Journal of the Egyptian Mathematical Society (2013) xxx, xxx–xxx

Egyptian Mathematical Society

Journal of the Egyptian Mathematical Society

www.etms-eg.orgwww.elsevier.com/locate/joems

ORIGINAL ARTICLE

Mizoguchi–Takahashi’s type common fixed point theorem

Muhammad Usman Ali

Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12 Islamabad, Pakistan

Received 29 March 2013; revised 2 July 2013; accepted 18 August 2013

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KEYWORDS

Fixed points;

Coincidence points;

Common fixed points;

T-weakly commuting maps;

Lower semi continuous

maps;

T-orbitally lower semi con-

tinuous maps

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Abstract Recently Kamran extended the result of Mizoguchi and Takahashi for closed multi-

valued mappings and proved a fixed point theorem. In this paper we further extend the result

concluded by Kamran and prove a common fixed point theorem by using the concept of lower

semi-continuity.

2000 MATHEMATICS SUBJECT CLASSIFICATION: 54H25

ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction

Let (X,d) be a metric space. For each x 2 X and A ˝ X,d(x,A) = {d(x,y):y 2 A}. We denote by K(X) the class of all

nonempty compact subset of X, by CB(X) the class of all non-empty closed and bounded subsets of X, by CL(X) the class ofall nonempty closed subsets of X. For every A, B 2 CL(X), let

HðA;BÞ¼maxfsup

x2Adðx;BÞ;sup

y2Bdðy;AÞg; if themaximumexists;

1; otherwise:

(

Such a map H is called generalized Hausdorff metric inducedby d. A point x 2 X is said to be a fixed point of T:X fi CL(X)

if x 2 Tx. A point x 2 X is said to be a coincidence point off:X fi X and T:X fi CL(X) if fx 2 Tx. A point x 2 X is saidto be a common fixed point of f:X fi X and T:X fi CL(X) if

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tian Mathematical Society.

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.U. Ali, Mizoguchi–Takahashi//dx.doi.org/10.1016/j.joems.20

ing by Elsevier B.V. on behalf of E

8.004

x = fx 2 Tx. Let T:X fi CL(X), then f:X fi X is said to beT-weakly commuting [1] at x 2 X, if ffx 2 Tfx. If for x0 2 X,there exists a sequence {xn} in X such that xn 2 Txn�1, thenO(T,x0) = {x0,x1,x2, . . . } is said to be orbit of T:X fi CL(X).

If for x0 2 X and f:X fi X there exists a sequence {fxn} in fXsuch that fxn 2 Txn�1, then Of(x0) = {fx1, fx2, fx3, . . . } is saidto be f-orbit of T:X fi CL(X). A mapping g : X! R is said

to be lower semi continuous at n if for every sequence {xn}in X such that xn fi n, implies g(n) 6 lim infn!1 g(xn). If {xn}is a sequence in O(T,x0) and xn fi n implies

g(n) 6 lim infn!1ng(xn), then g is said to be T-orbitally lowersemi continuous [2].

Nadler [3] extended the Banach contraction principle to

multi-valued mappings as follows:

Theorem 1.1 [3]. Let (X,d) be a complete metric space and T isa mapping from X into CB(X) such that

HðTx;TyÞ 6 rdðx; yÞ; for all x; y 2 X;

where r 2 [0,1). Then T has a fixed point.

Reich [4] extended the above result in the following way:

’s type common fixed point theorem, Journal of the Egyptian13.08.004

gyptian Mathematical Society.

2 M.U. Ali

Theorem 1.2 [4]. Let (X,d) be a complete metric space and

T:X fi K(X) is a mapping satisfying

HðTx;TyÞ 6 aðdðx; yÞÞdðx; yÞ; for each x; y 2 X;

where a is a function of (0,1) into [0,1) such that

lim supr!tþ

aðrÞ < 1; for each t 2 ð0;1Þ: ð1:1Þ

Then T has a fixed point.

Reich [4] raised the question: whether the range of T, K(X)

can be replaced by CB(X) or CL(X). Mizoguchi and Takahashi[5] gave the positive answer to the conjecture of Reich [4],when the inequality holds also for t= 0, in particular theyproved:

Theorem 1.3 [5]. Let (X,d) be a complete metric space andT:X fi CB(X) is a mapping satisfying

HðTx;TyÞ 6 aðdðx; yÞÞdðx; yÞ; for each x; y 2 X;

where a is a function of [0,1) into [0,1) such that

lim supr!tþ

aðrÞ < 1; for each t 2 ½0;1Þ:

Then T has a fixed point.

