intuitionistic fuzzy stability of a general mixed additive-cubic equation
TRANSCRIPT
Intuitionistic fuzzy stability of a general mixed additive-cubic equationTian Zhou Xu, John Michael Rassias, and Wan Xin Xu Citation: Journal of Mathematical Physics 51, 063519 (2010); doi: 10.1063/1.3431968 View online: http://dx.doi.org/10.1063/1.3431968 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/51/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Mappings on intuitionistic fuzzy soft classes AIP Conf. Proc. 1522, 1022 (2013); 10.1063/1.4801242 Complex intuitionistic fuzzy sets AIP Conf. Proc. 1482, 464 (2012); 10.1063/1.4757515 Linear programming using symmetric triangular intuitionistic fuzzy numbers AIP Conf. Proc. 1479, 2086 (2012); 10.1063/1.4756601 λ – Statistical convergence in intuitionistic fuzzy 2–normed space AIP Conf. Proc. 1479, 979 (2012); 10.1063/1.4756306 Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces J. Math. Phys. 51, 093508 (2010); 10.1063/1.3482073
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
Intuitionistic fuzzy stability of a general mixedadditive-cubic equation
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,Beijing 100081, People’s Republic of China2Pedagogical Department of E.E., Section of Mathematics and Informatics,National and Capodistrian 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 11 November 2009; accepted 28 April 2010; published online 29 June 2010�
We establish some stability results concerning the general mixed additive-cubicfunctional equation, f�kx+y�+ f�kx−y�=kf�x+y�+kf�x−y�+2f�kx�−2kf�x�,in intu-itionistic fuzzy normed spaces. In addition, we show under some suitable condi-tions that an approximately mixed additive-cubic function can be approximatedby a mixed additive and cubic mapping. © 2010 American Institute ofPhysics. �doi:10.1063/1.3431968�
I. INTRODUCTION
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. 2, 9, and 18�. 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. 13–17�.
The usual uncertainty principle of Werner Heisenberg leads to a generalized uncertainty prin-ciple, which has been motivated by string theory and noncommutative geometry. In strong quan-tum gravity regime space-time points are determined in a fuzzy manner. Thus impossibility ofdetermining the position of particles gives the space-time a fuzzy structure.13 Because of this fuzzystructure, position space representation of quantum mechanics breaks down and therefore a gen-eralized normed space of quasiposition eigenfunction is required.18 Hence, one needs to discuss ona new family of fuzzy norms. There are many situations where the norm of a vector is not possibleto be found and the concept of intuitionistic fuzzy norm12,20,23–26 seems to be more suitable in suchcases, that is, we can deal with such situations by modeling the inexactness through the intuition-istic fuzzy norm.
The study of stability problems for functional equations is related to a question of Ulam35
concerning the stability of group homomorphisms and affirmatively answered for Banach spacesby Hyers.5 Subsequently, the result of Hyers was generalized by Aoki1 for additive mappings andby Rassias21 for linear mappings by considering an unbounded Cauchy difference. In addition,Rassias22,27–30 generalized the Hyers stability result by introducing two weaker conditions con-trolled by a product of different powers of norms and a mixed product-sum of powers of norms,
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, 063519 �2010�
51, 063519-10022-2488/2010/51�6�/063519/21/$30.00 © 2010 American Institute of Physics
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
respectively. Recently, several further interesting discussions, modifications, extensions, and gen-eralizations of the original problem of Ulam have been proposed �see, e.g., Refs. 3, 4, 10, and 19and the references therein�. In particular, various fuzzy stability results concerning Cauchy, Jensen,and quadratic functional equations were discussed in Refs. 6–8. The stability problem for theJensen and cubic functional equations have been investigated in Refs. 9, 11, and 34 in the settingof intuitionistic fuzzy normed spaces �IFNSs�.
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.12 Functional equation �1.1�is called the mixed additive-cubic functional equation, since the function f�x�=ax3+bx is itssolution. Every solution of the mixed additive-cubic functional equation is said to be a mixedadditive-cubic mapping. In Ref. 36 the problem of generalized Hyers–Ulam stability of a generalmixed additive-cubic functional equation,
f�kx + y� + f�kx − y� = kf�x + y� + kf�x − y� + 2f�kx� − 2kf�x� , �1.2�
was investigated in quasi-Banach spaces.In this paper, we determine some stability results concerning the stability of general mixed
additive-cubic functional equation �1.2� in the setting of IFNSs.
II. PRELIMINARIES
In this section we recall some notations and basic definitions used in this paper.Lemma 2.1: �References 13 and 33� Consider the set L� and the order relation �L� defined by
L� = ��x1,x2�:�x1,x2� � �0,1�2,x1 + x2 � 1� ,
�x1,x2��L��y1,y2� ⇔ x1 � y1,x2 � y2, ∀ �x1,x2�,�y1,y2� � L�.
Then �L� ,�L�� is a complete lattice.We denote its units by 0L� = �0,1� and 1L� = �1,0�.Definition 2.2: �Reference 33� An intuitionistic fuzzy set A�,� in a universal set U is an object
A�,�= ���A�u� ,�A�u�� :u�U�, where, for all u�U, �A�u�� �0,1� and �A�u�� �0,1� are called themembership degree and the nonmembership degree, respectively, of u in A�,�, and, furthermore,they satisfy �A�u�+�A�u��1.
Classically, a triangular norm T=� on �0, 1� is defined as an increasing, commutative, asso-ciative mapping T : �0,1�2→ �0,1� satisfying T�1,x�=1�x=x for all x� �0,1�. A triangularconorm S=� is defined as an increasing, commutative, associative mapping S : �0,1�2→ �0,1�satisfying S�0,x�=0�x=x for all x� �0,1�.
Definition 2.3: �Reference 33� A triangular norm �t-norm� on L� is a mapping T : �L��2→L�
satisfying the following conditions:
�a� �∀x�L���T�x ,1L��=x� �boundary condition�;�b� �∀�x ,y�� �L��2��T�x ,y�=T�y ,x�� �commutativity�;�c� �∀�x ,y ,z�� �L��3��T�x ,T�y ,z��=T�T�x ,y� ,z�� �associativity�;�d� �∀�x ,x� ,y ,y��� �L��4��x�L�x� and y�L�y�⇒T�x ,y��L�T�x� ,y��� �monotonicity�.
Definition 2.4: �References 13 and 33� A continuous t-norm T on L� is said to be continuoust-representable if there exist a continuous t-norm � and a continuous t-conorm � on �0, 1� suchthat, for all x= �x1 ,x2� ,y= �y1 ,y2��L�,
T�x,y� = �x1 � y1,x2 � y2� .
Example 2.5: For all a= �a1 ,a2� ,b= �b1 ,b2��L�, consider
063519-2 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
T�a,b� = �a1b1,min�a2 + b2,1��
and
M�a,b� = �min�a1,b1�,max�a2,b2�� .
