Marichev OI. Fractional integrals and derivatives, Theor Appl. Yverdon:Gordan and Breach; 1993.

3. Podlubny I. Fractional differential equations, vol. 198 of mathematics in science and engineering.San Diego, CA: Academic Press; 1999.

4. Baleanu D, Diethelm K, Scalas E, Trujillo JJ. Fractional calculus: models and numerical methods,vol. 3 of series on complexity, nonlinearity and chaos. Boston, Mass: World ScientificPublishing; 2012.

5. Yang XJ. Advanced local fractional calculus and its applications. New York, NY: World SciencePublisher; 2012.

6. Mainardi F. Fractional calculus and waves in linear viscoelasticity: an introduction tomathematical models. London: Imperial College Press; 2010.

7. He JH, Ji FY. Two-scale mathematics and fractional calculus for thermodynamics. Therm Sci.2019; 23: 2131-2133.

8. He JH, Elagan SK, Li Z. explanation of the fractional complex transform and derivative chain rulefor fractional calculus. Phys Lett A. 2012; 376: 257-259.

9. He JH. Jin X, A short review on analytical methods for the capillary oscillator in a nanoscaledeformable tube, Math Meth Appl Sci., 2020; 43: doi: 10.1002/mma.6321.

10. Li F, Nadeem M. He-Laplace method for nonlinear vibration in shallow water waves. J Low FreqNoise Vib Act Cont. 2019; 38: 1305-1313.

11. Atangana A, Aguilar JFG. Numerical approximation of Riemann-Liouville definition of fractionalderivative: from Riemann-Liouville to Atangana-Baleanu. Numer Meth Partial Diff Eq. 2017; 34:doi: 10.1002/num.22195.

12. Akgul A, Baleanu D, Inc M, Tchier F. On the solutions of electrohydrodynamic flow withfractional differential equations by reproducing kernel method. Open Phys. 2017; 128: 218-223.

13. Aslan EC, Inc M. Soliton solutions of NLSE with quadratic-cubic nonlinearity and stabilityanalysis. Waves Rand Comp Media. 2017; 27: 594-601.

14. Islam MT, Akbar MA, Azad MAK. Traveling wave solutions to some nonlinear fractionalpartial differential equations through the rational -expansion method. J Ocean Engr Sci. 2018;3: 76-81.

15. Inan IE, Ugurlu Y, Inc M. New applications of the -expansion method. Acta Phys Pol A. 2015; 128:245-251.

16. Islam MT, Akbar MA, Azad MAK. Traveling wave solutions in closed form for some nonlinearfractional evolution equations related to conformable fractional derivative. AIMS Mathematics.2018; 3(4): 625-646.

17. Baleanu D, Ugurlu Y, Inc M, Kilic B. Improved -expansion method for the time fractionalBiological population model and Cahn-Hilliard equation, J Comput Nonlin Dynam. 2015; 10:051016.

AJMS27,2

168

18. Islam MT, Akbar MA, Azad, MAK. A Rational -expansion method and its application to themodified KdV-Burgers equation and the (2þ1)-dimensional Boussinesq equation. Nonlinear Stud.2015; 6: 1-11.

19. Guner O, Bekir A, Bilgil, H. A note on Exp-function method combined with complex transformmethod applied to fractional differential equations. Adv Nonlinear Anal. 2015; 4: 201-208.

20. Alzaidy JF. The fractional sub-equation method and exact analytical solutions for some fractionalPDEs. Amer J Math Anal. 2013; 1: 14-19.

21. Kurt A. New analytical and numerical results for fractional Bogoyavlensky-Konopelchenkoequation arising in fluid dynamics. Appl Math J Chinese Univ. 2020; 35: 101-112.

22. Martinez HY, Aguilar JFG. Atangana, A. First integral method for nonlinear differential equationswith conformable derivative. Math Model Nat Phenom. 2018; 13.

23. Inc, M, Inan IE, Ugurlu Y. New applications of the functional variable method. Optik. 2017; 136:374-381.

24. Bulut H, Baskonus HM, Pandir Y. The modified trial equation method for fractional waveequation and time fractional generalized Burgers equation. Abstr Appl Anal. 2013; 2013: 636802.

25. Taghizadeh N, Mirzazadeh M, Rahimian M, Akbari M. 2013. Application of the simplest equationmethod to some time fractional partial differential equations. Ain Shams Eng J.; 4: 897-902.

26. Chen C, Jiang YL. Lie group analysis method for two classes of fractional partial differentialequations. Commun. Nonlinear Sci Numer Simul. 2015; 26: 24-35.

27. Wu GC. A fractional characteristic method for solving fractional partial differential equations.Appl Math Lett. 2011; 24: 1046-1050.

28. Seadawy AR. Travelling-wave solutions of a weakly nonlinear two-dimensional higher-orderKadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves. Eur Phys J Plus.2017; 2017: 29, 132.

29. Akbulut A, Kaplan M, Bekir A. Auxiliary equation method for fractional differential equationswith modified Riemann–Liouville derivative. Int J Nonlinear Sci Numer Simul. 2016; 17: doi: 10.1515/ijnsns-2016-0023.

30. Deng W. Finite element method for the space and time fractional Fokker-Planck equation. SIAM JNumer Anal. 2008; 47: 204-226.

31. Momani S, Odibat Z, Erturk VS. Generalized differential transform method for solving a space-and time-fractional diffusion-wave equation. Phys Lett A. 2007; 370: 379-387.

32. Hu Y., Luo Y., Lu Z. Analytical solution of the linear fractional differential equation by Adomiandecomposition method. J Comput Appl Math. 2008; 215: 220-229.

33. El-Sayed AMA, Behiry SH, Raslan WE. Adomian’s decomposition method for solving anintermediate fractional advection-dispersion equation. Comput Math Appl. 2010; 59: 1759-1765.

34. Inc M. The approximate and exact solutions of the space- and time-fractional Burgers equationswith initial conditions by variational iteration method. J Math Anal Appl. 2008; 345: 476-484.

35. Gao GH., Sun ZZ, Zhang YN. A finite difference scheme for fractional sub-diffusion equations onan unbounded domain using artificial boundary conditions, J Comput Phys. 2012; 231: 2865-2879.

36. Gepreel KA. The homotopy perturbation method applied to nonlinear fractional Kadomtsev-Petviashvili-Piskkunov equations, Appl Math Lett. 2011; 24: 1458-1434.

37. Inc M. Some special structures for the generalized nonlinear Schrodinger equation with nonlineardispersion. Waves Rand Comp Media. 2013; 23: 77-88.

38. Tozar A, Kurt A, Tasbozan O. New wave solutions of an integrable dispersive wave equationwith a fractional time derivative arising in ocean engineering models. Kuwait J Sci. 2020;47: 22-33.

39. Kurt A, Tozar A, Tasbozan O. Applying the new extended direct algebraic method to solve theequation of obliquely interacting waves in shallow water. J Ocean Univ China. 2020; 19: 772-780.


mathematicalphysics

169

40. Tasbozan O. New analytical solutions for time fractional Benjamin-Ono Equation arising internalwaves in deep water. China Ocean Eng. 2019; 33: 593-600.

41. Ozkan O, Kurt A. Conformable fractional double laplace transform and its applications tofractional partial integro-differential equations. J Frac Cal Appl. 2020; 11: 70-81.

42. Islam MT, Akbar MA, Azad MAK. Closed-form travelling wave solutions to the nonlinear space-time fractional coupled Burgers’ equation. Arab J Basic Appl. Sci. 2019; 26: 1-11.

43. Khalil R, Al Horani M, Yousef A, Sababheh MAM. A new definition of fractional derivative. JComput Appl Math. 2014; 264: 65-70.

44. Atangana A, Baleanu D, Alsaedi A. New properties of conformable derivative. Open Math. 2015;13(1): 889-898.

45. Eslami M, Rezazadeh H. The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo. 2016; 53: 475-85.

46. Cenesiz Y, Kurt A. The new solution of time fractional wave equation with conformable fractionalderivative definition. J New Theory. 2015; 7: 79-85.

Corresponding authorTarikul Islam can be contacted at: [email protected]


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Generalized cyclic contractionsand coincidence points involving

a control function on partialmetric spacesSushanta Kumar Mohanta

Department of Mathematics, West Bengal State University, Kolkata, India

Abstract

Purpose – In this paper, we use the notion of cyclic representation of a nonempty set with respect to a pair ofmappings to obtain coincidence points and common fixed points for a pair of self-mappings satisfying somegeneralized contraction- type conditions involving a control function in partial metric spaces. Moreover, weprovide some examples to analyze and illustrate our main results.Design/methodology/approach – Theoretical method.Findings – We establish some coincidence points and common fixed point results in partial metric spaces.Originality/value – Results of this study are new and interesting.

Keywords Partial metric, Cyclic contraction, 0-Completeness, Coincidence point


1. IntroductionIn 1994, Matthews [1] introduced the concept of partial metric spaces as a part of the study ofdenotational semantics of dataflow networks and proved the well-known Banach contractionprinciple in this setting. Complete partial metric space is a useful framework to model severalcomplex problems in theory of computation. The works of [2–10] are viable and have openednew avenues for application in different fields of mathematics and applied sciences. It isinteresting to note that in partial metric spaces self-distance of an arbitrary point need not beequal to zero. Recently, many authors studied fixed points of cyclic mappings in severalspaces. In 2003, Kirk et al. [11] introduced the notion of cyclic mappings and proved somefixed-point theorems for thesemappings. Some results for cyclic contractions in partial metricspaces have been obtained in [12–16]. In many cases, new results are being obtained byconsidering contractive conditions that depend on control functions. Indeed, the auxiliaryfunctions which involved in contractive-type conditions are known as control functions. In2013, Shatanawi et al. [17] proved some common fixed-point theoremswith the help of controlfunctions, namely, altering distance functions due to Khan et al. [18]. After that, severalgeneralized control functions were used to obtain fixed-point results in various spaces. Theresults of [19–22] have become the source of motivation of this study. In this work, weintroduce the concept of cyclic representation of a nonempty set with respect to a pair ofmappings and use it to prove a coincidence point and common fixed-point result for a pair ofself-mappings satisfying some generalized contraction-type conditions involving a control

Generalizedcylic

contractions

171

MSC Classification — 54H25, 47H10© Sushanta Kumar Mohanta. Published in Arab Journal of Mathematical Sciences. Published by

Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (forboth commercial and non-commercial purposes), subject to full attribution to the original publicationand authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode

The author is grateful to the referees for their valuable comments.



Received 17 July 2020Revised 18 October 2020

11 November 2020Accepted 19 November 2020


Vol. 27 No. 2, 2021pp. 171-188


DOI 10.1108/AJMS-07-2020-0023

function in partial metric spaces. We also prove another common fixed point result for a pairof self-mappings satisfying a new contraction condition in this framework. Our results extendand unify several existing results in the literature. Finally, we give some examples to justifythe validity of our results.

2. Some basic conceptsIn this section, we present some basic facts and properties of partial metric spaces.

Definition 2.1. [1] A partial metric on a nonempty setX is a function p : X 3X →ℝþ suchthat for all x; y; z∈X:

ðp1Þ pðx; xÞ ¼ pðy; yÞ ¼ pðx; yÞ5x ¼ y;

ðp2Þ pðx; xÞ≤ pðx; yÞ;ðp3Þ pðx; yÞ ¼ pðy; xÞ;

ðp4Þ pðx; yÞ≤ pðx; zÞ þ pðz; yÞ � pðz; zÞ:

The pair ðX ; pÞ is called a partial metric space.

It is obvious that if pðx; yÞ ¼ 0, then from ðp1Þ and ðp2Þ, it follows that x ¼ y. However, x ¼ ydoes not imply pðx; yÞ ¼ 0.

Example 2.2. [1] LetX ¼ ½0; ∞Þand let pðx; yÞ ¼ maxfx; yg, for all x; y∈X. Then ðX ; pÞis a partial metric space.

Example 2.3. [1] Let X ¼ f½a; b� : a; b∈ℝ; a≤ bg and let pð½a; b�; ½c; d�Þ ¼ ðmaxfb; dg−minfa; cgÞ. Then ðX ; pÞ is a partial metric space.

Each partial metric p on X generates a T0 topology τp on X which has as a base the family ofopen p-balls fBpðx; eÞ : x∈X ; e > 0g, where Bpðx; eÞ ¼ fy∈X : pðx; yÞ < pðx; xÞ þ eg forall x∈X and e > 0.

Theorem 2.4. If U ∈ τp and x∈U, then there exists r > 0 such that Bpðx; rÞ⊆U.

Proof. Since U is an open set containing x, there exists an open p-ball, say Bpðy; eÞ such thatx∈Bpðy; eÞ⊆U. Then pðx; yÞ < pðy; yÞ þ e. Let us choose 0 < r < pðy; yÞ− pðx; yÞ þ e andconsider the open p-ball Bpðx; rÞ. Then it is easy to verify that Bpðx; rÞ⊆Bpðy; eÞ⊆U.

Remark 2.5. Let ðX ; pÞ be a partial metric space, ðxnÞ be a sequence in X and x∈X. ThenðxnÞ converges to x with respect to (w.r.t.) τp if and only if lim

n→∞pðxn; xÞ ¼ pðx; xÞ.

Let xn → xw.r.t. τp and e > 0. Then there exists a natural number n0 such that xn ∈Bpðx; eÞ forall n≥ n0. This gives that pðxn; xÞ− pðx; xÞ < e for all n≥ n0. Since pðxn; xÞ− pðx; xÞ≥ 0, itfollows that jpðxn; xÞ− pðx; xÞj < e for all n≥ n0. This proves that lim

n→∞pðxn; xÞ ¼ pðx; xÞ.

Conversely, suppose that limn→∞

pðxn; xÞ ¼ pðx; xÞ. We shall show that xn → xw.r.t. τp. Let

U ∈ τp and x∈U. Then there exists e > 0 such that x∈Bpðx; eÞ⊆U. By hypotheses, itfollows that

limn→∞

ðpðxn; xÞ � pðx; xÞÞ ¼ 0:

So, there exists n0 ∈ℕ such that pðxn; xÞ− pðx; xÞ < e for all n≥ n0. This ensures thatxn ∈Bpðx; eÞ for all n≥ n0, and hence xn ∈U for all n≥ n0. Therefore, ðxnÞconverges to xw.r.t.τp on X.

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Definition 2.6. [1] Let ðX ; pÞbe a partial metric space, and let ðxnÞbe a sequence inX. Then(1) ðxnÞ converges to a point x∈X if lim

n→∞pðxn; xÞ ¼ pðx; xÞ. This will be denoted as

limn→∞

xn ¼ x or xn → xðn→∞Þ.(2) ðxnÞ is called a Cauchy sequence if lim

n;m→∞pðxn; xmÞ exists and is finite, say l, that is,

corresponding to every e > 0, there exists n0 ∈ℕ such that jpðxn; xmÞ− lj < e;

∀n; m≥ n0.

(3) ðX ; pÞ is said to be complete if every Cauchy sequence ðxnÞ in X converges to a pointx∈X such that pðx; xÞ ¼ lim

n;m→∞pðxn; xmÞ.

Definition 2.7. [23] A sequence ðxnÞ in ðX ; pÞ is called 0-Cauchy if

limn;m→∞

pðxn; xmÞ ¼ 0:

The space ðX ; pÞ is said to be 0-complete if every 0-Cauchy sequence inX converges to a pointx∈X such that pðx; xÞ ¼ 0.

It is easy to verify that every closed subset of a 0-complete partial metric space is0-complete.

Lemma 2.8. Let ðX ; pÞ be a partial metric space.

(1) (see [24, 25]) If pðxn; zÞ→ pðz; zÞ ¼ 0 as n→∞, then pðxn; yÞ→ pðz; yÞ as n→∞ foreach y∈X.

(2) (see [23]) If ðX ; pÞ is complete, then it is 0-complete.

The converse assertion of ðbÞmay not hold, in general. The following example supports theabove remark.

Example 2.9. [23] The space X ¼ ½0; ∞Þ∩ℚwith the partial metric pðx; yÞ ¼ maxfx; ygis 0-complete, but it is not complete. Moreover, the sequence ðxnÞwith xn ¼ 1 for each n∈ℕ isa Cauchy sequence in ðX ; pÞ, but it is not a 0-Cauchy sequence.

Definition 2.10. [26] Let f and g be self-mappings of a setX. If y ¼ fx ¼ gx for some x inX,then x is called a coincidence point of f and g and y is called a point of coincidence of f and g.

Definition 2.11. [25] The mappings f ; g : X →X are weakly compatible, if for everyx∈X, the following holds:

f ðgxÞ ¼ gðfxÞwhenever gx ¼ fx:

Proposition 2.12. [26] Let f and g be weakly compatible self-maps of a nonempty setX. If fand g have a unique point of coincidence y ¼ fx ¼ gx, then y is the unique common fixed pointof f and g.

Let Ψ be a class of functions ψ : ½0; ∞Þ→ ½0; ∞Þ satisfying the following conditions:ðψ1Þ ψ is a nondecreasing function;ðψ2Þ

P∞

n¼1ψnðtÞ < ∞ for each t > 0, where ψn is the nth iterate of ψ .

Let us notice that the class Ψ is nonempty. Indeed, the functions ψðtÞ ¼ kt belong to Ψwhenever k∈ ½0; 1Þ.

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Remark 2.13. [27] For each ψ ∈Ψ, we see that the following assertions hold:

(1) limn→∞

ψnðtÞ ¼ 0, for all t > 0;

(2) ψðtÞ < t for each t > 0;

(3) ψð0Þ ¼ 0.

Lemma2.14. Let ðX ; pÞbe a partial metric space. Let ðxnÞ∞n¼0 ⊆X be a sequence andψ ∈Ψbe such that

(1) pðx0; x1Þ> 0;

(2) pðxn; xnþ1Þ≤ψðpðxn−1; xnÞÞ; for each n∈ℕ.

Then ðxnÞ∞n¼0 is a 0-Cauchy sequence.

Proof. By hypothesis ð2Þ, we havepðxn; xnþ1Þ≤ψðpðxn−1; xnÞÞ; for each n∈ℕ: (2.1)

By repeated use of condition (2.1) and ðψ1Þ, we getpðxn; xnþ1Þ≤ψnðpðx0; x1ÞÞ; for each n∈ℕ:

For m; n∈ℕwith m > n, we have

pðxn; xmÞ≤ pðxn; xnþ1Þ þ pðxnþ1; xnþ2Þ þ . . .þ pðxm−2; xm−1Þþ pðxm−1; xmÞ≤ψnðpðx0; x1ÞÞ þ ψnþ1ðpðx0; x1ÞÞ þ � � � þ ψm−2ðpðx0; x1ÞÞþ ψm−1ðpðx0; x1ÞÞ

¼Xm−1

i¼n

ψ iðpðx0; x1ÞÞ:

SinceP∞

n¼1ψnðtÞ < ∞ for each t > 0 and pðx0; x1Þ> 0, it follows that

limn;m→∞

pðxn; xmÞ ¼ 0:

This proves that ðxnÞ∞n¼0 is a 0-Cauchy sequence in X.

3. Main resultsIn this section, we prove our new results. Throughout the paper, we use the followingnotation:

Let ðX ; pÞ be a partial metric space, and f ; g : X →X be self mappings. Then,

Mðgx; gyÞ ¼ max

�pðgx; gyÞ; pðgx; fxÞ; pðgy; fyÞ; pðgx; fyÞ þ pðgy; fxÞ

2

�:

We begin with the following definition.

Definition 3.1. Let X be a nonempty set, q∈ℕ, and let f ; g : X →X be self- mappings.Then X ¼ ∪q

i¼1Ai is a cyclic representation of X with respect to the pair ðf ; gÞ if(1) Ai; i ¼ 1; 2; . . . ; q are nonempty subsets of X;

(2) f ðAiÞ⊆ gðAiþ1Þ for i ¼ 1; 2; . . . ; q, where Aqþ1 ¼ A1.

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Theorem 3.2. Let ðX ; pÞ be a 0-complete partial metric space, q∈ℕ, and letA1; A2; � � � ; Aq be nonempty subsets of X, and X ¼ ∪q

i¼1Ai. Suppose f ; g : X →X

are self- mappings, gðA1Þ; gðA2Þ; . . . ; gðAqÞ are closed subsets of ðX ; pÞ and X ¼ ∪qi¼1Ai

is a cyclic representation of X with respect to the pair ðf ; gÞ. If there exists ψ ∈Ψ such that

pðfx; fyÞ≤ψðMðgx; gyÞÞ (3.1)

for all x; y∈X with ðgx; gyÞ∈ gðAiÞ3 gðAiþ1Þ; i ¼ 1; 2; . . . ; q, where Aqþ1 ¼ A1, thenf and g have a unique point of coincidence u in ∩ q

i¼1gðAiÞ with pðu; uÞ ¼ 0. Moreover, iff and g are weakly compatible, then f and g have a unique common fixed pointin ∩q

i¼1gðAiÞ.Proof. Let x0 be an arbitrary element of X. Then there exists i0 ∈ f1; 2; . . . ; qg such that

x0 ∈Ai0. Since f ðAi0Þ⊆ gðAi0þ1Þ, there exists x1 ∈Ai0þ1 such that gx1 ¼ fx0. Continuing thisprocess, we can construct a sequence ðxnÞ such that gxn ¼ fxn−1; n ¼ 1; 2; 3; . . . ; wherexn ∈Ai0þn and Aqþk ¼ Ak. If pðgxn; gxnþ1Þ ¼ 0 for some n∈ℕ, then gxn ¼ gxnþ1 ¼ fxn andhence gxnþ1 is a point of coincidence of f and g.

Without loss of generality, we may assume that

pðgxn; gxnþ1Þ> 0; ∀n∈ℕ:

We note that for all n∈ℕ there exists i∈ f1; 2; . . . ; qg such that ðxn; xnþ1Þ∈Ai 3Aiþ1 andso, ðgxn; gxnþ1Þ∈ gðAiÞ3 gðAiþ1Þ. We first compute Mðgxn−1; gxnÞ. We have,

Mðgxn−1; gxnÞ ¼ max

8><>:

pðgxn−1; gxnÞ; pðgxn−1; gxnÞ; pðgxn; gxnþ1Þ;

pðgxn−1; gxnþ1Þ þ pðgxn; gxnÞ2

9>=>;

≤max

8><>:

pðgxn−1; gxnÞ; pðgxn; gxnþ1Þ;

pðgxn−1; gxnÞ þ pðgxn; gxnþ1Þ2

9>=>;

¼ maxfpðgxn−1; gxnÞ; pðgxn; gxnþ1Þg:By ðψ1Þ, it follows that

ψðMðgxn−1; gxnÞÞ≤ψðmaxfpðgxn−1; gxnÞ; pðgxn; gxnþ1ÞgÞ: (3.2)

For any natural number n, we have by applying conditions (3.1) and (3.2) that

pðgxn; gxnþ1Þ ¼ pðfxn−1; fxnÞ≤ψðMðgxn−1; gxnÞÞ≤ψðmaxfpðgxn−1; gxnÞ; pðgxn; gxnþ1ÞgÞ:

(3.3)

We shall show that ðgxnÞ is a Cauchy sequence in gðXÞ.If maxfpðgxn−1; gxnÞ; pðgxn; gxnþ1Þg ¼ pðgxn; gxnþ1Þ, then from condition (3.3) and

using ψðtÞ < t for each t > 0, we obtain

pðgxn; gxnþ1Þ≤ψðpðgxn; gxnþ1ÞÞ < pðgxn; gxnþ1Þ;which is a contradiction. Therefore,

maxfpðgxn−1; gxnÞ; pðgxn; gxnþ1Þg ¼ pðgxn−1; gxnÞ:

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Thus, we obtain from condition (3.3) that

pðgxn; gxnþ1Þ≤ψðpðgxn−1; gxnÞÞ; for all n∈ℕ:

By using Lemma 2.14, it follows that ðgxnÞ is a 0-Cauchy sequence in gðXÞ. SincegðXÞ ¼ ∪q

i¼1gðAiÞ, it follows that gðXÞ is a closed subset of the 0-complete partial metricspace ðX ; pÞ, and hence gðXÞ is 0-complete. So, ðgxnÞ converges to some point u∈ gðXÞ suchthat pðu; uÞ ¼ 0. Therefore,

limn→∞

pðgxn; uÞ ¼ pðu; uÞ ¼ 0: (3.4)

We shall prove that u∈∩qi¼1gðAiÞ.

As x0 ∈Ai0, it follows that the sequence ðgxnqÞn≥0⊆ gðAi0Þ. Since gðAi0Þ is closed, condition(3.4) ensures that u∈ gðAi0Þ. Again, we get ðgxnqþ1Þn≥0⊆ gðAi0þ1Þ, where Aqþk ¼ Ak.Proceeding as above, we obtain that u∈ gðAi0þ1Þ. Continuing in this way, we get

u∈∩qi¼1gðAiÞ: (3.5)

Now we shall show that u is a point of coincidence of f and g.Indeed, since u∈ gðXÞ, there exists t ∈X such that gt ¼ u. Now, if xn ∈Ai, then

ðgxn; gtÞ ¼ ðgxn; uÞ∈ gðAiÞ3 gðAiþ1Þ because u∈∩qi¼1gðAiÞ. Therefore Mðgxn; gtÞ is well

defined, and by applying (3.1), we obtain that for all n∈ℕ,

pðgxnþ1; ftÞ ¼ pðfxn; ftÞ≤ψðMðgxn; gtÞÞ; (3.6)

where

Mðgxn; gtÞ ¼ max

8><>:

pðgxn; gtÞ; pðgxn; fxnÞ; pðgt; ftÞ;

pðgxn; ftÞ þ pðgt; fxnÞ2

9>=>;

¼ max

8><>:

pðgxn; gtÞ; pðgxn; gxnþ1Þ; pðgt; ftÞ;

pðgxn; ftÞ þ pðgt; gxnþ1Þ2

9>=>;:

Suppose that pðgt; ftÞ≠ 0. Let e ¼ pðgt; ftÞ2 > 0. Since lim

n→∞pðgxn; gtÞ ¼ 0, there exists k∈ℕ

such that

pðgxn; gtÞ< e; for each n≥ k: (3.7)

Then, for each n≥ k

pðgxn; ftÞ≤ pðgxn; gtÞ þ pðgt; ftÞ � pðgt; gtÞ≤ pðgxn; gtÞ þ pðgt; ftÞ< 3e:

For n≥ k, we have

pðgxn; ftÞ þ pðgt; gxnþ1Þ2

<1

2ð3eþ eÞ ¼ 2e: (3.8)

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Moreover, for n≥ k, we have

pðgxn; gxnþ1Þ≤ pðgxn; gtÞ þ pðgt; gxnþ1Þ<2e: (3.9)

Thus, for n≥ k, it follows from conditions (3.7), (3.8) and (3.9) that

max

8<:

pðgxn; gtÞ; pðgxn; gxnþ1Þ; pðgt; ftÞ;pðgxn; ftÞ þ pðgt; gxnþ1Þ

2

9=; ¼ 2e ¼ pðgt; ftÞ:

Therefore, we obtain from (3.6) that

pðgxnþ1; ftÞ≤ψðpðgt; ftÞÞ; for each n≥ k: (3.10)

By using condition (3.10), for n≥ k, we have

pðgt; ftÞ≤ pðgt; gxnþ1Þ þ pðgxnþ1; ftÞ � pðgxnþ1; gxnþ1Þ≤ pðgt; gxnþ1Þ þ ψðpðgt; ftÞÞ:

Passing to the limit as n→∞, we get

pðgt; ftÞ≤ψðpðgt; ftÞÞ;which is a contradiction since ψðtÞ < t for each t > 0. Therefore, pðgt; ftÞ ¼ 0 and henceft ¼ gt ¼ u. Therefore, u is a point of coincidence of f and g such that u∈∩q

i¼1gðAiÞand pðu; uÞ ¼ 0.

For uniqueness, we assume that there is another point of coincidence v of f and g such thatv∈∩q

i¼1gðAiÞ and pðv; vÞ ¼ 0. By supposition, there exists x∈X satisfying v ¼ gx ¼ fx.

Since u; v∈∩qi¼1gðAiÞ and gx ¼ v; gt ¼ u; Mðgx; gyÞ is well defined, therefore applying

(3.1), we have

pðu; vÞ ¼ pðft; fxÞ

≤ψ�max

�pðgt; gxÞ; pðgt; ftÞ; pðgx; fxÞ; pðgt; fxÞ þ pðgx; ftÞ

2

��

¼ ψ�max

�pðu; vÞ; pðu; uÞ; pðv; vÞ; pðu; vÞ þ pðv; uÞ

2

��

¼ ψðpðu; vÞÞ:

(3.11)

If pðu; vÞ > 0, then from condition (3.11), we get

0 < pðu; vÞ≤ψðpðu; vÞÞ;which is a contradiction since ψðtÞ < t for each t > 0. So, it must be the case that pðu; vÞ ¼ 0,and hence u ¼ v. Thus, f and g have a unique point of coincidence u∈∩q

i¼1gðAiÞand pðu; uÞ ¼ 0.

If f and g are weakly compatible, then by proposition 2.12, f and g have a unique commonfixed point in ∩q

i¼1 gðAiÞ.Corollary 3.3. Let ðX ; pÞbe a 0-complete partial metric space, and let f ; g : X →X be suchthat f ðXÞ⊆ gðXÞ and gðXÞ a closed subset of ðX ; pÞ. If there exists ψ ∈Ψ such that

pðfx; fyÞ≤ψðMðgx; gyÞÞfor all x; y∈X, then f and g have a unique point of coincidence u in gðXÞsuch that pðu; uÞ ¼ 0.