The other proofs of Theorem 1.3 have been given by Dafferand Kaneko [6] and Chang [7]. Eldred et al. [8] claimed thatTheorem 1.3 is equivalent to Theorem 1.1. Suzuki producedan example [9, p. 753] to disprove their claim and showed that

Mizoguchi–Takahashi’s fixed point theorem is a real general-ization of Nadler’s fixed point theorem. Kamran [10] extendedthe result of Mizoguchi and Takahashi [5] for closed multi-

valued mappings and proved a fixed point theorem. In this pa-per we further extend the result concluded by Kamran [10] andprove a common fixed point theorem.

2. Main results

We begin this section with following lemma.

Lemma 2.1 [10]. Let (X,d) be a metric space and B 2 CL(X).Then for each x 2 X and q > 1 there exists an element b 2 B

such that

dðx; bÞ 6 qdðx;BÞ:

Now we are in position to start our result.

Theorem 2.2. Let (X,d) be a metric space, f:X fi X and

T:X fi CL(X) be two mappings such that TX ˝ fX, satisfying

dðfy;TyÞ 6 aðdðfx; fyÞÞdðfx; fyÞ; for each x

2 X; and fy 2 Tx: ð2:1Þ

Where a is a function of [0,1) into [0,1) such that

lim supr!tþaðrÞ < 1 for each t 2 ½0;1Þ: ð2:2Þ

Let (fX,d) be a complete metric space. Then,

(i) For each x0 2 X, there exists an f-orbit {fxn} of T andfn 2 fX such that limn fxn = fn.

Please cite this article in press as: M.U. Ali, Mizoguchi–TakahashMathematical Society (2013), http://dx.doi.org/10.1016/j.joems.20

(ii) Moreover, n is coincidence point of f and T if and only if

g(x) = d(fx,Tx) is lower semi continuous at n.(iii) Furthermore, if ffn = fn and f is T-weakly commuting at

n, then f and T have a common fixed point.

Proof. Let x0 2 X, since Tx0 ˝ fX, there exists x1 2 X such thatfx1 2 Tx0. If x0 = x1, then fx1 2 Tx1, i.e., x1 is a coincidence

point. Let x0 „ x1, by taking q ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðdðfx0 ;fx1ÞÞp , it follows that

there exists fx2 2 Tx1 such that

dðfx1; fx2Þ 61ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

aðdðfx0; fx1ÞÞp dðfx1;Tx1Þ: ð2:3Þ

Repeating the above argument we get a sequence {fxn} in Xsuch that

dðfxn; fxnþ1Þ 61ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

aðdðfxn�1; fxnÞÞp dðfxn;TxnÞ; ð2:4Þ

where fxn 2 Txn�1, n = 1, 2, 3, 4, . . .. We assume thatxn�1 „ xn, otherwise xn is a coincidence point of f and T. From

(2.1) and (2.4), we have

dðfxn; fxnþ1Þ 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðdðfxn�1; fxnÞÞ

pdðfxn�1; fxnÞ; ð2:5Þ

< dðfxn�1; fxnÞ: ð2:6Þ

Hence {d(fxn, fxn+1)} is a decreasing sequence, it converges tosome nonnegative real number b. We claim that b = 0, for

otherwise, by taking limit in (2.6), we get

b 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilimn!1aðdðfxn�1; fxnÞÞ

pb < b:

A contradiction to our assumption. From (2.6), we get

dðfxn;fxnþ1Þ6 ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðdðfxn�1; fxnÞÞ

p� � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðdðfx0; fx1ÞÞ

pÞdðfx0; fx1Þ:

ð2:7Þ

It follows from (2.2) that we may choose e > 0 and a 2 (0,1)such that

aðtÞ < a2 for t 2 ð0; �Þ: ð2:8Þ

Let N be such that

dðfxn�1; fxnÞÞ < � for n P N: ð2:9Þ

From (2.7), we have

dðfxn; fxnþ1Þ 6 an�ðN�1ÞðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðdðfxN�2; fxN�1ÞÞ

p� � �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

aðdðfx0; fx1ÞÞp

Þdðfx0; fx1Þ < an�ðN�1Þdðfx0; fx1Þ: ð2:10Þ

Therefore for any m 2 N, we have

dðfxn; fnþmÞ 6 dðfxn; fnþ1Þ þ � � � þ dðfxnþm�1; fnþmÞ;6 an�ðN�1Þð1þ aþ a2 þ � � � þ am�1Þdðfx0; fx1Þ;

6an�ðN�1Þ

1� adðfx0; fx1Þ:

ð2:11Þ

This shows that {fxn} is a Cauchy sequence in fX. Since fX iscomplete, there exists fn in fX such that fxn fi fn. Sincefxn 2 Txn�1, it follows form (2.1) that

dðfxn;TxnÞ 6 aðdðfxn�1; fxnÞÞdðfxn�1; fxnÞ;< dðfxn�1; fxnÞ:

ð2:12Þ

Letting n fi1 in (2.12), we have

i’s type common fixed point theorem, Journal of the Egyptian13.08.004

3

limn!1

dðfxn;TxnÞ ¼ 0: ð2:13Þ

Suppose g(x) = d(fx,Tx) is lower semi continuous at n, then

dðfn;TnÞ ¼ gðnÞ 6 lim infn!1gðxnÞ ¼ lim infn!1dðfxn;TxnÞ¼ 0:

Hence by the closedness of T, we have fn 2 Tn. Conversely, if nis a coincidence point of f and T, then g(n) = 0 6 lim inf g(xn).