Then T�a ,b� and M�a ,b� are continuous t-representable.Now, we define a sequence Tn recursively by T1=T and
Tn�x�1�, . . . ,x�n+1�� = T�Tn−1�x�1�, . . . ,x�n��,x�n+1��
for all n�2 and x�i��L�.Definition 2.6: A negator on L� is any decreasing mapping N :L�→L� satisfying N�0L��
=1L� and N�1L��=0L�. If N�N�x��=x for all x�L�, then N is called an involutive negator. Anegator on �0,1� is a decreasing mapping N : �0,1�→ �0,1� satisfying N�0�=1 and N�1�=0. Ns
denotes the standard negator on ��0,1� ,�� defined by Ns�x�=1−x for all x� �0,1�.Definition 2.7: �References 33 and 34� The triple �X ,P ,T� is said to be an IFNS if X is a
vector space, T is a continuous t-representable, and P is a mapping X� �0,��→L�, satisfying thefollowing conditions for all x ,y�X and t ,s�0:
�a� P�x , t��0L�;�b� P�x , t�=1L� if and only if x=0;�c� P��x , t�=P�x , t / ���� for all ��0;�d� P�x+y , t+s��L�T�P�x , t� ,P�y ,s��;�e� P�x , ·� : �0,��→L� is continuous;�f� limt→� P�x , t�=1L�.
In this case, P is called an intuitionistic fuzzy norm on X. Given and , membership andnonmembership degrees of an intuitionistic fuzzy set from X� �0,�� to �0,1�, such that �x , t�+�x , t��1 for all x�X and t�0, we write P,�x , t�= ��x , t� ,�x , t��.
Example 2.8: Let �X , � · �� be a normed space. Let T�a ,b�= �a1b1 ,min�a2+b2 ,1�� for all a= �a1 ,a2�, b= �b1 ,b2��L�, and , be membership and nonmembership degree of an intuitionisticfuzzy set defined by
P,�x,t� = ��x,t�,�x,t�� = t
t + �x�,
�x�t + �x�, ∀ t � R+.
Then �X ,P, ,T� is an IFNS.In Example 2.8, �x , t�+�x , t�=1 for all x�X. We present an example in which �x , t�
+�x , t��1 for x�0. This example is a modification of the example of Saadati and Park.32
Example 2.9: Let �X , � · �� be a normed space. Let T�a ,b�= �a1b1 ,min�a2+b2 ,1�� for all a= �a1 ,a2�, b= �b1 ,b2��L�, and , be membership and nonmembership degrees of an intuition-istic fuzzy set defined by
P,�x,t� = ��x,t�,�x,t�� = t
t + m�x�,
�x�t + �x�
for all t�R+ in which m�1. Then �X ,P, ,T� is an IFNS. Here, �x , t�+�x , t�=1 for x=0 and�x , t�+�x , t��1 for x�0.
Lemma 2.10: �Reference 33� Let P, be an intuitionistic fuzzy norm on X. Then P,�x , t� isnondecreasing with respect to t for all x�X.
The concepts of convergence and Cauchy sequences in an IFNS are studied in Ref. 31.Definition 2.11:
�1� Let �X ,P, ,T� be an IFNS. Then, a sequence �xn� is said to be intuitionistic fuzzy conver-
gent to a point x�X �denoted by xn→IF
x� if P,�xn−x , t�→1L� as n→� for every t�0.
063519-3 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
�2� Let �X ,P, ,T� be an IFNS. Then, a sequence �xn� is said to be intuitionistic fuzzy Cauchysequence if, for any ��0 and t�0, there exists n0�N, such that
P,�xn − xm,t��L��Ns���,��, ∀ n,m � n0,
where Ns is the standard negator.�3� Let �X ,P, ,T� be an IFNS. Then �X ,P, ,T� is said to be complete if every intuitionistic
fuzzy Cauchy sequence in �X ,P, ,T� is intuitionistic fuzzy convergent in �X ,P, ,T�. Acomplete IFNS is called an intuitionistic fuzzy Banach space.
III. INTUITIONISTIC FUZZY STABILITY
Throughout this section, assume that X is a linear space and �Y ,P, ,M� and �Z ,P,� ,M� bean intuitionistic fuzzy Banach space and an IFNS, respectively, M is defined as in Example 2.5�see Refs. 33 and 34�. We establish the following new stability for the general mixed additive-cubic functional equation �1.2� in the setting of INFSs. For convenience, we use the followingabbreviation for a given mapping f :X→Y:
Df�x,y� ª f�kx + y� + f�kx − y� − kf�x + y� − kf�x − y� − 2f�kx� + 2kf�x�
for all x ,y�X.We begin with the following important lemma.Lemma 3.1: (Reference 36) Let f :X→Y be a mapping with f�0�=0 and satisfying Eq. (1.2),
then the mapping G�x�ª f�2x�−8f�x� is additive and H�x�ª f�2x�−2f�x� is cubic.Theorem 3.2: Let 0���2 and :X�X→Z be a mapping such that
P,� � �2x,2y�,t��L�P,� �� �x,y�,t� �3.1�
and
limn→�
P,� � �2nx,2ny�,2nt� = 1L� �3.2�
for all x ,y�X and t�0. Let f :X→Y be a mapping with f�0�=0 and such that
P,�Df�x,y�,t��L�P,� � �x,y�,t� �3.3�
for all x ,y�X and t�0. Then there exists a unique additive mapping A :X→Y, such that
P,�f�2x� − 8f�x� − A�x�,t��L�P,� x,�2 − ��t
2 �3.4�
for all x�X and t�0, where
P,� �x,t� ª M31�P,� x
2,�2k + 1�x
2,
t
384k,P,� x
2,�2k − 1�x
2,
t
384k,
P,� x
2,3kx
2,
t
384,P,� 0,
�3k − 1�x2
,�k − 1�t
384k,P,� �x,x�,
t
96k2,
P,� x
2,kx
2,
t
96,P,� 0,
�k + 1�x2
,�k − 1�t
96k,P,� �0,�k − 1�x�,
�k − 1�t96k2 ,
P,� �0,kx�,t
96�k + 1�,P,� �x,�k + 1�x�,t
128,P,� �x,�k − 1�x�,
t
128,
P,� �0,x�,�k − 1�t
128,P,� �0,kx�,
�k − 1�t128k
,P,� �2x,x�,t
32,
063519-4 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,� �2x,kx�,t
16,P,� �x,�2k − 1�x�,
t
56,P,� �x,�2k + 1�x�,
t
56,
P,� �3x,x�,t
56,P,� �x,x�,
t
56,P,� �0,�k + 1�x�,
�k − 1�t56
,
P,� �0,�k − 1�x�,�k − 1�t
56,P,� �0,2kx�,
�k − 1�t56k
P,� �x,�2k + 1�x�,t
384k,
P,� �x,�2k − 1�x�,t
384k,P,� �x,3kx�,
t
384,P,� �0,�3k − 1�x�,
�k − 1�t384k
,
P,� �2x,2x�,t
96k2,P,� �x,kx�,t
96,P,� �0,�k + 1�x�,
�k − 1�t96k
,
P,� �0,2�k − 1�x�,�k − 1�t
96k2 ,P,� �0,2kx�,t
96�k + 1�,P,� �2x,2kx�,t
16� .
