Abstract and Applied AnalysisVolume 2012 (2012), Article ID 953938, 22 pagesdoi:10.1155/2012/953938

Research Article

Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces

M. Eshaghi Gordji,¹ H. Azadi Kenary,² H. Rezaei,² Y. W. Lee,³ and G. H. Kim⁴

¹Department of Mathematics, Semnan University, Semnan 35131-19111, Iran²Department of Mathematics, Yasouj University, Yasouj 75918-74831, Iran³Department of Computer Hacking and Information Security, Daejeon University, Youngwoondong Donggu, Daejeon 300-716, Republic of Korea⁴Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 19 November 2011; Revised 21 January 2012; Accepted 13 February 2012

Academic Editor: Gerd Teschke

Copyright © 2012 M. Eshaghi Gordji et al. 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.

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equation for fixed integers with in fuzzy Banach spaces.

1. Introduction and Preliminaries

The stability problem of functional equations was originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let be a group and let be a metric group with the metric . Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that for all , and for some . Then there exists a unique additive mapping such that for all . Moreover if is continuous in for each fixed , then is linear. In 1978, Rassias [3] provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. In 1991, Gajda [4] answered the question for the case , which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [5–17]).

The functional equation is related to a symmetric biadditive function. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.3) is said to be a quadratic function. It is well known that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all (see [6, 18]). The biadditive function is given by A Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof for functions , where is normed space and Banach space (see [19–22]). Borelli and Forti [23] generalized the stability result of quadratic functional equations as follows (cf. [24, 25]): let be an Abelian group, and a Banach space. Assume that a mapping satisfies the functional inequality: for all , and is a function such that for all . Then there exists a unique quadratic mapping with the property for all .

Now, we introduce the following functional equation for fixed integers with : with in a non-Archimedean space. It is easy to see that the function is a solution of the functional equation (1.8), which explains why it is called additive-quadratic functional equation. For more detailed definitions of mixed type functional equations, we can refer to [26–47].

Definition 1.1 (see [48]). Let be a real vector space. A function is called a fuzzy norm on if for all and all , (N1) for ; (N2) if and only if for all ; (N3) if ; (N4);(N5) is a nondecreasing function of and ; (N6)for , is continuous on .

The pair is called a fuzzy normed vector space.

Example 1.2. Let be a normed linear space and . Then is a fuzzy norm on .

Definition 1.3. 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 in and one denotes it by .

Definition 1.4. 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 , one has .

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.

Example 1.5. Let be a fuzzy norm on defined by The is a fuzzy Banach space. Let be a Cauchy sequence in , , and . Then there exist such that for all and all , one has So for all and all . Therefore is a Cauchy sequence in . Let as . Then for all .

We say that a mapping between fuzzy normed vector spaces and is continuous at a point if for each sequence converging to , the sequence converges to . If is continuous at each , then is said to be continuous on ([49]).

Definition 1.6. Let be a set. A function is called a generalized metric on if satisfies the following conditions:(1) if and only if for all ; (2) for all ; (3) for all .

Theorem 1.7. Let (X,d) be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then, for all , either for all nonnegative integers , or there exists a positive integer such that (1) for all ;(2)the sequence converges to a fixed point of ;(3) is the unique fixed point of in the set ;(4) for all .

We have the following theorem from [42], which investigates the solution of (1.8).

Theorem 1.8. A function with satisfies (1.8) for all if and only if there exist functions and , such that for all , where the function is symmetric biadditive and is additive.

2. A Fixed Point Method

Using the fixed point methods, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.8) in fuzzy Banach spaces. Throughout this paper, assume that is a vector space and that is a fuzzy Banach space.

Theorem 2.1. Let be a mapping such that there exists an with for all . Let be an odd function satisfying and for all and all . Then exists for all and defines a unique additive mapping such that for all and .

