Abstract and Applied AnalysisVolume 2011 (2011), Article ID 456182, 19 pagesdoi:10.1155/2011/456182

Research Article

Intuitionistic Fuzzy Stability of Functional Equations Associated with Inner Product Spaces

Zhihua Wang¹ and Themistocles M. Rassias²

¹School of Science, Hubei University of Technology, Wuhan, Hubei 430068, China²Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

Received 2 September 2011; Accepted 29 September 2011

Academic Editor: Gabriel Turinici

Copyright © 2011 Zhihua Wang and Themistocles M. Rassias. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In intuitionistic fuzzy normed spaces, we investigate some stability results for the functional equation which is said to be a functional equation associated with inner products space.

1. Introduction and Preliminaries

The aim of this article is to prove an intuitionistic fuzzy version of the Hyers-Ulam-Rassias stability for the functional equation: which is said to be a functional equation associated with inner product spaces. It was shown by Rassias [1] that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer it follows for all . Interesting new results concerning functional equations associated with inner product spaces have recently been obtained by Park et al. [2, 3] and Najati and Rassias [4] as well as for the fuzzy stability of a functional equation associated with inner product spaces [5].

Stability problem of a functional equation was first posed by Ulam [6] which was answered by Hyers [7] on approximately additive mappings and then generalized by Aoki [8] and Rassias [9] for additive mappings and linear mappings, respectively. Later there have been proved several new results on stability of various classes of functional equations in the Hyers-Ulam sense (cf. the following books and papers [10–18] and the references cited therein), as well as various fuzzy stability results concerning Cauchy, Jensen, quadratic and cubic functional equations (cf. [19–22]). Furthermore some stability results concerning Jensen, cubic, mixed-type additive and cubic functional equations were investigated (cf. [23–26]) in the spirit of intuitionistic fuzzy normed spaces, while the idea of intuitionistic fuzzy normed space was introduced in [27] and further studied in [28–35].

In this section, we recall some notations and basic definitions used in this paper as follows.

Definition 1.1 (cf. [36]). A binary operation is said to be a continuous -norm if it satisfies the following conditions:(a) is commutative and associative, (b) is continuous,(c) for all , (d) whenever and for each .

Definition 1.2 (cf. [36]). A binary operation is said to be a continuous -conorm if it satisfies the following conditions:(a’) is commutative and associative, (b’) is continuous,(c’) for all , (d’) whenever and for each . Using the notions of continuous -norm and continuous -conorm, Saadati and Park [27] have recently introduced the concept of intuitionistic fuzzy normed spaces as follows.

Definition 1.3. The five-tuple is said to be an intuitionistic fuzzy normed space (for short, IFNS) if is a vector space, is a continuous -norm, is a continuous -conorm, and are fuzzy sets on satisfying the following conditions. For every and ,(i), (ii), (iii) if and only if ,(iv) for each , (v), (vi) is continuous, (vii) and ,(viii), (ix) if and only if , (x) for each , (xi) , (xii) is continuous, (xiii) and . In this case is called an intuitionistic fuzzy norm.

Example 1.4 (cf. [37]). Letbe a normed space, andfor all . For all and everyand, consider Then is an IFNS.

The concepts of convergence and Cauchy sequences in an intuitionistic fuzzy normed space are studied in [27].

Let be an IFNS. Then, a sequence is said to be intuitionistic fuzzy convergent to if, for every and , there exists such that and for all . In this case we write . The sequence is said to be intuitionistic fuzzy Cauchy sequence if, for every and , there exists such that and for all . is said to be complete if every intuitionistic fuzzy Cauchy sequence in is intuitionistic fuzzy convergent in .

2. Intuitionistic Fuzzy Stability

Throughout this section, assume that , , and are linear space, IFNS, and intuitionistic fuzzy Banach space, respectively. For convenience, we use the following abbreviation for a given function :

We begin with the Hyers-Ulam-Rassias type theorem in IFNS for the functional (1.1) which is said to be a functional equation associated with inner product spaces.