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Moreover, if f and g are weakly compatible, then f and g have a unique common fixed pointin gðXÞ.Proof. The proof follows from Theorem 3.2 by taking A1 ¼ A2 ¼ � � � ¼ Aq ¼ X.

Corollary 3.4. Let ðX ; pÞ be a 0-complete partial metric space, and let f : X →X be amapping. Suppose there exists ψ ∈Ψ such that

pðfx; fyÞ≤ψ�max

�pðx; yÞ; pðx; fxÞ; pðy; fyÞ; pðx; fyÞ þ pðy; fxÞ

2

��

for all x; y∈X. Then f has a unique fixed point u in X such that pðu; uÞ ¼ 0.

Proof. Since the maps f and I, the identity map on X are weakly compatible, the prooffollows from Theorem 3.2 by taking A1 ¼ A2 ¼ � � � ¼ Aq ¼ X and g ¼ I.

Corollary 3.5. Let ðX ; pÞ be a 0-complete partial metric space, q∈ℕ, and letA1; A2; � � � ; Aq be nonempty subsets of X, and X ¼ ∪q

i¼1Ai. Suppose f ; g : X →X

are self-mappings, gðA1Þ; gðA2Þ; � � � ; gðAqÞ are closed subsets of ðX ; pÞ and X ¼ ∪qi¼1Ai

is a cyclic representation of X with respect to the pair ðf ; gÞ. If there exists k∈ ½0; 1Þ suchthat

pðfx; fyÞ≤ kmax

�pðgx; gyÞ; pðgx; fxÞ; pðgy; fyÞ; pðgx; fyÞ þ pðgy; fxÞ

2

�

for all x; y∈X with ðgx; gyÞ∈ gðAiÞ3 gðAiþ1Þ; i ¼ 1; 2; � � � ; q, where Aqþ1 ¼ A1, thenf and g have a unique point of coincidence u in ∩q

i¼1gðAiÞ with pðu; uÞ ¼ 0. Moreover, iff and g are weakly compatible, then f and g have a unique common fixed pointin ∩q

i¼1gðAiÞ.Proof. The proof follows fromTheorem 3.2 by takingψðtÞ ¼ kt for each t ≥ 0, where k∈ ½0; 1Þis a fixed number.

Corollary 3.6. Let ðX ; pÞbe a 0-complete partial metric space and f : X →X be amapping.If there exists k∈ ½0; 1Þ such that

pðfx; fyÞ≤ kmax

�pðx; yÞ; pðx; fxÞ; pðy; fyÞ; pðx; fyÞ þ pðy; fxÞ

2

�

for all x; y∈X, then f has a unique fixed point u in X with pðu; uÞ ¼ 0.

Proof. The result follows from Theorem 3.2 by taking A1 ¼ A2 ¼ � � � ¼ Aq ¼ X ; g ¼ I andψðtÞ ¼ kt for each t ≥ 0, where k∈ ½0; 1Þ is a fixed number.

Corollary 3.7. Let ðX ; pÞ be a 0-complete partial metric space, q∈ℕ, and letA1; A2; � � � ; Aq be nonempty subsets of X, X ¼ ∪q

i¼1Ai. Suppose f ; g : X →X are self-

mappings, gðA1Þ; gðA2Þ; � � � ; gðAqÞ are closed subsets of ðX ; pÞ and X ¼ ∪qi¼1Ai is a cyclic

representation of X with respect to the pair ðf ; gÞ. If there exists α; β; γ; δ≥ 0 withαþ β þ γ þ 2δ < 1 such that

pðfx; fyÞ≤ αpðgx; gyÞ þ βpðgx; fxÞ þ γpðgy; fyÞ þ δðpðgx; fyÞ þ pðgy; fxÞÞ (3.12)

for any ðgx; gyÞ∈ gðAiÞ3 gðAiþ1Þ; i ¼ 1; 2; � � � ; q with Aqþ1 ¼ A1, then f and g have aunique point of coincidence u in ∩q

i¼1gðAiÞwith pðu; uÞ ¼ 0. Moreover, if f and g are weaklycompatible, then f and g have a unique common fixed point in ∩q

i¼1gðAiÞ.

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Proof. From condition (3.12), we obtain

pðfx; fyÞ≤ α pðgx; gyÞ þ β pðgx; fxÞ þ γ pðgy; fyÞ þ δ ðpðgx; fyÞ þ pðgy; fxÞÞ≤ ðαþ β þ γ þ 2δÞMðgx; gyÞ¼ kMðgx; gyÞ;

where k ¼ ðαþ β þ γ þ 2δÞ∈ ½0; 1Þ. Now, corollary 3.5 can be applied to obtain the desiredresult.

Remark 3.8. It is worth mentioning that theorem 3.8 [28] can be obtained as a particularcase of Theorem 3.2. Moreover, we obtain various important fixed-point results in partialmetric spaces includingMatthews version of Banach contraction theorem [1] as a special caseof corollary 3.7.

We now present our second main theorem

Theorem 3.9. Let ðX ; pÞ be a 0-complete partial metric space, and let f ; T : X →X bemappings. Suppose there exists ψ ∈Ψ such that

pðfx; TyÞ≤ψðNðx; yÞÞ (3.13)

for all x; y∈X, where Nðx; yÞ ¼ max

�pðx; yÞ; pðx; fxÞ; pðy; TyÞ; pðx;TyÞþpðy; fxÞ

2

�. Then

f and T have a unique common fixed point u in X with pðu; uÞ ¼ 0.

Proof. We first prove that u is a fixed point of T if and only if u is a fixed point of f.Suppose that u is a fixed point of T, that is, Tu ¼ u. Then, by using condition (3.13), we

obtain

pðfu; uÞ ¼ pðfu; TuÞ≤ψðNðu; uÞÞ;

where

Nðu; uÞ ¼ max

�pðu; uÞ; pðu; fuÞ; pðu; TuÞ; pðu; TuÞ þ pðu; fuÞ

2

�

¼ max

�pðu; uÞ; pðu; fuÞ; pðu; uÞ þ pðu; fuÞ

2

�

¼ max fpðu; uÞ; pðu; fuÞg¼ pðu; fuÞ:

Therefore,

pðfu; uÞ≤ψðpðu; fuÞÞ:If pðu; fuÞ > 0, thenψðpðu; fuÞÞ < pðu; fuÞ, a contradiction. This gives that pðu; fuÞ ¼ 0andhence fu ¼ u.

Proceeding similarly, we can show that if u is a fixed point of f, then u is also a fixed pointof T.

Let x0 ∈X be arbitrary. We can construct a sequence ðxnÞ in X such that

xn ¼�fxn−1; if n is odd;Txn−1; if n is even:

Generalizedcylic

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179

We assume that xn ≠ xn−1 for every n∈ℕ. If x2n ¼ x2nþ1 for some n∈ℕ∪ f0g, then x2n ¼ fx2nand hence x2n is a fixed point of f. By our previous discussion, it follows that x2n is also a fixedpoint ofT. So, x2n becomes a common fixed point of f andT. The case x2nþ1 ¼ x2nþ2 for somen∈ℕ∪ f0g can be treated similarly to achieve our goal. Therefore, pðxn; xn−1Þ>0; ∀n∈ℕ.

By using condition (3.13), we obtain

pðx2nþ1; x2nþ2Þ ¼ pðfx2n; Tx2nþ1Þ≤ψðNðx2n; x2nþ1ÞÞ; (3.14)

where

Nðx2n; x2nþ1Þ ¼ max

8><>:

pðx2n; x2nþ1Þ; pðx2n; fx2nÞ; pðx2nþ1;Tx2nþ1Þ;

pðx2n;Tx2nþ1Þ þ pðx2nþ1; fx2nÞ2

9>=>;

¼ max

8><>:

pðx2n; x2nþ1Þ; pðx2nþ1; x2nþ2Þ;

pðx2n; x2nþ2Þ þ pðx2nþ1; x2nþ1Þ2

9>=>;

≤max

8><>:

pðx2n; x2nþ1Þ; pðx2nþ1; x2nþ2Þ;

pðx2n; x2nþ1Þ þ pðx2nþ1; x2nþ2Þ2

9>=>;

¼ max fpðx2n; x2nþ1Þ; pðx2nþ1; x2nþ2Þg:By using ðψ1Þ, it follows from (3.14) that

pðx2nþ1; x2nþ2Þ≤ψðmax fpðx2n; x2nþ1Þ; pðx2nþ1; x2nþ2ÞgÞ: (3.15)

If max fpðx2n; x2nþ1Þ; pðx2nþ1; x2nþ2Þg ¼ pðx2nþ1; x2nþ2Þ, then by using ψðtÞ < t for eacht > 0, we obtain from condition (3.15) that

pðx2nþ1; x2nþ2Þ≤ψðpðx2nþ1; x2nþ2ÞÞ < pðx2nþ1; x2nþ2Þ;which is a contradiction. Therefore,

max fpðx2n; x2nþ1Þ; pðx2nþ1; x2nþ2Þg ¼ pðx2n; x2nþ1Þ:Thus, condition (3.15) becomes

pðx2nþ1; x2nþ2Þ≤ψðpðx2n; x2nþ1ÞÞ; ∀n∈ℕ: (3.16)

Similarly, we can show that

pðx2n; x2nþ1Þ≤ψðpðx2n−1; x2nÞÞ; ∀n∈ℕ: (3.17)

Combining conditions (3.16) and (3.17), we get

pðxn; xnþ1Þ≤ψðpðxn−1; xnÞÞ; ∀n∈ℕ: (3.18)

By using Lemma 2.14, it follows that ðxnÞ is a 0-Cauchy sequence inX. As ðX ; pÞ is 0-complete,there exists u∈X such that xn → uwith pðu; uÞ ¼ 0, that is, lim

n→∞pðxn; uÞ ¼ pðu; uÞ ¼ 0. This

ensures that limn→∞

pðx2n; uÞ ¼ pðu; uÞ ¼ 0 and limn→∞

pðx2nþ1; uÞ ¼ pðu; uÞ ¼ 0.

AJMS27,2

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By using condition (3.13), we obtain

pðx2nþ1; TuÞ ¼ pðfx2n; TuÞ≤ψðNðx2n; uÞÞ; (3.19)

where

Nðx2n; uÞ ¼ max

8><>:

pðx2n; uÞ; pðx2n; fx2nÞ; pðu; TuÞ;

pðx2n; TuÞ þ pðu; fx2nÞ2

9>=>;

¼ max

8><>:

pðx2n; uÞ; pðx2n; x2nþ1Þ; pðu; TuÞ;

pðx2n; TuÞ þ pðu; x2nþ1Þ2

9>=>;:

Suppose that pðu; TuÞ≠ 0. Let e ¼ pðu;TuÞ2 > 0. Since lim

n→∞pðx2n; uÞ ¼ 0, there exists k1 ∈ℕ

such that

pðx2n; uÞ< e; for each n≥ k1: (3.20)

Then, for each n≥ k1

pðx2n; TuÞ≤ pðx2n; uÞ þ pðu; TuÞ � pðu; uÞ¼ pðx2n; uÞ þ pðu; TuÞ< 3e:

As limn→∞

pðx2nþ1; uÞ ¼ 0, there exists k2 ∈ℕ such that

pðx2nþ1; uÞ< e; for each n≥ k2:

Put k ¼ maxfk1; k2g. Then, for n≥ k, we have

pðx2n; TuÞ þ pðu; x2nþ1Þ2

<1

2ð3eþ eÞ ¼ 2e: (3.21)

Moreover, for n≥ k, we have

pðx2n; x2nþ1Þ≤ pðx2n; uÞ þ pðu; x2nþ1Þ< 2e: (3.22)

Thus, for n≥ k, it follows from conditions (3.20), (3.21) and (3.22) that

max

8<:

pðx2n; uÞ; pðx2n; x2nþ1Þ; pðu; TuÞ;pðx2n; TuÞ þ pðu; x2nþ1Þ

2

9=; ¼ 2e ¼ pðu; TuÞ:

Therefore, we obtain from (3.19) that

pðx2nþ1; TuÞ≤ψðpðu; TuÞÞ; for each n≥ k: (3.23)

By using condition (3.23), for n≥ k, we have

pðu; TuÞ≤ pðu; x2nþ1Þ þ pðx2nþ1; TuÞ � pðx2nþ1; x2nþ1Þ≤ pðu; x2nþ1Þ þ ψðpðu; TuÞÞ:

Passing to the limit as n→∞ and using Lemma 2.8 ðaÞ, we getpðu; TuÞ≤ψðpðu; TuÞÞ;

Generalizedcylic

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181

which is a contradiction, since ψðtÞ < t for each t > 0. Therefore, pðu; TuÞ ¼ 0 and henceTu ¼ u. This proves that u is a fixed point of T. By our previous discussion, u is also a fixedpoint of f. Thus, u is a common fixed point of f and T in X with pðu; uÞ ¼ 0.

For uniqueness, let v be another common fixed point of f and T in X with pðv; vÞ ¼ 0. Byapplying condition (3.13), we get

pðu; vÞ ¼ pðfu; TvÞ≤ψðNðu; vÞÞ; (3.24)

where

Nðu; vÞ ¼ max

�pðu; vÞ; pðu; fuÞ; pðv; TvÞ

�;pðu; TvÞ þ pðv; fuÞ

2

�

¼ maxfpðu; vÞ; 0; 0; pðu; vÞg¼ pðu; vÞ:

Thus, condition (3.24) becomes

pðu; vÞ≤ψðpðu; vÞÞ:If pðu; vÞ > 0, then

0 < pðu; vÞ≤ψðpðu; vÞÞ;which is a contradiction since ψðtÞ < t for each t > 0. So, it must be the case that pðu; vÞ ¼ 0and hence u ¼ v. Therefore, u is a unique common fixed point of f andT inXwith pðu; uÞ ¼ 0.

Corollary 3.10. Let ðX ; pÞ be a 0-complete partial metric space, and let the mappingsf ; T : X →X be such that

pðfx; TyÞ≤ kmax

�pðx; yÞ; pðx; fxÞ; pðy; TyÞ; pðx; TyÞ þ pðy; fxÞ

2

�

for all x; y∈X. , where k∈ ½0; 1Þ is a constant. Then f and T have a unique common fixedpoint u in X with pðu; uÞ ¼ 0.

Proof.The proof follows fromTheorem 3.9 by takingψðtÞ ¼ kt for each t ≥ 0, where k∈ ½0; 1Þis a fixed number.

Corollary 3.11. Let ðX ; pÞ be a 0-complete partial metric space, and let f : X →X be amapping. Suppose there exists ψ ∈Ψ such that

pðfx; fyÞ≤ψðN 0 ðx; yÞÞ

for all x; y∈X, where N0 ðx; yÞ ¼ max

�pðx; yÞ; pðx; fxÞ; pðy; fyÞ; pðx;fyÞþpðy;fxÞ

2

�. Then f has a

unique fixed point u in X with pðu; uÞ ¼ 0.

Proof. The proof follows from Theorem 3.9 by considering T ¼ f .

Corollary 3.12. Let ðX ; pÞ be a 0-complete partial metric space, and let the mappingsf ; T : X →X be such that

pðfx;TyÞ≤ α pðx; yÞ þ β pðx; fxÞ þ γ pðy;TyÞ þ δ ðpðx;TyÞ þ pðy; fxÞÞ (3.25)

for all x; y∈X, where α; β; γ; δ≥ 0 with αþ β þ γ þ 2δ < 1. Then f and T have a uniquecommon fixed point u in X with pðu; uÞ ¼ 0.

AJMS27,2

182

Proof. From condition (3.25), we obtain

pðfx; TyÞ≤ α pðx; yÞ þ β pðx; fxÞ þ γ pðy; TyÞ þ δ ðpðx; TyÞ þ pðy; fxÞÞ≤ ðαþ β þ γ þ 2δÞNðx; yÞ¼ kNðx; yÞ¼ ψðNðx; yÞÞ;

where k ¼ ðαþ β þ γ þ 2δÞ∈ ½0; 1ÞandψðtÞ ¼ kt for each t ≥ 0. Now applyingTheorem 3.9,we can obtain the desired result.

Remark3.13. The results of this study are obtained under theweaker assumption that theunderlying partial metric space is 0-complete. However, they also valid if the space iscomplete.

Finally we conclude this section by providing two applications of our main results.

Example 3.14. Let X ¼ f½3− 5−n; 3� : n∈ℕg∪ f½3; 3þ 5−n� : n∈ℕg∪ ff3gg, wheref3g ¼ ½3; 3�. We define p : X 3X →ℝþ by pð½a; b�; ½c; d�Þ ¼ max fb; dg−min fa; cg.Then ðX ; pÞ is a 0-complete partial metric space. Let A1 ¼ f½3− 5−n; 3� : n∈ℕg∪ ff3ggand A2 ¼ f½3; 3þ 5−n� : n∈ℕg∪ ff3gg. Obviously, X ¼ A1 ∪ A2. Define mappingsf ; g : X →X by

fx ¼8<:�3; 3þ 5−ðnþ2Þ�; if x ¼ ½3� 5−n; 3�;�3� 5−ðnþ2Þ; 3

�; if x ¼ ½3; 3þ 5−n�;

f3g; if x ¼ f3gand

gx ¼8<:�3� 5−ðnþ1Þ; 3

�; if x ¼ ½3� 5−n; 3�;�

3; 3þ 5−ðnþ1Þ�; if x ¼ ½3; 3þ 5−n�;f3g; if x ¼ f3g:

Then, f ðA1Þ⊆ gðA2Þ; f ðA2Þ⊆ gðA1Þ and so X ¼ A1 ∪ A2 is a cyclic representation of X withrespect to the pair ðf ; gÞ. Moreover, gðA1Þ; gðA2Þare closed subsets of ðX ; pÞ. We now verifycondition (3.1) with the control function ψ : ½0; ∞Þ→ ½0; ∞Þ given by ψðtÞ ¼ t

3. We now

consider the following cases:

Case-I. x ¼ ½3− 5−n; 3�∈A1; y ¼ ½3; 3þ 5−k�∈A2; n; k∈ℕwith n < k.

In this case, we have 5−k < 5−n and 5−k ≤ 5−ðnþ1Þ. Then,

pðfx; fyÞ ¼ p��3; 3þ 5−ðnþ2Þ�; �3� 5−ðkþ2Þ; 3

� ¼ 1

25

�5−n þ 5−k

<

2

25: 5−n;

pðgx; gyÞ ¼ p��3� 5−ðnþ1Þ; 3

�;�3; 3þ 5−ðkþ1Þ� ¼ 5−ðkþ1Þ þ 5−ðnþ1Þ

¼ 1

5: 5−k þ 5−ðnþ1Þ ≤

�1

5þ 1

�5−ðnþ1Þ ¼ 6

25: 5−n;

pðgx; fxÞ ¼ p��3� 5−ðnþ1Þ; 3

�;�3; 3þ 5−ðnþ2Þ� ¼ 5−ðnþ2Þ þ 5−ðnþ1Þ ¼ 6

25: 5−n;

pðgy; fyÞ ¼ p��3; 3þ 5−ðkþ1Þ�; �3� 5−ðkþ2Þ; 3

� ¼ 5−ðkþ1Þ þ 5−ðkþ2Þ <6

25: 5−n;

Generalizedcylic

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183

pðgx; fyÞ ¼ p��3� 5−ðnþ1Þ; 3

�;�3� 5−ðkþ2Þ; 3

� ¼ 5−ðnþ1Þ ¼ 1

5: 5−n;

pðfx; gyÞ ¼ p��3; 3þ 5−ðnþ2Þ�; �3; 3þ 5−ðkþ1Þ� ¼ 5−ðnþ2Þ ¼ 1

25: 5−n:

Now, pðgx; fyÞ þ pðfx; gyÞ2

¼ 1

2

�1

5: 5−n þ 1

25: 5−n

�¼ 3

25: 5−n <

6

25: 5−n:

Thus, Mðgx; gyÞ ¼ 625: 5−n. Therefore,

pðfx; fyÞ < 2

25: 5−n ¼ ψðMðgx; gyÞÞ:

Case-II. x ¼ ½3− 5−n; 3�∈A1; y ¼ ½3; 3þ 5−k�∈A2; n; k∈ℕwith n > k.

In this case, we have 5−k > 5−n and 5−n ≤ 5−ðkþ1Þ. Then, pðfx; fyÞ< 225:5

−k;pðgx; gyÞ ¼ 15

ð5−k þ 5−nÞ≤ 625:5

−k; pðgx; fxÞ ¼ 625:5

−n; pðgy; fyÞ ¼ 625:5

−k and pðgx; fyÞ ¼ 5−ðkþ2Þ ¼ 125:5

−k;

pðfx; gyÞ ¼ 5−ðkþ1Þ ¼ 15:5

−k. So, pðgx; fyÞþpðfx;gyÞ2 ¼ 6

25:5−k.

Thus, Mðgx; gyÞ ¼ 625:5

−k. Therefore,

pðfx; fyÞ< 2

25:5−k ¼ ψðMðgx; gyÞÞ:

Case-III. x ¼ ½3− 5−n; 3�∈A1; y ¼ ½3; 3þ 5−k�∈A2; n; k∈ℕwith n ¼ k.Then, pðfx; fyÞ¼ 2

25: 5−n;pðgx;gyÞ¼ 2

5: 5−n;pðgx; fxÞ¼ 6

25: 5−n;pðgy; fyÞ¼ 6

25: 5−n and pðgx; fyÞ¼

15: 5

−n;pðfx;gyÞ¼ 15: 5

−n. So, pðgx;fyÞþpðfx;gyÞ2 ¼ 1

5: 5−n.

Thus, Mðgx;gyÞ¼ 625: 5

−n. Therefore,

pðfx; fyÞ¼ 2

25: 5−n¼ψðMðgx;gyÞÞ:

Case-IV. x ¼ ½3− 5−n; 3�∈A1; n∈ℕ; y ¼ f3g∈A2.Then,

pðfx; fyÞ ¼ p��3; 3þ 5−ðnþ2Þ�; f3gÞ ¼ 5−ðnþ2Þ ¼ 1

25: 5−n;

pðgx; gyÞ ¼ p��3� 5−ðnþ1Þ; 3

�; f3gÞ ¼ 5−ðnþ1Þ ¼ 1

5: 5−n;

pðgx; fxÞ ¼ p��3� 5−ðnþ1Þ; 3

�;�3; 3þ 5−ðnþ2Þ� ¼ 5−ðnþ2Þ þ 5−ðnþ1Þ ¼ 6

25: 5−n;

pðgy; fyÞ ¼ pðf3g; f3gÞ ¼ 0; pðgx; fyÞ ¼ p��3� 5−ðnþ1Þ; 3

�; f3gÞ ¼ 1

5: 5−n;

pðfx; gyÞ ¼ p��3; 3þ 5−ðnþ2Þ�; f3gÞ ¼ 1

25: 5−n:Thus; Mðgx; gyÞ ¼ 6

25: 5−n:

Therefore; pðfx; fyÞ ¼ 1

25: 5−n <

2


AJMS27,2

184

Case-V. x ¼ f3g∈A1; y ¼ ½3; 3þ 5−n�∈A2; n∈ℕ.In this case, we have

pðfx; fyÞ ¼ 1

25: 5−n; Mðgx; gyÞ ¼ 6

25: 5−n:

Therefore,

pðfx; fyÞ ¼ 1

25: 5−n <

2


Case-VI. x ¼ y ¼ f3g is trivial.The other possibility is treated similarly. Moreover, f and g are weakly compatible. Thus,

all the conditions of Theorem 3.2 are fulfilled, and f3g is the unique common fixed point of fand g in gðA1Þ∩ gðA2Þwith pðf3g; f3gÞ ¼ 0.

Even if in the following example it is not hard to prove that 0 is the only commonfixed point, it can provide an alternative proof and can inspire other application ofTheorem 3.9.

Example 3.15. Let β∈ ð0; 1Þbe fixed, and letX ¼ A∪ f0g⊆ ‘1, where0 ¼ ð0Þ∞n¼1 and thesubset A of ‘1 defined by xq ¼ ðxqnÞ∞n¼1 ∈A if f

xqn ¼�0; if n < 2q; or n ¼ 2k� 1; k∈ℕ;βn; if n ¼ 2k≥ 2q;

for q ¼ 1; 2; 3; � � �. Define p : X 3X →ℝþ by pððxnÞ; ðynÞÞ ¼P∞

n¼1max fxn; yng for allðxnÞ; ðynÞ∈X. Then ðX ; pÞ is a 0-complete partial metric space. Let T; f : X →X be definedby

T

0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2q−1

; β2q; 0; β2qþ2; 0; � � �!

¼ 0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2qþ1

; β2qþ2; 0; β2qþ4; 0; � � �!;T0 ¼ 0

and

f

0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2q−1

; β2q; 0; β2qþ2; 0; � � �!

¼ 0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2qþ3

; β2qþ4; 0; β2qþ6; 0; � � �!; f0 ¼ 0:

Define ψ : ½0; ∞Þ→ ½0; ∞Þ by ψðtÞ ¼ βtWe now verify condition (3.13) for all x; y∈X.

Case-I. Take x ¼ 0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2q−1

; β2q; 0; β2qþ2; 0; � � �!∈A and

y ¼�0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2r−1

; β2r; 0; β2rþ2; 0; � � ��∈A:

Generalizedcylic

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185

Then

fx ¼ 0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2qþ3

; β2qþ4; 0; β2qþ6; 0; � � �!;

Ty ¼�0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2rþ1

; β2rþ2; 0; β2rþ4; 0; � � ��:

(1) If r≤ q, then

pðfx; TyÞ ¼ β2rþ2 þ β2rþ4 þ � � � þ β2qþ2 þ β2qþ4 þ β2qþ6 þ � � �

¼ β2�β2r þ β2rþ2 þ � � � þ β2q−2 þ β2q

��þ β2qþ4

1� β2;

pðx; yÞ ¼ β2r þ β2rþ2 þ � � � þ β2q−2 þ β2q þ β2qþ2 þ β2qþ4 þ � � �

¼ β2r þ β2rþ2 þ � � � þ β2q−2 þ β2q þ β2qþ2

1� β2:

Thus,pðfx; TyÞ ¼ β2pðx; yÞ < β pðx; yÞ≤ β Nðx; yÞ ¼ ψðNðx; yÞÞ:

(2) If r > q, then 2r þ 2 > 2qþ 202r þ 2≥ 2qþ 4 and

pðfx; TyÞ ¼ β2qþ4 þ β2qþ6 þ � � � þ β2r þ β2rþ2 þ β2rþ4 þ � � �

¼ β2qþ4 þ β2qþ6 þ � � � þ β2r−2 þ β2r þ β2rþ2

1� β2;

pðx; yÞ ¼ β2q þ β2qþ2 þ � � � þ β2r−2 þ β2r þ β2rþ2 þ β2rþ4 þ � � �

¼ β2q þ β2qþ2 þ � � � þ β2r−4 þ β2r−2 þ β2r

1� β2;

which implies that

pðfx; TyÞ < β2 pðx; yÞ < β pðx; yÞ≤ βNðx; yÞ ¼ ψðNðx; yÞÞ:

Case-II. Take x ¼ 0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2q−1

; β2q; 0; β2qþ2; 0; � � �!∈A and y ¼ 0.

Then

fx ¼ 0; � � � ; 0|fflfflfflfflffl{zfflfflfflfflffl}

2qþ3

; β2qþ4; 0; β2qþ6; 0; � � �!; Ty ¼ 0;

pðfx; TyÞ ¼ β2qþ4 þ β2qþ6 þ � � � ¼ β2qþ4

1� β2;

AJMS27,2

186

pðx; yÞ ¼ β2q þ β2qþ2 þ � � � ¼ β2q

1� β2:

Thus,

pðfx; TyÞ ¼ β4pðx; yÞ < βpðx; yÞ≤ βNðx; yÞ ¼ ψðNðx; yÞÞ:

The case x ¼ 0; y∈A can be treated similarly, and the case x ¼ y ¼ 0 is trivial.Thus, we have all the conditions of Theorem 3.9, and 0 is the unique common fixed point off and T in X with pð0; 0Þ ¼ 0.

References

1. Matthews S. Partial metric topology. Ann N.Y. Acad Sci. 1994 (728): 183-97.

2. Altun I, Acar O. Fixed point theorems for weak contractions in the sense of Berinde on partialmetric spaces. Topol Appl. 2012; 159: 2642-48.

3. Altun I, Sola F, Simsek H. Generalized contractions on partial metric spaces. Topol Appl. 2010;157: 2778-85.

4. Bukatin M, Kopperman R, Matthews S, Pajoohesh H. Partial metric spaces. Am Math Mon. 2009;116: 708-18.

5. Ciric L, Samet B, Aydi H, Vetro C. Common fixed points of generalized contractions on partialmetric spaces and an application. Appl Math Comput. 2011; 218: 2398-2406.

6. Heckmann R. Approximation of metric spaces by partial metric spaces. Appl Categ. Struct. 1999;7: 71-83.

7. Karapinar E. A note on common fixed point theorems in partial metric spaces. Miskolc MathNotes. 2011; 12: 185-91.

8. Mohanta SK. Common fixed point theorems via w-distance. Bull Math Anal Appl. 2011; 3: 182-89.

9. Mohanta SK, Mohanta S. A common fixed point theorem in G-metric spaces. Cubo, A Math J.2012; 14: 85-101.

10. Mohanta SK, Patra S. Coincidence points and common fixed points for hybrid pair of mappings inb-metric spaces endowed with a graph. J Lin Top Alg. 2017; 6: 301-21.

