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Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces Tian Zhou Xu, John Michael Rassias, and Wan Xin Xu Citation: Journal of Mathematical Physics 51, 093508 (2010); doi: 10.1063/1.3482073 View online: http://dx.doi.org/10.1063/1.3482073 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/51/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stability of multi-Jensen mappings in non-Archimedean normed spaces J. Math. Phys. 53, 023507 (2012); 10.1063/1.3684746 A note on the stability of a generalized trigonometric functional equation AIP Conf. Proc. 1309, 909 (2010); 10.1063/1.3525223 Intuitionistic fuzzy stability of a general mixed additive-cubic equation J. Math. Phys. 51, 063519 (2010); 10.1063/1.3431968 On the stability of general cubic-quartic functional equations in Menger probabilistic normed spaces J. Math. Phys. 50, 123301 (2009); 10.1063/1.3269920 Hyers–Ulam stability on a generalized quadratic functional equation in distributions and hyperfunctions J. Math. Phys. 50, 113519 (2009); 10.1063/1.3263147 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.174.255.116 On: Fri, 11 Jul 2014 05:45:29

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Stability of a general mixed additive-cubic functional equation in non-Archimedeanfuzzy normed spacesTian Zhou Xu, John Michael Rassias, and Wan Xin Xu

Citation: Journal of Mathematical Physics 51, 093508 (2010); doi: 10.1063/1.3482073 View online: http://dx.doi.org/10.1063/1.3482073 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/51/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stability of multi-Jensen mappings in non-Archimedean normed spaces J. Math. Phys. 53, 023507 (2012); 10.1063/1.3684746 A note on the stability of a generalized trigonometric functional equation AIP Conf. Proc. 1309, 909 (2010); 10.1063/1.3525223 Intuitionistic fuzzy stability of a general mixed additive-cubic equation J. Math. Phys. 51, 063519 (2010); 10.1063/1.3431968 On the stability of general cubic-quartic functional equations in Menger probabilistic normed spaces J. Math. Phys. 50, 123301 (2009); 10.1063/1.3269920 Hyers–Ulam stability on a generalized quadratic functional equation in distributions and hyperfunctions J. Math. Phys. 50, 113519 (2009); 10.1063/1.3263147

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Stability of a general mixed additive-cubic functionalequation in non-Archimedean fuzzy normed spaces

Tian Zhou Xu,1,a� John Michael Rassias,2,b� and Wan Xin Xu3,c�

1Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing100081, People’s Republic of China2Pedagogical Department E.E., Section of Mathematics and Informatics, National andCapodistrian University of Athens, 4, Agamemnonos Str., Aghia Paraskevi, Athens 15342,Greece3School of Communication and Information Engineering, University of Electronic Scienceand Technology of China, Chengdu 611731, People’s Republic of China

�Received 12 June 2010; accepted 16 July 2010; published online 20 September 2010�

We establish some stability results concerning the general mixed additive-cubicfunctional equation in non-Archimedean fuzzy normed spaces. In addition, weestablish some results of approximately general mixed additive-cubic mappings innon-Archimedean fuzzy normed spaces. The results improve and extend some re-cent results. © 2010 American Institute of Physics. �doi:10.1063/1.3482073�

I. INTRODUCTION

In 1897, Hensel discovered the p-adic numbers as a number theoretical analog of power seriesin complex analysis. The most important examples of non-Archimedean spaces are p-adic num-bers. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom: for allx ,y�0, there exists an integer n, such that x�ny. It turned out that non-Archimedean spaces havemany nice applications.10,31,36

During the last three decades, theory of non-Archimedean spaces has gained the interest ofphysicists for their research, in particular, in problems coming from quantum physics, p-adicstrings, and superstrings �cf. Ref. 10�. Although many results in the classical normed space theoryhave a non-Archimedean counterpart, but their proofs are essentially different and require anentirely new kind of intuition. One may note that �n��1 in each valuation field, every triangle isisosceles and there may be no unit vector in a non-Archimedean normed space �cf. Ref. 10�. Thesefacts show that the non-Archimedean framework is of special interest.

Fuzzy set theory is a powerful hand set for modeling uncertainty and vagueness in variousproblems arising in the field of science and engineering. It has also very useful applications invarious fields, e.g., population dynamics, chaos control, computer programming, nonlinear dy-namical systems, nonlinear operators, statistical convergence, etc. �cf. Refs. 3, 13, 32, and 33�. Thefuzzy topology proves to be a very useful tool to deal with such situations where the use ofclassical theories breaks down. The most fascinating application of fuzzy topology in quantumparticle physics arises in string and E-infinity theory of EI Naschie �cf. Refs. 18–22�.

A basic question in the theory of functional equations is as follows: when is it true that afunction, which approximately satisfies a functional equation, must be close to an exact solution ofthe equation?

If the problem accepts a unique solution, we say the equation is stable. The first stabilityproblem concerning group homomorphisms was raised by Ulam35 in 1940 and affirmatively

a�Author to whom correspondence should be addressed. Electronic addresses: [email protected] [email protected].

b�Electronic addresses: [email protected], [email protected], and [email protected]. URL: http://www.primedu.uoa.gr/~jrassias/.

c�Electronic mail: [email protected].

JOURNAL OF MATHEMATICAL PHYSICS 51, 093508 �2010�

51, 093508-10022-2488/2010/51�9�/093508/19/$30.00 © 2010 American Institute of Physics

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solved by Hyers.9 The result of Hyers was generalized by Aoki1 for approximate additive map-pings and by Rassias25 for approximate linear mappings by allowing the Cauchy difference op-erator CDf�x ,y�= f�x+y�− �f�x�+ f�y�� to be controlled by ���x�p+ �y�p�. In 1994, a generalizationof Rassias’ theorem was obtained by Găvruţa,8 who replaced ���x�p+ �y�p� by a general controlfunction ��x ,y�. In addition, Rassias27–30,37–39 generalized the Hyers stability result by introducingtwo weaker conditions controlled by a product of different powers of norms and a mixed product-sum of powers of norms, respectively. Recently, several further interesting discussions, modifica-tions, extensions, and generalizations of the original problem of Ulam have been proposed �see,e.g., Refs. 2, 4–7, 10–17, 23, 24, 26, 30, 34, and 37–40 and the references therein�. In particular,Mirmostafaee and Moslehian12 introduced a notion of a non-Archimedean fuzzy norm and studiedthe stability of the Cauchy equation in the context of non-Archimedean fuzzy spaces. Theypresented an interdisciplinary relation between the theory of fuzzy spaces, the theory of non-Archimedean spaces, and the theory of functional equations.