Theorem 2.1. Let be a function such that for some real number with . Suppose that an even function with satisfies the inequality for all and all . Then there exists a unique quadratic function such that for all and , where

Proof. Put in (2.2), and, using the evenness of , we obtain for all and . Interchanging with in (2.5) and using the evenness of , we get for all and . It follows from (2.5) and (2.6) that for all and . Putting in (2.2) and using the evenness of , we obtain for all and . Hence, we obtain from (2.7) and (2.8) that for all and . So for all and . Putting in (2.2), we obtain for all and . It follows from (2.10) and (2.11) that for all and . Letting in (2.8) and replacing by in the obtained inequality, we get for all and . It follows from (2.10), (2.11), (2.12), and (2.13) that for all and . Applying (2.12) and (2.14), we obtain for all and . Setting in (2.2), we obtain for all and . It follows from (2.15) and (2.16) that It follows that Define Then, by our assumption, Replacing by in (2.18) and applying (2.20), we get Thus for each , we have where . Let and be given. Since and , there exists some such that and . Since , there is some such that for each . It follows that for all . This shows that the sequence is Cauchy in . Since is intuitionistic fuzzy Banach space, converges to some point . Thus, we can define a mapping such that . Moreover, if we put in (2.22), we get Thus, Now, we will show that is quadratic. Setting and in (2.2), we obtain for all and . Letting in (2.26), we obtain for all and all . This means that satisfies the functional (1.1) and so it is quadratic (see Lemma 2.2 of [4]).Next, we approximate the difference between and in intuitionistic fuzzy sense. By (2.25), we have for every and large enough . To prove the uniqueness of , assume that is another quadratic mapping from to , which satisfies the required inequality. Then, for each and , Since and are quadratic, we have for all and . Since and , we Therefore and for all and . Hence, for all . This completes the proof of the theorem.

Theorem 2.2. Let be a function such that for some real number with . Suppose that an odd function satisfies the inequality for all and all . Then there exists a unique additive function such that for all and , where

Proof. Put in (2.32) and using the oddness of , we obtain for all and . Interchanging with in (2.35) and using the oddness of , we get for all and . It follows from (2.35) and (2.36) that for all and . Setting in (2.32) and using the oddness of , we get for all and . It follows from (2.37) and (2.38) that for all and . Putting in (2.32), we obtain for all and . It follows from (2.39) and (2.40) that for all and . Replacing and by and in (2.41); respectively, we have It follows that Define Then by the assumption Replacing by in (2.43) and using (2.45), we obtain Thus, for each , we have where , . Let and be given. Since and , there exists some such that and . Since , there is some such that for each . It follows that for all . This shows that the sequence is Cauchy in . Since is intuitionistic fuzzy Banach space, converges to some point . Thus, we can define a mapping such that . Moreover, if we put in (2.47), we get Thus, Next we will show that is additive. Putting and in (2.32), we obtain for all and . Letting in (2.51), we obtain for all and all . This means that satisfies the functional (1.1), and so it is additive (see Lemma  2.1 of [4]).Now, we approximate the difference between and in intuitionistic fuzzy sense. For every , and sufficiently large , by (2.50), we have To prove the uniqueness of , assume that is another additive mapping from to , which satisfies the required inequality. Then, for each and , Therefore, by the additivity of and , we have for all and . Since and , we get Therefore, and for all and . Hence, for all . This completes the proof of the theorem.

Theorem 2.3. Let be a function such that for some real number with . Suppose that a function with satisfies the inequality for all and all . Then there exists a unique quadratic function and a unique additive function such that for all and , where

Proof. Passing to the even part and odd part of , we deduce from (2.57) that On the other hand, Applying the proofs of Theorems 2.1 and 2.2, we get a unique quadratic function and a unique additive function satisfying Also, Therefore, This completes the proof of the theorem.

Acknowledgment

The authors are very grateful to the referees for their helpful comments and suggestions.

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