11. Kirk WA, Srinivasan PS, Veeramani P. Fixed points for mappings satisfying cyclical contractiveconditions. Fixed Point Theory. 2003; 4: 79-89.

12. Agarwal RP, Alghamdi MA, Shahzad N. Fixed point for cyclic generalized contractions in partialmetric spaces. Fixed Point Theory Appl. 2012; 2012(40): 1-11.

13. Abbas M, Nazir T, Romaguera S. Fixed point results for generalized cyclic contraction mappingsin partial metric spaces. Rev Real Acad Ciencias Exactas, Fis. Nat. 2012; 106: 287-97.

14. Bari CD, Vetro P. Fixed points for weak w-contractions on partial metric spaces. Int J ContempMath Sci. 2011; 1: 5-12.

15. Karapinar E, Yuce IS. Fixed point theory for cyclic generalized weak w-contraction on partialmetric spaces. Abs Appl Anal. 2012; 2012, 491542.

16. Karapinar E, Shobkolaei N, Sedghi S, Vaezpour SM. A common fixed point theorem for cyclicoperators in partial metric spaces. Filomat. 2012; 26: 407-14.

17. Shatanawi W, Postolache M. Common fixed point results for mappings under nonlinearcontraction of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2010; 2010, 493298.

18. Khan MS, Swaleh M, Sessa S.. Fixed point theorems by altering distances between the points. BullAust Math Soc. 1984; 30: 1-9.

Generalizedcylic

contractions

187

19. He F, Chen A. Fixed points for cyclic f-contractions in generalized metric spaces. Fixed PointTheory Appl. 2016; 2016(67): 1-12.

20. Nashine HK, Kadelburg Z. Cyclic contractions and fixed point results via control functions onpartial metric spaces. Int J Anal. 2013; 2013, 726387.

21. Pacurar M, Rus IA. Fixed point theory for cyclic f-contractions. Nonlinear Anal. 2010; 72:1181-87.

22. Yamaod O, Sintunavarat W, Cho YJ. Common fixed point theorems for generalized cycliccontraction pairs in b-metric spaces with applications. Fixed Point Theory Appl. 2015; 2015: 164.

23. Romaguera S. A Kirk type characterization of completeness for partial metric spaces. Fixed PointTheory Appl. 2010; 2010: 493298.

24. Abdeljawad T, Karapinar E, Tas K. Existence and uniqueness of a common fixed point on partialmetric spaces. Appl Math Lett. 2011; 24: 1900-04.

25. Jungck G. Common fixed points for noncontinuous nonself maps on non-metric spaces. Far East JMath Sci. 1996; 4: 199-215.

26. Abbas M, Jungck G. Common fixed point results for noncommuting mappings without continuityin cone metric spaces. J Math Anal Appl. 2008; 341: 416-20.

27. Kaushik P, Kumar S. Fixed point results forðα; ψ ; ξÞcontractive compatible multi-valuedmappings. J Nonlinear Anal Appl. 2016; 2016(2): 28-36.

28. Samet B, Turinici M. Fixed point theorems on a metric space endowed with an arbitrary binaryrelation and applications. Commun Math Anal. 2012; 13: 82-97.

Corresponding authorSushanta Kumar Mohanta can be contacted at: [email protected]


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Multistep-type constructionof fixed point for multivaluedρ-quasi-contractive-like mapsin modular function spaces

Hudson Akewe and Hallowed OlaoluwaDepartment of Mathematics, University of Lagos, Lagos, Nigeria

Abstract

Purpose – In this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequencesare introduced in the context of modular function spaces and proven to converge to the fixed point of amultivalued map T such that PT

ρ , an associate multivalued map, is a ρ-contractive-like mapping.Design/methodology/approach – The concepts of relative ρ-stability and weak ρ-stability are introduced,and conditions in which these multistep iterations are relatively ρ-stable, weakly ρ-stable and ρ-stable areestablished for the newly introduced strong ρ-quasi-contractive-like class of maps.Findings – Noor type, Ishikawa type and Mann type iterative sequences are deduced as corollaries inthis study.Originality/value – The results obtained in this work are complementary to those proved in normed andmetric spaces in the literature.

Keywords Multistep iterations, Modular function spaces, Strong ρ-contractions, Relative ρ-stability,Weakly ρ-stabilityPaper type Research paper

1. Introduction and preliminary definitionsModular function spaces are well-known generalizations of both function and sequencevariants of many important spaces such as Calderon–Lozanovskii, Kothe, Lebesgue, Lorentz,Musielak–Orlicz, Orlicz and Orlicz–Lorentz spaces. Their applications are also very useful.There is huge interest in quasi-contractive mappings in modular function spaces mainlybecause of the richness of structure of modular function spaces: apart from being F-spaces inamore general setting, they are equippedwithmodular equivalents of norm ormetric notionsand also endowed with convergence in submeasure. It is worthy to mention that modular-type conditions are far more natural as their assumptions can be easily verified than theircorresponding metrics or norms, especially when related to fixed-point results andapplications to integral-type operators. More so, there are some fixed-point results that canbe proved only using the framework of modular function spaces. Thus, results in fixed-pointtheory inmodular function spaces and those in normed andmetric spaces are complementary(see, e.g. [1]). Different researchers have proved very useful fixed-points results in the contextof modular function spaces (see [1–6] for details).

The following background definitions in [1, 3, 7] are useful in proving the main results inthis manuscript:

Multistep-typeconstruction of

fixed point...

189

JEL Classification — 47H10, 54H25© Hudson Akewe and Hallowed Olaoluwa. Published in Arab Journal of Mathematical Sciences.

Published by Emerald Publishing Limited. This article is published under the Creative CommonsAttribution (CCBY4.0) licence. Anyonemay reproduce, distribute, translate and create derivativeworksof this article (for both commercial and non-commercial purposes), subject to full attribution to theoriginal publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode



Received 23 July 2020Revised 29 October 2020

Accepted 11 November 2020


Vol. 27 No. 2, 2021pp. 189-213


DOI 10.1108/AJMS-07-2020-0026

Let Ω be a nonempty set and Σ be a nontrivial σ − algebra of subsets of Ω: Let P be aδ− ring of subsets ofΩsuch thatE ∩ A∈P for anyE ∈P andA∈Σ:Assume there exists anincreasing sequence ðKnÞn∈ℕ⊂P such that Ω ¼ ∪n∈ℕKn:

Let E represent the linear space of all simple functions with supports from P, that is,functions s ¼ Pn

k¼1

αkIAk, where ðαkÞk∈ℕ is a sequence of real numbers, ðAkÞk∈ℕ is a sequence of

disjoint sets in P and IA represents the characteristic function of the set A in Ω:Let M∞ represent the space of all extended measurable functions, that is, all functions

f : Ω→ ½−∞; ∞� such that there exists a sequence ðgnÞ⊂ E satisfying jgnj≤ jf j andgnðωÞ→ f ðωÞ for all ω∈Ω:

Definition 1.1. ([7]). Let ρ : M∞ → ½0; ∞� be a nontrivial, convex and even function. ρ issaid to be a regular convex function pseudomodular if:

(1) ρð0Þ ¼ 0;

(2) ρ is monotone, that is, jf j≤ jgj on Ω implies ρðf Þ≤ ρðgÞ; where f ; g∈M∞;

(3) ρ is orthogonally subadditive, that is, ρðfIA∪BÞ≤ ρðfIAÞ þ ρðfIBÞ for any A; B∈Ωsuch that A∩B≠f; with f ∈M∞;

(4) ρ has Fatou’s property, that is, jfnðωÞj↑jf ðωÞj for allω∈Ω implies ρðfnÞ↑ρðf Þ; wheref ∈M∞;

(5) ρ is order continuous in E, that is, ðgnÞ⊂ E and jgnðωÞj↓0for allω∈Ω implies ρðgnÞ↓0:

Concepts similar to those in measure spaces are defined for function pseudomodular ρ: a setA∈Σ is said to be ρ-null if ρðfIAÞ ¼ 0 ∀f ∈ E; a property is said to hold ρ-almost everywhere(ρ-a.e.) on Σ if the set for which it does not hold is ρ-null.

The following set is defined:

MðΩ; Σ; P; ρÞ ¼ ff ∈M∞ : jf j < ∞ ρ� a:e:g;where each f ∈M∞ is actually an equivalence class of functions equal ρ-a.e. We will writeMinstead of MðΩ; Σ; P; ρÞwhen no confusion arises.

Definition 1.2. ([1]). Let ρ be a regular function pseudomodular.

(1) ρ is said to be a regular function modular if ρðf Þ ¼ 0 implies f ¼ 0 ρ-a.e.

(2) ρ is said to be a regular function semimodular if ρðαf Þ ¼ 0 for every α > 0 impliesf ¼ 0 ρ-a.e.

A regular convex function modular ρ satisfies the following properties (see [3])

(1) ρðf Þ ¼ 0 if f ¼ 0ρ -a.e.

(2) ρðαf Þ ¼ ρðf Þ for every scalar α such that jαj ¼ 1, where f ∈M:

(3) ρðαf þ βgÞ≤ αρðf Þ þ βρðgÞ if αþ β ¼ 1, α; β≥ 0 and f ; g ∈M:

The class of all nonzero regular convex function modulars on Ω is denoted by ℜ.

Definition 1.3. ([7]). A convex functionmodular ρ defines themodular function space Lρ as

Lρ ¼ ff ∈M : ρðλf Þ→ 0 as λ→ 0g:

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190

Lρ is a normed linear space with respect to

jfρj ¼ inf

�α > 0 : ρ

�f

α

�≤ 1

�

which is known as the Luxemburg norm.

Definition 1.4. ([7]). Let Lρ be a modular space. The sequence ffng⊂Lρ is called:

(1) ρ−convergent to f ∈Lρ if ρðfn − f Þ→ 0 as n→∞;

(2) ρ−Cauchy, if ρðfn − fmÞ→ 0 as n; m→∞.

Remark 1.1. ρ−convergent sequence implies ρ−Cauchy sequence if and only if ρsatisfies the Δ2 – condition given in the definition below. However, ρ does not satisfy thetriangle inequality.

Definition 1.5. ([7]). A nonzero regular convex function ρ is said to satisfy the Δ2 −

condition, if supn≥1ρð2fn; DkÞ→ 0 as k→∞ whenever fDkg↓0== and supn≥1ρðfn; DkÞ→ 0as k→∞:

Definition 1.6. ([7]). Let Lρ be a modular space and D⊂Lρ.

The ρ-distance from f ∈Lρ to the set D is given by:

distρðf ; DÞ ¼ inffρðf � hÞ : h∈Dg:A subset D⊂Lρ is called:

(1) ρ−closed if the ρ− limit of a ρ−convergent sequence of D always belongs to D;

(2) ρ−a.e. closed if the ρ−a.e. limit of a ρ−a.e. convergent sequence ofD always belongsto D;

(3) ρ−compact if every sequence in D has a ρ−convergent subsequence in D;

(4) ρ−a.e. compact if every sequence in D has a ρ−a.e. convergent subsequence in D;

(5) ρ−bounded if diamρðDÞ ¼ supfρðf − gÞ : f ; g∈Dg< ∞:

(6) ρ−proximal if for each f ∈Lρ there exists an element g ∈D such thatρðf − gÞ ¼ distρðf ; DÞ.

The family of nonempty ρ-bounded ρ-proximal subsets of D is denoted by PρðDÞ; thefamily of nonempty ρ-closed ρ-bounded subsets of D by CρðDÞ and the family of ρ-compactsubsets of D by KρðDÞ:Definition 1.7. ([7]). Let Lρ be amodular space. A function f ∈Lρ is called a fixed point of amultivalued mappingT : Lρ →PρðDÞ if f ∈Tf . The set of all fixed points ofT is representedby FρðTÞ so that:

FρðTÞ ¼ ff ∈Lρ : f ∈Tfg:The following set is also defined:

PTρ ðf Þ ¼ fg ∈Tf : ρðf � gÞ ¼ distρðf ; Tf Þg:

Zamfirescu [8] in 1972 proved the following theorem as a generalization of the Banach fixed-point theorem:


fixed point...

191

Theorem1.1. ([8]). LetX be a complete metric space andT : X →X a Zamfirescu operatorsatisfying:

dðTx; TyÞ≤ hmax

�dðx; yÞ; dðx; TxÞ þ dðy; TyÞ

2;dðx; TyÞ þ dðy; TxÞ�

2

�; (1.1)

where 0≤ h < 1. Then, T has a unique fixed point and the Picard iteration converges to pfor any x0 ∈X.

Observe that in a Banach space setting, condition (1.1) implies

kTx� Tyk≤ δkx� yk þ 2δkx� Txk; δ ¼ max

�h;

h

2� h

�∈ ½0; 1Þ (1.2)

Osilike [9] used the following contractive definition: for each x; y∈X ; there exist δ∈ ½0; 1Þand L≥ 0 such that jjTx� Tyjj≤ δkx� yk þ Ljjx� Txjj: (1.3)

Imoru andOlatinwo [10] proved some stability results using the following general contractivedefinition: for each x; y∈X ; there exist δ∈ ½0; 1Þ and a monotone increasing functionw : ℝþ

→ℝþ with wð0Þ ¼ 0 such that

kTx� Tyk≤ δkx� yk þ wðjjx� TxjjÞ: (1.4)

Observe that (1.4) generalizes (1.3) and (1.2). The map T considered in (1.2)–(1.4) is single-valued. Now, we state the generalizations of (1.2)–(1.4) to multivalued mappings, asconformed to literature. (e.g. see [7]).

Let Hρð$; $Þ be the ρ−Hausdorff distance on the family CρðLρÞ of nonempty ρ-closedρ-bounded subsets of Lρ, that is,

HρðA; BÞ ¼ max�supf∈A

distρðf ; BÞ; supg∈B

distρðg; AÞ�; A; B∈CρðLρÞ:

A multivalued map T : D→CρðLρÞ is said to be a:

(1) ρ−contraction mapping if there exists a constant δ∈ ½0; 1Þ such that

HρðTf ; TgÞ≤ δρðf � gÞ; ∀f ; g ∈D: (1.5)

(2) ρ−Zamfirescu mapping if

HρðTf ; TgÞ≤ δρðf � gÞ þ 2δρðh� f Þ; ∀f ; g ∈D ∀h∈Tf : (1.6)

(3) ρ−quasi-contractive mapping if

HρðTf ; TgÞ≤ δρðf � gÞ þ Lρðh� f Þ; ∀f ; g ∈D ∀h∈Tf ; L≥ 0: (1.7)

(4) ρ−quasi-contractive-like mapping if

HρðTf ; TgÞ≤ δρðf � gÞ þ wðρðh� f ÞÞ; ∀f ; g ∈D ∀h∈Tf : (1.8)

where w : ℝþ→ℝþ is a monotone increasing function with wð0Þ ¼ 0:

Convergence and stability of fixed-point iterative sequences for single mappingT are twovery vital concepts in fixed-point theory and applications. Some of the results of colossalvalue in this work are those in [9–20]. Rhoades and Soltuz [21] introduced the multistepiteration and proved its equivalence with Mann and Ishikawa iterations. Olaleru and Akewe[22] proved convergence of multistep iteration for a pair of mappings ðS; TÞ

AJMS27,2

192

We now introduce the following iterative sequences in the framework of modular functionspaces and use them to prove new fixed-point theorems.

Let T : D→PρðDÞ be a multivalued mapping.The explicit multistep iterative sequence ffng∞n¼0 ⊂D is defined by:8>>>>><

>>>>>:

f0 ∈ D

fnþ1 ¼ ð1� αnÞfn þ αnv1n;

gin ¼ �1� βin

fn þ βinv

iþ1n ; i ¼ 1; 2; . . . ; k� 2

gk−1n ¼ �1� βk−1n

fn þ βk−1n un; n ¼ 0; 1; 2; . . .

(1.9)

where un ∈PTρ ðfnÞ, vin ∈PT

ρ ðginÞ, i ¼ 1; 2; . . . ; k− 1, and the sequences fαng∞n¼0 and fβing∞

n¼0,

i ¼ 1; 2; . . . ; k− 1; are in ½0; 1Þ such thatP∞

n¼0αn ¼ ∞:The explicit Noor iterative sequence ffng∞n¼0 ⊂D is defined by:8>>>>><

>>>>>:

f0 ∈ D


g1n ¼ �1� β1n

fn þ β1nv

2n;

g2n ¼ �1� β2n

fn þ β2nun; n ¼ 0; 1; 2; . . .

(1.10)

where un ∈PTρ ðfnÞ; v1n ∈PT

ρ ðg1nÞ; v2n ∈PTρ ðg2nÞ; and the sequences fαng∞n¼0, fβ1ng

∞

n¼0 and

fβ2ng∞

n¼0 are in ½0; 1Þ such thatP∞

n¼0αn ¼ ∞:The explicit Ishikawa iterative sequence ffng∞n¼0 ⊂D is defined by:8>><

>>:f0 ∈ D


g1n ¼ �1� β1n

fn þ β1nun; n ¼ 0; 1; 2; . . .

(1.11)


ρ ðg1nÞ; and the sequences fαng∞n¼0 and fβ1ng∞

n¼0 are in ½0; 1Þ suchthat

P∞

n¼0αn ¼ ∞:The explicit Mann iterative sequence ffng∞n¼0 ⊂D is defined by:�

f0 ∈ D

fnþ1 ¼ ð1� αnÞfn þ αnun; n ¼ 0; 1; 2; . . .(1.12)

where un ∈PTρ ðfnÞ; fαng∞n¼0 ⊂ ½0; 1Þ andP∞

n¼0αn ¼ ∞:

The explicit multistep-SP iterative sequence ffng∞n¼0 ⊂D is defined by:8>>>>><>>>>>:

f0 ∈ D

fnþ1 ¼ ð1� αnÞg1n þ αnv1n

gin ¼ �1� βin

giþ1n þ βinv

iþ1n ; i ¼ 1; 2; . . . ; k� 2

gk−1n ¼ �1� βk−1n

fn þ βk−1n un; n ¼ 0; 1; 2; . . .

(1.13)

where un ∈PTρ ðfnÞ; vin ∈PT

ρ ðginÞ, i ¼ 1; 2; . . . ; k− 1, and the sequences fαng∞n¼0 and fβing∞

n¼0,

i ¼ 1; 2; . . . ; k− 1, are in ½0; 1Þ such thatP∞

n¼0αn ¼ ∞:


fixed point...

193

The explicit SP iterative sequence ffng∞n¼0 ⊂D is defined by:

8>>>>><>>>>>:

f0 ∈ D

fnþ1 ¼ ð1� αnÞg1n þ αnv1n;

g1n ¼ �1� β1n

g2n þ β1nv

2n;

g2n ¼ �1� β2n

fn þ β2nun; n ¼ 0; 1; 2; . . .

(1.14)


ρ ðg1nÞ; v2n ∈PTρ ðg2nÞ; and the sequences fαng∞n¼0; fβ1ng

∞

n¼0, and

fβ2ng∞


0 αn ¼ ∞:The implicit multistep iterative sequence ffng∞n¼0 ⊂D is defined by:

8>>>>>><>>>>>>:

f0 ∈ D

fnþ1 ¼ ð1� αnÞf 1n þ αnunþ1;

f in ¼ �1� βin

f iþ1n þ βinu

in; i ¼ 1; 2; . . . ; k� 2

f k−1n ¼ �1� βk−1n

fn þ βk−1n uk−1n ; n ¼ 0; 1; 2; . . .

(1.15)

where unþ1 ∈PTρ ðfnÞ; uin ∈PT

ρ ðf inÞ; i ¼ 1; 2; . . . ; k− 1, and the sequences fαng∞n¼0 and

fβing∞

n¼0; i ¼ 1; 2; . . . ; k− 1, are in ½0; 1Þ such thatP∞

n¼0αn ¼ ∞:It should be noted that the implicit multistep iterative sequence exists if and only if T

satisfies the property (I) as follows:

ðIÞ : ∀h∈D ∀β∈ ð0; 1Þ ∃f ∈D ∃g ∈PTρ ðf Þ : f ¼ ð1� βÞhþ βg:

The implicit Noor iterative sequence ffng∞n¼0 ⊂D is defined by:

8>>>>>><>>>>>>:

f0 ∈ D


f 1n ¼ �1� β1n

f 2n þ β1nu

1n;

f 2n ¼ �1� β2n

fn þ β2nu

2n; n ¼ 0; 1; 2; . . .

(1.16)

where unþ1 ∈PTρ ðfnþ1Þ; u1n ∈PT

ρ ðf 1n Þ; u2n ∈PTρ ðf 2n Þ; and the sequences fαng∞n¼0; fβ1ng

∞

n¼0, and

fβ2ng∞


n¼0αn ¼ ∞:The implicit Ishikawa iterative sequence ffng∞n¼0 ⊂D is defined by:

8>><>>:

f0 ∈ D


f 1n ¼ �1� β1n

fn þ β1nu

1n; n ¼ 0; 1; 2; . . .

(1.17)

where unþ1 ∈ PTρ ðfnþ1Þ; u1n ∈PT

ρ ðf 1n Þ; fαng∞n¼0 ⊂ ½0; 1Þ, fβ1ng∞

n¼0 ⊂ ½0; 1Þ andP∞

n¼0αn ¼ ∞:

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194

The implicit Mann iterative sequence ffng∞n¼0 ⊂D is defined by:�f0 ∈ D

fnþ1 ¼ ð1� αnÞfn þ αnunþ1; n ¼ 0; 1; 2; . . .(1.18)

where unþ1 ∈PTρ ðfnþ1Þ; fαng∞n¼0 ⊂ ½0; 1Þ andP∞

n¼0αn ¼ ∞:

The following Lemmas will be needed in proving the main results.

Lemma 1.1. ([3]). Let T : D→PρðDÞ be a multivalued mapping and PTρ ðf Þ ¼ fg ∈Tf :

ρðf − gÞ ¼ distρðf ; Tf Þg: Then the following are equivalent:

(1) f ∈FρðTÞ; that is, f ∈Tf :

(2) PTρ ðf Þ ¼ ffg:

(3) f ∈FðPTρ ðf ÞÞ; that is, f ∈PT

ρ ðf Þ: Further FρðTÞ ¼ FðPTρ ðf ÞÞ where FðPT

ρ ðf ÞÞrepresent the set of fixed points of PT

ρ ðf Þ:

Lemma1.2. (see [13]). Let δ be a real number satisfying 0≤ δ < 1and fεng∞n¼0 and fτng∞n¼0two sequences of positive or zero numbers, less than 1, such that limn→∞εn ¼ 0 andP∞

n¼0τn ¼ ∞. Then any sequence of positive numbers fung∞n¼0 satisfying any of thefollowing properties converges to 0:

(1) unþ1 ≤ δun þ εn for all n ¼ 0; 1; 2; . . .

(2) unþ1 ≤ ð1− τnÞun for all n ¼ 0; 1; 2; . . .

(3) unþ1 ≤ εn þ ð1− τnÞun for all n ¼ 0; 1; 2; . . . if in addition, fτng∞n¼0 is bounded awayfrom 0.

2. Convergence results2.1 Strong convergence results for explicit multistep iterative sequences in modular functionspaces

Theorem 2.1. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ, and T : D→PρðDÞ be a multivalued mapping such that PT

ρ is a ρ−quasi-

contractive-like mapping, satisfying contractive-like condition (1.8). Suppose that FρðTÞ≠∅.Let f0 ∈D and ffng⊂Dbe defined by the explicit multistep iterative sequence (1.9), where the

sequences fαng∞n¼0; fβing∞

n¼0 ⊂ ½0; 1Þ, ði ¼ 1; 2; . . . ; k− 1Þ are such thatP∞

0 αn ¼ ∞:Thenthe explicit multistep iterative sequence (1.9) converges strongly to the fixed point of T:

Proof. Let f ∈FρðTÞ; from Lemma 1.1, PTρ ðf Þ ¼ ffg and FρðTÞ ¼ FðPT

ρ ðf ÞÞ.Using the explicit multistep iterative sequence (1.9) and the convexity of ρ, we obtain the

following estimate:

ρðfnþ1 � f Þ ¼ ρ½ð1� αnÞfn þ αnv1n � f

(2.1)

¼ ρ�ð1� αnÞðfn � f Þ þ αn

�v1n � f

≤ð1� αnÞρðfn � f Þ þ αnρ

�v1n � f

:

v1n ∈PTρ ðg1nÞ and PT

ρ ðf Þ ¼ ffg imply that:


fixed point...

195

ρ�v1n � f

¼ distρ�v1n; P

Tρ ðf Þ

≤Hρ

�PTρ ðg1n

; PT

ρ ðf ÞÞ;

which combined with (2.1) yields:

ρðfnþ1 � f Þ≤ ð1� αnÞρðfn � f Þ þ αnHρ

�PTρ ðg1n

; PT

ρ ðf ÞÞ: (2.2)

In (1.8), letting g ¼ g1n and noting that PTρ ðf Þ ¼ ffg and wð0Þ ¼ 0, we have:

Hρ

�PTρ ðg1n

; PT

ρ ðf ÞÞ≤ δρ�g1n � f

þ wðf � hÞ ∀h∈PTρ ðf Þ (2.3)

≤ δρ�g1n � f

:

Substituting (2.3) in (2.2), we obtain

ρðfnþ1 � f Þ≤ ð1� αnÞρðfn � f Þ þ δαnρ�g1n � f

: (2.4)

Similarly, from (1.9) and the convexity of ρ,

ρ�g1n � f

¼ ρ��1� β1n

fn þ β1nv

2n � f

(2.5)

¼ ρ��1� β1n

ðfn � f Þ þ β1n�v2n � f

≤�1� β1n

ρðfn � f Þ þ β1nρ

�v2n � f

:

v2n ∈PTρ ðg2nÞ and PT

ρ ðf Þ ¼ ffg imply that:

ρ�v2n � f

¼ distρ�v2n; P

Tρ ðf Þ

≤Hρ

�PTρ ðg2n

; PT

ρ ðf ÞÞ;


ρ�g1n � f

≤�1� β1n

ρðfn � f Þ þ β1nHρ

�PTρ ðg2n

; PT

ρ ðf ÞÞ: (2.6)

In (1.8), letting g ¼ g2n and noting that PTρ ðf Þ ¼ ffg and wð0Þ ¼ 0, we get:

ρ�g1n � f

≤�1� β1n

ρðfn � f Þ þ δβ1nρ

�g2n � f

: (2.7)

Similarly, an application of (1.8) and (1.9) gives

ρ�g2n � f

≤�1� β2n


�g3n � f

: (2.8)

Also, an application of (1.8) and (1.9) gives

ρ�g3n � f

≤�1� β3n


�g4n � f

: (2.9)

Substituting (2.9) in (2.8), (2.8) in (2.7) and (2.7) in (2.4), and simplifying, we obtain

ρðfnþ1 � f Þ≤ �1� ð1� δÞαn � ð1� δÞδαnβ1n � ð1� δÞδ2αnβ

1nβ

2n (2.10)

−ð1� δÞδ3αnβ1nβ

2nβ

3n

þ δ3αnβ1nβ

2nβ

3nρ�g4n � f

:

Continuing this process, an application of (1.8) and (1.9) gives

ρ�gk−2n � f

≤�1� βk−2n

ρðfn � f Þ þ δβk−2n ρ

�gk−1n � f

: (2.11)

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196

and

ρ�gk−1n � f

≤�1� βk−1n

ρðfn � f Þ þ δβk−1n ρðfn � f Þ: (2.12)

Substituting (2.12) and (2.11) in (2.10) inductively and simplifying, we obtain

ρðfnþ1 � f Þ≤"1� ð1� δÞαn �

Xk−1i¼1

ð1� δÞδiαnβ1nβ

2n . . . β

in (2.13)

þ δkαnβ1nβ

2nβ

3nβ

4n . . . β

k−1n

ρðfn � f Þ

≤½1� ð1� δÞαn�ρðfn � f Þ:From (2.13), we inductively obtain

ρðfnþ1 � f Þ≤"Ym¼0

n

ð1� ð1� δÞαmÞ#ρðf0 � f Þ: (2.14)

Using that fact that δ∈ ½0; 1Þ fαng∞n¼0 ⊂ ½0; 1Þ satisfying P∞

n¼0αn ¼ ∞, then from (2.14),we obtain

limn→∞

ρðfnþ1 � f Þ≤ limn→∞

Ym¼0

n

½1� ð1� δÞαm�ρðf0 � f Þ ¼ 0: (2.15)

Therefore, ffng ρ-converges to f ∈FρðTÞ: The proof is complete. ▪Since the explicit Noor (1.10), explicit Ishikawa (1.11), explicit Mann (1.12) iterative

sequences are special cases of the explicit multistep iterative sequence (1.9) (see [22] fordetails), then Theorem 2.1 leads to the following corollary:

Corollary 2.1. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ, and T : D→PρðDÞ be a multivalued mapping such that PT

ρ is a ρ−quasi-contractive-like mapping, satisfying contractive-like condition (1.8). Suppose thatFρðTÞ≠ 0==. Let f0 ∈D and ffng⊂D be defined by the explicit Noor (1.10), the explicitIshikawa (1.11) and the explicit Mann (1.12) iterative sequences respectively, where thesequences fαng∞n¼0; fβ1ng

∞

n¼0; fβ2ng∞

n¼0 ⊂ ½0; 1Þ are such thatP∞

0 αn ¼ ∞: Then:

(1) the explicit Noor iterative sequence (1.10) converges strongly to the fixed pointof T:

(2) the explicit Ishikawa iterative sequence (1.11) converges strongly to the fixed pointof T:

(3) the explicit Mann iterative sequence (1.12) converges strongly to the fixed point ofT:

2.2 Strong convergence results for explicit multistep-SP iterative sequences in modularfunction spaces

Theorem 2.2. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ, and T : D→PρðDÞ be a multivalued mapping such that PT

ρ is a ρ−quasi-contractive-like mapping, satisfying contractive-like condition (1.8). Suppose thatFρðTÞ≠ 0==. Let f0 ∈D and ffng⊂D be defined by the explicit multistep-SP iterativesequence (1.13), where the sequences fαng∞n¼0; fβing

∞

n¼0 ⊂ ½0; 1Þ, ði ¼ 1; 2; . . . ; k− 1Þ are


fixed point...