Proof: Letting x=0 in �3.3�, we get
P,�f�y� + f�− y�,t��L�P,� � �0,y�,�k − 1�t� �3.5�
for all y�X and t�0. Putting y=x in �3.3�, we have
P,�f��k + 1�x� + f��k − 1�x� − kf�2x� − 2f�kx� + 2kf�x�,t��L�P,� � �x,x�,t� �3.6�
for all x�X and t�0. Replacing x by 2x in �3.6�, we obtain
P,�f�2�k + 1�x� + f�2�k − 1�x� − kf�4x� − 2f�2kx� + 2kf�2x�,t��L�P,� � �2x,2x�,t�
�3.7�
for all x�X and t�0. Letting y=kx in �3.3�, we get
P,�f�2kx� − kf��k + 1�x� − kf�− �k − 1�x� − 2f�kx� + 2kf�x�,t��L�P,� � �x,kx�,t� �3.8�
for all x�X and t�0. Letting y= �k+1�x in �3.3�, we have
P,�f��2k + 1�x� + f�− x� − kf��k + 2�x� − kf�− kx� − 2f�kx� + 2kf�x�,t��L�P,� � �x,�k + 1�x�,t�
�3.9�
for all x�X and t�0. Letting y= �k−1�x in �3.3�, we have
P,�f��2k − 1�x� − �k + 2�f�kx� − kf�− �k − 2�x� + �2k + 1�f�x�,t��L�P,� � �x,�k − 1�x�,t�
�3.10�
for all x�X and t�0. Replacing x and y by 2x and x in �3.3�, respectively, we get
P,�f��2k + 1�x� + f��2k − 1�x� − 2f�2kx� − kf�3x� + 2kf�2x� − kf�x�,t��L�P,� � �2x,x�,t�
�3.11�
for all x�X and t�0. Replacing x and y by 3x and x in �3.3�, respectively, we get
P,�f��3k + 1�x� + f��3k − 1�x� − 2f�3kx� − kf�4x� − kf�2x� + 2kf�3x�,t��L�P,� � �3x,x�,t�
�3.12�
for all x�X and t�0. Replacing x and y by 2x and kx in �3.3�, respectively, we have
063519-5 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,�f�3kx� + f�kx� − kf��k + 2�x� − kf�− �k − 2�x� − 2f�2kx� + 2kf�2x�,t��L�P,� � �2x,kx�,t�
�3.13�
for all x�X and t�0. Setting y= �2k+1�x in �3.3�, we have
P,�f��3k + 1�x� + f�− �k + 1�x� − kf�2�k + 1�x� − kf�− 2kx� − 2f�kx� + 2kf�x�,t�
�L�P,� � �x,�2k + 1�x�,t� �3.14�
for all x�X and t�0. Letting y= �2k−1�x in �3.3�, we have
P,�f��3k − 1�x� + f�− �k − 1�x� − kf�− 2�k − 1�x� − kf�2kx� − 2f�kx� + 2kf�x�,t�
�L�P,� � �x,�2k − 1�x�,t� �3.15�
for all x�X and t�0. Letting y=3kx in �3.3�, we have
P,�f�4kx� + f�− 2kx� − kf��3k + 1�x� − kf�− �3k − 1�x� − 2f�kx� + 2kf�x�,t��L�P,� � �x,3kx�,t�
�3.16�
for all x�X and t�0. By �3.5�, �3.6�, �3.12�, �3.14�, and �3.15�, we get
P,�kf�2�k + 1�x� + kf�− 2�k − 1�x� + 6f�kx� − 2f�3kx� − kf�4x� + 2kf�3x� − 6kf�x�,t�
�L�M6�P,� �x,�2k − 1�x�,t
7,P,� �x,�2k + 1�x�,
t
7,P,� �3x,x�,
t
7,P,� �x,x�,
t
7,
P,� �0,�k + 1�x�,�k − 1�t
7,P,� �0,�k − 1�x�,
�k − 1�t7
,P,� �0,2kx�,�k − 1�t
7k�
�3.17�
for all x�X and t�0. By �3.5�, �3.9�, and �3.10�, we have
P,�f��2k + 1�x� + f��2k − 1�x� − kf��k + 2�x� − kf�− �k − 2�x� − 4f�kx� + 4kf�x�,t�
�L�M3�P,� �x,�k + 1�x�,t
4,P,� �x,�k − 1�x�,
t
4,
P,� �0,x�,�k − 1�t
4,P,� �0,kx�,
�k − 1�t4k
� �3.18�
for all x�X and t�0. It follows from �3.11� and �3.18� that
P,�kf��k + 2�x� + kf�− �k − 2�x� − 2f�2kx� + 4f�kx� − kf�3x� + 2kf�2x� − 5kf�x�,t�
�L�M4�P,� �x,�k + 1�x�,t
8,P,� �x,�k − 1�x�,
t
8,P,� �0,x�,
�k − 1�t8
,
P,� �0,kx�,�k − 1�t
8k,P,� �2x,x�,
t
2� �3.19�
for all x�X and t�0. By �3.13� and �3.19�, we have
063519-6 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,�f�3kx� − 4f�2kx� + 5f�kx� − kf�3x� + 4kf�2x� − 5kf�x�,t�
�L�M5�P,� �x,�k + 1�x�,t
16,P,� �x,�k − 1�x�,
t
16,P,� �0,x�,
�k − 1�t16
,
P,� �0,kx�,�k − 1�t
16k,P,� �2x,x�,
t
4,P,� �2x,kx�,
t
2� �3.20�
for all x�X and t�0. By �3.5�, �3.14�, and �3.16�, we have
P,�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�
�L�M3�P,� �x,�2k + 1�x�,t
4k,P,� �x,�2k − 1�x�,
t
4k,P,� �x,3kx�,
t
4,
P,� �0,�3k − 1�x�,�k − 1�t
4k� �3.21�
for all x�X and t�0. It follows from �3.5�–�3.8� and �3.21� that
P,�f�4kx� − 2f�2kx� − k3f�4x� + 2k3f�2x�,t�
�L�M8�P,� �x,�2k + 1�x�,t
24k,P,� �x,�2k − 1�x�,
t
24k,P,� �x,3kx�,
t
24,
P,� �0,�3k − 1�x�,�k − 1�t
24k,P,� �2x,2x�,
t
6k2,P,� �x,kx�,t
6,
P,� �0,�k + 1�x�,�k − 1�t
6k,P,� �0,2�k − 1�x�,
�k − 1�t6k2 ,P,� �0,2kx�,
t
6�k + 1���3.22�
for all x�X and t�0. Hence
P,�f�2kx� − 2f�kx� − k3f�2x� + 2k3f�x�,t�
�L�M8�P,� x
2,�2k + 1�x
2,
t
24k,P,� x
2,�2k − 1�x
2,
t
24k,P,� x
2,3kx
2,
t
24,
P,� 0,�3k − 1�x
2,
�k − 1�t24k
,P,� �x,x�,t
6k2,P,� x
2,kx
2,
t
6,
P,� 0,�k + 1�x
2,
�k − 1�t6k
,P,� �0,�k − 1�x�,�k − 1�t
6k2 ,P,� �0,kx�,t
6�k + 1���3.23�
for all x�X and t�0. By �3.8�, we have
P,�f�4kx� − kf�2�k + 1�x� − kf�− 2�k − 1�x� − 2f�2kx� + 2kf�2x�,t��L�P,� � �2x,2kx�,t�
�3.24�
for all x�X and t�0. From �3.22� and �3.24�, we have
063519-7 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,�kf�2�k + 1�x� + kf�− 2�k − 1�x� − k3f�4x� + �2k3 − 2k�f�2x�,t�
�L�M9�P,� �x,�2k + 1�x�,t
48k,P,� �x,�2k − 1�x�,
t
48k,P,� �x,3kx�,
t
48,
P,� �0,�3k − 1�x�,�k − 1�t
48k,P,� �2x,2x�,
t
12k2,P,� �x,kx�,t
12,
P,� �0,�k + 1�x�,�k − 1�t
12k,P,� �0,2�k − 1�x�,
�k − 1�t12k2 ,
P,� �0,2kx�,t
12�k + 1�,P,� �2x,2kx�,t
2� �3.25�
for all x�X and t�0. Also, from �3.17� and �3.25�, we get
P,�2f�3kx� − 6f�kx� + �k − k3�f�4x� − 2kf�3x� + �2k3 − 2k�f�2x� + 6kf�x�,t�
�L�M16�P,� �x,�2k − 1�x�,t
14,P,� �x,�2k + 1�x�,
t
14,P,� �3x,x�,
t
14,
P,� �x,x�,t
14,P,� �0,�k + 1�x�,
�k − 1�t14
,P,� �0,�k − 1�x�,�k − 1�t
14,
P,� �0,2kx�,�k − 1�t
14k,P,� �x,�2k + 1�x�,
t
96k,P,� �x,�2k − 1�x�,
t
96k,
P,� �x,3kx�,t
96,P,� �0,�3k − 1�x�,
�k − 1�t96k
,P,� �2x,2x�,t
24k2,
P,� �x,kx�,t
24,P,� �0,�k + 1�x�,
�k − 1�t24k
,P,� �0,2�k − 1�x�,�k − 1�t
24k2 ,
P,� �0,2kx�,t
24�k + 1�,P,� �2x,2kx�,t
4� �3.26�
for all x�X and t�0.On the other hand, it follows from �3.20� and �3.26� that
P,�8f�2kx� − 16f�kx� + �k − k3�f�4x� + �2k3 − 10k�f�2x� + 16kf�x�,t�
�L�M22�P,� �x,�k + 1�x�,t
64,P,� �x,�k − 1�x�,
t
64,P,� �0,x�,
�k − 1�t64
,
P,� �0,kx�,�k − 1�t
64k,P,� �2x,x�,
t
16,P,� �2x,kx�,
t
8,
P,� �x,�2k − 1�x�,t
28,P,� �x,�2k + 1�x�,
t
28,P,� �3x,x�,
t
28,
P,� �x,x�,t
28,P,� �0,�k + 1�x�,
�k − 1�t28
,P,� �0,�k − 1�x�,�k − 1�t
28,
P,� �0,2kx�,�k − 1�t
28kP,� �x,�2k + 1�x�,
t
192k,P,� �x,�2k − 1�x�,
t
192k,
P,� �x,3kx�,t
192,P,� �0,�3k − 1�x�,
�k − 1�t192k
,P,� �2x,2x�,t
48k2,
063519-8 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,� �x,kx�,t
48,P,� �0,�k + 1�x�,
�k − 1�t48k
,P,� �0,2�k − 1�x�,�k − 1�t
48k2 ,
P,� �0,2kx�,t
48�k + 1�,P,� �2x,2kx�,t
8� �3.27�
for all x�X and t�0. Therefore, by �3.23� and �3.27�, we get
P, f�4x� − 10f�2x� + 16f�x�,t
k3 − k�L�P,� �x,t� �3.28�
for all x�X and t�0, where
P,� �x,t� ª M31�P,� x
2,�2k + 1�x
2,
t
384k,P,� x
2,�2k − 1�x
2,
t
384k,
P,� x
2,3kx
2,
t
384,P,� 0,
�3k − 1�x2
,�k − 1�t
384k,P,� �x,x�,
t
96k2,
P,� x
2,kx
2,
t
96,P,� 0,
�k + 1�x2
,�k − 1�t
96k,P,� �0,�k − 1�x�,
�k − 1�t96k2 ,
P,� �0,kx�,t
96�k + 1�,P,� �x,�k + 1�x�,t
128,P,� �x,�k − 1�x�,
t
128,
P,� �0,x�,�k − 1�t
128,P,� �0,kx�,
�k − 1�t128k
,P,� �2x,x�,t
32,
P,� �2x,kx�,t
16,P,� �x,�2k − 1�x�,
t
56,P,� �x,�2k + 1�x�,
t
56,
P,� �3x,x�,t
56,P,� �x,x�,
t
56,P,� �0,�k + 1�x�,
�k − 1�t56
,
P,� �0,�k − 1�x�,�k − 1�t
56,P,� �0,2kx�,
�k − 1�t56k
P,� �x,�2k + 1�x�,t
384k,
P,� �x,�2k − 1�x�,t
384k,P,� �x,3kx�,
t
384,P,� �0,�3k − 1�x�,
�k − 1�t384k
,
P,� �2x,2x�,t
96k2,P,� �x,kx�,t
96,P,� �0,�k + 1�x�,
�k − 1�t96k
,
P,� �0,2�k − 1�x�,�k − 1�t
96k2 ,P,� �0,2kx�,t
96�k + 1�,P,� �2x,2kx�,t
16�
Now, let g :X→Y be the mapping defined by g�x�ª f�2x�−8f�x� for all x�X. Thus, by �3.28�and Lemma 2.10, we have
P,�g�2x� − 2g�x�,t��L�P,� �x,t� �3.29�
for all x�X and t�0. Replacing x by 2nx in �3.29� and using �3.1�, we obtain
P,�g�2n+1x� − 2g�2nx�,t��L�P,� �2nx,t��L�P,� x,t
�n �3.30�
for all x�X and t�0. Hence
063519-9 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,g�2n+1x�2n+1 −
g�2nx�2n ,
t
2n+1�L�P,� x,t
�nand so
P,g�2n+1x�2n+1 −
g�2nx�2n ,
�nt
2n+1�L�P,� �x,t� �3.31�
for all x�X , t�0 and all non-negative integers n. For all x�X , t�0 and all non-negative integersn and m with n�m, we have
P,g�2nx�2n −
g�2mx�2m ,
k=m
n−1�kt
2k+1 = P, k=m
n−1 �g�2k+1x�2k+1 −
g�2kx�2k �,
k=m
n−1�kt
2k+1�L�Mn−m−1P,g�2m+1x�
2m+1 −g�2mx�
2m ,�mt
2m+1, . . . ,P,g�2nx�2n
−g�2n−1x�
2n−1 ,�n−1t
2n �L�P,� �x,t� . �3.32�
Hence
P,g�2nx�2n −
g�2mx�2m ,t�L�P,� �x,
t
k=mn−1 �k
2k+1 � �3.33�
for all x�X , t�0 and m ,n�N with n�m.Since 0���2 and k=0
� �� /2�k��, then �g�2nx� /2n� is a Cauchy sequence in �Y ,P, ,M�for each x�X. Since �Y ,P, ,M� is an intuitionistic fuzzy Banach space, this sequence convergesto some point A�x��Y. So we can define the mapping A :X→Y by
A�x� ª limn→�
g�2nx�2n �3.34�
for all x�X. Fix x�X and put m=0 in �3.33� to obtain
P,g�2nx�2n − g�x�,t�L�P,� �x,
t
k=0n−1 �k
2k+1 � �3.35�
for all t�0 and so we have that
P,�g�x� − A�x�,t� = P,g�x� −g�2nx�
2n +g�2nx�
2n − A�x�,t�L�MP,g�x� −
g�2nx�2n ,
t
2,P,g�2nx�
2n − A�x�,t
2
�L�M�P,� �x,t
2 k=0n−1 �k
2k+1 �,P,g�2nx�2n − A�x�,
t
2� . �3.36�
Taking the limit as n→� in �3.36� and using �3.34� we get
063519-10 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,�f�2x� − 8f�x� − A�x�,t��L�P,� x,�2 − ��t
2