The generalized Hyers–Ulam stability for a mixed additive-cubic functional equation,

f�2x + y� + f�2x − y� = 2f�x + y� + 2f�x − y� + 2f�2x� − 4f�x� , �1.1�

in quasi-Banach spaces has been investigated by Najati and Eskandani.17 Functional equation �1.1�is called mixed additive-cubic functional equation, since the function f�x�=ax3+bx is its solution.Every solution of this mixed additive-cubic functional equation is said to be a mixed additive-cubic mapping.

In Refs. 37–39, we considered the following general mixed additive-cubic functional equa-tion:

f�kx + y� + f�kx − y� = kf�x + y� + kf�x − y� + 2f�kx� − 2kf�x� . �1.2�

It is easy to show that the function f�x�=ax3+bx satisfies functional equation �1.2�. We observethat in case k=2, Eq. �1.2� yields mixed additive-cubic equation �1.1�. Therefore, Eq. �1.2� is ageneralized form of the mixed additive-cubic equation. The main purpose of this paper is toestablish Ulam–Hyers stability for general mixed additive-cubic functional equation �1.2� in thesetting of non-Archimedean fuzzy normed spaces. The achieved results via this paper improve andextend some recent well-known pertinent results.

II. PRELIMINARIES

In this section we recall some notations and basic definitions used in this paper. The definitionof non-Archimedean fuzzy normed spaces was given in Ref. 12.

Definition 2.1: Let K be a field. A non-Archimedean absolute value on K is a function� · � :K→R, such that for any a ,b�K we have

�i� �a��0 and equality holds if and only if a=0;�ii� �ab�= �a��b�;�iii� �a+b��max��a� , �b�.

The condition �iii� is called the strong triangle inequality. Clearly, �1�= �−1�=1 and �n��1 for alln�N. We always assume in addition that � · � is non trivial, i.e., that

�iv� there is an a0�K, such that �a0��0,1.

The most important examples of non-Archimedean spaces are p-adic numbers.Example 2.2: Let p be a prime number. For any nonzero rational number x, there exists a

unique integer nx, such that x= ab pnx, where a and b are integers not divisible by p. Then �x�p

ªp−nx defines a non-Archimedean norm on Q. The completion of Q with respect to the metricd�x ,y�= �x−y�p is denoted by Qp which is called the p-adic number field. In fact, Qp is the set of

all formal series x= k�nx

akpk, where �ak�� p−1 are integers. The addition and multiplication be-

093508-2 Xu, Rassias, and Xu J. Math. Phys. 51, 093508 �2010�

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tween any two elements of Qp are defined naturally. The norm � k�nx

akpk�p= p−nx is a non-

Archimedean norm on Qp and it makes Qp a locally compact field �see Ref. 31�. Note that if p�2, then �2n�p=1 for each integer n but �2�2�1.

Now we give the definition of a non-Archimedean fuzzy normed space.Definition 2.3: Let X be a linear space over a non-Archimedean field K. A function N :X

R→ �0,1� is said to be a non-Archimedean fuzzy norm on X if for all x ,y�X and all s , t�R:

�N1� N�x ,c�=0 for all c�0;�N2� x=0 if and only if N�x ,c�=1 for all c�0;�N3� N�cx , t�=N�x , t

�c�� if c�0;

�N4� N�x+y ,max�s , t��min�N�x ,s� ,N�y , t�;�N5� lim

t→�

N�x , t�=1.

A non-Archimedean fuzzy normed space is a pair �X ,N�, where X is a linear space and N isa non-Archimedean fuzzy norm on X. If �N4� holds then so is �N6� N�x+y ,s+ t��min�N�x ,s� ,N�y , t�.

Recall that a classical vector space over the complex or real field satisfying �N1�–�N5� iscalled a fuzzy normed space in the literature. We repeatedly use the fact N�−x , t�=N�x , t�, x�X,t�0, which is deduced from �N3�. It is easy to see that �N4� is equivalent to the followingcondition:

�N7� N�x + y,t� � min�N�x,t�,N�y,t��x,y � X,t � R� .

Example 2.4: Let �X , � · �� be a non-Archimedean normed space. For all x�X, consider

N�x,t� = � t

t + �x�, t � 0

0, t � 0.�

Then �X ,N� is a non-Archimedean fuzzy normed space.Example 2.5: Let �X , � · �� be a non-Archimedean normed space. For all x�X, consider

N�x,t� = 0, t � �x�1, t � �x� .

�Then �X ,N� is a non-Archimedean fuzzy normed space.

Definition 2.6: Let �X ,N� be a non-Archimedean fuzzy normed space. Let �xn be a sequencein X. Then �xn is said to be convergent if there exists x�X, such that lim

n→�

N�xn−x , t�=1 for all

t�0. In that case, x is called the limit of the sequence �xn and we denote it by limn→�

xn=x.

A sequence �xn in X is said to be a Cauchy sequence if limn→�

N�xn+p−xn , t�=1 for all t�0 and

p=1,2 ,3 ,¯. Due to the fact that

N�xn − xm,t� � min�N�xj+1 − xj,t�:m � j � n − 1�n � m� ,

a sequence �xn is Cauchy if and only if limn→�

N�xn+1−xn , t�=1 for all t�0.

It is known that every convergent sequence in a non-Archimedean fuzzy normed space is aCauchy sequence. If every Cauchy sequence is convergent, then the non-Archimedean fuzzynormed space is called a non-Archimedean fuzzy Banach space.

093508-3 Stability of additive-cubicfunctional equation J. Math. Phys. 51, 093508 �2010�

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III. STABILITY OF THE FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FUZZYNORMED SPACES

Throughout this section, unless otherwise explicitly stated, we will assume that K is a non-Archimedean field, assume that X is a vector space over K, �Y ,N� is a non-Archimedean fuzzyBanach space over K, and �Z ,N�� is �an Archimedean or a non-Archimedean fuzzy� normed space.We establish the following new stability for general mixed additive-cubic functional equation �1.2�in non-Archimedean fuzzy Banach spaces.