197

such thatP∞

0 αn ¼ ∞:Then the explicit multistep-SP iterative sequence (1.13) ρ-converges toa fixed point of T:

Proof. Let f ∈FρðTÞ. From Lemma 1.1, we have that PTρ ðf Þ ¼ ffg and FρðTÞ ¼ FðPT

ρ ðf ÞÞ.Using the explicit multistep-SP iterative sequence (1.13) and the convexity of ρ, we obtain thefollowing estimate:

ρðfnþ1 � f Þ ¼ ρ½ð1� αnÞg1n þ αnv1n � f

(2.16)

¼ ρ�ð1� αnÞ

�g1n � f

þ αn

�v1n � f

≤ð1� αnÞρ

�g1n � f

þ αnρ�v1n � f

:

Since v1n ∈PTρ ðg1nÞ and PT

ρ ðf Þ ¼ ffg, we haveρ�v1n � f

¼ distρ�v1n; P

Tρ ðf Þ

≤Hρ

�PTρ ðg1n

; PT

ρ ðf ÞÞ;


ρðfnþ1 � f Þ≤ ð1� αnÞρ�g1n � f

þ αnHρ

�PTρ ðg1n

; PT

ρ ðf ÞÞ: (2.17)

In (1.8), letting g ¼ g1n and noting that PTρ ðf Þ ¼ ffg and wð0Þ ¼ 0, we get

Hρ

�PTρ ðg1n

;PT

ρ ðf ÞÞ≤ δρ�g1n � f

þ wð0Þ ¼ δρ�g1n � f

: (2.18)


ρðfnþ1 � f Þ≤ ð1� αnÞρ�g1n � f

þ δαnρ�g1n � f

(2.19)

¼ ½1� ð1� δÞαn�ρ�g1n � f

:

Next, from (1.13) and the convexity of ρ,

ρ�g1n � f

¼ ρ��1� β1n

g2n þ β1nv

2n � f

(2.20)

¼ ρ��1� β1n

�g2n � f

þ β1n�v2n � f

≤�1� β1n

ρ�g2n � f

þ β1nρ�v2n � f

:

Since v2n ∈PTρ ðg2nÞ and PT

ρ ðf Þ ¼ ffg, we haveρ�v2n � f

¼ distρ�v2n; P

Tρ ðf Þ

≤Hρ

�PTρ ðg2n

; PT

ρ ðf ÞÞ;


ρ�g1n � f

≤�1� β1n

ρ�g2n � f

þ β1nHρ

�PTρ ðg2n

; PT

ρ ðf ÞÞ: (2.21)

Using (1.8) with g ¼ g2n in (2.21) and noting that wð0Þ ¼ 0 and PTρ ðf Þ ¼ ffg, then we get the

following:

ρ�g1n � f

≤�1� β1n

ρ�g2n � f

þ δβ1nρ�g2n � f

(2.22)

¼ �1� ð1� δÞβ1nρ�g2n � f

:


ρ�g2n � f

≤�1� β2n

ρ�g3n � f


(2.23)

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198

¼ �1� ð1� δÞβ2nρ�g3n � f

:

Also, an application of (1.8) and (1.13) gives

ρ�g3n � f

≤�1� β3n

ρ�g4n � f


(2.24)

¼ �1� ð1� δÞβ3nρ�g4n � f

:

Continuing this process, an application of (1.8) and (1.13) gives

ρ�gk−2n � f

≤�1� βk−2n

ρ�gk−1n � f

þ δβk−2n ρ�gk−1n � f

(2.25)

¼ �1� ð1� δÞβk−2n

ρ�gk−1n � f

:

and ρ�gk−1n � f

≤�1� βk−1n

ρðfn � f Þ þ δβk−1n ρðfn � f Þ (2.26)

¼ �1� ð1� δÞβk−1n

ρðfn � f Þ:

Substituting (2.22)–(2.26) in (2.19) inductively and simplifying, we obtain

ρðfnþ1 � f Þ≤ ½1� ð1� δÞαn�

Yi¼1

k 1 �1� ð1� δÞβin

!ρðfn � f Þ (2.27)

≤½1� ð1� δÞαn�ρðfn � f Þ:From (2.27), we inductively obtain

ρðfnþ1 � f Þ≤Ym¼0

n

½1� ð1� δÞαm�ρðf0 � f Þ: (2.28)

Using that fact that δ∈ ½0; 1Þ fαng∞n¼0 ⊂ ½0; 1Þ satisfyingP∞

n¼0αn ¼ ∞, then from (2.28), weobtain

limn→∞


Ym¼0

n

½1� ð1� δÞαm�ρðf0 � f Þ ¼ 0: (2.29)

Therefore, limn→∞ρðfn − f Þ ¼ 0, where f ∈FρðTÞ: The proof is complete. ▪Theorem 2.2 leads to the following corollary:

Corollary 2.2. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ, and T : D→PρðDÞ be a multivalued mapping such that PT

ρ is a ρ−quasi-contractive-like mapping, satisfying contractive-like condition (1.8). Suppose that FρðTÞ≠∅.Let f0 ∈D and ffng⊂D be defined by the explicit SP iterative sequence (1.14), with thesequences fαng∞n¼0; fβ1ng

∞

n¼0; fβ2ng∞

n¼0 ⊂ ½0; 1Þ such thatP∞

0 αn ¼ ∞: Then, the explicit SPiterative sequence (1.14) ρ-converges strongly to a fixed point of T:

2.3 Strong convergence results for implicit multistep iterative sequences inmodular functionspaces

Theorem 2.3. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ. Let T : D→PρðDÞ be a multivalued mapping satisfying property (I)and such that PT

ρ is a ρ−quasi-contractive-like mapping, satisfying contractive-like condition (1.8). Suppose that FρðTÞ≠ 0==. Let f0 ∈D and ffng⊂D be defined by theimplicit multistep iterative sequence (1.15), where the sequences fαng∞n¼0; fβing

∞

n¼0 ⊂ ½0; 1Þ


fixed point...

199

ði ¼ 1; 2; . . . ; k− 1Þ are such thatP∞

0 αn ¼ ∞: Then, the implicit multistep iterativesequence (1.15) ρ-converges strongly to a fixed point of T:

Proof. Let f ∈FρðTÞ. From Lemma 1.1, we have that PTρ ðf Þ ¼ ffg and FρðTÞ ¼ FðPT

ρ ðf ÞÞ.Using implicit multistep iterative sequence (1.15) and the convexity of ρ, we obtain the

following estimate:

ρðfnþ1 � f Þ ¼ ρ½ð1� αnÞf 1n þ αnunþ1 � f

(2.30)

¼ ρ�ð1� αnÞ

�f 1n � f

þ αnðunþ1 � f Þ≤ð1� αnÞρ

�f 1n � f

þ αnρðunþ1 � f Þ

Since unþ1 ∈PTρ ðfnþ1Þ and PT

ρ ðf Þ ¼ ffg,ρðunþ1 � f Þ≤ distρ

�unþ1; P

Tρ ðf Þ

≤Hρ

�PTρ ðfnþ1Þ; PT

ρ ðf ÞÞ;

which combined with (2.30) gives

ρðfnþ1 � f Þ≤ ð1� αnÞρ�f 1n � f

þ αnHρ


ρ ðf ÞÞ: (2.31)

In (1.8), by letting g ¼ fnþ1 and noting that wð0Þ ¼ 0 and PTρ ðf Þ ¼ ffg, we get:

Hρ


ρ ðf ÞÞ≤ δρðfnþ1 � f Þ þ wρð0Þ ¼ δρðfnþ1 � f Þ: (2.32)


ρðfnþ1 � f Þ≤ ð1� αnÞρ�f 1n � f

þ δαnρðfnþ1 � f Þ

That is,ρðfnþ1 � f Þ≤

�1� αn

1� δαn

ρ�f 1n � f

: (2.33)

Next, from (1.15) and the convexity of ρ, we have

ρ�f 1n � f

¼ ρ��1� β1n f

2n þ β1n

u1n � f

(2.34)

¼ ρ��1� β1n

�f 2n � f

þ β1n�u1n � f

¼ �1� β1n

ρ�f 2n � f

þ β1nρ�u1n � f

:

Since u1n ∈PTρ ðf 1n Þ and PT

ρ ðf Þ ¼ ffg,ρ�u1n; f

¼ distρ�u1n; P

Tρ ðf Þ

≤Hρ

�PTρ ðf 1n

; PT

ρ ðf ÞÞ;

which combined with (2.34) gives:

ρ�f 1n � f

≤�1� β1n

ρ�f 2n � f

þ β1nHρ

�PTρ ðf 1n

; PT

ρ ðf ÞÞ: (2.35)

By letting g ¼ f 1n in (1.8) and noting that wð0Þ ¼ 0 and PTρ ðf Þ ¼ ffg, we get:

Hρ

�PTρ ðf 1n

; PT

ρ ðf ÞÞ≤ δρ�f 1n � f

þ wρð0Þ ¼ δρ�f 1n � f

(2.36)


ρ�f 1n � f

≤�1� β1n

ρ�f 2n � f

þ δβ1nρ�f 1n � f

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200

That is,

ρ�f 1n � f

≤

�1� β1n1� δβ1n

ρ�f 2n � f

: (2.37)


ρ�f 2n � f

≤

�1� β2n1� δβ2n

ρ�f 3n � f

: (2.38)

ρ�f 3n � f

≤

�1� β3n1� δβ3n

ρ�f 4n � f

: (2.39)

..

.

ρ�f k−2n � f

�≤

�1� βk−2n

1� δβk−2n

ρ�f k−1n � f

�: (2.40)

ρ�f k−1n � f

�≤

�1� βk−1n

1� δβk−1n

ρðfn � f Þ: (2.41)

Substituting (2.37)–(2.40) in (2.33) inductively and simplifying, we obtain

ρðfnþ1 � f Þ≤�1� αn

1� δαn

�Yi¼1

k 1 1� βin1� δβin

ρðfn � f Þ: (2.42)

Observe that

1� αn

1� δαn

≤ 1� αn þ δαn;

�1� βin1� δβin

≤ 1� βin þ δβin; i ¼ 1; . . . ; k� 1 (2.43)

Substituting (2.43) in (2.42) and simplifying, we obtain

ρðfnþ1 � f Þ≤ ½1� ð1� δÞαn�ρðfn � f Þ: (2.44)

From (2.44), we inductively obtain

ρðfnþ1 � f Þ≤Ym¼0

n

½1� ð1� δÞαm�ρðf0 � f Þ: (2.45)

Using that fact that δ∈ ½0; 1Þ fαng∞n¼0 ⊂ ½0; 1Þ satisfyingP∞

n¼0αn ¼ ∞, then from (2.45), weobtain

limn→∞


Ym¼0

n

½1� ð1� δÞαm�ρðf0 � f Þ ¼ 0: (2.46)

Therefore, limn→∞ρðfn − f Þ ¼ 0, with f ∈FρðTÞ: The proof is complete. ▪Theorem 2.3 leads to the following corollary:

Corollary 2.3. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ. Let T : D→PρðDÞbe a multivalued mapping satisfying property (I), such


fixed point...

201

that PTρ is a ρ−quasi-contractive-like mapping, satisfying contractive-like condition (1.8).

Suppose that FρðTÞ≠∅. Let f0 ∈D and ffng⊂D be defined by the implicit Noor (1.16),implicit Ishikawa (1.17) and implicit Mann (1.18) iterative sequences respectively,

where the sequences fαng∞n¼0, fβ1ng∞

n¼0, fβ2ng∞

n¼0 ⊂ ½0; 1Þ are such thatP∞

0 αn ¼ ∞: Then:

(1) the implicit Noor iterative sequence (1.16) converges strongly to the fixed point of T:

(2) the implicit Ishikawa iterative sequence (1.17) converges strongly to the fixed pointof T:

(3) the implicit Mann iterative sequence (1.18) converges strongly to the fixed point ofT:

3. Stability results for strong ρ-quasi-contractive-like mapsIn this section, conditions for some stability types of the explicit and implicit multistepiterative sequences are stated and backed by proofs in the framework of modular functionspaces.

The first important result on T − stable single mappings was proved by Ostrowski [18]for Picard iteration. Berinde [13], presented useful explanation on how to obtain the stabilityof various iterative sequences. Okeke and Khan [7] gave a similar version of stability resultsfor multivalued mapping in modular function spaces.

In this paper, we introduce two other versions of ρ-stability and attempt to relate themwith the concept of ρ-stability in literature.

Definition 3.1. Let D be a nonempty ρ−closed, ρ−bounded and convex subset of aρ−complete modular space Lρ, and T : D→PρðDÞ be a multivalued mapping withFρðTÞ≠∅. Suppose that a fixed-point iterative sequence defined by

fnþ1 ¼ FðT; fnÞ (3.1)

with initial guess f0 ∈D and F is a given function, converges to a fixed point f of T: Letfhng∞n¼0 be an arbitrary sequence in D. The fixed-point iterative sequence is said to be:

(1) ρ-stable with respect to T if and only if

limn→∞

εn ¼ 00 limn→∞

hn ¼ f ; where εn ¼ ρðhnþ1 � FðT; hnÞÞ: (3.2)

(2) relatively ρ-stable with respect to T if and only if

limn→∞

δn ¼ 00 limn→∞

hn ¼ f ; where δn ¼ ρðhnþ1 � f Þ � ρðFðT; hnÞ � f Þ: (3.3)

(3) weakly ρ-stable with respect to T if and only if

supλ∈ð0;1�

λρ�hnþ1 � FðT; hnÞ

λ

�→ 00 inf

λ∈½1;∞Þλρ�hn � f

λ

�→ 0: (3.4)

The term “relatively” in (2) is employed because the premise of the convergence of fhng to f ishinged to the fact that ρðhnþ1 − f Þand ρðFðT; hnÞ− f Þget closer to each other as n increases.It is not known if this concept is directly related to ρ-stability as defined in [7]. If ρ satisfies thetriangular inequality (an unwanted condition in this paper), the relation between relativelyρ-stability and ρ-stability is as follows: (1) a relative ρ-stable fixed-point iteration is ρ-stable if

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202

δn > 0 for n sufficiently big since jδnj≤ εn; (2) a ρ-stable fixed-point iteration is relativelyρ-stable if for n sufficiently big, δn < 0 and jδnj≤ εn.

However, a ρ-stable fixed-point iteration is weakly ρ-stable, hence the term “weakly.”In this sequel, we also introduce the following concepts of strong quasi-contractions

particular to modular function spaces and compatible in some sense to the newly introducedstability notions.

Definition 3.2. LetHρð$; $Þbe the ρ-Hausdorff distance on the family CρðLρÞ of nonemptyρ-closed ρ-bounded subsets of Lρ, that is,

HρðA; BÞ ¼ max�supf∈A

distρðf ; BÞ ; supg∈B

distρðg; AÞ�; A; B∈CρðLρÞ:

A multivalued map T : D→CρðLρÞ is said to be an:

(1) m-strong ρ−contraction mapping, where m∈ℕ, if there exists a constant δ∈ ½0; 1Þsuch that

HρðTf ; TgÞ≤mδρ�f � g

m

�; ∀f ; g ∈D; (3.5)

(If δ ¼ 1 in (3.5), T is said to be an m-strong ρ-nonexpansive mapping)

(2) m-strong ρ−quasi-contractive mapping, where m∈ℕ, if


m

�þ Lρðh� f Þ; ∀f ; g ∈D ∀h∈Tf ; L≥ 0; (3.6)

(If δ ¼ 1 in (3.6), T is said to be an m-strong ρ-quasi-contractive mapping)

(3) m-strong ρ−quasi-contractive-like mapping, where m∈ℕ, if


m

�þ wðρðh� f ÞÞ; ∀f ; g ∈D ∀h∈Tf : (3.7)

wherew : ℝþ→ℝþ is amonotone increasing functionwithwð0Þ ¼ 0: (If δ ¼ 1 in (3.7),T is

said to be a m-strong ρ-quasi-contractive-like mapping).Given any m∈ℕ, an m-strong ρ-contraction (resp. ρ-quasi-contractive mapping,

or a ρ-quasi-contractive-like mapping) is a ρ-contraction (resp. ρ-quasi-contractivemapping, or a ρ-quasi-contractive-like mapping), thus, the convergence results in theprevious section hold for m-strong ρ-quasi-contractive-like mappings. The converse istrivial when m ¼ 1.

3.1 Stability results for explicit multistep iterative sequences in modular function spaces

Theorem 3.1. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ, andT : D→PρðDÞbe a multivalued mapping such that PT

ρ is anm-strong

ρ-quasi-contractive-like mapping, satisfying contractive-like condition (3.7), where m∈ℕ.Suppose that FρðTÞ≠∅. Let f0 ∈D and ffng⊂D be defined by the explicit multistep

iterative sequence (1.9), where the sequences fαng∞n¼0; fβing∞

n¼0 ⊂ ½0; 1Þ ði ¼ 1; 2; . . . ; k− 1Þare such that fαng∞n¼0 is bounded away from 0. Then, (1.9) is:

(1) relatively ρ-stable with respect to T if m ¼ 1;

(2) weakly ρ-stable with respect to T if m > 1.


fixed point...

203

(3) ρ-stable with respect toT ifm > 1 and ∀g ∈D ρðg − f Þ ¼ mρ�g − fm

�where f ∈FρðTÞ

(in this case, PTρ is a ρ-quasi-contractive-like map).

Proof. Let fαng∞n¼0; fβing∞

n¼0 ⊂ ½0; 1Þ ði ¼ 1; 2; . . . ; k− 1Þbe sequences such that fαng∞n¼0 isbounded away from 0.

Let fhng∞n¼0 be an arbitrary sequence in D and set:

εn ¼ ρ�hnþ1 � ð1� αnÞhn � αnz

1nÞ

δn ¼ ρðhnþ1 � f Þ � ρ��1� αnÞhn þ αnz

1n � f Þ

γn ¼ supλ∈ð0;1�

λρ�hnþ1 � ð1� αnÞhn � αnz

1n

λ

�

sin ¼�1� βin

hn þ βinz

iþ1n ; i ¼ 1; 2; . . . ; k� 2

sk−1n ¼ �1� βk−1n

hn þ βk−1n wn; n ¼ 0; 1; 2; . . .

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

(3.8)

where wn ∈PTρ ðhnÞ and zin ∈PT

ρ ðsinÞ; i ¼ 1; 2; . . . ; k− 1.Let:

rn;m ¼ ð1� αnÞρ�hn � f

m

�þ αn

mρ�z1n � f

: (3.9)

By the convexity of ρ, we have:

ρðhnþ1 � f Þ ¼ δn þ ρ��1� αnÞhn þ αnz

1n � f Þ

¼ δn þ ρ��1� αnÞðhn � f Þ þ αn

�z1n � f

Þ≤ δn þ rn;1:

(3.10)

If m > 1, we have:

ρ�hnþ1 � f

m

�¼ ρ�αn

m

hnþ1 � ð1� αnÞhn � αnz1n

αn

þ ð1� αnÞhn � f

mþ αn

m

�z1n � f

�

≤αn

mρ�hnþ1 � ð1� αnÞhn � αnz

1n

αn

�þ rn;m

≤γnmþ rn;m:

(3.11)

and if in addition ∀g∈D ρðg− f Þ ¼mρ�g− fm

�;

ρ�hnþ1 � f

m

�¼ ρ�hnþ1 � ð1� αnÞhn � αnz

1n

mþ 1� αn

mðhn � f Þ þ αn

m

�z1n � f

�

≤1

mεn þ rn;m:

(3.12)

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204

Since z1n ∈PTρ ðs1nÞ, then ρðz1n − f Þ ¼ distρðz1n; PT

ρ ðf ÞÞ≤HρðPTρ ðs1nÞ; PT

ρ ðf ÞÞ hence:

rn;m ≤ ð1� αnÞρ�hn � f

m

�þ αn

mHρ

�PTρ ðs1n

; PT

ρ ðf ÞÞ: (3.13)

Using (3.7) and (3.8), and noting that wð0Þ ¼ 0, we get the following:


m

�þ δαnρ

�s1n � f

m

�(3.14)

Using the convexity of ρ in (3.8), and the fact that z2n ∈PTρ ðs2nÞ, we have

ρ�s1n � f

m

�≤�1� β1n

ρ�hn � f

m

�þ β1nρ

�z2n � f

m

�(3.15)

≤�1� β1n

ρ�hn � f

m

�þ β1n

mdistρ

�z2n; P

Tρ ðf Þ

≤�1� β1n

ρ�hn � f

m

�þ β1n

mHρ

�PTρ ðs2n

; PT

ρ ðf ÞÞ:

Using (3.7) and noting that wð0Þ ¼ 0, then we get the following:

ρ�s1n � f

m

�≤�1� β1n

ρ�hn � f

m

�þ δβ1nρ

�s2n � f

m

�: (3.16)

Substituting (3.16) in (3.15), then in (3.14), we obtain


m

�þ δαnρ

�s1n � f

m

�(3.17)

≤ð1� αnÞρ�hn � f

m

�þ δαn

�1� β1n

ρ�hn � f

m

�þ δ2αnβ

1nρ�s2n � f

m

�

≤½1� ð1� δÞαn � αnβ1nδ�ρ

�hn � f

m

�þ δ2αnβ

1nρ�s2n � f

m

�:

Similarly, successive applications of (1.8) and (3.3) give:

ρ�s2n � f

m

�≤�1� β2n

ρ�hn � f

m

�þ δβ2nρ

�s3n � f

m

�

ρ�s3n � f

m

�≤�1� β3n

ρ�hn � f

m

�þ δβ3nρ

�s4n � f

m

�

ρ�sk−2n � f

m

�≤�1� βk−2n

ρ�hn � f

m

�þ δβk−2n ρ

�sk−1n � f

m

�

ρ�sk−1n � f

m

�≤�1� βk−1n

ρ�hn � f

m

�þ δβk−1n ρ

�hn � f

m

�

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

(3.18)


fixed point...

205

Substituting (3.18) in (3.17), and simplifying, we obtain

rn;m ≤ ½1� ð1� δÞαn�ρ�hn � f

m

�: (3.19)

Hence we have the equations:

ρðhnþ1 � f Þ≤ δn þ ½1� ð1� δÞαn�ρðhn � f Þ (3.20)

and if m > 1,

ρ�hnþ1 � f

m

�≤γnm

þ ½1� ð1� δÞαn�ρ�hn � f

m

�: (3.21)

and if in addition ∀g ∈D ρðg − f Þ ¼ mρ�g − fm

�,

ρðhnþ1 � f Þ≤ 1

mεn þ ½1� ð1� δÞαn�ρðhn � f Þ: (3.22)

(1) If m ¼ 1, then from (3.20) and Lemma 1.2, limn→∞δn ¼ 00hn → f . Thus, the fixed-point iteration (1.9) is relatively ρ-stable.

(2) Suppose now that m > 1 and that limn→∞γn ¼ 0.

Then by (3.21) and Lemma 1.2, ρ�hn − fm

�→ 0andmρ

�hn − fm

�→ 0. Thus, the fixed-point

iteration (1.9) is weakly ρ-stable.

(3) Suppose thatm > 1 and that ∀g ∈D ρðg − f Þ ¼ mρ�g − fm

�: If limn→∞εn ¼ 0, then by

(3.22) and Lemma 1.2, hn → f . Thus, the fixed-point iteration (1.9) is ρ-stable. ∎

Theorem 3.1 leads to the following corollary:

Corollary 3.1. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ, andT : D→PρðDÞbe a multivalued mapping such that PT

ρ is anm-strong

ρ-quasi-contractive-like mapping, satisfying contractive-like condition (3.7), where m∈ℕ.Suppose that FρðTÞ≠ 0==. Let f0 ∈D and ffng⊂D be the explicit Noor (1.10), the explicitIshikawa (1.11) or the explicit Mann (1.12) iterative sequence, where the sequences

fαng∞n¼0; fβ1ng∞

n¼0; fβ2ng∞

n¼0 ⊂ ½0; 1Þ are such that fαng∞n¼0 is bounded away from 0. Thenffng is






3.2 Stability results for explicit multistep-SP iterative sequences in modular function spaces

Theorem 3.2. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ, andT : D→PρðDÞbe a multivalued mapping such that PT

ρ is anm-strong

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206

ρ-quasi-contractive-like mapping, satisfying contractive-like condition (3.7), where m∈ℕ.Suppose that FρðTÞ≠∅. Let f0 ∈D and ffng⊂D be defined by the explicit multistep

iterative sequence (1.13), where the sequences fαng∞n¼0; fβing∞

n¼0 ⊂ ½0; 1Þ ði ¼ 1; 2; . . . ; k− 1Þare such that fαng∞n¼0 is bounded away from 0. Then, (1.13) is:



(3) ρ-stable with respect toT ifm > 1and ∀g∈D ρðg − f Þ ¼ mρ�g − fm



Proof. The method of proof is similar to that of Theorem 3.1. ▪Theorem 3.2 leads to the following corollary:

Corollary 3.2. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ, andT : D→PρðDÞbe a multivalued mapping such that PT

ρ is anm-strong

ρ-quasi-contractive-like mapping, satisfying contractive-like condition (3.7), where m∈ℕ.Suppose that FρðTÞ≠∅. Let f0 ∈D and ffng⊂D be defined by the explicit SP iterative

sequence (1.14), with the sequences fαng∞n¼0, fβ1ng∞

n¼0, fβ2ng∞

n¼0 ⊂ ½0; 1Þ such that fαng∞n¼0 isbounded away from 0. Then (1.14) is:


(2) weakly ρ-stable with respect to T if m > 1;




3.3 Stability results for implicit multistep iterative sequences in modular function spaces

Theorem 3.3. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ. Let T : D→PρðDÞ be a multivalued mapping satisfying property (I),

such that PTρ is an m-strong ρ-quasi-contractive-like mapping, satisfying contractive-

like condition (3.7), where m∈ℕ. Suppose that FρðTÞ≠ 0==. Let f0 ∈D and ffng⊂D bedefined by the implicit multistep iterative sequence (1.15), where the sequences

fαng∞n¼0; fβing∞

n¼0 ⊂ ½0; 1Þ ði ¼ 1; 2; . . . ; k− 1Þ are such that fαng∞n¼0 is bounded awayfrom 0. Then, (1.15) is:






Proof.

Let fαng∞n¼0; fβing∞

n¼0 ⊂ ½0; 1Þ be sequences such that fαng∞n¼0 is bounded away from 0.Suppose f ∈FρðTÞ. Let fhng∞n¼0 is an arbitrary sequence and set:


fixed point...

207

εn ¼ ρ�hnþ1 � ð1� αnÞh1n � αnznþ1Þ;

δn ¼ ρðhnþ1 � f Þ � ρ��1� αnÞh1n þ αnznþ1 � f Þ;

γn ¼ supλ∈ð0;1�

λρ

hnþ1 � ð1� αnÞh1n � αnznþ1

λ

!

hin ¼�1� βin

hiþ1n þ βinz

in; i ¼ 1; 2; . . . ; k� 2

hk−1n ¼ �1� βk−1n

hn þ βk−1n zk−1n ; n ¼ 0; 1; 2; . . . ;

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

(3.23)

where znþ1 ∈PTρ ðhnþ1Þ; zin ∈PT

ρ ðhinÞ; i ¼ 1; 2; . . . ; k− 1:

Let:

rn;m ¼ ð1� αnÞρ h1n � f

m

!þ αn

mρðznþ1 � f Þ: (3.24)

By the convexity of ρ, we have:

ρðhnþ1 � f Þ ¼ δn þ ρ��1� αnÞh1n þ αnznþ1 � f Þ

¼ δn þ ρ��1� αnÞ

�h1n � f

þ αnðznþ1 � f ÞÞ≤ δn þ rn;1:

(3.25)

If m > 1, we have:

ρ�hnþ1� f

m

�¼ ρ

αn

m

hnþ1�ð1�αnÞh1n�αnznþ1

αn

þð1�αnÞh1n� f

mþαn

mðznþ1� f Þ

!

≤αn

mρ


αn

!þ rn;m

≤γnmþ rn;m:

(3.26)

and if in addition ∀g∈D ρðg− f Þ¼mρ�g− fm

�;

ρ�hnþ1� f

m

�¼ ρ


mþ1�αn

m

�h1n� f

þαn

mðznþ1� f Þ

!