for all x�X and t�0, which shows that A satisfies �3.4�.Now, we will show that A is an additive mapping. It follows from �3.34� that
limn→�
P,A�2x� −g�2n+1x�
2n ,t = 1L�, limn→�
P,A�x� −g�2n+1x�
2n+1 ,t = 1L�
for all x�X and t�0. Therefore,
P,�A�2x� − 2A�x�,t� = P,A�2x� −g�2n+1x�
2n +g�2n+1x�
2n − 2A�x�,t�L�MP,A�2x� −
g�2n+1x�2n ,
t
2,P,g�2n+1x�
2n+1 − A�x�,t
4 ,
and so we have that
A�2x� = 2A�x� �3.37�
for all x�X. Replacing x ,y by 2nx ,2ny in �3.3� to get
P, 1
2nDf�2nx,2ny�,t�L�P,� � �2nx,2ny�,2nt�
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,
P,�DA�x,y�,t� = P,�A�kx + y� + A�kx − y� − kA�x + y� − kA�x − y� − 2A�kx� + 2kA�x�,t�
= P,�A�kx + y� −g�2n�kx + y��
2n � + �A�kx − y� −g�2n�kx − y��
2n � − k�A�x
+ y� −g�2n�x + y��
2n � − k�A�x − y� −g�2n�x − y��
2n � − 2�A�kx� −g�2nkx�
2n �+ 2k�A�x� −
g�2nx�2n � +
1
2nDg�2nx,2ny�,t = P,�A�kx + y�
−g�2n�kx + y��
2n � + �A�kx − y� −g�2n�kx − y��
2n � − k�A�x + y�
−g�2n�x + y��
2n � − k�A�x − y� −g�2n�x − y��
2n � − 2�A�kx� −g�2nkx�
2n �+ 2k�A�x� −
g�2nx�2n � +
1
2n �Df�2n+1x,2n+1y� − 8Df�2nx,2ny��,t�L�M7�P,A�kx + y� −
g�2n�kx + y��2n ,
t
8,P,A�kx − y�
−g�2n�kx − y��
2n ,t
8,P,A�x + y� −
g�2n�x + y��2n ,
t
8k,P,A�x − y�
063519-11 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
−g�2n�x − y��
2n ,t
8k,P,A�kx� −
g�2nkx�2n ,
t
16,P,A�x�
−g�2nx�
2n ,t
16k,P, 1
2n+1Df�2n+1x,2n+1y�,t
16,P, 1
2nDf�2nx,2ny�,t
64�
�L�M7�P,A�kx + y� −g�2n�kx + y��
2n ,t
8,P,A�kx − y�
−g�2n�kx − y��
2n ,t
8,P,A�x + y� −
g�2n�x + y��2n ,
t
8k,P,A�x − y�
−g�2n�x − y��
2n ,t
8k,P,A�kx� −
g�2nkx�2n ,
t
16,P,A�x�
−g�2nx�
2n ,t
16k,P,� � �2n+1x,2n+1y�,2n−3t�,P,� � �2nx,2ny�,2n−6t��
for all x ,y�X and t�0. The first six terms on the right hand side of the above inequality tend to1L� as n→� by �3.34� and the seventh and eighth terms tend to 1L� as n→� by �3.2�. Therefore,P,�DA�x ,y� , t�=1L� for all t�0, we conclude that A fulfills �1.2�, and so by Lemma 3.1, we seethat the mapping x→A�2x�−8A�x� is additive. Equation �3.37� implies that the mapping A isadditive.
To prove the uniqueness of the mapping A subject to �3.4�, assume that there exists anotheradditive mapping T :X→Y which satisfies �3.4�. Fix x�X. Clearly A�2nx�=2nA�x� and T�2nx�=2nT�x� for all n�N. It follows from �3.4� that
P,�A�x� − T�x�,t� = P,A�2nx�2n −
T�2nx�2n ,t�L�M�P,A�2nx�
2n −f�2n+1x�
2n
+ 8f�2nx�
2n ,t
2,P, f�2n+1x�
2n − 8f�2nx�
2n −T�2nx�
2n ,t
2�
�L�P,� x,�2 − ��2nt
4�n for each x�X, t�0, and n�N. Since 0���2, limn→��2 /��n=�. Thus the right hand side of theabove inequality tends to 1L� as n→�. Therefore, P,�A�x�−T�x� , t�=1L� for all t�0, whenceA�x�=T�x�. This completes the proof. �
Corollary 3.3: Let �X , � · �X� be a normed space, r ,s be non-negative real numbers such that�ªr+s�1, and let z0�Z. Suppose that a mapping f :X→Y with f�0�=0 and such that
P,�Df�x,y�,t��L�P,� ���x�Xr �y�X
s + ��x�Xr+s + �y�X
r+s��z0,t�
for all x ,y�X and t�0. Then there exists a unique additive mapping A :X→Y, such that
P,�f�2x� − 8f�x� − A�x�,t��L�P,� �x�X�z0,
�2 − 2��t768k2�3k��
for all x�X and t�0.Proof: Let :X�X→Z be defined by �x ,y�= ��x�X
r �y�Xs + ��x�X
r+s+ �y�Xr+s��z0 for all x ,y�X.
Then the corollary is followed from Theorem 3.2 by �=2�. �
One can prove a similar result for the case ��2.Theorem 3.4: Let ��2 and :X�X→Z be a mapping with the following property:
P,� x
2,y
2,t�L�P,� � �x,y�,�t� �3.38�
063519-12 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
and
limn→�
P,� �2n �2−nx,2−ny�,t� = 1L� �3.39�
for all x ,y�X and t�0. Let f :X→Y be a mapping with f�0�=0 and such that
P,�Df�x,y�,t��L�P,� � �x,y�,t� �3.40�
for all x ,y�X and t�0. Then there exists a unique additive mapping A :X→Y, such that
P,�f�2x� − 8f�x� − A�x�,t��L�P,� x,�� − 2�t
2 �3.41�
for all x�X and t�0, where P,� �x , t� is defined as in Theorem 3.2.Corollary 3.5: Let �X , � · �X� be a normed space, r ,s be non-negative real numbers such that
�ªr+s�1, and let z0�Z. Suppose that a mapping f :X→Y with f�0�=0 and such that
P,�Df�x,y�,t��L�P,� ���x�Xr �y�X
s + ��x�Xr+s + �y�X
r+s��z0,t�
for all x ,y�X and t�0. Then there exists a unique additive mapping A :X→Y, such that
P,�f�2x� − 8f�x� − A�x�,t��L�P,� �x�X�z0,
�2� − 2�t768k2�3k��
for all x�X and t�0.Proof: Let :X�X→Z be defined by �x ,y�= ��x�X
r �y�Xs + ��x�X
r+s+ �y�Xr+s��z0 for all x ,y�X.