For convenience, given f :X→Y, we define the difference operator,

Df�x,y� = f�kx + y� + f�kx − y� − kf�x + y� − kf�x − y� − 2f�kx� + 2kf�x� ,

for fixed integers k with k�0, 1 and for all x ,y�X.Theorem 3.1: Let � :XX→Z be a mapping and for some ��0 with �2���,

N����2−1x,2−1y�,t� � N����x,y�,�t� �3.1�

for all x ,y�X and t�0. Let f :X→Y be a mapping with f�0�=0, satisfying condition

N�Df�x,y�,t� � N����x,y�,t� �3.2�

for all x ,y�X and t�0. Then there exists a unique additive mapping A :X→Y, such that

N�f�2x� − 8f�x� − A�x�,t� � N1�x,��k3 − k�t� �3.3�

for all x�X and t�0, where

N1�x,t� = min N����x,�k + 1�x�,1

�2�t�,N����x,�k − 1�x�,

1

�2�t�,N����2x,x�,

1

�2�t�,

N����2x,kx�,1

�2�t�,N����0,x�,

�k − 1��2�

t�,N����0,kx�,�k − 1��2k�

t�,N����x,�2k + 1�x�,t�,

N����x,�2k − 1�x�,t�,N����3x,x�,t�,N����x,x�,t�,N����0,�k + 1�x�, �k − 1�t�,

N����0,�k − 1�x�, �k − 1�t�,N����0,2kx�,�k − 1�

�k�t�,N����x,3kx�,t�,N����x,kx�,t�,

N����2x,2x�,1

�k2�t�,N����2x,2kx�,t�,N����0,�3k − 1�x�,

�k − 1��k�

t�,N����0,kx�,1

�8k + 8�t�

N����0,2�k − 1�x�,�k − 1�

�k2�t�,N����0,2kx�,

1

�k + 1�t�,N���� x

2,�2k + 1�x

2�,

1

�8k�t�,

N���� x

2,�2k − 1�x

2�,

1

�8k�t�,N���� x

2,3kx

2�,

1

�8�t�,N���� x

2,kx

2�,

1

�8�t�,

N����x,x�,1

�8k2�t�,N����0,

�3k − 1�x2

�,�k − 1��8k�

t�,N����0,�k + 1�x

2�,

�k − 1��8k�

t�,

N����0,�k − 1�x�,�k − 1��8k2�

t�� .

Proof: Letting x=0 in �3.2�, we get

N�f�y� + f�− y�,t� � N����0,y�, �k − 1�t� �3.4�

for all y�X and t�0. Putting y=x in �3.2�, we have

093508-4 Xu, Rassias, and Xu J. Math. Phys. 51, 093508 �2010�

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N�f��k + 1�x� + f��k − 1�x� − kf�2x� − 2f�kx� + 2kf�x�,t� � N����x,x�,t� �3.5�

for all x�X and t�0. Hence

N�f�2�k + 1�x� + f�2�k − 1�x� − kf�4x� − 2f�2kx� + 2kf�2x�,t� � N����2x,2x�,t� �3.6�

for all x�X and t�0. Letting y=kx in �3.2�, we get

N�f�2kx� − kf��k + 1�x� − kf�− �k − 1�x� − 2f�kx� + 2kf�x�,t� � N����x,kx�,t� �3.7�

for all x�X and t�0. Letting y= �k+1�x in �3.2�, we have

N�f��2k + 1�x� + f�− x� − kf��k + 2�x� − kf�− kx� − 2f�kx� + 2kf�x�,t� � N����x,�k + 1�x�,t�

�3.8�

for all x�X and t�0. Letting y= �k−1�x in �3.2�, we have

N�f��2k − 1�x� − �k + 2�f�kx� − kf�− �k − 2�x� + �2k + 1�f�x�,t� � N����x,�k − 1�x�,t�

�3.9�

for all x�X and t�0. Replacing x and y by 2x and x in �3.2�, respectively, we get

N�f��2k + 1�x� + f��2k − 1�x� − 2f�2kx� − kf�3x� + 2kf�2x� − kf�x�,t� � N����2x,x�,t�

�3.10�

for all x�X and t�0. Replacing x and y by 3x and x in �3.2�, respectively, we get

N�f��3k + 1�x� + f��3k − 1�x� − 2f�3kx� − kf�4x� − kf�2x� + 2kf�3x�,t� � N����3x,x�,t�

�3.11�

for all x�X and t�0. Replacing x and y by 2x and kx in �3.2�, respectively, we have

N��f�3kx� + f�kx� − kf��k + 2�x� − kf�− �k − 2�x� − 2f�2kx� + 2kf�2x�,t�� � N����2x,kx�,t�

�3.12�

for all x�X and t�0. Setting y= �2k+1�x in �3.2�, we have

N�f��3k + 1�x� + f�− �k + 1�x� − kf�2�k + 1�x� − kf�− 2kx� − 2f�kx� + 2kf�x�,t�

� N����x,�2k + 1�x�,t� �3.13�

for all x�X and t�0. Letting y= �2k−1�x in �3.2�, we have

N�f��3k − 1�x� + f�− �k − 1�x� − kf�− 2�k − 1�x� − kf�2kx� − 2f�kx� + 2kf�x�,t�

� N����x,�2k − 1�x�,t� �3.14�

for all x�X and t�0. Letting y=3kx in �3.2�, we have

N�f�4kx� + f�− 2kx� − kf��3k + 1�x� − kf�− �3k − 1�x� − 2f�kx� + 2kf�x�,t� � N����x,3kx�,t�

�3.15�

for all x�X and t�0. By �3.4�, �3.5�, �3.11�, �3.13�, and �3.14�, we get

093508-5 Stability of additive-cubicfunctional equation J. Math. Phys. 51, 093508 �2010�

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N�kf�2�k + 1�x� + kf�− 2�k − 1�x� + 6f�kx� − 2f�3kx� − kf�4x� + 2kf�3x� − 6kf�x�,t�

� min N����x,�2k + 1�x�,t�,N����x,�2k − 1�x�,t�,N����3x,x�,t�,N����x,x�,t�,

N����0,�k + 1�x�, �k − 1�t�,N����0,�k − 1�x�, �k − 1�t�,N����0,2kx�,�k − 1�

�k�t��

�3.16�

for all x�X and t�0. By �3.4�, �3.8�, and �3.9�, we have

N�f��2k + 1�x� + f��2k − 1�x� − kf��k + 2�x� − kf�− �k − 2�x� − 4f�kx� + 4kf�x�,t�

� min N����x,�k + 1�x�,t�,N����x,�k − 1�x�,t�,N����0,x�, �k − 1�t�,

N����0,kx�,�k − 1�

�k�t�� �3.17�

for all x�X and t�0. It follows from �3.10� and �3.17� that

N�kf��k + 2�x� + kf�− �k − 2�x� − 2f�2kx� + 4f�kx� − kf�3x� + 2kf�2x� − 5kf�x�,t�

� min N����x,�k + 1�x�,t�,N����x,�k − 1�x�,t�,N����2x,x�,t�,N����0,x�, �k − 1�t�,

N����0,kx�,�k − 1�

�k�t�� �3.18�

for all x�X and t�0. By �3.12� and �3.18�, we have

N�f�3kx� − 4f�2kx� + 5f�kx� − kf�3x� + 4kf�2x� − 5kf�x�,t�

� min N����x,�k + 1�x�,t�,N����x,�k − 1�x�,t�,N����2x,x�,t�,

N����2x,kx�,t�,N����0,x�, �k − 1�t�,N����0,kx�,�k − 1�

�k�t�� �3.19�

for all x�X and t�0. By �3.4� and �3.13�–�3.15�, we have

N�kf�− �k + 1�x� − kf�− �k − 1�x� − k2f�2�k + 1�x� + k2f�− 2�k − 1�x� + k2f�2kx� − �k2 − 1�f�− 2kx�

+ f�4kx� − 2f�kx� + 2kf�x�,t� � min N����x,�2k + 1�x�,1

�k�t�,

N����x,�2k − 1�x�,1

�k�t�,N����x,3kx�,t�,N����0,�3k − 1�x�,

�k − 1��k�

t�� �3.20�

for all x�X and t�0. It follows from �3.4�, �3.6�, �3.7�, and �3.20� that

N�f�4kx� − 2f�2kx� − k3f�4x� + 2k3f�2x�,t�

� min N����x,�2k + 1�x�,1

�k�t�,N����x,�2k − 1�x�,

1

�k�t�,N����x,3kx�,t�,N����x,kx�,t�,

N����2x,2x�,1

�k2�t�,N����0,�3k − 1�x�,

�k − 1��k�

t�,N����0,�k + 1�x�,�k − 1�

�k�t�,

093508-6 Xu, Rassias, and Xu J. Math. Phys. 51, 093508 �2010�

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N����0,2�k − 1�x�,�k − 1�

�k2�t�,N����0,2kx�,

1

�k + 1�t�� �3.21�

for all x�X and t�0. Thus,

N�f�2kx� − 2f�kx� − k3f�2x� + 2k3f�x�,t�

� min N���� x

2,�2k + 1�x

2�,

1

�k�t�,N���� x

2,�2k − 1�x

2�,

1

�k�t�,N���� x

2,3kx

2�,t�,

N���� x

2,kx

2�,t�,N����x,x�,

1

�k2�t�,N����0,

�3k − 1�x2

�,�k − 1�

�k�t�,

N����0,�k + 1�x

2�,

�k − 1��k�

t�,N����0,�k − 1�x�,�k − 1�

�k2�t�,N����0,kx�,

1

�k + 1�t��

�3.22�

for all x�X and t�0. By �3.7�, we have

N�f�4kx� − kf�2�k + 1�x� − kf�− 2�k − 1�x� − 2f�2kx� + 2kf�2x�,t� � N����2x,2kx�,t�

�3.23�

for all x�X and t�0. From �3.21� and �3.23�, we have

N�kf�2�k + 1�x� + kf�− 2�k − 1�x� − k3f�4x� + �2k3 − 2k�f�2x�,t�

� min N����x,�2k + 1�x�,1

�k�t�,N����x,�2k − 1�x�,

1

�k�t�,N����x,3kx�,t�,

N����x,kx�,t�,N����2x,2x�,1

�k2�t�,N����2x,2kx�,t�,N����0,�3k − 1�x�,

�k − 1��k�

t�,

N����0,�k + 1�x�,�k − 1�

�k�t�,N����0,2�k − 1�x�,

�k − 1��k2�

t�,N����0,2kx��,1

�k + 1���3.24�

for all x�X and t�0. Also, from �3.16� and �3.24�, we get

N�2f�3kx� − 6f�kx� + �k − k3�f�4x� − 2kf�3x� + �2k3 − 2k�f�2x� + 6kf�x�,t�

� min N����x,�2k + 1�x�,t�,N����x,�2k − 1�x�,t�,N����3x,x�,t�,N����x,x�,t�,

N����0,�k + 1�x�, �k − 1�t�,N����0,�k − 1�x�, �k − 1�t�,N����0,2kx�,�k − 1�

�k�t�,

N����x,3kx�,t�,N����x,kx�,t�,N����2x,2x�,1

�k2�t�,N����2x,2kx�,t�,

N����0,�3k − 1�x�,�k − 1�

�k�t�,N����0,2�k − 1�x�,

�k − 1��k2�

t�,N����0,2kx�,1

�k + 1����3.25�

for all x�X and t�0.On the other hand, it follows from �3.19� and �3.25� that

093508-7 Stability of additive-cubicfunctional equation J. Math. Phys. 51, 093508 �2010�

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N�8f�2kx� − 16f�kx� + �k − k3�f�4x� + �2k3 − 10k�f�2x� + 16kf�x�,t�

� min N����x,�k + 1�x�,1

�2�t�,N����x,�k − 1�x�,

1

�2�t�,N����2x,x�,

1

�2�t�,

N����2x,kx�,1

�2�t�,N����0,x�,

�k − 1��2�

t�,N����0,kx�,�k − 1��2k�

t�,N����x,�2k + 1�x�,t�,

N����x,�2k − 1�x�,t�,N����3x,x�,t�,N����x,x�,t�,N����0,�k + 1�x�, �k − 1�t�,

N����0,�k − 1�x�, �k − 1�t�,N����0,2kx�,�k − 1�

�k�t�,N����x,3kx�,t�,N����x,kx�,t�,

N����2x,2x�,1

�k2�t�,N����2x,2kx�,t�,N����0,�3k − 1�x�,

�k − 1��k�

t�,

N����0,2�k − 1�x�,�k − 1�

�k2�t�,N����0,2kx�,

1

�k + 1�t�� �3.26�

for all x�X. Hence by �3.22� and �3.26�, we get

N��k3 − k��f�4x� − 10f�2x� + 16f�x��,t�

� min N����x,�k + 1�x�,1

�2�t�,N����x,�k − 1�x�,

1

�2�t�,N����2x,x�,

1

�2�t�,

N����2x,kx�,1

�2�t�,N����0,x�,

�k − 1��2�

t�,N����0,kx�,�k − 1��2k�

t�,N����x,�2k + 1�x�,t�,

N����x,�2k − 1�x�,t�,N����3x,x�,t�,N����x,x�,t�,N����0,�k + 1�x�, �k − 1�t�,

N����0,�k − 1�x�, �k − 1�t�,N����0,2kx�,�k − 1�

�k�t�,N����x,3kx�,t�,N����x,kx�,t�,

N����2x,2x�,1

�k2�t�,N����2x,2kx�,t�,N����0,�3k − 1�x�,

�k − 1��k�

t�,N����0,kx�,1

�8k + 8�t�,

N����0,2�k − 1�x�,�k − 1�

�k2�t�,N����0,2kx�,

1

�k + 1�t�,N���� x

2,�2k + 1�x

2�,

1

�8k�t�,

N���� x

2,�2k − 1�x

2�,

1

�8k�t�,N���� x

2,3kx

2�,

1

�8�t�,N���� x

2,kx

2�,

1

�8�t�,N����x,x�,

1

�8k2�t�,

N����0,�3k − 1�x

2�,

�k − 1��8k�

t�,N����0,�k + 1�x

2�,

�k − 1��8k�

t�,N����0,�k − 1�x�,�k − 1��8k2�

t�� .