Proof. Note that and for all since is an odd function. Putting in (2.2), we get for all and all . Replacing by in (2.4), we have for all and all . Consider the set and introduce the generalized metric on : where, as usual, . It is easy to show that is complete (see [50]). We consider the mapping as follows: for all . Let be given such that . Then for all and all . Hence for all and all . So implies that . This means that for all . It follows from (2.5) that By Theorem 1.7, there exists a mapping satisfying the following.(1) is a fixed point of , that is, for all . The mapping is a unique fixed point of in the set . This implies that is a unique mapping satisfying (2.11) such that there exists a satisfying for all .(2) as . This implies the equality , for all .(3) , which implies the inequality This implies that the inequality (2.3) holds.It follows from (2.1) and (2.2) that for all , all , and all . So for all , all , and all . Since for all and all , we obtain that for all and all . Hence the mapping is additive, as desired.

Corollary 2.2. Let and let be a real positive number with . Let be a normed vector space with norm . Let be an odd mapping satisfying for all and all . Then the limit exists for each and defines a unique additive mapping such that for all and all .

Proof. The proof follows from Theorem 2.1 by taking for all . Then we can choose and we get the desired result.

Theorem 2.3. Let be a mapping such that there exists an with for all . Let be an odd mapping satisfying and (2.2). Then the limit exists for all and defines a unique additive mapping such that for all and all .

Proof. Let be the generalized metric space defined as in the proof of Theorem 2.1.Consider the mapping by for all . Let be given such that . Then for all and all . Hence for all and all . So implies that . This means that for all . It follows from (2.5) that for all and all . Therefore So . By Theorem 1.7, there exists a mapping satisfying the following.(1) is a fixed point of , that is, for all . The mapping is a unique fixed point of in the set . This implies that is a unique mapping satisfying (2.26) such that there exists satisfying for all and .(2) as . This implies the equality for all .(3) with , which implies the inequality This implies that the inequality (2.20) holds.The rest of proof is similar to the proof of Theorem 2.1.

Corollary 2.4. Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.17). Then exists for each and defines a unique additive mapping such that for all and all .

Proof. The proof follows from Theorem 2.3 by taking for all . Then we can choose and we get the desired result.

Theorem 2.5. Let be a function such that there exists an with for all . Let be an even mapping with and satisfying (2.2). Then exists for all and defines a unique quadratic mapping such that for all and all .

Proof. Replacing by in (2.2), we get for all and all . Putting and replacing by in (2.32), we have for all and all . By (2.33), , and , we get for all and all . Consider the set and introduce the generalized metric on : where, as usual, . It is easy to show that is complete (see [50]). Now we consider the linear mapping such that for all . Proceeding as in the proof of Theorem 2.1, we obtain that implies that . This means that for all . It follows from (2.34) that By Theorem 1.7, there exists a mapping such that one has the folowing.(1) is a fixed point of , that is, for all . The mapping is a unique fixed point of in the set . This implies that is a unique mapping satisfying (2.38) such that there exists a satisfying for all .(2) as . This implies the equality for all .(3), which implies the inequality . This implies that the inequality (2.31) holds.The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.6. Let and let be a real positive number with . Let be a normed vector space with norm . Let be an even mapping with and satisfying (2.17). Then the limit exists for each and defines a unique quadratic mapping such that for all and all .

Proof. The proof follows from Theorem 2.5 by taking for all . Then we can choose and we get the desired result.

Theorem 2.7. Let be a function such that there exists an with for all . Let be an even mapping with and satisfying (2.2). Then the limit exists for all and defines a unique quadratic mapping such that for all and .

Proof. Let be the generalized metric space defined as in the proof of Theorem 2.5. It follows from (2.34) that for all and . So The rest of the proof is similar to the proofs of Theorems 2.1 and 2.3.

Corollary 2.8. Let and let be a real number with . Let be a normed vector space with norm . Let be an even mapping with and satisfying (2.17). Then exists for each and defines a unique quadratic mapping such that for all and all .

Proof. It follows from Theorem 2.7 by taking for all . Then we can choose and we get the desired result.