Then the corollary is followed from Theorem 3.4 by �=2�. �
Theorem 3.6: Let 0���8 and :X�X→Z be a mapping with the following property:
P,� � �2x,2y�,t��L�P,� �� �x,y�,t� �3.42�
and
limn→�
P,� � �2nx,2ny�,8nt� = 1L� �3.43�
for all x ,y�X and t�0. Let f :X→Y be a mapping with f�0�=0 and such that
P,�Df�x,y�,t��L�P,� � �x,y�,t� �3.44�
for all x ,y�X and t�0. Then there exists a unique cubic mapping C :X→Y, such that
P,�f�2x� − 2f�x� − C�x�,t��L�P,� x,�8 − ��t
2 �3.45�
for all x�X and t�0, where P,� �x , t� is defined as in Theorem 3.2.Proof: As in the proof of Theorem 3.2, we have
P,�f�4x� − 10f�2x� + 16f�x�,t��L�P,� �x,t� �3.46�
for all x�X and t�0, where P,� �x , t� is defined as in Theorem 3.2. Let h :X→Y be the mappingdefined by
h�x� = f�2x� − 2f�x�
for all x�X. By �3.46�, we have
063519-13 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,�h�2x� − 8h�x�,t��L�P,� �x,t� �3.47�
for all x�X and t�0. Replacing x by 2nx in �3.47� and using �3.42�, we get
P,h�2n+1x�8n+1 −
h�2nx�8n ,
t
8n+1 = P,�h�2n+1x� − 8h�2nx�,t��L�P,� �2nx,t��L�P,� x,t
�n�3.48�
for all x�X , t�0 and all non-negative integers n. Hence
P,h�2n+1x�8n+1 −
h�2nx�8n ,
�nt
8n+1�L�P,� �x,t� �3.49�
for all x�X , t�0 and all non-negative integers n. For all x�X , t�0 and all non-negative integersn and m with n�m, we have
P,h�2nx�8n −
h�2mx�8m ,
k=m
n−1�kt
8k+1 = P, k=m
n−1 �h�2k+1x�8k+1 −
h�2kx�8k �,
k=m
n−1�kt
8k+1�L�Mn−m−1P,h�2m+1x�
8m+1 −h�2mx�
8m ,�mt
8m+1, . . . ,P,h�2nx�8n
−h�2n−1x�
8n−1 ,�n−1t
8n �L�P,� �x,t� . �3.50�
Hence
P,h�2nx�8n −
h�2mx�8m ,t�L�P,� �x,
t
k=mn−1 �k
8k+1 � �3.51�
for all x�X , t�0 and m ,n�N with n�m.Since 0���8 and k=0
� � �8
�k��, then �h�2nx� /8n� is a Cauchy sequence in �Y ,P, ,M� for
each x�X. Since �Y ,P, ,M� is an intuitionistic fuzzy Banach space this sequence convergent tosome point C�x��Y. So we can define the mapping C :X→Y by
C�x� ª limn→�
h�2nx�8n �3.52�
for all x�X. Fix x�X and put m=0 in �3.51� to obtain
P,h�2nx�2n − h�x�,t�L�P,� �x,
t
k=0n−1 �k
8k+1 � �3.53�
for all t�0 and so we have that
063519-14 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,�h�x� − C�x�,t� = P,h�x� −h�2nx�
8n +h�2nx�
8n − C�x�,t�L�MP,h�x� −
h�2nx�8n ,
t
2,P,h�2nx�
8n − C�x�,t
2
�L�M�P,� �x,t
2 k=0n−1 �k
8k+1 �,P,h�2nx�8n − C�x�,
t
2� . �3.54�
Taking the limit as n→� in �3.54� and using �3.52�, we get
P,�f�2x� − 2f�x� − C�x�,t��L�P,� x,�8 − ��t
2
for all x�X and t�0, which shows that C satisfies �3.45�.Now, we will show that the mapping C is cubic. It follows from �3.52� that
limn→�
P,C�2x� −h�2n+1x�
8n ,t = 1L�, limn→�
P,C�x� −h�2n+1x�
8n+1 ,t = 1L�
for all x�X and t�0. Therefore,
P,�C�2x� − 8C�x�,t� = P,C�2x� −h�2n+1x�
8n +h�2n+1x�
8n − 8C�x�,t�L�MP,C�2x� −
h�2n+1x�8n ,
t
2,P,h�2n+1x�
8n+1 − C�x�,t
16
and so we have that
C�2x� = 8C�x� �3.55�
for all x�X. Replacing x ,y by 2nx ,2ny in �3.44� to get
P, 1
8nDf�2nx,2ny�,t�L�P,� � �2nx,2ny�,8nt�
for all x, y�X, and t�0. Clearly Dh�x ,y�=Df�2x ,2y�−2Df�x ,y� for all x ,y�X. Therefore,
P,�DC�x,y�,t� = P,�C�kx + y� + C�kx − y� − kC�x + y� − kC�x − y� − 2C�kx� + 2kC�x�,t�
= P,�C�kx + y� −h�2n�kx + y��
8n � + �C�kx − y� −h�2n�kx − y��
8n � − k�C�x
+ y� −h�2n�x + y��
8n � − k�C�x − y� −h�2n�x − y��
8n � − 2�C�kx� −h�2nkx�
8n �+ 2k�C�x� −
h�2nx�8n � +
1
8nDh�2nx,2ny�,t = P,�C�kx + y�
−h�2n�kx + y��
8n � + �C�kx − y� −h�2n�kx − y��
8n � − k�C�x + y�
−h�2n�x + y��
8n � − k�C�x − y� −h�2n�x − y��
8n � − 2�C�kx� −h�2nkx�
8n �
063519-15 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
+ 2k�C�x� −h�2nx�
8n � +1
8n �Df�2n+1x,2n+1y� − 2Df�2nx,2ny��,t�L�M7�P,C�kx + y� −
h�2n�kx + y��8n ,
t
8,P,C�kx − y�
−h�2n�kx − y��
8n ,t
8,P,C�x + y� −
h�2n�x + y��8n ,
t
8k,P,C�x − y�
−h�2n�x − y��
8n ,t
8k,P,C�kx� −
h�2nkx�8n ,
t
16,P,C�x�
−h�2nx�
8n ,t
16k,P, 1
8n+1Df�2n+1x,2n+1y�,t
64,P, 1
8nDf�2nx,2ny�,t
16�
�L�M7�P,C�kx + y� −h�2n�kx + y��
8n ,t
8,P,C�kx − y�
−h�2n�kx − y��
8n ,t
8,P,C�x + y� −
h�2n�x + y��8n ,
t
8k,P,C�x − y�
−h�2n�x − y��
8n ,t
8k,P,C�kx� −
h�2nkx�8n ,
t
16,P,C�x�
−h�2nx�
8n ,t
16k,P,� �2n+1x,2n+1y�,
8n+1t
64,P,� �2nx,2ny�,
8nt
16�
for all x, y�X, and t�0. The first six terms on the right hand side of the above inequality tend to1L� as n→� by �3.52� and the seventh and eighth terms tend to 1L� as n→� by �3.43�. Therefore,P,�DC�x ,y� , t�=1L� for all t�0, we conclude that C fulfills �1.2�, and so by Lemma 3.1, we seethat the mapping x→C�2x�−2C�x� is cubic. Equation �3.55� implies that the mapping C is cubic.
To prove the uniqueness of the mapping C subject to �3.45�, assume that there exists anothercubic mapping S :X→Y which satisfies �3.45�. Fix x�X. Clearly C�2nx�=8nA�x� and T�2nx�=8nT�x� for all n�N. It follows from �3.45� that
P,�C�x� − S�x�,t� = P,C�2nx�8n −
S�2nx�8n ,t�L�M�P,C�2nx�
8n −f�2n+1x�
8n
+ 2f�2nx�
8n ,t
2,P, f�2n+1x�
8n − 2f�2nx�
8n −S�2nx�
8n ,t
2�
�L�P,� x,�8 − ��8nt
4�n for each x�X, t�0, and n�N. Since 0���8, limn→��8 /��n=�. Thus, the right hand side ofthe above inequality tends to 1L� as n→�. Therefore, P,�C�x�−S�x� , t�=1L� for all t�0, whenceC�x�=S�x�. This completes the proof. �
Corollary 3.7: Let �X , � · �X� be a normed space, r ,s be non-negative real numbers such that�ªr+s�3, and let z0�Z. Suppose that a mapping f :X→Y with f�0�=0 and such that
P,�Df�x,y�,t��L�P,� ���x�Xr �y�X
s + ��x�Xr+s + �y�X
r+s��z0,t�
for all x y�X and t�0. Then there exists a unique cubic mapping C :X→Y such that
P,�f�2x� − 2f�x� − C�x�,t��L�P,� �x�X�z0,
�8 − 2��t768k2�3k��
for all x�X and t�0.