�3.27�

Therefore,

N�f�4x� − 10f�2x� + 16f�x�,t� � N1�x, �k3 − k�t� �3.28�

for all x�X and t�0, where

093508-8 Xu, Rassias, and Xu J. Math. Phys. 51, 093508 �2010�

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N1�x,t� = min N����x,�k + 1�x�,1

�2�t�,N����x,�k − 1�x�,

1

�2�t�,N����2x,x�,

1

�2�t�,

N����2x,kx�,1

�2�t�,N����0,x�,

�k − 1��2�

t�,N����0,kx�,�k − 1��2k�

t�,N����x,�2k + 1�x�,t�,

N����x,�2k − 1�x�,t�,N����3x,x�,t�,N����x,x�,t�,N����0,�k + 1�x�, �k − 1�t�,

N����0,�k − 1�x�, �k − 1�t�,N����0,2kx�,�k − 1�

�k�t�,N����x,3kx�,t�,N����x,kx�,t�,

N����2x,2x�,1

�k2�t�,N����2x,2kx�,t�,N����0,�3k − 1�x�,

�k − 1��k�

t�,N����0,kx�,1

�8k + 8�t�

N����0,2�k − 1�x�,�k − 1�

�k2�t�,N����0,2kx�,

1

�k + 1�t�,N���� x

2,�2k + 1�x

2�,

1

�8k�t�,

N���� x

2,�2k − 1�x

2�,

1

�8k�t�,N���� x

2,3kx

2�,

1

�8�t�,N���� x

2,kx

2�,

1

�8�t�,

N����x,x�,1

�8k2�t�,N����0,

�3k − 1�x2

�,�k − 1��8k�

t�,N����0,�k + 1�x

2�,

�k − 1��8k�

t�,

N����0,�k − 1�x�,�k − 1��8k2�

t�� . �3.29�

Let g :X→Y is the mapping defined by g�x�ª f�2x�−8f�x� for all x�X. From �3.28�, we have

N�g�2x� − 2g�x�,t� � N1�x, �k3 − k�t� �3.30�

for all x�X and t�0. Replacing x by 2−n−1x in �3.30� and using inequality �3.1�, we get

N�g�2−nx� − 2g�2−�n+1�x�,t� � N1�x,�n+1�k3 − k�t� �3.31�

for all x�X and t�0. Hence

N�2ng�2−nx� − 2n+1g�2−�n+1�x�,t� � N1�x,�n+1

�2�n�k3 − k�t� �3.32�

for all x�X, t�0 and all non-negative integers n.Since limn→� N1�x , �n+1

�2�n �k3−k�t�=1, inequality �3.32� shows that �2ng�2−nx� is a Cauchy se-quence in the non-Archimedean fuzzy Banach space �Y ,N� for each x�X. Hence we can definethe mapping A :X→Y by

A�x� ª limn→�

2ng�2−nx� �3.33�

for all x�X. Thus,

limn→�

N�2ng�2−nx� − A�x�,t� = 1 �3.34�

for all x�X and t�0. For each n�1, x�X, and t�0,

093508-9 Stability of additive-cubicfunctional equation J. Math. Phys. 51, 093508 �2010�

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N�g�x� − 2ng�2−nx�,t� = N�i=0

n−1

�2ig�2−ix� − 2i+1g�2−�i+1�x��,t�� min �

i=0

n−1

�N�2ig�2−ix� − 2i+1g�2−�i+1�x�,t�

� N1�x,��k3 − k�t� . �3.35�

It follows from �3.34� and �3.35� that for each x�X, t�0, and large enough n, we have

N�g�x� − A�x�,t� � min�N�g�x� − 2ng�2−nx�,t�,N�2ng�2−nx� − A�x�,t� � N1�x,��k3 − k�t� .

�3.36�

This proves �3.3�.Now, we will show that A is an additive mapping. It follows from �3.34� that

limn→�

N�2ng�2−n+1x� − A�2x�,t� = 1, limn→�

N�A�x� − 2n−1g�2−n+1x�,t� = 1

for all x�X and t�0. Therefore,

N�A�2x� − 2A�x�,t� = N�A�2x� − 2ng�2−n+1x� + 2ng�2−n+1x� − 2A�x�,t�

� min�N�A�2x� − 2ng�2−n+1x�,t�,N�2ng�2−n+1x� − 2A�x�,t�

= min N�A�2x� − 2ng�2−n+1x�,t�,N�2n−1g�2−n+1x� − A�x�,t

�2���→ 1�as n → �� ,

and so �N2� implies that

A�2x� = 2A�x� �3.37�

for all x�X. Replacing x , y by 2−nx , 2−ny, respectively, in �3.2� and using �N3�, we have

N�2nDf�2−nx,2−ny�,t� � N����2−nx,2−ny�,t

�2n��for all x ,y�X and t�0. On the other hand, it can be easily verified that

Dg�x,y� = Df�2x,2y� − 8Df�x,y�

for all x ,y�X. Hence

N�DA�x,y�,t�

= N�A�kx + y� + A�kx − y� − kA�x + y� − kA�x − y� − 2A�kx� + 2kA�x�,t�

= N��A�kx + y� − 2ng�2−n�kx + y��� + �A�kx − y� − 2ng�2−n�kx − y���

− k�A�x + y� − 2ng�2−n�x + y��� − k�A�x − y� − 2ng�2−n�x − y��� − 2�A�kx� − 2ng�2−nkx��

+ 2k�A�x� − 2ng�2−nx�� + 2n�Df�2−n+1x,2−n+1y� − 8Df�2−nx,2−ny��,t�

� min N�A�kx + y� − 2ng�2−n�kx + y��,t�,N�A�kx − y� − 2ng�2−n�kx − y��,t�,

N�A�x + y� − 2ng�2−n�x + y��,t

�k��,N�A�x − y� − 2ng�2−n�x − y��,t

�k��,

093508-10 Xu, Rassias, and Xu J. Math. Phys. 51, 093508 �2010�

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N�A�kx� − 2ng�2−nkx�,t

�2��,N�A�x� − 2ng�2−nx�,t

�2k��,

N����x,y�,�n−1t

�2n� �,N����x,y�,�nt

�2n+3���for all x ,y�X and t�0. The first six terms on the right hand side of the above inequality tend to1 as n→� by �3.34� and the seventh and eighth terms tend to 1 as n→� by �2��� and �N5�.Therefore, N�DA�x ,y� , t�=1 for all t�0. By �N2�, we infer that

A�kx + y� + A�kx − y� − kA�x + y� − kA�x − y� − 2A�kx� + 2kA�x� = 0

for all x ,y�X, and so by Ref. 39, Lemma 3.1, we see that the mapping x→A�2x�−8A�x� isadditive. Equation �3.37� implies that the mapping A is additive.