3. Direct Method

In this section, using direct method, we prove the Hyers-Ulam stability of functional equation (1.8) in fuzzy Banach spaces. Throughout this section, we assume that is a linear space, is a fuzzy Banach space, and is a fuzzy normed space. Moreover, we assume that is a left continuous function on .

Theorem 3.1. Assume that a mapping is an odd mapping with satisfying the inequality for all , , and is a mapping for which there is a constant satisfying such that for all and all . Then there exists a unique additive mapping satisfying (1.8) and the inequality for all and all .

Proof. It follows from (3.2) that for all and all . Putting in (3.1) and then replacing by , we get for all and all . Replacing by in (3.5), we have for all , all , and all integer . So which yields for all , , and all integers , . So for all , , and any integers , . Hence one can obtain for all , , and any integers , . Since the series is a convergent series, we see by taking the limit in the last inequality that the sequence is a Cauchy sequence in the fuzzy Banach space and so it converges in . Therefore a mapping defined by is well defined for all . This means that for all and all . In addition, it follows from (3.10) that for all and all . So for sufficiently large and for all , , and with . Since is arbitrary and is left continuous, we obtain for all and . It follows from (3.1) that for all and all . Therefore, we obtain in view of (3.11) for all and all , which implies that Hence the mapping is additive, as desired.To prove the uniqueness, let there be another mapping which satisfies the inequality (3.3). Since for all , we have for all . Therefore for all . This completes the proof.

Corollary 3.2. Let be a normed space and let be a fuzzy Banach space. Assume that there exist real numbers and such that an odd mapping with satisfies the following inequality: for all and . Then there is a unique additive mapping satisfying (1.8) and the inequality

Proof. Let and . Applying Theorem 3.1, we get desired results.

Theorem 3.3. Let be an odd mapping with satisfying the inequality (3.1) and let be a mapping for which there exists a constant satisfying such that for all and all . Then there exists a unique additive mapping satisfying (1.8) and the following inequality: for all and all .

Proof. It follows from (3.5) that for all and all . Replacing by in (3.41), we obtain So for all and all . Proceeding as in the proof of Theorem 3.1, we obtain that for all , all , and any integer . So The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.4. Let be a normed space and let be a fuzzy Banach space. Assume that there exist real numbers and such that an odd mapping with satisfies (3.19). Then there exists a unique additive mapping satisfying (1.8) and the inequality

Proof. Let and . Applying Theorem 3.3, we get the desired results.

Theorem 3.5. Let be an even mapping with satisfying the inequality (3.1) and let be a mapping for which there exists a constant such that and that for all and all . Then there exists a unique quadratic mapping satisfying (1.8) and the inequality for all and all .

Proof. Replacing by in (3.1), we get for all and all . Putting and replacing by in (3.31), we have for all and all . Replacing by in (3.32), we find for all and all . Also, replacing by in (3.33), we obtain So for all and all . Proceeding as in the proof of Theorem 3.1, we obtain that for all , all , and any integer . So The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.6. Let be a normed space and let be a fuzzy Banach space. Assume that there exist real numbers and such that an even mapping with satisfies the inequality (3.19). Then there exists a unique quadratic mapping satisfying (1.8) and the inequality

Proof. Let and . Applying Theorem 3.5, we get the desired results.

Theorem 3.7. Assume that an even mapping with satisfies the inequality (3.1) and is a mapping for which there is a constant satisfying such that for all and all . Then there exists a unique quadratic mapping satisfying (1.8) and the following inequality for all and all .

Proof. It follows from (3.32) that for all and all . Replacing by in (3.41), we obtain for all and all . So for all and all . So The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.8. Let be a normed space and let be a fuzzy Banach space. Assume that there exist real numbers and such that an even mapping with satisfies (3.19). Then there is a unique quadratic mapping satisfying (1.8) and the inequality for all , all .

Proof. Let and . Applying Theorem 3.7, we get the desired results.

Acknowledgment

The fifth author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2010-0010243).

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