063519-16 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
Proof: Let :X�X→Z be defined by �x ,y�= ��x�Xr �y�X
s + ��x�Xr+s+ �y�X
r+s��z0 for all x ,y�X.Then the corollary is followed from Theorem 3.6 by �=2�. �
One can prove a similar result for the case ��8.Theorem 3.8: Let ��8 and :X�X→Z be a mapping with the following property:
P,� x
2,y
2,t�L�P,� � �x,y�,�t�
and
limn→�
P,� �8n �2−nx,2−ny�,t� = 1L�
for all x, y�X and t�0. Let f :X→Y be a mapping with f�0�=0 and such that
P,�Df�x,y�,t��L�P,� � �x,y�,t�
for all x y�X and t�0. Then there exists a unique cubic mapping C :X→Y, such that
P,�f�2x� − 2f�x� − C�x�,t��L�P,� x,�� − 8�t
2
for all x�X and t�0, where P,� �x , t� is defined as in Theorem 3.2.Corollary 3.9: Let �X , � · �X� be a normed space, r ,s be non-negative real numbers such that
�ªr+s�3, and let z0�Z. Suppose that a mapping f :X→Y with f�0�=0 and such that
P,�Df�x,y�,t��L�P,� ���x�Xr �y�X
s + ��x�Xr+s + �y�X
r+s��z0,t�
for all x y�X and t�0. Then there exists a unique cubic mapping C :X→Y, such that
P,�f�2x� − 2f�x� − C�x�,t��L�P,� �x�X�z0,
�2� − 8�t768k2�3k��
for all x�X and t�0.Theorem 3.10: Let 0���2 and :X�X→Z be a mapping with the following property:
P,� � �2x,2y�,t��L�P,� �� �x,y�,t� �3.56�
and
limn→�
P,� � �2nx,2ny�,2nt� = 1L� �3.57�
for all x, y�X, and t�0. Suppose that a mapping f :X→Y with f�0�=0 satisfies the inequality
P,�Df�x,y�,t��L�P,� � �x,y�,t� �3.58�
for all x, y�X, and t�0. Then there exist a unique additive mapping A :X→Y and a unique cubicmapping C :X→Y, such that
P,�f�x� − A�x� − C�x�,t��L�P,� x,3�2 − ��t
2 �3.59�
for all x�X and t�0, where P,� �x , t� is defined as in Theorem 3.2.Proof: By Theorems 3.2 and 3.6, there exist an additive mapping A1 :X→Y and a cubic
mapping C1 :X→Y, such that
063519-17 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,�f�2x� − 8f�x� − A1�x�,t��L�P,� x,�2 − ��t
2 �3.60�
and
P,�f�2x� − 2f�x� − C1�x�,t��L�P,� x,�8 − ��t
2 �3.61�
for all x�X and t�0, where P,� �x , t� is defined as in Theorem 3.2. Therefore from �3.60� and�3.61� we get
P, f�x� +1
6A1�x� −
1
6C1�x�,t = P,1
6�f�2x� − 2f�x� − C1�x�� −
1
6�f�2x� − 8f�x� − A1�x��,t
�L�M�P,1
6�f�2x� − 2f�x� − C1�x��,
t
2,P,1
6�f�2x� − 8f�x�
− A1�x��,t
2� = M�P,�f�2x� − 2f�x� − C1�x�,3t�,P,�f�2x�
− 8f�x� − A1�x�,3t��
�L�M�P,� x,3�2 − ��t
2,P,� x,
3�8 − ��t2
�= P,� x,
3�2 − ��t2
�3.62�
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�3.62� that
P,�f�x� − A�x� − C�x�,t��L�P,� x,3�2 − ��t
2
for all x�X and t�0.To prove the uniqueness of A and C, let A0 ,C0 :X→Y be another additive and cubic mapping
satisfying �3.59�. Set A=A−A0 and C=C−C0. So
P,�A�x� + C�x�,t� = P,�A�x� + C�x� − f�x� + f�x� − A0�x� − C0�x�,t��L�M�P,A�x� + C�x�
− f�x�,t
2,P, f�x� − A0�x� − C0�x�,
t
2��L�P,� x,
3�2 − ��t4
�3.63�
for all x�X and t�0. Hence
P,�C�x�,t� = P,�C�x� + A�x� − A�x�,t��L�M�P,C�x� + A�x�,t
2,P,A�x�,
t
2�
�L�M�P,� x,3�2 − ��t
8,P,A�x�,
t
2� �3.64�
for all x�X and t�0. By A�2x�=2A�x�, C�2x�=8C�x�, and �3.56� and �3.64�, we get
063519-18 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
P,�C�x�,t� = P,�C�2nx�,8nt��L�M�P,� 2nx,3�2 − ��8nt
8,P,A�2nx�,
8nt
2�
�L�M�P,� x,3�2 − ��8nt
8�n ,P,A�x�,4nt
2� �3.65�
for all x�X and t�0. From 0���2 and �3.65�, we get P,�C�x� , t�=1 for all t�0 and so we
have that C=0. Therefore, it follows from �3.63� that
P,�A�x�,t��L�P,� x,3�2 − ��t
4
for all x�X and t�0. Thus, by the additivity of A and A0, we have
P,�A�x�,t� = P,�A�2nx�,2nt��L�P,� 2nx,3�2 − ��2nt
4�L�P,� x,
3�2 − ��2nt
4�n for all x�X and t�0. Since limn→�3�2−��2nt /4�n=�, by the definition of P,� �x , t�, we get
limn→�
P,� x,3�2 − ��2nt
4�n = 1L�
for all x�X. So P,�A�x� , t�=1L� for all t�0, whence A=0. This completes the proof. �
We present a result similar to Theorem 3.10 for the case where ��8.Theorem 3.11: Let ��8 and :X�X→Z be a mapping with the following property:
P,� x
2,y
2,t�L�P,� � �x,y�,�t�
and
limn→�
P,� �8n �2−nx,2−ny�,t� = 1L�
for all x, y�X, and t�0. Suppose that a mapping f :X→Y with f�0�=0 satisfies the inequality
P,�Df�x,y�,t��L�P,� � �x,y�,t�
for all x, y�X, and t�0. Then there exist a unique additive mapping A :X→Y and a unique cubicmapping C :X→Y, such that
P,�f�x� − A�x� − C�x�,t��L�P,� x,3�� − 8�t
2
for all x�X and t�0, where P,� �x , t� is defined as in Theorem 3.2.Proof: The proof is similar to the proof of Theorem 3.5 and the result follows from Theorems
3.4 and 3.8. �
Theorem 3.12: Let 2�� ,��8 and :X�X→Z be a mapping with the following property:
P,� x
2,y
2,t�L�P,� � �x,y�,�t�, P,� � �2x,2y�,t��L�P,� �� �x,y�,t�
and
limn→�
P,� �8n �2−nx,2−ny�,t� = 1L� limn→�
P,� � �2nx,2ny�,8nt� = 1L�
063519-19 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
for all x ,y�X and t�0. Suppose that a mapping f :X→Y with f�0�=0 and satisfying the inequal-ity
P,�Df�x,y�,t��L�P,� � �x,y�,t�
for all x ,y�X and t�0. Then there exist a unique additive mapping A :X→Y and a unique cubicmapping C :X→Y, such that
P,�f�x� − A�x� − C�x�,t��L�M�P,� x,3�� − 2�t
2,P,� x,
3�8 − ��t2
�for all x�X and t�0, where P,� �x , t� is defined as in Theorem 3.2.