≤1

mεnþ rn;m:

(3.27)

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Since znþ1 ∈PTρ ðhnþ1Þ, from (3.24) and (3.7) we have that:

rn;m ¼ ð1� αnÞρ h1n � f

m

!þ αn

mρðznþ1 � f Þ

¼ ð1� αnÞρ h1n � f

m

!þ αn

mdistρ

�znþ1; P

Tρ ðf Þ

≤ ð1� αnÞρ h1n � f

m

!þ αn

mHρ

�PTρ ðhnþ1Þ; PT

ρ ðf ÞÞ

≤ ð1� αnÞρ h1n � f

m

!þ δαnρ

�hnþ1 � f

m

�:

(3.28)

Using the convexity of ρ in (3.23), and the fact that z1n ∈PTρ ðh1nÞ, we have

ρ

h1n � f

m

!≤�1� β1n

ρ

h2n � f

m

!þ β1nρ

�z1n � f

m

�

≤�1� β1n

ρ

h2n � f

m

!þ β1n

mdistρ

�z1n; P

Tρ ðf Þ

≤�1� β1n

ρ

h2n � f

m

!þ β1n

mHρ

�PTρ ðh1n

; PT

ρ ðf ÞÞ

≤�1� β1n

ρ

h2n � f

m

!þ δβ1nρ

h1n � f

m

!:

Thus:

ρ

h1n � f

m

!≤

�1� β1n1� δβ1n

ρ

h2n � f

m

!: (3.29)

Similarly, we have the following:

ρ

h2n � f

m

!≤

�1� β2n1� δβ2n

ρ

h3n � f

m

!(3.30)

..

.

ρ

hk−2n � f

m

!≤

�1� βk−2n

1� δβk−2n

ρ

hk−1n � f

m

!(3.31)


fixed point...

209

ρ

hk−1n � f

m

!≤

�1� βk−1n

1� δβk−1n

ρ�hn � f

m

�(3.32)

Substituting (3.29) – (3.32), and simplifying, we obtain

rn;m ≤ ð1� αnÞ"Ym

i¼1

1� βi−1n

1� δβi−1n

#ρ�hn � f

m

�þ δαnρ

�hnþ1 � f

m

�

≤ ð1� αnÞρ�hn � f

m

�þ δαnρ

�hnþ1 � f

m

�:

(3.33)

Hence, substituting (3.33) in (3.25)–(3.27), we have the equations:

ρðhnþ1 � f Þ≤ δn þ�1� αn

1� δαn

�ρðhn � f Þ (3.34)

and if m > 1,ρ�hnþ1 � f

m

�≤γnm

þ�1� αn

1� δαn

�ρ�hn � f

m

�: (3.35)

and if in addition ∀g ∈D ρðg − f Þ ¼ mρ�g − fm

�,

ρ�hnþ1 � f

m

�≤εnm

þ�1� αn

1� δαn

�ρ�hn � f

m

�: (3.36)

(1) If m ¼ 1, then from (3.34) and Lemma 1.2, limn→∞δn ¼ 00hn → f . Thus the fixed-point iteration (1.15) is relatively ρ-stable.

(2) Suppose now that m > 1 and that limn→∞γn ¼ 0.

Then by (3.35) and Lemma 1.2, ρ�hn − fm

�→ 0. Thus mρ

�hn − fm

�→ 0. Thus, the fixed-

point iteration (1.15) is weakly ρ-stable.

(3) Suppose thatm > 1 and that ∀g ∈D ρðg − f Þ ¼ mρ�g − fm

�: If limn→∞εn ¼ 0, then by

(3.36) and Lemma 1.2, hn → f . Thus, the fixed-point iteration (1.15) is ρ-stable. ▪

Theorem 3.3 leads to the following corollary:

Corollary 3.3. Let D be a ρ−closed, ρ−bounded and convex subset of a ρ−completemodular space Lρ. Let T : D→PρðDÞbe a multivalued mapping satisfying property (I), suchthatPT

ρ is anm-strong ρ-quasi-contractive-likemapping, satisfying contractive-like condition(3.7), wherem∈ℕ. Suppose that FρðTÞ≠∅. Let f0 ∈D and ffng⊂Dbe defined by the implicitNoor (1.16), implicit Ishikawa (1.17), implicit Mann (1.18) iterative sequence respectively,where the sequences fαng∞n¼0; fβ1ng

∞

n¼0; fβ2ng∞

n¼0 ⊂ ½0; 1Þ are such that fαng∞n¼0 is boundedaway from 0. Then, (1.16)–(1.18) are:



(3) ρ-stable with respect toT ifm > 1and ∀g ∈D ρðg− f Þ ¼ mρ�g − fm



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210

3.3.1 Numerical example. Let M ½0; 1� be the collection of all real-valued measurable

functions on ½0; 1� and ρ : M ½0; 1�→ℝ a convex function modular defined by ρðf Þ ¼ R 1

0jf j∀f ∈M ½0; 1�. LetD ¼ ff ∈Lρ : 0≤ f ðxÞ≤ 2 ∀x∈ ½0; 1�gbe a subset of the modular functionspace Lρ ¼ M ½0; 1� defined by ρ. D is nonempty, closed and convex.

Define map T : D→PρðDÞ by Tf ¼ fδf g, where δ ¼ 0:9. T satisfies property (I), has a

unique fixed point f ¼ 0 (since 0∈Tð0Þ), andPTρ is aρ-contraction,withPT

ρ ðf Þ ¼ fTfg ∀f ∈D.

In fact, PTρ is an m-strong ρ-strong contraction for all m∈ℕ, since ρðgÞ ¼ mρ

�gm

�.

We present the results of convergence to f ¼ 0 of a multistep iterative sequence (1.9), anexplicit multistep-SP iterative sequence (1.13) and an implicit multistep iterative sequence (1.15)using MATLAB. The parameters used are the following: f0ðxÞ ¼ 0:5xþ 0:95 ∀x∈ ½0; 1�,αn ¼ 1

4 þ 1nþ2, β

in ¼ 1

nþ2 for i ¼ 1; 2; . . . ; k− 1, where k ¼ 11 and n ¼ 1; 2; . . . ; 100 (see

Tables 1 and 2).

N Explicit multistep fnðxÞ Explicit multistep-SP fnðxÞ Implicit multistep fnðxÞ0 0.5000x þ 0.9500 0.5000x þ 0.9500 0.5000x þ 0.95001 0.4583x þ 0.8708 0.3470x þ 0.6593 0.3904x þ 0.7418... ..

. ... ..

.

16 0.2461x þ 0.4676 0.0443x þ 0.0842 0.0695x þ 0.132017 0.2383x þ 0.4527 0.0410x þ 0.0779 0.0648x þ 0.1230... ..

. ... ..

.

24 0.1917x þ 0.3642 0.0252x þ 0.0480 0.0417x þ 0.079225 0.1860x þ 0.3534 0.0237x þ 0.0450 0.0394x þ 0.0748... ..

. ... ..

.

60 0.0690x þ 0.1311 0.0043x þ 0.0081 0.0080x þ 0.015261 0.0671x þ 0.1276 0.0041x þ 0.0078 0.0077x þ 0.0146... ..

. ... ..

.

77 0.0435x þ 0.0827 0.0022x þ 0.0042 0.0042x þ 0.008078 0.0424x þ 0.0805 0.0021x þ 0.0040 0.0041x þ 0.007779 0.0412x þ 0.0783 0.0020x þ 0.0039 0.0039x þ 0.0075... ..

. ... ..

.

101 0.0229x þ 0.0435 0.0009x þ 0.0017 0.0018x þ 0.0035

N Explicit multistep Explicit multistep-SP Implicit multistep

0 1.2 1.2 1.2... ..

. ... ..

.

16 0.5906 0.1064 0.166717 0.5719 0.0984 0.1554... ..

. ... ..

.

24 0.4601 0.0606 0.100125 0.4464 0.0569 0.0945... ..

. ... ..

.

60 0.1656 0.0103 0.019261 0.1611 0.0099 0.0184... ..

. ... ..

.

77 0.1044 0.0053 0.010178 0.1016 0.0051 0.009879 0.0990 0.0049 0.0094... ..

. ... ..

.

101 0.0550 0.0022 0.0044

Table 1.Convergence

Table 2.Approximates

ρðfn − f Þ


fixed point...

211

For this example, the explicit multistep-SP sequence seems to converge to the fixed pointf ¼ 0 slightly faster than the implicit multistep sequence, with approximates ρðfn − f Þunder10−2 at n ¼ 17 and n ¼ 25 respectively, while the explicit multistep sequence is considerablyslower, with ρðfn − f Þ < 10−2 only from n ¼ 79.

4. ConclusionIn Theorems 2.1–2.3, the fixed points of multivalued maps T with a ρ-contractive-likeassociate map PT

ρ in modular spaces are successfully approximated, with supporting proofsand a numerical example, via the explicit multistep (1.9), the explicit multistep-SP (1.13) andthe implicit multistep (1.15) iterative sequences. These sequences involve more steps (k≥ 1)than the iterations considered in [6, 7].

In an attempt to prove the stability of these iterations, a new approach is used tomatch theconvexity structure of ρ: the concepts of relative ρ-stability (3.3) and weak ρ-stability (3.4) areintroduced for the first time in literature, as well as the notions of m-strong ρ-quasi-contraction types (3.5)–(3.7), wherem∈ℕ, which coincide with quasi-contraction types whenρ is nonnegative homogeneous. Theorems 3.1–3.3 then state conditions under which schemes(1.9), (1.13) and (1.15) are ρ-stable, relatively ρ-stable and weakly ρ-stable, when PT

ρ is an m-strong ρ-quasi-contractive-like mapping. The proofs of this theorem are fundamentallydifferent from those of parallel results in metric spaces as they elegantly cut out the use oftriangle inequality.

References

1. Khamsi MA, Kozlowski WM. Fixed point theory in modular function spaces: SpringerInternational Publishing 2015.

2. Khan K. Approximating fixed points of ðλ; ρÞ− firmly nonexpansive mappings in modularfunction spaces. Arab J Math. 2018; 7. doi: 10.1007/s40065-018-0204-x.

3. Khan SH, Abbas M. Approximating fixed points of multivalued €I -nonexpansive mappings inmodular function spaces. Fixed Point Theory Appl. 2014; 34: 9.

4. Khan SH, Abbas M, Ali S. Fixed point approximation of multivalued €I -quasi-nonexpansivemappings in modular function spaces. J Nonlinear Sci Appl. 2017; 10: 3168-179.

5. Kutbi MA, Latif A. Fixed points of multivalued mappings in modular function spaces. Fixed PointTheory Appl. 2009; 2009: 12.

6. Okeke GA, Bishop SA and Khan SH. Iterative approximation of fixed point of multivalued€I-quasinonexpansive mappings in modular function spaces with applications. J Fun Spaces. 2018;2018: 9.

7. Okeke GA, Khan SH. Approximation of fixed point of multivalued ρ−quasi-contractivemappings in modular function spaces. Arab J Mat Sci. 2019; 26(1/2): 75-93 (accessed 3February 2019).

8. Zamfirescu T. Fixed point theorems in metric spaces. Arch Math. 1972; 23: 292-98.

9. Osilike MO. Stability results for Ishikawa fixed point iteration procedure. Indian J Pure ApplMath. 1995/96; 26(10): 937-41.

10. Imoru CO, Olatinwo MO. On the stability of Picard and Mann iteration. Carpathian J Mat. 2003;19: 155-60.

11. Akewe H. Approximation of fixed and common fixed points of generalized contractive-likeoperators. Ph.D. Thesis, Lagos, Nigeria: University of Lagos. 2010: 112.

12. Akewe H, Okeke GA, Olayiwola A. Strong convergence and stability of Kirk-multistep-typeiterative schemes for contractive-type operators. Fixed Point Theory Appl. 2014 (45): 24.

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13. Berinde V. On the stability of some fixed point procedures, Buletinul Stiintific al Universitatii dinBaia Mare. Seria B. Fascicola Mathematica-Informatica. 2002; XVIII(1): 7-14.

14. Berinde V. Iterative approximation of fixed points, Baia Mare: Efemeride. 2002.

15. Berinde V. On the convergence of the Ishikawa iteration in the class of quasi-contractiveoperators. Acta Math Univ Comen. 2004; LXXIII(1): 119-26.

16. Chugh R, Malik P, and Kumar V. On a new faster implicit fixed point iterative scheme in convexmetric space. J Fun Spaces. 2015; 2015: 11.

17. Harder AM and Hicks TL. Stability results for fixed point iteration procedures. Math Japonica.1988; 33(5): 693-706.

18. Ostrowski AM. The round-off stability of iterations. Zeilschrift fur Angewandte Mathemalik undMechanik. 1967; 47: 77-81.

19. Rhoades BE. Fixed point theorems and stability results for fixed point iteration procedures.Indian J Pure Appl Math. 1990; 21: 1-9.

20. Rhoades BE. Fixed point theorems and stability results for fixed point iteration procedures II.Indian J Pure Appl Math. 1993; 24(11): 691-703.

21. Rhoades BE and Soltuz SM. The equivalence between Mann-Ishikawa iterations and multi-stepiteration. Nonlinear Anal. 2004; 58: 219-28.

22. Olaleru JO and Akewe H. On the convergence of Jungck-type iterative schemes for generalizedcontractive-like operators. Fas Mat. 2010; 45: 87-98.

Corresponding authorHallowed Olaoluwa can be contacted at: [email protected]



fixed point...

213


Fixed point theorems for Geraghty-type mappings applied to solvingnonlinear Volterra-Fredholmintegral equations in modular

G-metric spacesGodwin Amechi Okeke and Daniel Francis

Department of Mathematics, School of Physical Sciences,Federal University of Technology, Owerri, Nigeria

Abstract

Purpose – The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preorderedmodularG-metric spaces. The authors apply the results in solvingnonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and includeseveral known results as special cases.Design/methodology/approach – The results of this paper are theoretical and analytical in nature.Findings – The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinearVolterra-Fredholm-type integral equations. The results extend, generalize, compliment and include severalknown results as special cases.Research limitations/implications – The results are theoretical and analytical.Practical implications – The results were applied to solving nonlinear integral equations.Social implications – The results has several social applications.Originality/value – The results of this paper are new.

Keywords Fixed point, Preordered, Modular G-metric spaces, Contractive mapping, Existence and

uniqueness, Nonlinear Volterra-Fredholm integral equations


1. IntroductionIn 1973, Geraghty [1] introduced an interesting generalization of Banach contractionmapping principle using the concept of class S of functions, that is α : ℝþ → ½0; 1Þwith thecondition that αðtnÞ→ 10tn → 0 where ℝþ is the set of all nonnegative real numbers andt ∈ℝþ for all n∈N. In 2012, Gordji et al. [2] proved some fixed point theorems for generalizedGeraghty contraction in partially ordered complete metric spaces. Bhaskar and

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214

JEL Classification — 47H09, 47H10, 06A75.© Godwin Amechi Okeke and Daniel Francis. Published in Arab Journal of Mathematical Sciences.


The authors wish to thank the editor and the referees for their comments and suggestions. Thispaper was completed while the first author was visiting the Abdus Salam School of MathematicalSciences (ASSMS), Government College University Lahore, Pakistan as a postdoctoral fellow.

Authors Contributions: All authors contributed equally to the writing of this paper.Conflicts of Interest: The authors declare no conflict of interests.



Received 26 October 2020Revised 19 December 202029 December 2020Accepted 30 December 2020

Arab Journal of MathematicalSciencesVol. 27 No. 2, 2021pp. 214-234Emerald Publishing Limitede-ISSN: 2588-9214p-ISSN: 1319-5166DOI 10.1108/AJMS-10-2020-0098

Lakshmikantham [3] proved a fixed point theorem for amixedmonotonemapping in ametricspace endowedwith partial order, using a weak contractivity type of assumption. Yolacan [4]established some new fixed point theorems in 0-complete ordered partial metric spaces. Healso remarked on coupled generalized Banach contraction mapping. Faraji et al. [5] extendedsome fixed point theorems for Geraghty contractive mappings in b-complete b-metric spaces.

Furthermore, Gupta et al. [6], established some fixed point theorems in an orderedcomplete metric space using distance function. Chaipunya et al. [7] proved some fixed pointtheorems of Geraghty-type contractions concerning the existence and uniqueness of fixedpoints under the setting of modular metric spaces which also generalized the results in Gordjiet al. [2] under the influence of a modular metric space.

Geraghty-type contractive mappings in metric spaces was generalized to the concept ofpreordered G-metric spaces in [8] and the authors in [8] obtained unique fixed point results.Furthermore, other interesting fixed point results in G-metric spaces can be found in [9] andthe references therein.

In 2010, an essential study by Chistyakov [10] introduced an aspect of metric calledmodular metric spaces or parameterized metric space with the time parameter λ (say)and his purpose was to define the notion of a modular on an arbitrary set, develop thetheory of metric spaces generated by modulars, called modular metric spaces and, onthe basis of it, defined new metric spaces of (multi-valued) functions of boundedgeneralized variation of a real variable with values in metric semigroups and abstractconvex cones.

In the same year, Chistyakov [11], as an application presented an exhausting descriptionof Lipschitz continuous and some other classes of superposition (or Nemytskii) operators,acting in thesemodular metric spaces. He developed the theory of metric spaces generated bymodulars and extended the results given by Nakano [12], Musielak and Orlicz [13], Musielak[14] to modular metric spaces. Modular spaces are extensions of Lebesgue, Riesz and Orliczspaces of integrable functions.

Modular theories on linear spaces can be found in Nakano [12, 15], where he developed aspectral theory in semi-ordered linear spaces (vector lattices) and established the integralrepresentation for projections acting in this modular space.

Nakano [12] established some modulars on real linear spaces which are convexfunctionals. Non-convex modulars and the corresponding modular linear spaces wereconstructed by Musielak and Orlicz [13]. Orlicz spaces and modular linear spaces havealready become classical tools in modern nonlinear functional analysis.

Furthermore, the development of theory of metric spaces generated by modulars,called modular metric spaces attracted the attention of several mathematicians (see, e.g.[16–19]).

Okeke et al. [20] established some convergence results for three multi-valued ρ-quasi-nonexpansive mappings using a three step iterative scheme. Moreover, these fixed pointresults are applicable to nonlinear integral and differential equations see [19, 21–26] and thereferences therein, while [7] deals with application to partial differential equation in modularmetric spaces.

In 2013, Azadifar et al. [27] introduced the notion of modular G-metric space and provedsome fixed point theorems for contractive mappings defined on modular G-metric spaces.Based on definitions given in [27], we intend to extend the fixed point theorems obtained in[7] to preordered modular G-metric spaces in this paper. Furthermore, we prove some fixedpoint theorems for Geraghty-type contraction mappings in the setting of preorderedmodularG-metric spaces.We apply our results in proving the existence of a unique solutionfor a system of nonlinear Volterra-Fredholm integral equations in modular G-metricspaces, XωG :

ModularG-metricspaces

215

2. PreliminariesWebegin this sectionwith the following results and definitionswhichwill be useful in this paper.

Theorem 2.1.[28] If fangn∈N; fbngn∈N; fcngn∈N are three sequences in ℝ such that

(1) limn→∞

an ¼ limn→∞

bn ¼ ‘;

(2) for some positive integer N ; an ≤ cn ≤ bn for all n≥N.Then lim

n→∞cn ¼ ‘.

Definition 2.1. [29] A preorder set X is a relation 6 that is both,

(1) transitive i.e; x6 y and y6 z implies x6 z and,

(2) reflexive i.e; x6 x:

A preordered set is a pair ðX ;6Þ consisting of a set X and a preorder 6 on X :

Remark 2.1. If a preorder6 is antisymmetric i.e; x6 y and y6 x implies x ¼ y, then6 iscalled a partial order.

Definition 2.2. [1] Let S be the family of all Geraghty functions, that is functionsα : ½0; ∞Þ→ ½0; 1Þ satisfying the condition fαðtnÞg→ 10ftng→ 0.

For the rest of this paper, we denote the the class of all Geraghty functions by SGer: SuchGeraghty class was discussed in [7].

Definition 2.3. [7] Let S be the family of all Geraghty functions, that is functionsβi : ℝþ∪f∞g→ ½0; 1Þ satisfying the condition βiðtkÞ→ 1

n0ftkg→ 0 for all i.

Definition 2.4. [7] Let Ψ be the class of functions ψ : ℝþ →ℝþ such that the followingconditions hold;

(1) ψ is decreasing,

(2) ψ is continuous,

(3) ψðtÞ ¼ 0 if and only if t ¼ 0:

Extension of Definition 2.2 above is as follows:

Definition 2.5. [7] Let Ψ be the class of functions ψ : ℝþ∪f∞g→ℝþ∪f∞g such that thefollowing conditions hold;

(1) ψ is subadditive,

(2) ψðtÞ is finite for 0 < t < ∞;

(3) ψ jℝþ ∈Ψ:

Definition 2.6. [27] Let X be a nonempty set, and let ωG : ð0; ∞Þ3 X 3 X 3 X → ½0; ∞�be a function satisfying;

(1) ωGλ ðx; y; zÞ ¼ 0 for all x; y∈X and λ > 0 if x ¼ y ¼ z;

(2) ωGλ ðx; x; yÞ > 0 for all x; y∈X and λ > 0 with x≠ y;

(3) ωGλ ðx; x; yÞ≤ωG

λ ðx; y; zÞ for all x; y; z∈X and λ > 0 with z≠ y;

(4) ωGλ ðx; y; zÞ ¼ ωG

λ ðx; z; yÞ ¼ ωGλ ðy; z; xÞ ¼ . . . for all λ > 0 (symmetry in all three

variables),

AJMS27,2

216

(5) ωGλþμðx; y; zÞ≤ωG

λ ðx; a; aÞ þ ωGμ ða; y; zÞ, for all x; y; z; a∈X and λ; ν > 0,

then the function ωGλ is called a modular G-metric on X.

Remarks 2.1.

(1) The pair ðX ; ωGÞ is called a modular G-metric space, and without any confusion wewill take XωG as a modular G-metric space.From condition (5), if ωG is convex, then we have a strong form as,

(2) ωGλþμðx; y; zÞ≤ωG

λλþμðx; a; aÞ þ ωG

μλþμða; y; zÞ,

(3) If x ¼ a, then (5) above becomes ωGλþμða; y; zÞ≤ωG

μ ða; y; zÞ,(4) Condition (5) is called rectangle inequality.

Definition2.7. [27] Let ðX ; ωGÞbe amodularG-metric space. The sequence fxngn∈N inX isωG-convergent to x, if it converges to x in the topology τðωG

λ Þ.A function T : XωG →XωG at x∈XωG is called ωG-continuous if ωG

λ ðxn; x; xÞ→ 0 then

ωGλ ðTxn; Tx; TxÞ→ 0, for all λ > 0:

Remark 2.2. fxngn∈Nmodular G-converges to x as n→∞, if limn→∞

ωGλ ðxn; xm; xÞ ¼ 0. That

is for all e > 0 there exists n0 ∈N such that ωGλ ðxn; xm; xÞ < e for all n; m≥ n0. Here we say

that x is modular G-limit of fxngn∈N.Definition 2.8. [27] Let ðX ; ωGÞ be a modular G-metric space, then fxngn∈N ⊆ XωG is saidto be ωG-Cauchy if for every e > 0, there exists ne ∈N such that ωG

λ ðxn; xm; xlÞ < e for alln;m; l ≥ ne and λ > 0.A modular G-metric space XωG is said to be ωG-complete if every ωG-Cauchy sequence in XωG

is ωG-convergent in XωG .

Proposition 2.2. [27] Let ðX ; ωGÞ be a modular G-metric space, for any x; y; x; a∈X, itfollows that:

(1) If ωGλ ðx; y; zÞ ¼ 0 for all λ > 0, then x ¼ y ¼ z:

(2) ωGλ ðx; y; zÞ≤ωG

λ=2ðx; x; yÞ þ ωGλ=2ðx; x; zÞ for all λ > 0:

(3) ωGλ ðx; y; yÞ≤ 2ωG

λ=2ðx; x; yÞ for all λ > 0:


λ=2ðx; a; zÞ þ ωGλ=2ða; y; zÞ for all λ > 0:

(5) ωGλ ðx; y; zÞ≤ 2

3

�ωGλ=2ðx; y; aÞ þ ωG

λ=2ðx; a; zÞ þ ωGλ=2ða; y; zÞ

�for all λ > 0:


λ=2ðx; a; aÞ þ ωGλ=2ðy; a; aÞ þ ωG

λ=2ðz; a; aÞ for all λ > 0:

Proposition 2.3. [27] Let ðX ; ωGÞbe a modular G-metric space and fxngn∈Nbe a sequencein XωG : Then the following are equivalent:

(1) fxngn∈N is ωG-convergent to x,

(2) ωGλ ðxn; xÞ→ 0as n→∞; i.e; fxngn∈N converges to x relative to modular metricωG

λ ð:Þ,(3) ωG

λ ðxn; xn; xÞ→ 0 as n→∞ for all λ > 0;

(4) ωGλ ðxn; x; xÞ→ 0 as n→∞ for all λ > 0;

(5) ωGλ ðxm; xn; xÞ→ 0 as m; n→∞ for all λ > 0:

We give the following definition which will be useful in our results.


217

Definition 2.9. An ordered modularG-metric space is a triple ðX ; ωG; 6Þwhere ðX ;ωÞ isa modular metric space and 6 is a partial order on XωG : If 6 is a preorder on XωG , thenðX ; ωG; 6Þ is a preordered modular G-metric space.

3. Main results

Theorem3.1. Let ðX ;ωGÞbe a complete modularG-metric space with a preorder,6and anondecreasing self-mapping T : XωG →XωG on XωG such that for each λ > 0, there isνðλÞ∈ ½0; λÞ such that the following conditions hold:

(1)

ψ�ωG

λ ðTx; Ty; TyÞ�≤ α

�ψ�ωG

λ ðx; y; yÞ��

ψ�ωG

λþνðλÞðx; y; yÞ�

þ β�ψ�ωG


ψ�ωG

λ ðx; Tx; TxÞ�þ γ

�ψ�ωG


ψ�ωG

λ ðy; Ty; TyÞ�;

(3.1)

where ψ ∈Ψ and fα; β; γg∈SGer with αðtÞ þ 2maxfsupt≥0βðtÞ; supt≥0γðtÞg < 1; anddistinct x; y∈XωG :Assuming that if a nondecreasing sequence fxngn∈N converges to x�, thenxn 6 x* for each n∈N,

(2) if ψ is subadditive and for any x; y∈XωG , there exists z∈XωG with z6Tz and

ωGλ ðz; Tz; TzÞ is finite for all λ > 0 such that z is comparable to both x and y. Then

T has a fixed point u∈XωG and the sequence define by fTnx0gn≥1 converges to u.Moreover, the fixed point of T is unique.

Proof. Let x0 ∈XωG be such that x0 6Tx0 and let xn ¼ Txn−1 ¼ Tnx0 for all n∈N:Regardingthat T is nondecreasing mapping, we have that x0 6Tx0 ¼ x1, which implies thatx1 ¼ Tx0 6Tx1 ¼ x2. Inductively, we have

x0 6 x1 6 x2 6 . . .6 xn−1 6 xn 6 xnþ1 6 . . . : (3.2)

Assume that there exists n0 ∈N such that xn0 ¼ xn0þ1. Since xn0 ¼ xn0þ1 ¼ Txn0, then xn0 isthe fixed point ofT. Now suppose that xnLxnþ1 for all n∈N, thus by inequality (3.2), we havethat

x0 a x1a x2a . . .a xn−1a xna xnþ1a . . . : (3.3)

Now for each λ > 0; and x0 aTx0 for all n∈N implies thatωGλ ðx0; Tx0; Tx0Þ > 0:Again, let

x0 ∈XωG such that ωGλ ðx0; Tx0; Tx0Þ < ∞ ∀ λ > 0.

First, we show that for all n∈N, the sequence ωGλ ðTnx0; T

nþ1x0; Tnþ1x0Þ ¼ 0 for all

λ > 0, as n→∞. Assume that, for each n∈N, there exists λn > 0 such thatωGλnðTnx0; T

nþ1x0; Tnþ1x0Þ≠ 0. Otherwise the proof is complete. Suppose not, for each

n≥ 1, if 0 < λ < λn, then we have that ωGλ ðTnx0; T

nþ1x0; Tnþ1x0Þ≠ 0. Since Tnx06Tnþ1x0,

from inequality (3.1) we can see that ψ�ωGλnðTnx0; Tnþ1x0; T

nþ1x0Þ�≤ψ

�ωGλ ðTnx0;

Tnþ1x0; Tnþ1x0Þ

�¼ ψ

�ωGλ ðTTn−1x0; TT

nx0; TTnx0Þ

�. Take x ¼ Tn−1x0 and y ¼ Tnx0,

then inequality (3.1) becomes;

AJMS27,2

218

ψ�ωG

λnðTnx0; T

nþ1x0; Tnþ1x0

��≤ψ

�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��≤ α

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3 ψ

�ωG

λþνðλÞ�Tn−1x0; T

nx0; Tnx0

��þ β

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3 ψ

�ωG

λ

�Tn−1x0; TT

n−1x0; TTn−1x0

��þ γ

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3 ψ

�ωG

λ ðTnx0; TTnx0; TT

nx0Þ�

¼ α�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3 ψ

�ωG


nx0; Tnx0

��þ β

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3 ψ

�ωG

λ

�Tn−1x0;T

nx0;Tnx0

��þ γ

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3 ψ

�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��≤ α

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3 ψ

�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��þ β

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3 ψ

�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��þ γ

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3 ψ

�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��;

(3.4)

for which we have that

ψ�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��≤ δψ

�ωG

λ ðTn−1x0; Tnx0; T

nx0��

≤ψ�ωG


nx0��

..