To prove the uniqueness of the mapping A, let B :X→Y be another additive mapping, suchthat N�f�2x�−8f�x�−B�x� , t��N1�x ,��k3−k�t�. Then for each x�X and all t�0, we have

N�A�x� − B�x�,t� = N�A�x� − f�2x� + 8f�x� + f�2x� − 8f�x� − B�x�,t�

� min�N�A�x� − f�2x� + 8f�x�,t�,N�f�2x� − 8f�x� − B�x�,t�

� N1�x,��k3 − k�t� .

Hence by the above inequality, �3.1�, �N3�, and the additivity of A and B, we get

N�A�x� − B�x�,t� = N�A�2−nx� − B�2−nx�,t

�2n�� � N1�2−nx,�

�2n��k3 − k�t� � N1�x,

�n+1

�2n��k3 − k�t�

for all x�X, t�0, and n�N. Since �2���, limn→�� �

�2��n=�. Thus, the right hand side of the above

inequality tends to 1 as n→�. So A�x�=B�x� for all x�X. This completes the proof. �

Similar to Theorem 3.1, one can prove the following result.Theorem 3.2: Let � :XX→Z be a mapping and for some ��0 with �8���,

N����2−1x,2−1y�,t� � N����x,y�,�t� �3.38�

for all x ,y�X and t�0. Let f :X→Y be a mapping with f�0�=0, satisfies condition

N�Df�x,y�,t� � N����x,y�,t� �3.39�

for all x ,y�X and t�0. Then there exists a unique cubic mapping C :X→Y, such that

N�f�2x� − 2f�x� − C�x�,t� � N1�x,��k3 − k�t� �3.40�

for all x�X and t�0, where N1�x , t� is defined as in Theorem 3.1.Proof: Similar to the proof of Theorem 3.1, let h :X→Y be the mapping defined by h�x�

ª f�2x�−2f�x� for all x�X. From �3.28�, we have

N�h�2x� − 8h�x�,t� � N1�x, �k3 − k�t� �3.41�

for all x�X and t�0. Replacing x by 2−n−1x in �3.41� and using inequality �3.38�, we get

N�h�2−nx� − 8h�2−�n+1�x�,t� � N1�x,�n+1�k3 − k�t� �3.42�

for all x�X and t�0. Hence

N�8nh�2−nx� − 8n+1h�2−�n+1�x�,t� � N1�x,�n+1

�8�n�k3 − k�t� �3.43�

for all x�X, t�0, and all non-negative integers n.

093508-11 Stability of additive-cubicfunctional equation J. Math. Phys. 51, 093508 �2010�

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From �8���, we conclude that limn→� N1�x , �n+1

�8�n �k3−k�t�=1. Then inequality �3.43� showsthat �8nh�2−nx� is a Cauchy sequence in the non-Archimedean fuzzy Banach space �Y ,N� for eachx�X. Hence we can define the mapping C :X→Y by

C�x� ª limn→�

8nh�2−nx� �3.44�

for all x�X. Thus,

limn→�

N�8nh�2−nx� − C�x�,t� = 1 �3.45�

for all x�X and t�0. For each n�1, x�X, and t�0,

N�h�x� − 8nh�2−nx�,t� = N�i=0

n−1

�8ih�2−ix� − 8i+1h�2−�i+1�x��,t�� min �

i=0

n−1

�N�8ih�2−ix� − 8i+1h�2−�i+1�x�,t�

� N1�x,��k3 − k�t� . �3.46�

It follows from �3.45� and �3.46� that for each x�X, t�0, and large enough n, we have

N�h�x� − C�x�,t� � min�N�h�x� − 8nh�2−nx�,t�,N�8nh�2−nx� − C�x�,t� � N1�x,��k3 − k�t� .

�3.47�

This proves �3.40�.The rest of the proof is similar to the proof of Theorem 3.1. �

Theorem 3.3: Let � :XX→Z be a mapping and for some ��0 with �2���,

N����2−1x,2−1y�,t� � N����x,y�,�t� �3.48�

for all x ,y�X and t�0. Let f :X→Y be a mapping with f�0�=0, satisfies condition

N�Df�x,y�,t� � N����x,y�,t� �3.49�

for all x ,y�X and t�0. Then there exist an additive mapping A :X→Y and a cubic mappingC :X→Y, such that

N�f�x� − A�x� − C�x�,t� � N1�x,��6��k3 − k�t� �3.50�

for all x�X and t�0, where N1�x , t� is defined as in Theorem 3.1.Proof: Clearly �8�� �2���. By Theorems 3.1 and 3.2, there exist a unique additive mapping

A1 :X→Y and a unique cubic mapping C1 :X→Y, such that

N�f�2x� − 8f�x� − A1�x�,t� � N1�x,��k3 − k�t� �3.51�

and

N�f�2x� − 2f�x� − C1�x�,t� � N1�x,��k3 − k�t� �3.52�

for all x�X and t�0, where N1�x , t� is defined as in Theorem 3.1. Therefore, from �3.51� and�3.52�, we get

093508-12 Xu, Rassias, and Xu J. Math. Phys. 51, 093508 �2010�

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N� f�x� +1

6A1�x� −

1

6C1�x�,t�

= N�1

6�f�2x� − 2f�x� − C1�x�� −

1

6�f�2x� − 8f�x� − A1�x��,t�

� min N�1

6�f�2x� − 2f�x� − C1�x��,t�,N�1

6�f�2x� − 8f�x� − A1�x��,t��

= min�N�f�2x� − 2f�x� − C1�x�, �6�t�,N�f�2x� − 8f�x� − A1�x�, �6�t�

� N1�x,��6��k3 − k�t� �3.53�

for all x�X and t�0. Letting A�x�=− 16A1�x� and C�x�= 1

6C1�x� for all x�X, it follows from�3.53� that

N�f�x� − A�x� − C�x�,t� � N1�x,��6��k3 − k�t�

for all x�X and t�0. This completes the proof. �

IV. APPLICATIONS OF FUZZY STABILITY

In this section, we investigate applications of fuzzy stability to the stability of general mixedadditive-cubic functional equation in non-Archimedean normed spaces.

