- Research Article
- Open Access
Fuzzy Stability of Additive Functional Inequalities with the Fixed Point Alternative
- Choonkil Park^{1}Email author
https://doi.org/10.1155/2009/410576
© Choonkil Park. 2009
- Received: 12 October 2009
- Accepted: 30 November 2009
- Published: 24 January 2010
Abstract
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy additive functional inequality and of the Cauchy-Jensen additive functional inequality in fuzzy Banach spaces.
Keywords
- Additive Mapping
- Unique Fixed Point
- Functional Inequality
- Normed Vector Space
- Quadratic Functional Equation
1. Introduction and Preliminaries
Katsaras [1] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [2–4]. In particular, Bag and Samanta [5], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [7]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [8].
We use the definition of fuzzy normed spaces given in [5, 9, 10] to investigate a fuzzy version of the generalized Hyers-Ulam stability for the Cauchy additive functional inequality and for the Cauchy-Jensen additive functional inequality in the fuzzy normed vector space setting.
Definition 1.1 (see [5, 9–11]).
Let be a real vector space. A function is called a m on if for all and all ,
is a nondecreasing function of and ;
The pair is called a fuzzy normed vector space.
The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [10, 11].
Definition 1.2 (see [5, 9–11]).
Let be a fuzzy normed vector space. A sequence in is said to be convergent or converge if there exists an such that for all . In this case, is called the limit of the sequence and we denote it by - .
Definition 1.3 (see [5, 9, 10]).
Let be a fuzzy normed vector space. A sequence in is called Cauchy if for each and each there exists an such that for all and all , we have .
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping between fuzzy normed vector spaces and is continuous at a point if for each sequence converging to in , then the sequence converges to . If is continuous at each , then is said to be continuous on (see [8]).
The stability problem of functional equations originated from a question of Ulam [12] concerning the stability of group homomorphisms. Hyers [13] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [14] for additive mappings and by Th. M. Rassias [15] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [15] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by G vruta [16] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.
The functional equation
is called a quadratic mapping equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [17] for mappings , where is a normed space and is a Banach space. Cholewa [18] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [19] proved the generalized Hyers-Ulam stability of the quadratic functional equation.
In [20], Jun and Kim considered the following cubic functional equation:
which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping. In [21], Lee et al. considered the following quartic functional equation:
which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping. Quartic functional equations have been investigated in [22, 23].
Surveys of expository results on related advances both in single variables and in multivariables have been given in [24, 25]. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [26–33]).
Gilányi [34] showed that if satisfies the functional inequality
then satisfies the Jordan-von Neumann functional equation
See also [35]. Fechner [36] and Gilányi [37] proved the generalized Hyers-Ulam stability of the functional inequality (1.4). Park et al. [38] investigated the Cauchy additive functional inequality
and the Cauchy-Jensen additive functional inequality
and proved the generalized Hyers-Ulam stability of the functional inequalities (1.6) and (1.7) in Banach spaces.
Let be a set. A function is called a generalized metric on if satisfies
We recall a fundamental result in fixed point theory.
for all nonnegative integers or there exists a positive integer such that
(2)the sequence converges to a fixed point of ;
(3) is the unique fixed point of in the set ;
In 1996, Isac and Th. M. Rassias [41] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [42–47]).
The generalized Hyers-Ulam stability of different functional equations in random normed spaces and in probabilistic normed spaces has been recently studied in [48–52].
In [53], Park et al. proved the generalized Hyers-Ulam stability of the functional inequalities (1.6) and (1.7) in fuzzy Banach spaces in the spirit of Hyers, Ulam, and Th. M. Rassias.
This paper is organized as follows. In Section 2, using the fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy additive functional inequality (1.6) in fuzzy Banach spaces. In Section 3, using fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy-Jensen additive functional inequality (1.7) in fuzzy Banach spaces.
Throughout this paper, assume that is a vector space and that is a fuzzy Banach space.
2. Fuzzy Stability of the Cauchy Additive Functional Inequality
In this section, using the fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy additive functional inequality (1.6) in fuzzy Banach spaces.
Theorem 2.1.
Proof.
Consider the set
where, as usual, . It is easy to show that is complete. (See the proof of Lemma ??2.1 of [49].)
Now we consider the linear mapping such that
It follows from (2.4) that
This implies that the inequality (2.3) holds.
By (2.2),
for all and all . By [53, Lemma ?2.1], the mapping is Cauchy additive, as desired.
Corollary 2.2.
Proof.
for all . Then we can choose and we get the desired result.