Since f is T-weakly commuting at n, then ffn 2 Tfn. Also, weassume that ffn = fn. Hence fn = ffn 2 Tfn. h

Corollary 2.3 [10]. Let (X,d) be a complete metric space andT:X fi CL(X) is a mapping satisfying

dðy;TyÞ 6 aðdðx; yÞÞdðx; yÞ; for each x 2 X and y 2 Tx;

where a is a function of [0,1) into [0,1) such that

lim supr!tþ

aðrÞ < 1; for each t 2 ½0;1Þ:

Then,

(i) For each x0 2 X, there exists an orbit {xn} of T and n 2 Xsuch that limnxn = n;

(ii) n is a fixed point of T if and only if the functionf(x):¼d(x,Tx) is T-orbitally lower semi continuous at n.

Proof. This corollary follows from Theorem 2.2, by consider-

ing f= I. h

Example 2.4. Let X= (�1,0] endowed with the usual metricd. Define f:X fi X and T:X fi CL(X) by

fx ¼ x=2 for each x 2 X; ð2:14Þ

and

Tx ¼ ð�1; x=4� for each x 2 X: ð2:15Þ

For each x 2 X and fy 2 Tx= (�1,x/4], then we have

dðfy;TyÞ ¼ 0 61

2dðfx; fyÞ:

By taking aðtÞ ¼ 12for all t P 0, we see that all condition of

Theorem 2.2 are satisfied. Then there exists x 2 X such that

fx 2 Tx, which is 0. Also, ff0 = f0 and f is T-weakly commut-ing at 0. Therefore f and T have a common fixed point.

Example 2.5. Let X ¼ 1n

: n 2 N� �

[ f0g endowed with theusual metric d. Define T:X fi CL(X) and f:X fi X by

Tx ¼

f0g if x ¼ 0

1nþ2 ;

1nþ3

n oif x ¼ 1

n: 1 6 n 6 6

1nþ1 ; 0n o

if x ¼ 1n

: n > 6;

8>>><>>>:

Please cite this article in press as: M.U. Ali, Mizoguchi–TakahashiMathematical Society (2013), http://dx.doi.org/10.1016/j.joems.20

and

fx ¼1

nþ1 if x ¼ 1n

: n 2 N

0 if x ¼ 0:

�

Define a: [0,1) fi [0,1) by

aðtÞ ¼

45

if 0 6 t 6 156

34

if 156< t 6 1

24

23

if 124< t 6 1

612

if t > 16:

8>>><>>>:

One can check that for each x 2 X and fy 2 Tx, we have

dðfy;TyÞ 6 aðdðfx; fyÞÞdðfx; fyÞ

and all other conditions of Theorem 2.2 are satisfied. Thenthere exists x 2 X such that fx 2 Tx. Also, ff0 = f0 and f isT-weakly commuting at 0. Therefore f and T have a common

fixed point.

Acknowledgements

Author is grateful to referees for their suggestions and cautious

reading.

References

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contractions, J. Math. Anal. Appl. 299 (2004) 235–241.

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Math. Japon. 24 (1979) 327–330.

[3] S.B. Nadler Jr., Multivalued contraction mappings, Pacific J.

Math. 30 (1969) 475–488.

[4] S. Reich, Fixed points of contractive functions, Boll. Unione

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[5] N. Mizoguchi, W. Takahashi, Fixed point theorems for

multivalued mappings on complete metric space, J. Math.

Anal. Appl. 141 (1989) 177–188.

[6] P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive

multi-valued mappings, J. Math. Anal. Appl. 192 (1995) 655–

666.

[7] T.H. Chang, Common fixed point theorems for multi-valued

mappings, Math. Japon. 41 (1995) 311–320.

[8] A.A. Eldred, J. Anuradha, P. Veeramani, On equivalence of

generalized multi-valued contractions and Nadler’s fixed point

theorem, J. Math. Anal. Appl. (2007), http://dx.doi.org/10.1016/

j.jmaa.2007.01.087.

[9] T. Suzuki, Mizoguchi–Takahashi’s fixed point theorem is a real

generalization of Nadler’s, J. Math. Anal. Appl. (2007), http://

dx.doi.org/10.1016/j.jmaa.2007.08.022.

[10] T. Kamran, Mizoguchi–Takahashi’s type fixed point theorem,

Comput. Math. Appl. 57 (2009) 507–511.

’s type common fixed point theorem, Journal of the Egyptian13.08.004