.

≤ψ�ωG

λ ðx0; Tx0; Tx0ÞÞ< ∞;

(3.5)

where

δ :¼ α�ψðωG

λ

�Tn−1x0; T

nx0; Tnx0

��þ β�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��1� γ

�ψ�ωG

λ

�Tn−1x0; Tnx0; Tnx0

�� : (3.6)

Therefore, fψðωGλ ðTnx0; T

nþ1x0; Tnþ1x0ÞÞgn≥1 is nonincreasing and bounded below, so the

sequence fψðωGλ ðTnx0; T

nþ1x0; Tnþ1x0ÞÞgn≥1 converges to some real number k≥ 0. Assume

k > 0;we can see clearly that by using inequality 3, inequality 3 becomes

ψ�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��≤�α�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��þ β

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��þ γ�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��(3.7)

as n→∞; we get

1≤ limn→∞

inf�α�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0


λ

�Tn−1x0; T

nx0; Tnx0

��þ γ

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��(3.8)

So, we have that

ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

�� ¼ 0; (3.9)

hence

limn→∞

ωGλ

�Tn−1x0; T

nx0; Tnx0

� ¼ 0 (3.10)


219

for all λ > 0, which is a contradiction to our assumption. Therefore,

limn→∞

ψ�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

�� ¼ 0; (3.11)

so,

limn→∞

ωGλ

�Tnx0;T

nþ1x0;Tnþ1x0

� ¼ 0 (3.12)

for all λ > 0: This shows that ωGλ ðTnx0; T

nþ1x0; Tnþ1x0Þ ¼ 0 for all λ > 0, n≥ 1:

Next, we show that fTnx0gn≥1 is a modular G-Cauchy sequence. Suppose, if possible thatfTnx0gn≥1 not a modular G-Cauchy sequence , then there exists real numbers, λ0 > 0, e > 0and also there exists two subsequences fTnkx0gk≥1 and fTmkx0gk≥1 of the sequence

fTnx0gn≥1 such that, for nk > mk > k, we have that ωGλ0ðTmkx0; T

nkx0; Tnkx0Þ≥ e, but

ωGλ0ðTmkx0; T

nk−1x0; Tnk−1x0Þ < e. Now, since Tmkx06Tnkx0, we have that e≤ωG

λ0ðTmkx0;

Tnkx0; Tnkx0Þwhich implies thatψðeÞ≤ψðωG

λ0ðTmkx0; T

nkx0; Tnkx0ÞÞ ¼ ψ ðωG

λ0ðTTmk−1x0;

TTnk−1x0; TTnk−1x0ÞÞ. Set x ¼ Tmk−1x0 and y ¼ Tnk−1x0 into inequality (3.1), then we have

ψðeÞ≤ψ�ωG

λ0ðTmkx0; T

nkx0; Tnkx0Þ

�≤ α

�ψ�ωG

λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG

λ0þνðλ0Þ�Tmk−1x0; T

nk−1x0; Tnk−1x0


λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG

λ0

�Tmk−1x0; TT

mk−1x0; TTmk−1x0


λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG

λ0

�Tnk−1x0; TT

nk−1x0; TTnk−1x0

�� ¼ α�ψ�ωG

λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG


nk−1x0; Tnk−1x0


λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0


λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG

λ0

�Tnk−1x0; T

nkx0; Tnkx0

��≤ α

�ψ�ωG

λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��þ ψ

�ωG

λ0

�Tmkx0; T

nk−1x0; Tnk−1x0


λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0


λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG

λ0

�Tnk−1x0; T

nkx0; Tnkx0

��≤ψ

�ωG

νðλ0Þ�Tmk−1x0;T

mkx0;Tmkx0

��þ ψ

�ωG

λ0

�Tmkx0; T

nk−1x0; Tnk−1x0

��þ ψ�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0

��þ ψ

�ωG

λ0

�Tnk−1x0; T

nkx0; Tnkx0

��≤ψ

�ωG

νðλ0Þ�Tmk−1x0; T

mkx0; Tmkx0

��þ ψðeÞ þ ψ

�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0

��þ ψ�ωG

λ0

�Tnk−1x0; T

nkx0; Tnkx0

��;

(3.13)

as k→∞, we obtain

ψðeÞ≤ limk→∞

ψ�ωG

λ0ðTmkx0; T

nkx0; Tnkx0Þ

�≤ψðeÞ; (3.14)

so that

AJMS27,2

220

limk→∞

ψ�ωG

λ0ðTmkx0; T

nkx0; Tnkx0Þ

�¼ ψðeÞ: (3.15)

Hence

limk→∞

ωGλ0ðTmkx0; T

nkx0; Tnkx0Þ ¼ e: (3.16)

Again, using condition 5 of Definition 2.6, we get

ψ�ωG

λ0ðTmkx0; T

nkx0; Tnkx0Þ

�≤ α

�ψ�ωG

λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG


nk−1x0; Tnk−1x0


λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0


λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��3ψ

�ωG

λ0

�Tnk−1x0; T

nkx0; Tnkx0

��≤ψ

�ωG


nk−1x0; Tnk−1x0

��þ ψ

�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0

��þ ψ�ωG

λ0

�Tnk−1x0; T

nkx0; Tnkx0

��≤ψ

�ωG

λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��þ ψ�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0

��þ ψ

�ωG

λ0

�Tnk−1x0; T

nkx0; Tnkx0

��≤ψ

�ωG

λ0

�Tmk−1x0;T

nk−1x0;Tnk−1x0

��þ ψ

�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0

��þ ψ�ωG

λ02

�Tnk−1x0; T

mkx0; Tmkx0

��þ ψ

�ωG

λ02

ðTmkx0; Tnkx0; T

nkx0Þ�≤ψ

�ωG

λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��þ ψ

�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0

��þ ψ�ωG

λ0

�Tnk−1x0; T

mkx0; Tmkx0

��þ ψ

�ωG

λ0ðTmkx0; T

nkx0; Tnkx0Þ

�≤ψ

�ωG

λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��þ ψ

�ωG

λ0

�Tmk−1x0; T

mkx0; Tmkx0

��þ ψ�ωG

λ0

�Tnk−1x0; T

mkx0; Tmkx0

��þ ψ

�ωG

λ0ðTmkx0; T

nkx0; Tnkx0Þ

�;

(3.17)

as k→∞; we have

ψðeÞ≤ limk→∞

ψ�ωG

λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

��≤ψðeÞ; (3.18)

so that

limk→∞

ψ�ωG

λ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

�� ¼ ψðeÞ: (3.19)

Hence

limk→∞

ωGλ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

� ¼ e: (3.20)

Thus, it follows that


221

1≤ limk→∞

inf�α�ψ�ωG

λ0

�Tmk−1x0;T

nk−1x0;Tnk−1x0

��: (3.21)

Therefore, we conclude that

limk→∞

ωGλ0

�Tmk−1x0; T

nk−1x0; Tnk−1x0

� ¼ 0 ∀ λ > 0: (3.22)

This is a contradiction. Therefore, it follows that fTnx0gn≥1 is a modular G-Cauchy sequencein XωG . Since XωG is complete modular G-metric space, there exists u∈XωG such thatTnx0 → u∈XωG . Now we show that u is a fixed point of T for any arbitrary λ > 0, usingcondition 5 of Definition 2.6 and inequality (3.1), we have that

ψ�ωG

λ ðTnx0; Tu; TuÞ�≤ψ

�ωG

λ2

�Tnþ1x0; Tu; Tu

��þ ψ

�ωG

λ2

�Tnx0; T

nþ1x0; Tnþ1x0

��¼ ψ

�ωG

λ=2

�Tnþ1x0; Tu; Tu

��þ ψ

�ωG

λ=2

�Tnx0; T

nþ1x0; Tnþ1x0

��≤ψ

�ωG

λ

�Tnþ1x0; Tu; Tu

��þ ψ

�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��≤α

�ψ�ωG

λ ðTnx0; u; uÞ��ψ�ωG

λþνðλÞðTnx0; u; uÞ�

þ β�ψ�ωG



nx0Þ�

þ γ�ψ�ωG


λ ðu; Tu; TuÞ�

þ ψ�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��¼ α

�ψ�ωG


λþνðλÞðTnx0; u; uÞ�

þ β�ψ�ωG


λ

�Tnx0; T

nþ1x0; Tnþ1x0

��þ γ

�ψ�ωG


λ ðu; Tu; TuÞ�

þ ψ�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��;

(3.23)

as n →∞, we have that

ψ�ωG

λ ðu; Tu; TuÞ�≤ γð0Þψ

�ωG

λ ðu; Tu; TuÞ�

(3.24)

for all λ > 0, which implies that

ð1� γð0ÞÞωGλ ðu;Tu;TuÞ≤ 0 ∀ λ > 0: (3.25)

Therefore,

ωGλ ðu;Tu;TuÞ ¼ 0 ∀ λ > 0; (3.26)

where 1− γð0Þ < 1: Hence, u is a fixed point of T for all λ > 0, i.e; Tu ¼ u:Finally, for the uniqueness, we can see from above that T has a fixed point u∈XωG .

Suppose that there is another fixed point of T i.e; Tv ¼ v, for v∈XωG , thus condition (2) of

AJMS27,2

222

Theorem 3.1 tells us that if z∈XωG with z ≤ Tzand it is comparable to both u and v andTnz isalso comparable to u and v for each n∈N. Now for λ > 0, then ψðωG

λ ðTnþ1z; u; uÞÞ andψðωG

λ ðTnþ1z; v; vÞÞ are finite. Claim : u ¼ v. Indeed, using inequality (3.1), we have by takingx ¼ Tnz and y ¼ u. First consider ψðωG

λ ðTnþ1z; u; uÞÞ<∞; so that we have the followinginequality by using condition 6 of Proposition 2.2

ψ�ωG

λ

�Tnþ1z; u; u

�� ¼ ψ�ωλ

�Tnþ1z; Tu; Tu

��¼ ψ

�ωG

λ ðTTnz; Tu; TuÞ�≤ α

�ψ�ωG

λ ðTnz; u; uÞ��ψ�ωGλþνðλÞðTnz; u; uÞ�

þ β�ψ�ωG

λ ðTnz; u; uÞ��ψ�ωGλ ðTnz; TTnz; TTnzÞ�

þ γ�ψ�ωG

λ ðTnz; u; uÞ��ψ�ωGλ ðu; Tu; TuÞ

�¼ α

�ψ�ωG


þ β�ψ�ωG

λ ðTnz; u; uÞ��ψ�ωGλ

�Tnz; Tnþ1z; Tnþ1z

��≤ α

�ψ�ωG


þ ψ�ωG

λ=4

�Tnþ1z; u; u

��þ ψ�ωG

λ4

�Tnþ1z; u; u

��¼ α

�ψ�ωG


þ β�ψ�ωG

λ ðTnz; u; uÞ��ψ�ωGλ=2ðTnz; u; uÞ

�þ 2ψ

�ωG

λ=4

�Tnþ1z; u; u

��≤ α

�ψ�ωG

λ ðTnz; u; uÞ��ψ�ωGλ ðTnz; u; uÞ��

þ β�ψ�ωG

λ ðTnz; u; uÞ��ψ�ωGλ ðTnz; u; uÞ�

þ 2ψ�ωG

λ

�Tnþ1z; u; u

��¼ α

�ψ�ωG

λ ðTnz; u; uÞ��ψ�ωGλ ðTnz; u; uÞ��

þ β�ψ�ωG


þ 2β�ψ�ωG


�Tnþ1z; u; u

��;

(3.27)

so, we have that

ψ�ωG

λ

�Tnþ1z; u; u

��≤α�ψ�ωG

λ ðTnz; u; uÞ��þ β�ψ�ωG

λ ðTnz; u; uÞ��1� 2β

�ψ�ωG

λ ðTnz; u; uÞ�� ψ�ωG

λ ðTnz; u; uÞ�

≤ψ�ωG

λ ðTnz; u; uÞ�≤ψ

�ωG

λ

�Tn−1z; u; u

��...

≤ψ�ωG

λ ðz; u; uÞ�

< ∞:

(3.28)


223

Therefore, fψðωGλ ðTnþ1z; u; uÞÞgn≥1 is nonincreasing sequence which is bounded below and

converges to some real number ‘∈ ½0; ∞Þ Assume that ‘ > 0; using the fact thatlimn→∞ωG

λ ðTnx0; Tnþ1x0; T

nþ1x0Þ ¼ 0 for all λ > 0, from inequality 3, we have that

ψ�ωG

λ

�Tnþ1z; u; u

��≤ α

�ψ�ωG


þ β�ψ�ωG



��≤ α

�ψ�ωG


þ β�ψ�ωG



��:

(3.29)

Using inequality 3 and letting n→∞, inequality 3 becomes

1≤ limn→∞

inf α�ψ�ωG

λ ðTnz; u; uÞ��: (3.30)

Thus, by condition 4 of Proposition 2.3 we have that

limn→∞

ωGλ ðTnz; u; uÞ ¼ 0 (3.31)

for all λ > 0: Therefore, Tnz→ u as n→∞.Secondly consider ψðωG

λ ðTnþ1z; v; vÞÞ<∞; from inequality (3.1), we have by takingx ¼ Tnz and y ¼ v, so that we have the following inequality by using condition 6 ofProposition 2.2

ψ�ωG

λ

�Tnþ1z; v; v

�� ¼ ψ�ωλ

�Tnþ1z; Tv; Tv

��¼ ψ

�ωG

λ ðTTnz; Tv; TvÞ�≤ α

�ψ�ωG

λ ðTnz; v; vÞ��ψ�ωGλþνðλÞðTnz; v; vÞ�

þ β�ψ�ωG

λ ðTnz; v; vÞ��ψ�ωGλ ðTnz; TTnz; TTnzÞ�

þ γ�ψ�ωG

λ ðTnz; v; vÞ��ψ�ωGλ ðv; Tv; TvÞ

�¼ α

�ψ�ωG


þ β�ψ�ωG

λ ðTnz; v; vÞ��ψ�ωGλ


��≤α

�ψ�ωG


þ β�ψ�ωG

λ ðTnz; v; vÞ��ψ�ωGλ=2ðTnz; v; vÞ

��þ ψ

�ωG

λ=4

�Tnþ1z; v; v

��þ ψ�ωG

λ=4

�Tnþ1z; v; v

��¼ α

�ψ�ωG


þ β�ψ�ωG

λ ðTnz; v; vÞ��ψ�ωGλ=2ðTnz; v; vÞ

�þ 2ψ

�ωG

λ=4

�Tnþ1z; v; v

��≤ α

�ψ�ωG

λ ðTnz; v; vÞ��ψ�ωGλ ðTnz; v; vÞ��

þ β�ψ�ωG

λ ðTnz; v; vÞ��ψ�ωGλ ðTnz; v; vÞ�

þ 2ψ�ωG

λ

�Tnþ1z; v; v

�� ¼ α�ψ�ωG

λ ðTnz; v; vÞ��ψ�ωGλ ðTnz; v; vÞ��

þ β�ψ�ωG


þ 2β�ψ�ωG


�Tnþ1z; v; v

��:

(3.32)

AJMS27,2

224

Therefore, we have

ψ�ωG

λ

�Tnþ1z; v; v

��≤α�ψ�ωG

λ ðTnz; v; vÞ��þβ�ψ�ωG

λ ðTnz; v; vÞ��1�2β

�ψ�ωG

λ ðTnz; v; vÞ�� ψ�ωG

λ ðTnz; v; vÞ�

≤ψ�ωG

λ ðTnz; v; vÞ�≤ψ

�ωG

λ

�Tn−1z; v; v

��...

≤ψ�ωG

λ ðz; v; vÞ�

<∞:

(3.33)

Hence, fψðωGλ ðTnþ1z; v; vÞÞgn≥1 is nonincreasing sequence which is bounded below and

converges to some real number ‘0 ∈ ½0; ∞Þ Suppose that ‘0 > 0; using the fact thatlimn→∞ωG

λ ðTnx0; Tnþ1x0; T

nþ1x0Þ ¼ 0 for all λ > 0. From inequality 3, we have that

ψ�ωG

λ

�Tnþ1z; v; v

��≤ α

�ψ�ωG


þ β�ψ�ωG



��≤ α

�ψ�ωG


þ β�ψ�ωG



��:

(3.34)

Using inequality 3 and letting n→∞, inequality 3 becomes

1≤ limn→∞

inf α�ψ�ωG

λ ðTnz; v; vÞ��: (3.35)

Thus, by condition 4 of Proposition 2.3 we have that

limn→∞

ωGλ ðTnz; v; vÞ ¼ 0 (3.36)

for all λ > 0: Therefore, Tnz→ v as n→∞.Suppose, if possible, that lim

n→∞Tnzexists and not unique. Let lim

n→∞Tnz ¼ uand lim

n→∞Tnz ¼ v

as we have seen above, where u≠ v. For each λ > 0, u≠ v0ωGλ ðu; v; vÞ > 0. If we take

ψðe1Þ ¼ 13ψðωλðu; v; vÞÞ>0; then for λ > 0, lim

n→∞Tnz ¼ u0 given e1 > 0; ∃ m1 ∈N such

that ψðωGλ=2ðu; Tnz; TnzÞÞ<ψðe1Þ for n > m1. Again, lim

n→∞Tnz ¼ v0 given e1 > 0;

∃ m2 ∈N such that ψðωGλ=4ðv; Tnz; TnzÞÞ<ψðe1Þ for n > m2. Set m ¼ maxfm1; m2g, then

for n≥m, by condition 6 of Proposition 2.2, we have

ψ�ωG

λ ðu; v; vÞ�≤ψ

�ωG

λ=2ðu; Tnz; TnzÞ

þ 2ωGλ=4ðv; Tnz; TnzÞ

�≤ψ

�ωG

λ=2ðu; Tnz; TnzÞ�

þ 2ψ�ωG

λ=4ðv; Tnz; TnzÞ�<ψðe1Þ þ 2ψðe1Þ

¼ 3ψðe1Þ;


225

which shows that ψðωGλ ðu; v; vÞÞ<ψðωG

λ ðu; v; vÞÞ for all λ > 0. This is a contradiction.Hence, u ¼ v. Therefore the fixed point of T is unique and the proof is complete.,

We shall give an example to support Theorem 3.1 above.

Example 3.1. Let X ¼ ℝ, define modular G-metric by ωGλ ðx; y; yÞ ¼ ∞ if λ≤ 2jx− yj and

ωGλ ðx; y; yÞ ¼ 0 if λ > 2jx− yj, and ωG

λ ðx; y; yÞ ¼ Gðx; y; yÞλ , where Gðx; y; yÞ ¼ 2jx− yj or

Gðx; y; zÞ ¼ jx− yj þ jy− zj þ jx− zj for x; y; z∈ℝ. We can see that XωG ¼ ℝ. So it followsfrom Theorem 3.1 that ℝ is a complete preordered modular G-metric space. Now define a

map T : ℝ→ℝ by Tx ¼ x3

1þx2: For x; y∈ℝ, then ωG

λ ðx; y; yÞ ¼ ∞ if λ≤ 2 jx− yj, so

inequality (3.1) is satisfied. Again, if λ > 2 jx− yj and x; y∈ℝ, then

GðTx; Ty; TyÞ ¼ 2

�� x3

1þx2−

y3

1þy2

��≤4jx−yj< 2λ, therefore, ωGλ ðTx; Ty; TyÞ ¼ 2ωG

λ ðx; y; yÞ≤0:

We can take ψðtÞ ¼ t; αðtÞ ¼ βðtÞ ¼ γðtÞ ¼ 12 : But T has a fixed point at x¼ 0:

Corollary 3.2. Let ðX ; ωGÞbe a completemodularG-metric space with a preorder,6and anondecreasing self-mapping T : XωG →XωG on XωG such that for each λ > 0, there isνðλÞ∈ ½0; λÞ such that the following conditions hold:

(1)

ψ�ωG

λ ðTx; Ty; TyÞ�≤ α

�ψ�ωG

λ ðx; y; yÞ��ψ�ωG

λþνðλÞðx; y; yÞ�; (3.37)

where ψ ∈Ψand α∈SGer and distinct x; y∈XωG :Assuming that if a nondecreasing sequencefxngn∈N converges to x�, then xn6x� for each n∈N,

(2) if ψ is subadditive and for any x; y∈XωG , there exists z∈XωG with z6Tz andωGλ ðz; Tz; TzÞ is finite for all λ > 0 such that z is comparable to both x and y. ThenT

has a fixed point u∈XωG and the sequence define by fTnx0gn≥1 converges to u.Moreover, the fixed point of T is unique.

Proof: Let x0 ∈XωG be such that x06Tx0 and let xn ¼ Txn−1 ¼ Tnx0 for all n∈N:Regarding that T is nondecreasing mapping, we have that x06Tx0 ¼ x1, which implies thatx1 ¼ Tx06Tx1 ¼ x2. Inductively, we have

x0 6 x1 6 x2 6 � � �6 xn−1 6 xn 6 xnþ1 6 � � � : (3.38)

Assume that there exists n0 ∈N such that xn0 ¼ xn0þ1. Since xn0 ¼ xn0þ1 ¼ Txn0, then xn0 isthe fixed point of T. Now suppose that xnLxnþ1 for all n∈N, thus by inequality (3.38), wehave that

x0 a x1 a x2 a � � �a xn−1 a xna xnþ1 a � � � : (3.39)

Now for each λ > 0; and x0aTx0 for all n∈N implies that ωGλ ðx0; Tx0; Tx0Þ > 0:Again, let




λ > 0, as n→∞.Assume that for each n∈N, there exists λn > 0 such that ωG

λnðTnþ1x0; T

nx0; Tnx0Þ≠ 0.

Otherwise we are done. Suppose that for each n≥ 1, if 0 < λ < λn, then we haveωGλ ðTnþ1x0; T

nx0; Tnx0Þ≠ 0. Since Tnx06Tnþ1x0, we have from inequality (3.37) that

ψðωGλnðTnþ1x0; T

nx0; Tnx0ÞÞ≤ψðωG

λ ðTnþ1x0; Tnx0; T

nx0ÞÞ ¼ ψðωGλ ðTTnx0; TT

n−1x0;

TTn−1x0ÞÞ. Take x ¼ Tnx0 and y ¼ Tn−1x0, then inequality (3.37) becomes;

AJMS27,2

226

ψ�ωG

λ

�Tnþ1x0; T

nx0; Tnx0

��≤α

�ψ�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��3ψ

�ωG

λþνðλÞ�Tnx0; T

n−1x0; Tn−1x0

��≤α

�ψ�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��3ψ

�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��< ψ

�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��:

(3.40)

Therefore, fψðωGλ ðTnþ1x0; T

nx0; Tnx0ÞÞgn≥1 is nonincreasing and bounded below and

converges to some real number τ≥ 0. Assume that τ > 0: In such a case,

τ < ψ�ωG

λ

�Tnþ1x0; T

nx0; Tnx0

��≤ α

�ψ�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��ψ�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��< ψ

�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��;

(3.41)

which implies that

1 <ψ�ωG

λ

�Tnþ1x0; T

nx0; Tnx0

��τ

≤α�ψ�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��ψ�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��τ

<ψ�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��τ

:

(3.42)

Letting, n→∞, Theorem 2.1 ensure that�αðψðωG

λ ðTnx0; Tn− 1x0; T

n− 1x0ÞÞÞ�n≥1

→ 1 orfrom inequality 3,

1≤ limn→∞

inf α�ψ�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��: (3.43)

As α∈SGer, then

limn→∞

ψ�ωG

λ

�Tnx0; T

n−1x0; Tn−1x0

��→ 0 (3.44)

for all λ > 0, which contradicts the fact that τ > 0: Thus τ ¼ 0, so that

limn→∞

ωGλ

�Tnþ1x0; T

nx0; Tnx0

� ¼ 0: (3.45)

Hence ωGλ ðTnþ1x0; T

nx0; Tnx0Þ ¼ 0 for all λ > 0, n≥ 1. Following the proof of Theorem 3.1

above, we see that T has a unique fixed point in XωG :,

Theorem3.3. Let ðX ; ωGÞbe a completemodularG-metric spacewith a preorder,6, and anondecreasing self-mapping T : XωG →XωG on XωG such that for each λ > 0, there isνðλÞ∈ ½0; λÞ such that the following conditions hold:

(1)

ψ�ωG

λ ðTmx; Tmy; TmyÞ�≤ α�ψ�ωG


λþνðλÞðx; y; yÞ�; (3.46)

where ψ ∈Ψand α∈SGer and distinct x; y∈XωG :Assuming that if a nondecreasing sequencefxngn∈N converges to x�, then xn6x� for each n∈N,

(2) if ψ is subadditive and for any x; y∈XωG , there exists z∈XωG with z6Tz andωGλ ðz; Tz; TzÞ is finite for all λ > 0 such that z is comparable to both x and y.ThenT


227

has a fixed point u∈XωG for some positive integer m and the sequence define byfTnx0gn≥1 converges to u. Moreover, the fixed point of T is unique.

Proof: By Corollary 3.2,Tm has a fixed point say u∈XωG for some positive integerm≥ 1. NowTmðTuÞ ¼ Tmþ1u ¼ TðTmuÞ ¼ Tu, soTu is a fixed point ofTm. By the uniqueness of fixedpoint of Tm, we have Tu ¼ u. Therefore, u is a fixed point of T. Since fixed point of T is alsofixed point of Tm, hence T has a unique fixed point in XωG . ,

Theorem3.4. Let ðX ; ωGÞbe a complete modularG-metric space with a preorder,6and anondecreasing self-mapping T: XωG →XωG on XωG such that for each λ > 0, there isνðλÞ∈ ½0; λÞ such that the following conditions hold:

(1)

ψ�ωG

λ ðTmx; Tmy; TmyÞ�≤ α�ψ�ωG


λþνðλÞðx; y; yÞ�þ β

�ψ�ωG

λ ðx; y; yÞ�

3�ψ�ωG

λ ðx; Tmx; TmxÞ�þ γ�ψ�ωG


λ ðy; Tmy; TmyÞ�;(3.47)

where ψ ∈Ψ and fα; β; γg∈SGer with αðtÞ þ 2maxfsupt≥0βðtÞ; supt≥0γðtÞg < 1; anddistinct x; y∈XωG : Assuming that if a nondecreasing sequence fxngn∈N converges to x�,then xn6x� for each n∈N,

(2) ifψ is subadditive and for any distinct x; y∈XωG , there exists z∈XωG with z6TzandωGλ ðz; Tz; TzÞ is finite for all λ > 0 such that z is comparable to both x and y. ThenT

has a fixed point u∈XωG for some positive integerm≥ 1 and the sequence define byfTnx0gn≥1 converges to u. Moreover, the fixed point of T is unique.

Proof: By Theorem 3.1, Tm has a fixed point say u� ∈XωG for some positive integer m≥ 1.Now TmðTu�Þ ¼ Tmþ1u� ¼ TðTmu�Þ ¼ Tu�, so Tu� is a fixed point of Tm. By theuniqueness of fixed point of Tm, we have Tu� ¼ u�. Therefore, u� is a fixed point of T. Sincefixed point of T is also fixed point of Tm, hence T has a unique fixed point in XωG . ,

Theorem3.5. Let ðX ; ωGÞbe a complete modularG-metric space with a preorder,6and anondecreasing self-mapping T : XωG →XωG on XωG such that for each λ > 0, there isνðλÞ∈ ½0; λÞ such that the following conditions hold:

(1)

ψ�ωG

λ ðTx;Ty;TzÞ�≤α

�ψ�ωG

λ ðx; y; zÞ��ψ�ωG

λþνðλÞðx; y; zÞ�þβ

�ψ�ωG

λ ðx; y; zÞ�

3�ψ�ωG

λ ðx;Tx;TxÞ�þ γ

�ψ�ωG


λ ðy;Ty;TyÞ�þδ

�ψ�ωG

λ ðx; y; zÞ�

3�ψ�ωG

λ ðz;Tz;TzÞ�; (3.48)

where ψ∈Ψand fα; β; γ; δg∈SGerwith αðtÞþ2maxfsupt≥0βðtÞ; supt≥0γðtÞ; supt≥0δðtÞg< 1;and x; y; z∈XωG : Assuming that if a nondecreasing sequence fxngn∈N converges to x, thenxn6x for each n∈N,

(2) if ψ is subadditive and for any x; y; z∈XωG , there exists w∈XωG with w6Tw andωGλ ðw; Tw; TwÞ is finite for all λ > 0 such that w is comparable to both x; y and z.

ThenT has a fixed point u∈XωG and the sequence define by fTnx0gn≥1 converges tou. Moreover, the fixed point of T is unique.

Proof. Let x0 ∈XωG be such that x06Tx0 and let xn ¼ Txn−1 ¼ Tnx0 for all n∈N:Regardingthat T is nondecreasing mapping, we have that x06Tx0 ¼ x1, which implies thatx1 ¼ Tx06Tx1 ¼ x2. Inductively, we have

AJMS27,2

228

x0 6 x1 6 x2 6 � � �6 xn−1 6 xn 6 xnþ1 6 � � � : (3.49)

Assume that there exists n0 ∈N such that xn0 ¼ xn0þ1. Since xn0 ¼ xn0þ1 ¼ Txn0, then xn0 isthe fixed point of T. Now suppose that xnLxnþ1 for all n∈N, thus by inequality (3.38),we have that

x0 a x1 a x2 a � � � a xn−1 a xn a xnþ1 a � � � : (3.50)

Now for each λ > 0; and x0aTx0 for all n∈N implies that ωGλ ðx0; Tx0; Tx0Þ > 0:Again, let




λ > 0, as n→∞.Assume that, for each n∈N, there exists λn > 0 such thatωG

λnðTnx0; T

nþ1x0; Tnþ1x0Þ≠ 0.