Theorem 2.3.
Proof.
Let be the generalized metric space defined in the proof of Theorem 2.1.
Consider the linear mapping such that
It follows from (2.4) that
This implies that the inequality (2.24) holds.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.4.
Proof.
3. Fuzzy Stability of the Cauchy-Jensen Additive Functional Inequality
In this section, using the fixed point method, we prove the generalized Hyers-Ulam stability of the Cauchy-Jensen additive functional inequality (1.7) in fuzzy Banach spaces.
Theorem 3.1.
Proof.
Consider the set
where, as usual, . It is easy to show that is complete. (See the proof of Lemma ?2.1 of [49].)
Now we consider the linear mapping such that
It follows from (3.4) that
This implies that the inequality (3.3) holds.
The rest of proof is similar to the proof of Theorem 2.1.
Corollary 3.2.
Proof.
for all . Then we can choose and we get the desired result.
Theorem 3.3.
Proof.
Let be the generalized metric space defined in the proof of Theorem 3.1.
Consider the linear mapping such that
It follows from (3.4) that
This implies that the inequality (3.21) holds.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 3.4.
Proof.
Declarations
Acknowledgment
This work was supported by the Hanyang University in 2009.
Authors’ Affiliations
References
- Katsaras AK: Fuzzy topological vector spaces. II. Fuzzy Sets and Systems 1984,12(2):143–154. 10.1016/0165-0114(84)90034-4MathSciNetView ArticleMATHGoogle Scholar
- Felbin C: Finite-dimensional fuzzy normed linear space. Fuzzy Sets and Systems 1992,48(2):239–248. 10.1016/0165-0114(92)90338-5MathSciNetView ArticleMATHGoogle Scholar
- Krishna SV, Sarma KKM: Separation of fuzzy normed linear spaces. Fuzzy Sets and Systems 1994,63(2):207–217. 10.1016/0165-0114(94)90351-4MathSciNetView ArticleMATHGoogle Scholar
- Xiao J, Zhu X: Fuzzy normed space of operators and its completeness. Fuzzy Sets and Systems 2003,133(3):389–399. 10.1016/S0165-0114(02)00274-9MathSciNetView ArticleMATHGoogle Scholar
- Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. Journal of Fuzzy Mathematics 2003,11(3):687–705.MathSciNetMATHGoogle Scholar
- Cheng SC, Mordeson JN: Fuzzy linear operators and fuzzy normed linear spaces. Bulletin of the Calcutta Mathematical Society 1994,86(5):429–436.MathSciNetMATHGoogle Scholar
- Kramosil I, Michálek J: Fuzzy metrics and statistical metric spaces. Kybernetika 1975,11(5):336–344.MathSciNetMATHGoogle Scholar
- Bag T, Samanta SK: Fuzzy bounded linear operators. Fuzzy Sets and Systems 2005,151(3):513–547. 10.1016/j.fss.2004.05.004MathSciNetView ArticleMATHGoogle Scholar
- Mirmostafaee AK, Mirzavaziri M, Moslehian MS: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 2008,159(6):730–738. 10.1016/j.fss.2007.07.011MathSciNetView ArticleMATHGoogle Scholar
- Mirmostafaee AK, Moslehian MS: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets and Systems 2008,159(6):720–729. 10.1016/j.fss.2007.09.016MathSciNetView ArticleMATHGoogle Scholar
- Mirmostafaee AK, Moslehian MS: Fuzzy approximately cubic mappings. Information Sciences 2008,178(19):3791–3798. 10.1016/j.ins.2008.05.032MathSciNetView ArticleMATHGoogle Scholar
- Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.Google Scholar
- Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
- Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
- Gavruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar
- Skof F: Proprieta' locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar
- Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar
- Czerwik St: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1992, 62: 59–64. 10.1007/BF02941618MathSciNetView ArticleMATHGoogle Scholar
- Jun K-W, Kim H-M: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. Journal of Mathematical Analysis and Applications 2002,274(2):267–278.MathSciNetView ArticleGoogle Scholar
- Lee SH, Im SM, Hwang IS: Quartic functional equations. Journal of Mathematical Analysis and Applications 2005,307(2):387–394. 10.1016/j.jmaa.2004.12.062MathSciNetView ArticleMATHGoogle Scholar
- Chung JK, Sahoo PK: On the general solution of a quartic functional equation. Bulletin of the Korean Mathematical Society 2003,40(4):565–576.MathSciNetView ArticleMATHGoogle Scholar
- Rassias JM: Solution of the Ulam stability problem for quartic mappings. Glasnik Matematicki 1999,34(54)(2):243–252.MathSciNetMATHGoogle Scholar
- Agarwal RP, Xu B, Zhang W: Stability of functional equations in single variable. Journal of Mathematical Analysis and Applications 2003,288(2):852–869. 10.1016/j.jmaa.2003.09.032MathSciNetView ArticleMATHGoogle Scholar
- Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae 1995,50(1–2):143–190. 10.1007/BF01831117MathSciNetView ArticleMATHGoogle Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle Scholar
- Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.View ArticleMATHGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
- Park C, Cui J: Generalized stability of -ternary quadratic mappings. Abstract and Applied Analysis 2007, 2007:-6.Google Scholar
- Park C, Najati A: Homomorphisms and derivations in -algebras. Abstract and Applied Analysis 2007, 2007:-12.Google Scholar
- Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000,251(1):264–284. 10.1006/jmaa.2000.7046MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000,62(1):23–130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM, Šemrl P: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proceedings of the American Mathematical Society 1992,114(4):989–993. 10.1090/S0002-9939-1992-1059634-1MathSciNetView ArticleMATHGoogle Scholar
- Gilányi A: Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequationes Mathematicae 2001,62(3):303–309. 10.1007/PL00000156MathSciNetView ArticleGoogle Scholar
- Rätz J: On inequalities associated with the Jordan-von Neumann functional equation. Aequationes Mathematicae 2003,66(1–2):191–200. 10.1007/s00010-003-2684-8MathSciNetView ArticleMATHGoogle Scholar
- Fechner W: Stability of a functional inequality associated with the Jordan-von Neumann functional equation. Aequationes Mathematicae 2006,71(1–2):149–161. 10.1007/s00010-005-2775-9MathSciNetView ArticleMATHGoogle Scholar
- Gilányi A: On a problem by K. Nikodem. Mathematical Inequalities & Applications 2002,5(4):707–710.MathSciNetView ArticleMATHGoogle Scholar
- Park C, Cho YS, Han M-H: Functional inequalities associated with Jordan-von Neumann-type additive functional equations. Journal of Inequalities and Applications 2007, 2007:-13.Google Scholar
- Cadariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003.,4(1, article 4):Google Scholar
- Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bulletin of the American Mathematical Society 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleMATHGoogle Scholar
- Isac G, Rassias ThM: Stability of -additive mappings: applications to nonlinear analysis. International Journal of Mathematics and Mathematical Sciences 1996,19(2):219–228. 10.1155/S0161171296000324MathSciNetView ArticleMATHGoogle Scholar
- Cadariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory, Grazer Mathematische Berichte. Volume 346. Karl-Franzens-Universität Graz, Graz, Austria; 2004:43–52.Google Scholar
- Cadariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, 2008:-15.Google Scholar
- Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation. Bulletin of the Brazilian Mathematical Society 2006,37(3):361–376. 10.1007/s00574-006-0016-zMathSciNetView ArticleMATHGoogle Scholar
- Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory and Applications 2007, 2007:-15.Google Scholar
- Park C: Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, 2008:-9.Google Scholar
- Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003,4(1):91–96.MathSciNetMATHGoogle Scholar
- Alsina C: On the stability of a functional equation arising in probabilistic normed spaces. In General Inequalities, 5 (Oberwolfach, 1986), Internationale Schriftenreihe zur Numerischen Mathematik. Volume 80. Birkhäuser, Basel, Switzerland; 1987:263–271.Google Scholar
- Mihet D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. Journal of Mathematical Analysis and Applications 2008,343(1):567–572. 10.1016/j.jmaa.2008.01.100MathSciNetView ArticleMATHGoogle Scholar
- Mihet D, Saadati R, Vaezpour SM: The stability of the quartic functional equation in random normed spaces. to appear in Acta Applicandae Mathematicae to appear in Acta Applicandae MathematicaeGoogle Scholar
- Mihet D, Saadati R, Vaezpour SM: The stability of an additive functional equation in Menger probabilistic -normed spaces. Mathematica Slovaca. In press Mathematica Slovaca. In pressGoogle Scholar
- Saadati R, Vaezpour SM, Cho YJ: Erratum: a note to paper "On the stability of cubic mappings and quadratic mappings in random normed spaces". Journal of Inequalities and Applications 2009, 2009:-6.Google Scholar
- Park C, Cho J, Saadati R: Fuzzy stability of functional inequalities. preprint preprintGoogle Scholar
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