Proof: The proof is similar to the proof of Theorem 3.10 and the result follows from Theorems3.4 and 3.8. �
Corollary 3.13: Let �X , � · �X� be a normed space, r ,s be non-negative real numbers such that�ªr+s� �0,1�� �1,3�� �3,��, and let z0�Z. Suppose that a mapping f :X→Y with f�0�=0and such that
P,�Df�x,y�,t��L�P,� ���x�Xr �y�X
s + ��x�Xr+s + �y�X
r+s��z0,t�
for all x ,y�X and t�0. Then there exist a unique additive mapping A :X→Y and a unique cubicmapping C :X→Y, such that
P,�f�x� − A�x� − C�x�,t��L��P,� �x�X
�z0,�2 − 2��t
256k2�3k�� , � � �0,1�
P,� �x�X�z0,
�2� − 2�t256k2�3k�� , � � �1,log2 5�
P,� �x�X�z0,
�8 − 2��t256k2�3k�� , � � �log2 5,3�
P,� �x�X�z0,
�2� − 8�t256k2�3k�� , � � �3,�� ,
�for all x�X and t�0.
Proof: Define �x ,y�= ��x�Xr �y�X
s + ��x�Xr+s+ �y�X
r+s��z0 for all x ,y�X, and apply Theorems 3.10–3.12. �
ACKNOWLEDGMENTS
The authors would like to thank the referees for giving useful suggestions for the improve-ment of this paper. The first author was supported by the National Natural Science Foundation ofChina �Grant No. 10671013�
1 Aoki, T., “On the stability of the linear transformation in Banach spaces,” J. Math. Soc. Jpn. 2, 64 �1950�.2 Atanassov, K. T., “Intuitionistic fuzzy sets,” Fuzzy Sets Syst. 20, 87 �1986�.3 Brzdek, J., “On a method of proving the Hyers-Ulam stability of functional equations on restricted domains,” Aust. J.Math. Anal. Appl. 6, 1 �2009�.
4 Ciepliński, K., “Stability of the multi-Jensen equation,” J. Math. Anal. Appl. 363, 249 �2010�.5 Hyers, D. H., “On the stability of the linear functional equation,” Proc. Natl. Acad. Sci. U.S.A. 27, 222 �1941�.6 Mirmostafaee, A. K., Mirzavaziri, M., and Moslehian, M. S., “Fuzzy stability of the Jensen functional equation,” FuzzySets Syst. 159, 730 �2008�.
7 Mirmostafaee, A. K. and Moslehian, M. S., “Fuzzy versions of Hyers-Ulam-Rassias theorem,” Fuzzy Sets Syst. 159, 720�2008�.
8 Mirmostafaee, A. K. and Moslehian, M. S., “Fuzzy approximately cubic mappings,” J. Inf. Sci. 178, 3791 �2008�.9 Mohiuddine, S. A., “Stability of Jensen functional equation in intuitionistic fuzzy normed space,” Chaos, SolitonsFractals 42, 2989 �2009�.
10 Moszner, Z., “On the stability of functional equations,” Aequ. Math. 77, 33 �2009�.11 Mursaleen, M. and Mohiuddine, S. A., “On stability of a cubic functional equation in intuitionistic fuzzy normed spaces,”
063519-20 Xu, Rassias, and Xu J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52
Chaos, Solitons Fractals 42, 2997 �2009�.12 Najati, A. and Eskandani, G. Z., “Stability of a mixed additive and cubic functional equation in quasi-Banach spaces,”
J. Math. Anal. Appl. 342, 1318 �2008�.13 EI Naschie, M. S., “A review of applications and results of E-infinity theory,” Int. J. Nonlinear Sci. Numer. Simul. 8, 11
�2007�.14 EI Naschie, M. S., “Fuzzy dodecahedron topology and E-infinity space-time as a model for quantum physics,” Chaos,
Solitons Fractals 30, 1025 �2006�.15 EI Naschie, M. S., “On a fuzzy Kähler-like manifold which is consistent with two-slit experiment,” Int. J. Nonlinear Sci.
Numer. Simul. 6, 95 �2005�.16 EI Naschie, M. S., “On two new fuzzy Kähler manifolds, Klein modular space and Hooft holographic principles,” Chaos,
Solitons Fractals 29, 876 �2006�.17 EI Naschie, M. S., “The idealized quantum two-slit Gedanken experiment revisited-criticism and reinterpretation,”
Chaos, Solitons Fractals 27, 9 �2006�.18 Nozari, K. and Fazlpour, B., “Some consequences of space-time fuzziness,” Chaos, Solitons Fractals 34, 224 �2007�.19 Paneah, B., “A new approach to the stability of linear functional operators,” Aequ. Math. 78, 45 �2009�.20 Park, J. H., “Intuitionistic fuzzy metric spaces,” Chaos, Solitons Fractals 22, 1039 �2004�.21 Rassias, Th. M., “On the stability of the linear mapping in Banach spaces,” Proc. Am. Math. Soc. 72, 297 �1978�.22 Rassias, J. M., “On approximation of approximately linear mappings by linear mappings,” J. Funct. Anal. 46, 126 �1982�.23 Rassias, J. M., “On the Heisenberg-Pauli-Weyl inequality,” J. Inequ. Pure and Appl. Math. 5, 1 �2004�.24 Rassias, J. M., “On the Heisenberg-Weyl inequality,” J. Inequ. Pure and Appl. Math. 6, 8 �2005�.25 Rassias, J. M., “On the refined Heisenberg-Weyl type inequality,” J. Inequ. Pure and Appl. Math. 6, 11 �2005�.26 Rassias, J. M., “On the sharpened Heisenberg-Weyl type inequality,” J. Inequ. Pure and Appl. Math. 7, 1 �2006�.27 Rassias, J. M., “Solution of the Ulam stability problem for cubic mapping,” Glas. Math. 36, 63 �2001�.28 Rassias, J. M., “Solution of a problem of Ulam,” J. Approx. Theory 57, 268 �1989�.29 Rassias, J. M. and Rassias, M. J., “Asymptotic behavior of alternative Jensen and Jensen type functional equations,”
Bull. Sci. Math. 129, 545 �2005�.30 Ravi, K., Arunkumar, M., and Rassias, J. M., “Ulam stability for the orthogonally general Euler-Lagrange type functional
equation,” Intern. J. Math. Stat. 3, 36 �2008�.31 Saadati, R. and Park, J. H., “On the intuitionistic fuzzy topological spaces,” Chaos, Solitons Fractals 27, 331 �2006�.32 Saadati, R. and Park, J. H., “Intuitionistic fuzzy Euclidean normed spaces,” Commun. Math. Anal. 1, 85 �2006�.33 Saadati, R., “A note on “Some results on the IF-normed spaces”,” Chaos, Solitons Fractals 41, 206 �2009�.34 Shakeri, S., “Intuitionnistic fuzzy stability of Jensen type mapping,” J. Nonlinear Sci. 2, 105 �2009�.35 Ulam, S. M., A Collection of the Mathematical Problems �Interscience, New York, 1960�.36 Xu, T. Z., Rassias, J. M., and Xu, W. X. �unpublished�.
063519-21 Stability of mixed additive-cubic equation J. Math. Phys. 51, 063519 �2010�
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:
193.61.135.80 On: Fri, 19 Dec 2014 11:55:52