Otherwise there is nothing to prove. Suppose that for each n≥ 1, if 0 < λ < λn, thenwe have thatωGλ ðTnx0; T

nx0; Tnþ1x0Þ≠ 0. Since Tnx06Tnþ1x0, we have from inequality (3.5) that

ψðωGλnðTnx0;T

nþ1x0;Tnþ1x0ÞÞ≤ψðωG

λ ðTnx0;Tnþ1x0;T

nþ1x0ÞÞ ¼ ψðωGλ ðTTn−1x0; TT

nx0;

TTnx0ÞÞ. Take x ¼ Tn−1x0 and y ¼ Tnx0 ¼ z, then inequality (3.5) becomes;

ψ�ωG

λn

�Tnx0; T

nþ1x0; Tnþ1x0

��≤ψ

�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��≤ α

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG


nx0; Tnx0

��þ β

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG

λ

�Tn−1x0; TT

n−1x0; TTn−1x0

��þ γ

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG


nx0Þ�

þ δ�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG


nx0Þ�

¼ α�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG


nx0; Tnx0

��þ β

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��þ γ

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��þ δ

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��≤ α

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��þ β

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��þ γ

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��þ δ

�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��3ψ

�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��;

(3.51)

which implies that

ψ�ωG

λ

�Tnx0; T

nþ1x0; Tnþ1x0

��≤ ρψ

�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��≤ψ

�ωG

λ

�Tn−1x0; T

nx0; Tnx0

��...

≤ψ�ωG

λ ðx0; Tx0; Tx0Þ�<∞;

(3.52)


229

where

ρ :¼ α�ψ�ωG

λ

�Tn−1x0; T

nx0; Tnx0


λ

�Tn−1x0; T

nx0; Tnx0

��1� �

γ�ψ�ωG

λ


��þ δ�ψ�ωG

λ


��: (3.53)

Therefore, fψðωGλ ðTnx0; T

nþ1x0; Tnþ1x0ÞÞgn≥1 is nonincreasing and bounded below, hence

converges to some real number s≥ 0. We can also see clearly that by taking θð:Þ :¼γðψðωG


nx0ÞÞÞ þ δðψðωGλ ðTn−1x0; T

nx0; Tnx0ÞÞÞ; as y ¼ Tnx0 ¼ z, then

following Theorem 3.1, T has a unique fixed point in XωG : This complete the proof. ,

Theorem3.6. Let ðX ; ωGÞbe a complete modularG-metric space with a preorder,6and anondecreasing self-mapping T : XωG →XωG on XωG such that for each λ > 0, there isνðλÞ∈ ½0; λÞ such that the following conditions hold:

(1)

ψ�ωG

λ ðTmx; Tmy; TmzÞ�≤ α�ψ�ωG


λþνðλÞðx; y; zÞ�

þ β�ψ�ωG


λ ðx; Tmx; TmxÞ�þ γ�ψ�ωG


λ ðy; Tmy; TmyÞ�þ δ

�ψ�ωG


λ ðz; Tmz; TmzÞ�;(3.54)

where ψ ∈Ψ and fα; β; γ; δg∈SGer with αðtÞ þ 2maxfsupt≥0βðtÞ; supt≥0γðtÞ; supt≥0δðtÞg< 1; and x; y; z∈XωG :Assuming that if a nondecreasing sequence fxngn∈N converges to x,then xn6x for each n∈N;

(2) if ψ is subadditive and for any x; y; z∈XωG , there exists w∈XωG with w6Tw andωGλ ðw; Tw; TwÞ is finite for all λ > 0 such that w is comparable to both x; y and z.

Then T has a fixed point u∈XωG for some positive integer m≥ 1 and the sequencedefine by fTnx0gn≥1 converges to u. Moreover, the fixed point of T is unique.

Proof: Take y ¼ z and fðÞ ¼ γðψðωGλ ðx; y; zÞÞÞ þ δðψðωG

λ ðx; y; zÞÞÞ, then Theorem 3.5 tellsus thatTm has a fixed point say u∈XωG for some positive integerm≥ 1. Therefore, Theorem3.4 shows that T has a unique fixed point in XωG . ,

4. Applications to nonlinear Volterra-Fredholm-type integral equationsIn this section, we construct a system of nonlinear integral equation that satisfies theconditions of Theorem 3.1. We consider the following general nonlinear Volterra-Fredholm-type integral equations.

uðt; xÞ ¼ hðt; xÞ þZ t

0

ZB

Fðt; x; s; y; uðs; yÞ; ðL�uÞðs; yÞÞdyds; (4.1)

and

vðt; xÞ ¼ eðt; xÞ þZ t

0

ZB

Gðt; x; s; y; vðs; yÞ; ðL*vÞðs; yÞÞdsdy; (4.2)

AJMS27,2

230

where

ðL�uÞðt; xÞ ¼Z t

0

ZB

Kðt; x; τ; z; uðτ; zÞÞdzdτ; (4.3)

and

ðL�vÞðt; xÞ ¼Z t

0

ZB

Kðt; x; τ; z; vðτ; zÞÞdzdτ; (4.4)

h; F ; K and e; G; L are given functions and u; v are the unknown functions. We assume thath; e∈Cðℝþ3B; ℝnÞ, K ∈CðΩ3ℝn; ℝnÞ, F ∈CðΩ3ℝn 3ℝn; ℝnÞ, G∈CðΩ3ℝn 3ℝn;ℝnÞandΩ ¼ fðt; x; s; yÞ : 0≤ s≤ t < ∞; x; y∈Bg ,B ¼ Q n

j¼1½aj; bj�; bj > aj:Take supðt;xÞ∈ℝþ 3Bgðs; yÞ≤ 1

tQ n

j¼1½aj ; bj�

, where ðt; xÞ∈ℝþ 3B:

Let α; β; γ > 0 with αðtÞ þ 2maxt∈Ωfsupt≥0βðtÞ; supt≥0γðtÞgh1 such that

kFðt; x; s; y; uðs; yÞ; ðL�uÞðs; yÞÞ−Gðt; x; s; y; vðs; yÞ; ðL�vÞðs; yÞÞk≤gðs; yÞfαðku− vkÞku− vk þ βðku− vkÞmðu; L�uÞ þ γðku− vkÞrðv; L�vÞgLet F; G : CðΩ3ℝn 3ℝn; ℝnÞ→ℝn be such that Fu; Gv ∈CðΩ3ℝn 3ℝn; ℝnÞ and let

Fu ¼Z t

0

ZB

Fðt; x; s; y; uðs; yÞ; ðL*uÞðs; yÞÞdyds; (4.5)

and

Gv ¼Z t

0

ZB

Gðt; x; s; y; vðs; yÞ; ðL�vÞðs; yÞÞdsdy; (4.6)

for F ∈CðΩ3ℝn 3ℝn; ℝnÞ, G∈CðΩ3ℝn 3ℝn; ℝnÞ and u; v are the unknown functions.Now for any λ > 0, we define

ωGλ ðx; y; zÞ :¼

1

2ð1þ λÞ supðt;xÞ∈ℝþ3B

fkxðtÞ � yðtÞk þ kyðtÞ � zðtÞk þ kxðtÞ � zðtÞkg; (4.7)

so that

ωGλ ðx; y; yÞ :¼

1

ð1þ λÞ supðt;xÞ∈ℝþ3B

fkxðtÞ � yðtÞkg: (4.8)

In fact Eqns. (4.7) and (4.8) satisfies all the conditions in Definition 2.6 endowedwith XωG ¼ ðX ; ωGÞ ¼ CðΩ3ℝn 3ℝn; ℝnÞ


231

Now, take A ¼ 11þλ supðt;xÞ∈ℝþ3BfkFu −Gv þ hðt; xÞ− eðt; xÞkg, so that

A≤1

1þλsup

ðt;xÞ∈ℝþ3B

Z t

0

ZB

kFðt; x; s; y; uðs; yÞ;ðL*uÞðs; yÞÞ

�Gðt; x; s; y; vðs; yÞ;ðL*vÞðs; yÞÞk3dyds

≤1

1þλsup

ðt;xÞ∈ℝþ3B

Z t

0

ZB

gðs; yÞfαðku�vkÞku�vk

þβðku�vkÞmðu;L*uÞþγðku�vkÞrðv;L*vÞgdyds

≤1

1þλsup

ðt;xÞ∈ℝþ3B

ffαðku�vkÞku�vkþβðku�vkÞmðu;L*uÞ

þ γðku�vkÞ3rðv;L*vÞgg≤

1

1þλsup

ðt;xÞ∈ℝþ3B

fαðku�vkÞku�vkg

þ 1

1þλsupt∈Ω

fβðku�vkÞmðu;L*uÞþγðku�vkÞ3rðv;L*vÞg

≤1

1þλsup

ðt;xÞ∈ℝþ3B

fαðku�vkÞku�vkg

þ 1

1þλsup

ðt;xÞ∈ℝþ3B

fβðku�vkÞmðu;L�uÞg

þ 1

1þλsup

ðt;xÞ∈ℝþ3B

fγðku�vkÞrðv;L�vÞg;

(4.9)

where m; r∈Cðℝþ3B3ℝn;ℝnÞmðu;L�uÞðt; xÞ¼ sup

ðt;xÞ∈ℝþ3B

kFuðt; xÞþhðt; xÞ�uðt; xÞk; (4.10)

rðv;L�vÞðt; xÞ¼ supðt;xÞ∈ℝþ3B

kGvðt; xÞþeðt; xÞ�vðt; xÞk (4.11)

Theorem4.1. LetXωG ¼ CðΩ3ℝn 3ℝn; ℝnÞbe a complete modularG-metric space andωG : ð0; ∞Þ3XωG 3XωG 3XωG →ℝn

þ∪ f∞g be defined by

ωGλ ðu; v; vÞ :¼

1

1þ λsup

ðt;xÞ∈ℝþ3B

nnormuðt; xÞ � vðt; xÞ; λ > 0 (4.12).

and u6v5uðt; xÞ≤ vðt; xÞ ∀ ðt; xÞ∈ℝþ3B. Let Fu; Gv : CðΩ3ℝn3ℝn; ℝnÞ→ℝn aresuch that Fu; Gv ∈XωG for each u; v∈XωG and Fu, Gv satisfying Eqns. (4.5) and (4.6),respectively, for all ðt; xÞ∈ℝþ3B. Suppose that there exists nonnegative reals α; β; γ > 0with αðtÞ þ 2maxt∈Ωfsupt≥0βðtÞ; supt≥0γðtÞg < 1such that inequality 4 is satisfied for everyu; v∈XωG :Moreover ifψ is subadditive and for any u; v∈XωG , there existsw0; w1 ∈XωG withw06w1 and ωG

λ ðw0; w1; w1Þ is finite for all λ > 0 such that w is comparable to both u and v.Then the system of integral Eqns (4.1) and (4.2) have a unique solution in XωG :

AJMS27,2

232

Proof. Define the mappingT : XωG →XωG byTu ¼ Fu þ eandTv ¼ Gv þ h. Then for λ > 0,ωGλ ðTu; Tv; TvÞ ¼ 1

1þλsupðt;xÞ∈ℝþ3BnnormFuðt; xÞ−Gvðt; xÞ þ eðt; xÞ− hðt; xÞ, ωGλ ðu; Tu;

TuÞ ¼ 11þλsupðt;xÞ∈ℝþ3BnnormFuðt; xÞ þ eðt; xÞ− uðt; xÞ and ωG

λ ðv; Tv; TvÞ ¼ 11þλsupðt;xÞ

∈ℝþ3BnnormGvðt; xÞ þ hðt; xÞ− vðt; xÞ. So from inequality 4, we get by noticing that ψ iscontinuous and subadditive and there exists ν∈ ½0; λÞ such that

ψ�ωG

λ ðTu; Tv; TvÞ�≤ α

�ψ�ωG

λ ðu; v; vÞ��ψ�ωG

λþνðλÞðu; v; vÞ�

þ β�ψ�ωG


λ ðu; Tu; TuÞ�þ γ

�ψ�ωG


λ ðv; Tv; TvÞ�;

(4.13)

where ψ ∈Ψ. By Theorem 3.1, we conclude that the system of nonlinear Volterra-Fredholmintegral Eqns (4.1) and (4.2) have a unique solution in XωG :,

References

1. Geraghty M. On contractive mapping. Proc Amer Math Soc. 1973; 40: 604-8.

2. Gordji ME, Cho YJ, Pirbavafa S. A generalization of Geraghty’s theorem in partially orderedmetric spaces and application to ordinary differential equations. Fixed Point Theor Appl. 2012:74; 1687-812.

3. Bhaskar IG, Lakshmikantham V. Fixed point theorems in partially ordered metric spaces andapplications. Nonlinear Anal. 2006; 65: 1379-93.

4. Yolacan E. Fixed point theorems for Geraghty type contraction mappings and coupled fixed pointresults in 0-complete ordered partial metric spaces. Int J Math Sci. 2016: 2016; 5.

5. Faraji H, Savic D, Radenovic S. Fixed point theorems for Geraghty contraction type mappings inb-metric spaces and application. Axioms. 2019; 8: 34. doi: 10.3390/axioms/8010034.

6. Gupta V, Shatanawi W, Mani N. Fixed point theorems forðψ ; βÞ-Geraghty contraction-type mapsin ordered metric spaces and some applications to integral and ordinary differential equations. JFixed Point Theor Appl. 2017; 19: 1251-1267.

7. Chaipunya P, Cho YJ, Kumam P. Geraghty-type theorems in modular metric spaces withapplication to partial differential equation. Adv Differ Equ. 2012; 83: 1687-847.

8. Agarwal RP, Karapinar E, O’Regan D, Roldan-Lopez-de-Hierro AF. Fixed point theory inmetrictype spaces, New York Dordrecht London: Springer Cham Heildelberg. 2015.

9. Abbas M, Nazir T, Shatanawi W, Mustafa Z. Fixed and related fixed point theorems for threemaps in G-metric spaces. Hacet J Math Stat. 2012; 41(2): 291-06.

10. Chistyakov VV. Metric modular spaces, I basic concepts. Nonlinear Anal Theor Meth Appl. 2010;72: 1-14.

11. Chistyakov VV. Metric modular spaces, II Applications to superposition operators. NonlinearAnal Theor Meth Appl. 2010; 72: 15-30.

12. Nakano H. Modulared semi-ordered linear spaces. Vol 1, Tokyo: Maruzen. 1950.

13. Orlicz W. Collected papers. Vols I, II, Warszawa: PWN. 1988.

14. Musielak J. Orlicz spaces and modular spaces, Lecture Notes in Math. Vol. 1034, Berlin: Springer-verlag. 1983.

15. Nakano H. Topology of linear topological spaces. Vol 3, Tokyo: Maruzen. 1951.

16. Chistyakov VV. A fixed point theorem for contractions in metric modular spaces. arXiv:1112.5561.2011; 65-92.

17. Mutlu A, Ozkan K, Gurdal U. A new fixed point theorem in modular metric spaces. Int J AnalAppl. 2018; 4: 472-83.


233

18. Okeke GA, Kim JK. Approximation of common fixed point of three multi-valued ρ-quasi-nonexpansive mappings in modular function spaces. Nonlinear Func Anal Appl. 2019:24(4): 651-64.

19. Rhoades BE. Two fixed point theorems for mappings satisfying a general contractive condition inintegral-type. Int J Math Sci. 2003: 63; 4007-13.

20. Okeke GA, Khan SH. Approximation of fixed point of multivalued ρ-quasi-contractive mappingsin modular function spaces. Arab J Math Sci. 2020: 26(1/2); 75-93.

21. Bishop SA, Okeke GA, Eke K. Mild solutions of evolution quantum stochastic differentialequations with nonlocal conditions. Math Meth Appl Sci. 2020: 1-8. doi: 10.1002/mma.6368.

22. Chlebowicz A. Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type. Adv. Nonlinear Anal. 2020 9(1): 1187-204.

23. Liu Z, Li X, Kang SM, Cho YJ. Fixed point theorems for mappings satisfying contractiveconditions of integral-type and applications. Fixed Point Theor Appl. 2011; 64: 1687-812.

24. Okeke GA, Francis D, de la Sen M. Some fixed point theorems for mappings satisfying rationalinequality in modular metric spaces with applications. Heliyon 2020: 6; e04785.

25. Papageorgiou NS, Radulescu VR, Repovs DD. Nonlinear analysis - theory and methods, Cham:Springer Monographs in Mathematics. Springer. 2019.

26. Zhao HY. Pseudo almost periodic solutions for a class of differential equation with delaysdepending on state. Adv Nonlinear Anal. 2020: 9(1); 1251-58.

27. Azadifar B., Maramaei M., Sadeghi G.. On the modular G-metric spaces and fixed point theorems.J Nonlinear Sci Appl. 2013; 6: 293-304.

28. Bali NP. Golden real analysis, New Delhli: Laxmi Publications. 1986.

29. Schechter E. Handbook of analysis and its foundations. New York: Academic Press. 1997.

Further reading

30. Amini-Harandi A. A fixed point theory for generalized quasicontraction maps in vector modularspaces. Comput Math Appl. 2011; 61: 1891-97.

31. Mustafa Z, Khandagji M, Shatanawi W. Fixed point results on complete G-metric spaces. Stud SciMath Hung. 2011: 48(3); 304-19.

32. Nakano H. On the stability of functional equations. Aequationes Math. 2009: 77; 33-88.

Corresponding authorGodwin Amechi Okeke can be contacted at: [email protected]


AJMS27,2

234


Existence of positive solutions forp-Laplacian systems involving leftand right fractional derivatives

Samira Ramdane�Ecole Normale Sup�erieure d’Enseignement Technologique de Skikda,

Skikda, Algeria, and

Assia Guezane-LakoudLaboratory of Advanced Material, Faculty of Sciences,Badji Mokhtar-Annaba University, Annaba, Algeria

Abstract

Purpose –The paper deals with the existence of positive solutions for a coupled system of nonlinear fractionaldifferential equations with p-Laplacian operator and involving both right Riemann–Liouville and left Caputo-type fractional derivatives. The existence results are obtained by the help of Guo–Krasnosel’skii fixed-pointtheorem on a cone in the sublinear case. In addition, an example is included to illustrate the main results.Design/methodology/approach – Fixed-point theorems.Findings – No finding.Originality/value – The obtained results are original.

Keywords Fractional derivatives, Integral condition, Existence of solutions, Fixed point theorem


1. IntroductionIn this paper, we consider the following coupled system of nonlinear fractional differentialequations with p-Laplacian operator:8>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>:

RDα1−fp

�CD

β10þuðtÞ

�þ a1ðtÞf1ðuðtÞ; vðtÞÞ ¼ 0; t ∈ ½0; 1�;

RDα1−fp

�CD

β20þvðtÞ

�þ a2ðtÞf2ðuðtÞ; vðtÞÞ ¼ 0; t ∈ ½0; 1�;

fp

�CD

β10þuð1Þ

�¼ 0; u

0 ð0Þ ¼ 0; η1uð1Þ � uð0Þ ¼Z1

0

g1ðs; uðsÞ; vðsÞÞds;

fp

�CD

β20þvð1Þ

�¼ 0; v

0 ð0Þ ¼ 0; η2vð1Þ � vð0Þ ¼Z10

g2ðs; uðsÞ; vðsÞÞds:

Positivesolutions forp-Laplacian

systems

235

JEL Classification — 34A08, 34B15©Samira Ramdane andAssia Guezane-Lakoud. Published inArab Journal ofMathematical Sciences.


The authors are grateful to the anonymous referees for their valuable comments and suggestions,which helped to improve the quality of the paper.



Received 10 October 2020Revised 16 December 2020

Accepted 26 December 2020


Vol. 27 No. 2, 2021pp. 235-248


DOI 10.1108/AJMS-10-2020-0086

where 0 < α < 1; 1 < βi < 2; ηi > 1; ði ¼ 1; 2Þ and fpðsÞ ¼ jsjp−2s; p > 1; fq ¼ ðfpÞ−1;1pþ 1

q¼ 1; ℝDα

1− the right Riemann–Liouville fractional derivative, CDβi0þ denotes the left

Caputo fractional derivative of order βi; the functions ai ∈Cð½0; 1�; ℝþÞ; fi ∈Cðℝþ 3ℝþ;ℝþÞ; gi ∈Cð½0; 1�3ℝþ 3ℝþ; ℝþÞ for i ¼ 1; 2:

Fractional differential equations arise inmany engineering and scientific disciplines as themathematical modeling of systems and processes in the fields of physics, chemistry,aerodynamics, electrodynamics of a complex medium, polymer rheology, etc. Fractionaldifferential equations also serve as an excellent tool for the description of hereditaryproperties of variousmaterials and processes. For the basic theory and recent development ofsubject, see [1, 2, 3]. Recently, a linear boundary value problem involving both the rightCaputo and the left Riemann–Liouville fractional derivatives have been studied by manyauthors [4, 5] Many people pay attention to the existence and multiplicity of solutions orpositive solutions for boundary value problems of nonlinear fractional differential equationsby means of some fixed-point theorems [6–13].

In [14], by applying Guo–Krasnosel’ski�ı’s fixed-point theorem, Guezane-Lakoud andAshyralyev discussed the existence of positive solutions for the following fractional BVP8>>><

>>>:Dq

0þuðtÞ ¼ f ðt; uðtÞÞ; t ∈ ½0; 1�; 1 < q < 2

u0 ð0Þ ¼ 0; uð0Þ � αuð1Þ ¼

Z10

gðs; uðsÞÞds:

where f : ½0; 1�3ℝ→ℝ is a given function, α ∈ ℝþ; Dq0þ denotes the Caputo’s fractional

derivative of order q.On the other hand, the study of coupled systems involving fractional differential

equations is also important as such systems occur in various problems, see [13, 15, 16] and thereferences therein.

In the interesting paper [17], Liu studied by the help of Picard iterative method andSchaefer’s fixed-point theorem, the existence of solutions for four classes of boundary valueproblems for impulsive fractional differential equations.

In [12], relying on the Guo–Krasnosel’ski�ı’s fixed-point theorem, Li and Wei discussedexistence of positive solutions for the following coupled system of mixed higher-ordernonlinear singular fractional differential equations with integral boundary conditions8>>>>>>>>><

>>>>>>>>>:

Dα10þuðtÞ þ a1ðtÞf1ðt; uðtÞ; vðtÞÞ ¼ 0; t ∈ ½0; 1�

Dα20þvðtÞ þ a2ðtÞf2ðt; uðtÞÞ ¼ 0; t ∈ ½0; 1�

uðjÞð0Þ ¼ vðkÞð0Þ ¼ 0; 0 ≤ j ≤ n1 � 2; 0 ≤ k ≤ n2 � 2

uð1Þ ¼Z10

h1ðsÞuðsÞds; vð1Þ ¼Z10

h2ðsÞvðsÞds

where ni − 1 < αi < ni; ni ≥ 3; Dαi0þ are the standard Riemann–Liouville fractional

derivative, aiðtÞ∈C½0; 1� may be singular at t ¼ 0; and/or t ¼ 1; hi ∈ L1½0; 1� arenonnegative ði ¼ 1; 2Þ.

On the other hand, differential equations with p-Laplacian operator have been widelystudied owing to its importance in theory and application of mathematics and physics, suchin non-Newtonian mechanics, nonlinear elasticity and glaciology, population biology,

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236

nonlinear flow laws. There are a very large number of papers devoted to the existence ofsolutions of the p-Laplacian operator, see for example [18–25].

In [26] G. Q. Chai, studied the existence of positive solutions for the boundary-valueproblem of nonlinear fractional differential equations with p-Laplacian operator(

Dβ0þfp

�Dα

0þuðtÞ�þ f ðt; uðtÞÞ ¼ 0; 0 < t < 1;

Dα0þuð0Þ ¼ 0; Dα

0þuð1Þ þ σDγ0þuð1Þ ¼ 0; uð0Þ ¼ 0:

where 1 < α < 2; 0 < β < 1; fpðsÞ ¼ jsjp−2s; p > 1; fq ¼ ðfpÞ−1; 1pþ 1

q¼ 1; Dα

0þ ; Dβ0þ are

the standard Riemann–Liouville fractional derivatives, 0 < γ ≤ 1; The functionf : ½0; 1�3ℝþ

→ℝþ is continuous.The rest of the paper is organized as follows. In Section 2, we present preliminaries and

lemmas. Section 3, we investigate the existence of a solution for the corresponding fractionallinear boundary value problem. Finally, Section 4 is devoted to the existence of positivesolutions under some sufficient conditions on the nonlinear terms, thenwe give an example toillustrate our results.

2. PreliminariesIn this section, we recall the basic definitions and lemmas from fractional calculus theory,see [2, 3], for more details.

Let α > 0; ½a; b� be a finite interval ofℝ and g a real function on ða; bÞ:The left and rightRiemann–Liouville fractional integral of the function g are defined, respectively, by

Iαaþ f ðtÞ ¼1

ΓðαÞZ t

a

ðt � sÞα−1gðsÞds; Iαb− f ðtÞ ¼1

ΓðαÞZb

t

ðs� tÞα−1gðsÞds;

provided that the right-hand side exists.The right Riemann–Liouville fractional derivative and the left Caputo fractional

derivative of order α > 0 of g are, respectively

RDαb− f ðtÞ ¼

�−d

dt

�n

I n−αb− gðtÞ; CDαaþ f ðtÞ ¼ I n−αaþ gðnÞðtÞ;

where n < α < nþ 1; n ¼ ½α� þ 1; provided that the right-hand side exists.For the properties of Riemann–Liouville fractional derivative and Caputo fractional

derivative, we obtain the following statement. Let u∈L1ð0; 1Þ then

I αR1− Dα1−uðtÞ ¼ uðtÞ þ

Xni¼1

aið1� tÞα−i (2.1)

IαC0þ Dα0þuðtÞ ¼ uðtÞ þ

Xn−1k¼0

bktk (2.2)

where ai; bk ∈ℝ; i ¼ 0; . . . n; and k ¼ 0; . . . n− 1:We also need the following lemma and theorem to obtain our results.