Theorem 4.1: Let K be a non-Archimedean field, X be a linear space over K, �Y , � · �Y� be acomplete non-Archimedean normed space over K, and f :X→Y with f�0�=0, such that

�Df�x,y��Y � ��x,y�

for all x ,y�X, where � :XX→ �0,��. Suppose that

��2−1x,2−1y� �1

���x,y�

for all x ,y�X, where � is a positive real number with �2���. Then there exists a unique additivemapping A :X→Y, such that

�f�2x� − 8f�x� − A�x��Y �1

�M�x�

for all x�X, where

M�x� =1

�k3 − k�max �2���x,�k + 1�x�, �2���x,�k − 1�x�, �2���2x,x�, �2���2x,kx�,

�2��k − 1�

��0,x�,

�2k��k − 1�

��0,kx�,��x,�2k + 1�x�,��x,�2k − 1�x�,��3x,x�,��x,x�,1

�k − 1���0,�k + 1�x�,

1

�k − 1���0,�k − 1�x�,

�k��k − 1�

��0,2kx�,��x,3kx�,��x,kx�, �k2���2x,2x�,��2x,2kx�,

�k��k − 1�

��0,�3k − 1�x�, �8k + 8���0,kx�,�k2�

�k − 1���0,2�k − 1�x�, �k + 1���0,2kx�,

�8k��� x

2,�2k + 1�x

2�, �8k��� x

2,�2k − 1�x

2�, �8��� x

2,3kx

2�, �8��� x

2,kx

2�, �8k2���x,x�,

�8k��k − 1�

��0,�3k − 1�x

2�,

�8k��k − 1�

��0,�k + 1�x

2�,

�8k2��k − 1�

��0,�k − 1�x�� .

093508-13 Stability of additive-cubicfunctional equation J. Math. Phys. 51, 093508 �2010�

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Proof: Let Z=R with the fuzzy norm,

N��x,t� = � t

t + �x�, t � 0, x � R

0, t � 0, x � R ,�

and define

N�y,t� = � t

t + �y�Y, t � 0, y � Y

0, t � 0, y � Y .�

Then N is a non-Archimedean fuzzy norm on Y and N� is a fuzzy norm on R. The result followsfrom Theorem 3.1. �

Corollary 4.2: Let K be a non-Archimedean field, �X , � · �X� be a non-Archimedean normedspace over K, and �Y , � · �Y� be a complete non-Archimedean normed space over K. Let �0, 0�r�1, �2��1, and f :X→Y with f�0�=0, such that

�Df�x,y��Y � ��x�Xr + �y�X

r �

for all x ,y�X. Then there exists a unique additive mapping A :X→Y, such that

�f�2x� − 8f�x� − A�x��Y � �x�X

r

�k3 − k��2�2rmax 2,1

�k − 1��for all x�X.

Proof: Let � :XX→ �0,�� be defined by ��x ,y�= ��x�Xr + �y�X

r � for all x ,y�X. Then thecorollary is followed from Theorem 4.1 by �= �2�r. �

Corollary 4.3: Let K be a non-Archimedean field, �X , � · �X� be a non-Archimedean normedspace over K, and �Y , � · �Y� be a complete non-Archimedean normed space over K. Let �0,�2��1, and f :X→Y with f�0�=0, such that

�Df�x,y��Y � ��x�Xr �y�X

s + ��x�Xr+s + �y�X

r+s�� �x,y � X�

where r ,s be non-negative real numbers, such that �ªr+s�1. Then there exists a unique addi-tive mapping A :X→Y, such that

�f�2x� − 8f�x� − A�x��Y � �x�X

�k3 − k��2�2�max 3,1

�k − 1��for all x�X.

Proof: Let � :XX→ �0,�� be defined by ��x ,y�= ��x�Xr �y�X

s + ��x�Xr+s+ �y�X

r+s�� for all x ,y�X. Then the corollary is followed from Theorem 4.1 by �= �2��. This mixed product-sum sta-bility function � was introduced by Rassias30 �in 2008�. �

Ulam–Gavruta–Rassias stability product of powers of norm, introduced by Rassias28,29 �in1982 and 1989�. See also Refs. 37–39.

The following example shows that the assumption �2��1 cannot be omitted in Corollaries 4.2and 4.3. This example is a modification of the example of Ref. 11.

Example 4.4: Let p�2 be a prime number and f :Qp→Qp be defined by f�x�=2. By Example2.2, �2n�p=1 for all n�Z. Then for ��0,

�Df�x,y��p = �0�p = 0 � � �x,y � Qp� .

However,

093508-14 Xu, Rassias, and Xu J. Math. Phys. 51, 093508 �2010�

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�2ng�2−nx� − 2n+1g�2−�n+1�x��p = �2n+1�p�7�p = �7�p

for all x�Qp and n�N. Hence �2ng�2−nx� is not a Cauchy sequence, where g�x�= f�2x�−8f�x��see the proof of Theorem 3.1�.

Theorem 4.5: Let K be a non-Archimedean field, X be a linear space over K, �Y , � · �Y� be acomplete non-Archimedean normed space over K, and f :X→Y with f�0�=0, such that

�Df�x,y��Y � ��x,y�

for all x ,y�X, where � :XX→ �0,��. Suppose that

��2−1x,2−1y� �1

���x,y�

for all x ,y�X, where � is a positive real number with �8���. Then there exists a unique cubicmapping C :X→Y, such that

�f�2x� − 2f�x� − C�x��Y �1

�M�x�

for all x�X, where M�x� is defined as in Theorem 4.1.Proof: Similar to the proof of Theorem 4.1. The result follows from Theorem 3.2. �

Corollary 4.6: Let K be a non-Archimedean field, �X , � · �X� be a non-Archimedean normedspace over K, and �Y , � · �Y� be a complete non-Archimedean normed space over K. Let �0, 0�r�3, �2��1, and f :X→Y with f�0�=0, such that

�Df�x,y��Y � ��x�Xr + �y�X

r �

for all x ,y�X. Then there exists a unique cubic mapping C :X→Y, such that

�f�2x� − 2f�x� − C�x��Y � �x�X

r

�k3 − k��2�2rmax 2,1

�k − 1��for all x�X.

Proof: Let � :XX→ �0,�� be defined by ��x ,y�= ��x�Xr + �y�X

r � for all x ,y�X. Then thecorollary is followed from Theorem 4.5 by replacing �= �2�r. �

Corollary 4.7: Let K be a non-Archimedean field, �X , � · �X� be a non-Archimedean normedspace over K, and �Y , � · �Y� be a complete non-Archimedean normed space over K. Let �0,�2��1, and f :X→Y with f�0�=0, such that

�Df�x,y��Y � ��x�Xr �y�X

s + ��x�Xr+s + �y�X

r+s�� �x,y � X�

where r ,s be non-negative real numbers, such that �ªr+s�3. Then there exists a unique cubicmapping C :X→Y, such that

�f�2x� − 2f�x� − C�x��Y � �x�X

�k3 − k��2�2�max 3,1

�k − 1��for all x�X.

Proof: Let � :XX→ �0,�� be defined by ��x ,y�= ��x�Xr �y�X

s + ��x�Xr+s+ �y�X

r+s�� for all x ,y�X. Since �2��� �8�, we see that all conditions of Theorem 4.5 hold for �= �2��. Then the corollaryis followed from Theorem 4.5. �

The following example shows that the assumption �2��1 cannot be omitted in Corollaries 4.6and 4.7.

Example 4.8: Let p�2 be a prime number and f :Qp→Qp be defined by f�x�=2. By Example2.2, �2n�p=1 for all n�Z. Then for ��0,

093508-15 Stability of additive-cubicfunctional equation J. Math. Phys. 51, 093508 �2010�

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�Df�x,y��p = �0�p = 0 � ��x,y � Qp� .

However,

�8nh�2−nx� − 8n+1h�2−�n+1�x��p = �23n+1�p�7�p = �7�p

for all x�Qp and n�N. Therefore, �8nh�2−nx� is not a Cauchy sequence, where h�x�= f�2x�−2f�x� �see the proof of Theorem 3.2�.

Theorem 4.9: Let K be a non-Archimedean field, X be a linear space over K, �Y , � · �Y� be acomplete non-Archimedean normed space over K, and f :X→Y with f�0�=0, such that

�Df�x,y��Y � ��x,y�

for all x ,y�X, where � :XX→ �0,��. Suppose that

��2−1x,2−1y� �1

���x,y�

for all x ,y�X, where � is a positive real number with �2���. Then there exist an additivemapping A :X→Y and a cubic mapping C :X→Y, such that

�f�x� − A�x� − C�x��Y �1

�6��M�x�

for all x�X, where M�x� is defined as in Theorem 4.1.Proof: Similar to the proof of Theorem 4.1. The result follows from Theorem 3.3. �

The following results are due to Xu et al. �Ref. 38, see also Ref. 39�.Corollary 4.10: Let �X , � · �X� be a normed space, �Y , � · �Y� be a Banach space, and let , r be

non-negative real numbers, such that r� �0,1�� �1,3�� �3,��. Let f :X→Y be a mapping withf�0�=0, satisfying the inequality

�Df�x,y��Y � ��x�Xr + �y�X

r �

for all x ,y�X. Then there exist a unique additive mapping A :X→Y and a unique cubic mappingC :X→Y, such that

�f�x� − A�x� − C�x��Y ��768k2 �x�X

r

�k3 − k��2 − 2r�, r � �0,1�

768 · 3r−1kr+1 �x�Xr

�k3 − k��2r − 8�, r � �3,��

768 · 3r−1kr+1 �x�Xr

�k3 − k��2r − 2�, r � �1,

ln 3

ln 2�

768 · 3r−1kr+1 �x�Xr

�k3 − k��8 − 2r�, r � � ln 3

ln 2,3� ,

�for all x�X.

Corollary 4.11: Let �X , � · �X� be a normed space, �Y , � · �Y� be a Banach space and let , r, ands be non-negative real numbers, such that �ªr+s� �0,1�� �1,3�� �3,��. Let f :X→Y be amapping with f�0�=0, satisfying the inequality

�Df�x,y��Y � ��x�Xr �y�X

s + ��x�Xr+s + �y�X

r+s��

for all x ,y�X. Then there exist a unique additive mapping A :X→Y and a unique cubic mappingC :X→Y, such that

093508-16 Xu, Rassias, and Xu J. Math. Phys. 51, 093508 �2010�

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�f�x� − A�x� − C�x��Y ��768 · 3k2 �x�X

2�k3 − k��2 − 2�p�, � � �0,1�;

768 · 3�k�+1 �x�X�

2�k3 − k��2� − 8�, � � �3,��

768 · 3�k�+1 �x�X�

2�k3 − k��2� − 2�, � � �1,

ln 3

ln 2�

768 · 3�k�+1 �x�X�

2�k3 − k��8 − 2��, � � � ln 3

ln 2,3� ,

�for all x�X.

Note that A�x�=− 16 limn→� 2nj�f�2−nj+1x�−8f�2−njx�� and C�x�= 1

6 limn→� 8nj�f�2−nj+1x�−2f�2−njx�� for x�X, where j� �1,−1 �Ref. 38, see also Ref. 39�. The following example showsthat Corollaries 4.10 and 4.11 are not true in non-Archimedean normed spaces.

Example 4.12: Let p�2 be a prime number and f :Qp→Qp be defined by f�x�=2 for all x�Qp. Since �2n�p=1 for all n�Z, then for ��0,

�Df�x,y��p = �0�p = 0 � �

for all x ,y�Qp. However, �2nj�f�2−nj+1x�−8f�2−njx�� and �8nj�f�2−nj+1x�−2f�2−njx�� are notCauchy sequences for j=1 or �1. In fact, by using the fact that �2n�p=1�n�Z�, we have

�2nj�f�2−nj+1x� − 8f�2−njx�� − 2�n+1�j�f�2−�n+1�j+1x� − 8f�2−�n+1�jx���p = �7�p

and

�8nj�f�2−nj+1x� − 2f�2−njx�� − 8�n+1�j�f�2−�n+1�j+1x� − 2f�2−�n+1�jx���p = �7�p

for all x�Qp and n�N. Hence the sequences �2nj�f�2−nj+1x�−8f�2−njx�� and �8nj�f�2−nj+1x�−2f�2−njx�� are not converge in Qp.

However, we have the following version of Corollaries 4.10 and 4.11 for non-Archimedeannormed spaces.

Corollary 4.13: Let K be a non-Archimedean field, �X , � · �X� be a non-Archimedean normedspace over K, and �Y , � · �Y� be a complete non-Archimedean normed space over K. Let �0, 0�r�1, �2��1, and f :X→Y with f�0�=0, such that

�Df�x,y��Y � ��x�Xr + �y�X

r �

for all x ,y�X. Then there exist an additive mapping A :X→Y and a cubic mapping C :X→Y,such that

�f�x� − A�x� − C�x��Y � �x�X

r

�6��k3 − k��2�2rmax 2,1

�k − 1��for all x�X.

Proof: Similar to the proof of Corollary 4.2. The result follows from Theorem 4.9. �

Corollary 4.14: Let K be a non-Archimedean field, �X , � · �X� be a non-Archimedean normedspace over K, and �Y , � · �Y� be a complete non-Archimedean normed space over K. Let �0,�2��1, and f :X→Y with f�0�=0, such that

�Df�x,y��Y � ��x�Xr �y�X

s + ��x�Xr+s + �y�X

r+s�� �x,y � X�

where r , s be non-negative real numbers, such that �ªr+s�1. Then there exist an additivemapping A :X→Y and a cubic mapping C :X→Y, such that

093508-17 Stability of additive-cubicfunctional equation J. Math. Phys. 51, 093508 �2010�

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�f�x� − A�x� − C�x��Y � �x�X

�6��k3 − k��2�2�max 3,1

�k − 1��for all x�X.

Proof: Similar to the proof of Corollary 4.3. The result follows from Theorem 4.9. �

Remark: Example 4.12 shows that the assumption �2��1 cannot be omitted in Corollaries4.13 and 4.14.

ACKNOWLEDGMENTS

The first author was supported by the National Natural Science Foundation of China �GrantNo. 10671013�.

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