Lemma 2.1. [26] Let c > 0; γ > 0: for any x; y∈ ½0; c�we have(1) if γ > 1; then jxγ − yγ j ≤ γcγ−1jx− yj;(2) if 0 < γ ≤ 1; then jxγ − yγ j ≤ jx− yjγ :


systems

237

Theorem2.1. [27] (Guo–Krasnoselski�ı’s) LetE be a Banach space, and letK ⊂E, be a cone.AssumeΩ1 andΩ2 are open subsets ofEwith 0∈Ω1; Ω1 ⊂ Ω2 and letT : K ∩ ðΩ2nΩ1Þ→K,be a completely continuous operator such that

(1) kTuk≤ kuk; u ∈ K ∩ vΩ1, and jjTujj≥ kuk; u∈K ∩ vΩ2; or

(2) kTuk≥ kuk; u∈K ∩ vΩ1 and jjTujj≤ kuk; u∈K ∩ vΩ2:

Then T has a fixed point in K ∩ ðΩ2nΩ1Þ

3. Linear boundary value problem

Lemma 3.1. Assume that y∈Cð0; 1Þ∩L1ð0; 1Þ and 1 < βi < 2; i ¼ 1; 2, the uniquesolution of the boundary value problem

CDβi0þuðtÞ þ yðtÞ ¼ 0; t ∈ ½0; 1�; (3.1)

u0 ð0Þ ¼ 0; ηiuð1Þ � uð0Þ ¼

Z10

giðsÞds (3.2)

is given by

uðtÞ ¼Z10

Giðt; sÞyðsÞdsþ 1

ηi � 1

Z1

0

giðsÞds (3.3)

where

Giðt; sÞ ¼ 1

ΓðβiÞ

8><>:

ηiηi � 1

ð1� sÞβi−1 � ðt � sÞβi−1; 0 ≤ s ≤ t ≤ 1:

ηiηi � 1

ð1� sÞβi−1; 0 ≤ t ≤ s ≤ 1:

(3.4)

Proof. We apply (2.2) to equation (3.1) to get

uðtÞ ¼ −Iβi0þyðtÞ þ c1 þ c2t; t ∈ ½0; 1� (3.5)

thanks to boundary condition (3.2) we obtain c2 ¼ 0; and

c1 ¼ 1

ηi � 1

24 ηiΓðβiÞ

Z1

0

ð1� sÞβi−1yðsÞdsþZ1

0

giðsÞds35:

So, the unique solution of the problem (3.1) is

uðtÞ ¼ 1

ΓðβiÞ

24�

Z t

0

ðt � sÞβi−1yðsÞdsþ ηiηi � 1

Z10

ð1� sÞβi−1yðsÞds35þ 1

ηi � 1

Z10

giðsÞds

¼Z10

Giðt; sÞ yðsÞdsþ 1

ηi � 1

Z10

giðsÞds:

The proof is completed. ▪

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238

Lemma 3.2. If y ∈ Cð0; 1Þ∩L1ð0; 1Þ, then the boundary value problem

RDα1−fp

�CD

βi0þuðtÞ

�þ yðtÞ ¼ 0; 0 ≤ t ≤ 1 (3.6)

fp

�CD

βi0þuð1Þ

�¼ 0 (3.7)


Z10

giðsÞds (3.8)

has an unique solution

uðtÞ ¼Z1

0

Giðt; sÞfq

0@Z1

s

ðτ � sÞα−1ΓðαÞ yðτÞdτ

1Adsþ 1

ηi � 1

Z10

giðsÞds

where Giðt; sÞ is defined as (3.4).Proof. From Eqs (3.6) and (2.1), we have

fp

�CD

βi0þuðtÞ

�¼ −Iα1−yðtÞ þ C1ð1� tÞα−1; C1 ∈ℝ: (3.9)

By the boundary conditions (3.7) we get C1 ¼ 0; consequently,

fp

�CD

βi0þ uðtÞ

�¼ −I α1−yðtÞ

and then

CDβi0þuðtÞ þ fq

0@ 1

ΓðαÞZ1t

ðs� tÞα−1yðsÞds1A ¼ 0; t ∈ ½0; 1�: (3.10)

Thus, the fractional boundary value problem (3.1)–(3.2) is equivalent to the following problem

CDβi0þuðtÞ þ fq

0@ 1

ΓðαÞZ1

s

ðs� tÞα−1yðsÞds1A ¼ 0; t ∈ ½0; 1�


Z1

0

giðsÞds:

Lemma 3.1 implies that the problem (3.6), (3.7) and (3.8) has an unique solution

uðtÞ ¼Z10

Giðt; sÞfq

0@ 1

ΓðαÞZ1

s

ðτ � sÞα−1yðτÞdτ1Adsþ 1

ηi � 1

Z10

giðsÞds;

the proof is achieved. ▪

Lemma 3.3. The functions Giðt; sÞ; i ¼ 1; 2 are continuous on ½0; 1� 3 ½0; 1� and satisfythe following properties:

(1) Giðt; sÞ > 0 for t; s∈ ½0; 1Þ; i ¼ 1; 2

(2) 1ηiGiðs; sÞ ≤ Giðt; sÞ ≤ Giðs; sÞ; i ¼ 1; 2 for ðt; sÞ ∈ ½0; 1Þ3 ½0; 1Þ.


systems

239

Proof. (1) Observing the expression of Giðt; sÞ, it is easy to see that Giðt; sÞ > 0;for t; s∈ ½0; 1Þ; i ¼ 1; 2

(2) First, Giðt; sÞ ≤ Giðs; sÞ for t; s ∈ ½0; 1ÞSecond, setting

gi;1ðt; sÞ ¼ ηiðηi � 1ÞΓðβiÞ

ð1� sÞβi−1 � ðt � sÞβi−1ΓðβiÞ

; s≤ t

gi;2ðsÞ ¼ ηiðηi � 1ÞΓðβiÞ

ð1� sÞβi−1; t ≤ s

for given s∈ ½0; 1Þ; gi;1ðt; sÞ is decreasing as a function of t, then,

gi;1ðt; sÞ≥ gi;1ð1; sÞ

¼ 1

ðηi � 1ÞΓðβiÞð1� sÞβi−1

≥1

ηiGiðs; sÞ;

and gi;2ðsÞ ≥ 1ηiGiðs; sÞ: ▪

4. Existence of positive solutionsWe need to introduce some notations for the forthcoming discussion. LetX ¼ C½0; 1� 3 C½0; 1� be the Banach space endowed with the norm

kðx1; x2Þk ¼ maxðkxik∞; i ¼ 1; 2Þwhere kxik∞ ¼ max

t∈½0; 1�jxiðtÞj

Define the cone P ⊂X by

P ¼�ðx1; x2Þ∈X : xiðtÞ≥ 0; t ∈ ½0; 1�; min

t∈½0;1�xiðtÞ≥ 1

ηikxik∞; i ¼ 1; 2

(4.1)

Let us introduce the following notations

Aδ;i ¼ limðjujþjvjÞ→δ

fiðu; vÞðjuj þ jvjÞp−1; ðδ ¼ 0þ or þ∞Þ;

Ei ¼Z10

Giðs; sÞds;

Fi ¼ aq−1i

ðΓðαÞÞq−1Z10

Giðs; sÞ�Z1

s

ðτ � sÞα�1dτ�q−1

ds; where ai ¼ maxt∈½0:1�

aiðtÞ

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240

By simple calculation, we get

Ei ¼ ηiðηi � 1ÞΓðβi þ 1Þ

Fi ¼ ηiaq−1i

ðηi � 1ÞðΓðαþ 1ÞÞq−1ΓðβiÞ½αðq� 1Þðβi � 1Þ þ 1�; i ¼ 1; 2

We make the following assumption:(H): There exist two nonnegative functions c1; c2 ∈L1½0; 1� and two constants b1; b2 > 0

such that

g1ðt; u; vÞ ≤ b1c1ðtÞðuþ vÞ; g2ðt; u; vÞ ≤ b2c2ðtÞðuþ vÞ;for ðu; vÞ∈ℝþ 3ℝþ; with kcikL1 ≤ ηi − 1

2bi; i ¼ 1; 2:

Lemma 4.1. The system ðSÞ has a positive solution ðu; vÞ if and only if ðu; vÞ is a positivesolution for the following system of integral equations:8>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>:

uðtÞ ¼Z1

0

G1ðt; sÞ0@ 1

ΓðαÞZ1

s

ðτ � sÞα�1a1ðτÞf1ðuðτÞ; vðτÞÞdτ

1A

q−1

ds

þ 1

η1 � 1

Z10

g1ðs; uðsÞ; vðsÞÞds

vðtÞ ¼Z10

G2ðt; sÞ0@ 1

ΓðαÞZ1s

ðτ � sÞα�1a2ðτÞf2ðuðτÞ; vðτÞÞdτ

1A

q−1

ds

þ 1

η2 � 1

Z10

g2ðs; uðsÞ; vðsÞÞds:

(4.2)

Proof. Easily obtained by Lemma 3.2, then we omit it. ▪

Define the operator

T : P→C½0; 1�3C½0; 1�Tðu; vÞ ¼ ðT1ðu; vÞ;T2ðu; vÞÞ; (4.3)

where Ti : P→C½0; 1� and

Tiðu; vÞ ¼Z10

Giðt; sÞ0@ 1

ΓðαÞZ1s

ðτ � sÞα�1aiðτÞfiðuðτÞ; vðτÞÞdτ

1A

q−1

ds

þ 1

ηi � 1

Z10

giðs; uðsÞ; vðsÞÞds:

(4.4)


systems

241

Then, by Lemma 4.1, the existence of solutions for problem ðSÞ is translated into the existenceof fixed points for Tðu; vÞ ¼ ðu; vÞ, thus the fixed point of the operator T coincides with thesolution of problem ðSÞ.

Lemma 4.2. Let T : P→X be the operator defined by (4.3). Then T is completelycontinuous and TP ⊂P.

Proof. First, we shall show that TP ⊂P:We have for each t ∈ ½0; 1�;

jTiðuðtÞ; vðtÞÞj≤Z10

Giðt; sÞ0@ 1

ΓðαÞZ1s


1A

q−1

ds

þ 1

ηi � 1

Z10

giðs; uðsÞ; vðsÞÞds

Taking the supremum over ½0; 1�, we get

kTiðu; vÞk∞≤

Z10

Giðt; sÞ0@ 1

ΓðαÞZ1s


1A

q−1

ds

þ 1

ηi � 1

Z1

0

giðs; uðsÞ; vðsÞÞds:

On the other side, we have

TiðuðtÞ; vðtÞÞ ≥ 1

ηi

Z1

0

Giðt; sÞ0@ 1

ΓðαÞZ1

s


1A

q−1

ds

þ 1

ηi � 1

Z10


Since ηi > 1, then,


ηikTiðu; vÞk∞:

That is TP ⊂P:Second, we shall proof that T is completely continuous that will be done in two steps.

Step 1: By the continuity of the functions fi and gi it yields for n ≥ N ;

jfiðunðτÞ; vnðτÞÞ � fiðuðτÞ; vðτÞÞj < ε;

jgiðs; unðsÞ; vnðτÞÞ � giðs; uðsÞ; vðτÞÞj < ε:

AJMS27,2

242

(1) If 1 < q≤ 2; then from Lemma 2.10@Z1

s


1A

q−1

�0@Z1

s


1A

q−1

≤

0@Z1

s

ðτ � sÞα�1aiðτÞjfiðunðτÞ; vðτÞÞ � fiðuðτÞ; vðτÞÞjdτ

1A

q−1

<hεαai

iq−1:

Then,

jTiðun; vnÞ � Tiðu; vÞj < aq−1i εq−1

ðΓðαþ 1ÞÞq−1Z10

Giðs; sÞdsþ εηi � 1

¼ aq−1i εq−1

ðΓðαþ 1ÞÞq−1 Ei þ 1

ηi � 1ε:

Hence

jjTiðun; vnÞ � Tiðu; vÞjj∞≤

aq−1i Ei

ðΓðαþ 1ÞÞq−1 þ1

ηi � 1

!εq−1: (4.5)

(2) If q > 2; then from Lemma 2.1, we have0@ 1

ΓðαÞZ1

s

ðτ � sÞα�1aiðτÞfiðunðτÞ; vðτÞÞdτ

1A

q−1

�0@ 1

ΓðαÞZ1s


1A

q−1

≤ðq� 1ÞðcÞq−2

ΓðαÞZ1

s

ðτ � sÞα−1aiðτÞjfiðunðτÞ; vðτÞÞ � fiðuðτÞ; vðτÞÞjdτ

<ðq� 1Þcq−2aiΓðαþ 1Þ ε:

So,

jTiðun; vnÞ � Tiðu; vÞj <0@ðq� 1Þcq−2ai

Γðαþ 1ÞZ1

0

Giðs; sÞdsþ 1

ηi � 1

1Aε:


systems

243

Hence

kTiðun; vnÞ � Tiðu; vÞk∞ <

�ðq� 1Þcq−2aiΓðαþ 1Þ Ei þ 1

ηi � 1

�ε: (4.6)

From (4.5)–(4.6) it follows that jjTðun; vnÞ−Tðu; vÞjj→ 0 as n→∞; thus T is continuous.

Step 2: The operator T is uniformly bounded on P:LetΩbe an open bounded set in P. Set

Li ¼ max fiðuðtÞ; vðtÞÞ < ∞

ðu;vÞ∈Ω; li ¼ max giðt; uðtÞ; vðtÞÞ

ðt;u;vÞ∈½0;1�3Ω:

Then for ðt; u; vÞ∈ ½0; 1�3Ω; we have

jTiðuðtÞ; vðtÞÞj ≤Z10

Giðt; sÞ0@ 1

ΓðαÞZ1s


1A

q−1

ds

þ 1

ηi � 1

Z1

0


≤

�Liai

ðΓðαþ 1ÞÞ�q−1 Z1

0

Giðs; sÞdsþ li

ðηi � 1Þ

¼�

Liai

ðΓðαþ 1ÞÞ�q−1

Ei þ li

ηi � 1< ∞

thus TðΩÞ is uniformly bounded.Now we prove that TðΩÞ equicontinuous, Let ðu; vÞ∈Ω; 0≤ t1 ≤ t2 ≤ 1. We have

jTiðuðt1Þ; vðt1ÞÞ � Tiðuðt2Þ; vðt2ÞÞj

≤

Zt10

jGiðt2; sÞ � Giðt1; sÞj0@ 1

ΓðαÞZ1s


1A

q−1

ds

þZ1t2


ΓðαÞZ1s


1A

q−1

ds

þZt2t1


ΓðαÞZ1s


1A

q−1

ds

≤

�Liai

ðΓðαþ 1ÞÞ�q−1 jt2 � t1jβi

Γðβi þ 1Þ :

Consequently, jTiðuðt1Þ; vðt1ÞÞ−Tiðuðt2Þ; vðt2ÞÞj→ 0, when t2 → t1: Hence TðΩÞ isequicontinuous. Finally, by Arzela–Ascoli’s theorem, it follows that T is completelycontinuous mapping on Ω: ▪

AJMS27,2

244

Theorem 4.1. Assume that the condition ðHÞ is satisfied, then the system ðSÞ has at leastone nontrivial positive solution ðu; vÞ in the coneP, in the caseA0;i ¼ 0andA∞;i ¼ ∞; i ¼ 1; 2:

Proof. From A0;i ¼ 0; i ¼ 1; 2; we deduce that for

0 < ε ≤ mini¼1;2

8><>:��

1� bi

ηi � 1kcikL1

�1

Fi

� 1q−1

9>=>;;

there exist ρ1 > 0; such that if 0 < uþ v ≤ ρ1, then

fiðu; vÞ≤ εðjuj þ jvjÞp−1

Let Ω1 ¼ fðu; vÞ∈X ; kðu; vÞk < ρ1g:Assume that ðu; vÞ ∈ P ∩ vΩ1; then

TiðuðtÞ; vðtÞÞ≤Z10

Giðs; sÞ0@ 1

ΓðαÞZ1s

ðτ � sÞα�1aiðτÞεðjuj þ jvjÞp�1

dτ

1A

q−1

ds

þ 1

ηi � 1

Z10

biciðsÞðjuj þ jvjÞds:

≤

� εΓðαÞ

�q−1 Z10

Giðs; sÞ

3

0@Z1

s

ðτ � sÞα�1aiðτÞðkuk∞ þ kvk

∞Þp�1

dτ

1A

q−1

ds

þ bi

ηi � 1

Z10

ciðsÞðkuk∞ þ kvk∞Þds:

¼ jjðu; vÞjj�εq−1Fi þ bi

ηi � 1kcikL1

�

≤ kðu; vÞk:Hence

kTðu; vÞk ≤ jjðu; vÞjj; forðu; vÞ ∈ vΩ1 ∩P

Since A∞;i ¼ ∞; i ¼ 1; 2; so for

μ ≥ maxi¼1;2

8>><>>:�η2ΓðαÞ

ξi

� 1q−1

9>>=>>;; ξi ¼

Z10

Giðs; sÞ0@Z1

s

ðτ � sÞα�1aiðτÞdτ

1A

q−1

ds;


systems

245

there exists ρ > 0; such that if uþ v ≥ ρ, then

fiðu; vÞ ≥ μðjuj þ jvjÞp−1:

Let ρ2 ¼ maxð32 ρ1; ηρÞ; η ¼ maxðη1; η2Þ; and set Ω2 ¼ fðu; vÞ∈X ; kðu; vÞk < ρ2g; it iseasy to see that Ω1 ⊂Ω2:Assume that ðu; vÞ ∈ P∩ vΩ2; then


ηi

Z10

Giðs; sÞ0@ 1

ΓðαÞZ1s

ðτ � sÞα�1aiðτÞμðjuj þ jvjÞp�1

dτ

1A

q−1

ds

≥1

ηi

� μΓðαÞ

�q−1 Z10

Giðs; sÞ

3

0@Z1

s

ðτ � sÞα�1aiðτÞ

�1

η1kuk

∞þ 1

η2kvk

∞

�p�1

dτ

1A

q−1

ds

≥1

η2

� μΓðαÞ

�q−1ξikðu; vÞk ≥ kðu; vÞk;

thus

kTðu; vÞk≥ kðu; vÞk; ðu; vÞ∈ vΩ2 ∩ P:

By Guo–Krasnosel’skii fixed-point theorem, we conclude that T has a fixed pointðu; vÞ∈P ∩ ðΩ2nΩ1Þ: This means that the system ðSÞ has at least one positivesolution ðu; vÞ. ▪

Example 4.1. Consider the system ðSÞ, withf1ðu; vÞ ¼ ðuþ vÞ3; f2ðu; vÞ ¼ eðuþvÞ2 � 1

a1ðtÞ ¼ et; a2ðtÞ ¼ 1

g1ðt; u; vÞ ¼ ð1� tÞðuþ vÞ23uþ 4v

; g2ðt; u; vÞ ¼ t

9u

where α ¼ 12; β1 ¼ β2 ¼ 4

3; p ¼ 2; η1 ¼ 3

2; η2 ¼ 5

4: We check easily that A0;i ¼ 0; A∞;i ¼ ∞;

i ¼ 1; 2: Clearly,

g1ðt; u; vÞ ≤ 1� t

3ðuþ vÞ; g2ðt; u; vÞ ≤ t

5ðuþ vÞ

So, the assumption ðHÞ hold. Thus the system ðSÞ has at least one positive solution byTheorem 4.1.

References

[1] Guo DJ, Lakshmikantham V. Nonlinear problems in abstract cones in: notes and reports inmathematics in science and engineering, Boston, Mass: Academic Press; 1988; 5.

[2] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differentialequations. North-Holland mathematics studies, Amsterdam: Elsevier Science B.V., 2006; 204.

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[3] Podlubny I. Fractional differential equations, San Diego: Academic Press; 1999.

[4] Guezane Lakoud A, Khaldi R, Kılıçman A. Existence of solutions for a mixed fractionalboundary value problem. Adv Differ Equ. 2017; 2017, 164.

[5] Khaldi R, Guezane-Lakoud A. Higher order fractional boundary value problems for mixed typederivatives. J Nonlinear Funct Anal. 2017; 2017, 30.

[6] Bashir A, Nieto JJ. Existence of solutions for anti-periodic boundary value problems involvingfractional differential equations via Leray-Schauder degree theory. Topol Methods NonlinearAnal. 2010; 35: 295-304.

[7] Cabada A, Wang G. Positive solutions of nonlinear fractional differential equations with integralboundary value conditions. J Math Anal Appl. 2012; 389: 403-11.

[8] Ding Y, Xu J and Fu Z. Positive solutions for a system of fractional integral boundary valueproblems of Riemann–Liouville type involving Semipositone nonlinearities. Mathematics. 2019; 7:970. 10.3390.

[9] Guezane-Lakoud A, Khaldi R. Positive solutions for multiorder nonlinear fractional systems. Int JAnal Appl. 2017; 15: 18-22.

[10] Guezane-Lakoud A, Ramdane S. Existence of solutions for a system of mixed fractionaldifferential equations. J Taibah Univ Sci. 2018; 12. doi: 10.1080/16583655.2018.1477414.

[11] Khaldi R, Guezane-Lakoud A. Solvability of a boundary value problem with a Nagumo condition.J Taibah Univ Sci. 2018. doi: 10.1080/16583655.2018.1489025.

[12] Li Y, Wei Z. Positive solutions for a coupled system of mixed higher-order nonlinear singularfractional differential equations. F Point Theory. 2014; 15(1): 167-78.

[13] Yang W. Positive solutions for a coupled system of nonlinear fractional differential equationswith integral boundary conditions. Comput Math Appl. 2012; 63: 288-97.

[14] Guezane-Lakoud A, Ashyralyev A. Fixed point theorem applied to a fractional boundary valueproblem. Pure App Math Letters. 2014; 2: 1-6.

[15] Ahmad B, Nieto JJ. Existence results for a coupled system of nonlinear fractional differentialequations with three-point boundary conditions. Comput Math Appl. 2009; 58(9): 1838-43.

[16] Bai C, Fang J. The existence of a positive solution for a singular coupled system of nonlinearfractional differential equations. Appl Math Comput. 2004; 150(2): 611-21.

[17] Liu Y. A new method for converting boundary value problems for impulsive fractionaldifferential equations to integral equations and its applications. Adv Nonlinear Anal. 2019; 8(1):386-54.

[18] Agarwal RP, Liu HS and O’Regan D. Existence theorems for the one-dimensional singularp-Laplacian equation with sign changing nonlinearities. Appl Math Comput. 2003; 143: 15-38.

[19] Chen T, Liu WB. An anti-periodic boundary value problem for the fractional differential equationwith a p-Laplacian operator. Appl Math Lett. 2012; 25: 1671-75.

[20] Chen T, Liu WB and Hu ZG. A boundary value problem for fractional differential equation withp-Laplacian operator at resonance. Non-linear Anal. 2012; 75: 3210-17.

[21] Han Z, Lu H, Sun S, Yang D. Positive solutions to boundary -value problems of P-Laplacianfractional differential equations with a parameter in boundary. Elec Jou Diff Equa. 2012;2012(213): 1-14.

[22] Mahmudo NI, Unul S. Existence of solutions of fractional boundary value problems with p-Laplacian operator. Bound. Value Probl. 2015; 99. doi 10.1186/s13661-015-0358-9.

[23] Tian Y, Bai Z, Sun S. Positive solutions for a boundary value problem of fractional differentialequation with p-Laplacian operator. Tian et al. Adv Differ Equ. 2019; 349.

[24] Wang J, Xiang H. Upper and lower solutions method for a class of singular fractional boundary-value problems with p-Laplacian operator. Abs Appl Anal. 2010; 2010, 971824; 1-12.


systems

247

[25] Wang J, Xiang H, Liu Z. Existence of concave positive solutions for boundary-value problem ofnonlinear fractional differential equation with p-Laplacian operator. Int J Math Math Sci. 2010:2010, 495138: 1-17.

[26] Chai GQ. Positive solutions for boundary value problem of fractional differential equation withp-Laplacian operator, Bound Value Probl. 2012; 2012: 18. doi: 10.1186/1687-2770-2012-18.

[27] Krasnosel’skii MA. Positive solutions of operator equations: Groningen; 1964.

Further reading

[28] Agrawal OP. Formulation of Euler–Lagrange equations for fractional variational problems. JMath Anal Appl. 2002; 272: 368-79.

[29] Guezane-Lakoud A, Khaldi R, Torres DFM. On a fractional oscillator equation with naturalboundary conditions. Prog Frac Diff Appl. 2017; 3: 191-7.

[30] Podlubny I. Geometric and physical interpretation of fractional integration and fractionaldifferentiation. Fract Calc Appl, Anal. 2002; 5: 367-86.

[31] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives: theory andapplications. Yverdon: Gordon and Breach; 1993.

[32] Xie S. Positive solutions for a system of higher-order singular nonlinear fractional differentialequations with nonlocal boundary conditions. E. J. Qualitative Theory of Diff Equa. 2015; (18):1-17.

Corresponding authorSamira Ramdane can be contacted at: [email protected]


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A direct computation of a certainfamily of integrals

Lorenzo Fornari, Enrico Laeng and Vittorino PataDepartment of Mathematics, Politecnico di Milano, Milan, Italy

Abstract

Purpose – The authors propose a rather elementary method to compute a family of integrals on the half line,involving positive powers of sin x and negative powers of x, depending on the integer parameters n≥ q≥ 1.Design/methodology/approach – Combinatorics, sine and cosine integral functions.Findings – The authors prove an explicit formula to evaluate sinc-type integrals.Originality/value – The proof is not present in the current literature, and it could be of interest for a largeaudience.

Keywords Integral, Sinc function, SinIntegral and CosIntegral functions


In this note, let n≥ q≥ 1be any two given integers. The symbol b:cwill stand, as usual, for theinteger part. We consider the family of integrals

In;q ¼Z ∞

0

ðsin xÞnxq

dx:

Theorem 1. The following formulae hold

(i) If nþ q is even, then

In;q ¼ ð−1Þq−n2 π2nðq� 1Þ!

Xbn−12 c

k¼0

ð−1Þk�n

k

�ðn� 2kÞq−1:

(ii) If nþ q is odd and q≥ 2, then

In;q ¼ ð−1Þq−nþ12

2n−1ðq� 1Þ!Xbn−12 c

k¼0

ð−1Þk�n

k

�ðn� 2kÞq−1 logðn� 2kÞ:

The formulae above are recorded in the Wolfram MathWorld web page titled Sinc Function[1], which refers to the result as “amazing” and “spectacular”. However, the web page omitsthe proof, citing a 20-year-old online paper that seems not to be available any longer. Nor theproof is reported anywhere else, to the best of our knowledge. Nonetheless, particularinstances of In;q are discussed in several textbooks, typically by means of complex analysistools (see, e.g. Ref. [2]).

A family ofintegrals

249

© Lorenzo Fornari, Enrico Laeng and Vittorino Pata. Published in Arab Journal of MathematicalSciences. Published by Emerald Publishing Limited. This article is published under the CreativeCommons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and createderivative works of this article (for both commercial and non-commercial purposes), subject to fullattribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode



Received 16 January 2021Revised 21 January 2021

Accepted 21 January 2021


Vol. 27 No. 2, 2021pp. 249-252


DOI 10.1108/AJMS-01-2021-0019

The remaining of the paper is devoted to our proof of Theorem 1. To this end, form≥ 0, let

PmðxÞ ¼Xmk¼0

xk

k!

denote the Maclaurin polynomial of ex of order m. We agree to set P−1 ¼ 0. Let QðxÞ be theMaclaurin polynomial of ðsin xÞn of order q− 2, withQ ¼ 0 if q ¼ 1. Since ðsin xÞn has a zero oforder n at x ¼ 0, it follows that QðxÞ≡ 0 for all n≥ q≥ 1. On the other hand, as

ðsin xÞn ¼ 1

ð2iÞnXn

k¼0

ð−1Þk�n

k

�eiðn−2kÞx;

we immediately conclude that

QðxÞ ¼ 1

ð2iÞnXn

k¼0

ð−1Þk�n

k

�Pq−2ðiðn� 2kÞxÞ ¼ 0: (1)

Subtracting the two sums, we obtain

In;q ¼ 1

ð2iÞnXbn−12 c

k¼0

ð−1Þk�n

k

�Z ∞

0

eiðn−2kÞx � Pq−2ðiðn� 2kÞxÞxq

dx

þ ð−1Þnð2iÞn

Xbn−12 c

k¼0

ð−1Þk�n

k

�Z ∞

0

e−iðn−2kÞx � Pq−2ð−iðn� 2kÞxÞxq

dx:

(2)

Remark 2. From (1), we also deduce that the equality

Xbn−12 c

k¼0

ð−1Þk�n

k

�ðn� 2kÞq−1 ¼ 0; (3)

holds for every n > q≥ 2 , whenever nþ q is odd.We now start from formula (2) but considering the integral on ðε; ∞Þand only at the endwe

will take the limit ε→ 0. This allowsus tomove the integral inside the sum. Inwhat followsωðεÞwill denote a generic function of ε, vanishing at 0 as ε→ 0. Moreover, for α≠ 0, let us define

EεðαÞ ¼Z ∞

ε

eiαx

xdx:

Lemma 3. For every q≥ 1, every ε > 0 and every α≠ 0, we have

Z ∞

ε

eiαx � Pq−2ðiαxÞxq

dx ¼ cqαq−1 þ ðiαÞq−1ðq� 1Þ! EεðαÞ þ ωðεÞ;

where cq ¼ iq−1

ðq− 1Þ!P q−2

k¼01

kþ1for q≥ 2 and c1 ¼ 0.

Proof: The proof goes by induction on q. If q ¼ 1, equality holds with ωðεÞ ¼ 0. Then, weprove the formula for qþ 1, assuming it true for q≥ 1. Since P 0

q−1 ¼ Pq−2, an integration byparts yields

AJMS27,2

250

Z ∞

ε

eiαx � Pq−1ðiαxÞxqþ1

dx ¼ eiαε � Pq−1ðiαεÞqεq

þ iαq

Z ∞

ε


dx:

By the inductive hypothesis,

iαq

Z ∞

ε


dx ¼ icq

qαq þ ðiαÞq

q!EεðαÞ þ ωqðεÞ;

for some function ωq vanishing at 0. Noting that

ϖqðεÞ ¼ −ðiαÞqq! q

þ eiαε � Pq−1ðiαεÞqεq

→ 0 as ε→ 0;

we end up with the equality

Z ∞

ε

eiαx � Pq−1ðiαxÞxqþ1

dx ¼�iq

q!qþ icq

q

�αq þ ðiαÞq

q!EεðαÞ þ ωqðεÞ þϖqðεÞ:

The final observation that iq

q!qþ icq

q¼ cqþ1 completes the proof. ,

Proof of Theorem 1 for the case nþ q even. Substituting the expression given by Lemma 3into (2) and noting that

Eεðn� 2kÞ � Eεð−ðn� 2kÞÞ ¼ 2iSiððn� 2kÞεÞ;

where

SiðtÞ ¼Z ∞

t

sin x

xdx

is the SinIntegral function, we obtain

In;q ¼ ð−1Þq−n22n−1ðq� 1Þ!

Xbn−12 c

k¼0

ð−1Þk�n

k

�ðn� 2kÞq−1Siððn� 2kÞεÞ þ ωðεÞ:

Since

Siððn� 2kÞεÞ→ Sið0Þ ¼ π2

as ε→ 0;

the result follows. ,Proof of Theorem 1 for the case nþ q odd. Again, we substitute the expression given byLemma 3 into (2). Using (3) and noting that

Eεðn� 2kÞ þ Eεð−ðn� 2kÞÞ ¼ 2Ciððn� 2kÞεÞ;

where

CiðtÞ ¼Z∞t

cos x

xdx

A family ofintegrals

251

is the CosIntegral function, we obtain

In;q ¼ ð−1Þq−n−12

2n−1ðq� 1Þ!Xbn−12 c

k¼0

ð−1Þk�n

k

�ðn� 2kÞq−1Ciððn� 2kÞεÞ þ ωðεÞ:

By a further use of (3), we can replace Ciððn− 2kÞεÞwithCiððn� 2kÞεÞ � CiðεÞ→ � logðn� 2kÞ as ε→ 0;

and a final limit ε→ 0 completes the argument. ,

References

[1] Weisstein ES. Sinc function. Available from: https://mathworld.wolfram.com/SincFunction.html.

[2] Ahlfors LV. Complex analysis. New York: McGraw-Hill; 1978.

Corresponding authorVittorino Pata can be contacted at: [email protected]


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