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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 235359, 14 pages
http://dx.doi.org/10.1155/2012/235359
Research Article

On the Stability of a Parametric Additive Functional Equation in Quasi-Banach Spaces

1Department of Mathematics, Payame Noor University, P.O. BOX 19395-3697, Tehran, Iran
2Department of Mathematics, Urmia University, Urmia, Iran
3Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

Received 22 January 2012; Revised 16 February 2012; Accepted 16 February 2012

Academic Editor: Narcisa C. Apreutesei

Copyright © 2012 A. Ebadian 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

We investigate the generalized Hyers-Ulam stability of the following functional equation for a fixed positive integer with in quasi-Banach spaces.

1. Introduction

It is of interest to consider the concept of stability for a functional equation arising when we replace the functional equation by an inequality which acts as a perturbation of the equation.

The first stability problem was raised by Ulam [1] during his talk at the University of Wisconsin in 1940. The stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation? If the answer is affirmative, we would say that the equation is stable.

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. Aoki [3], Bourgin [4] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [5] provided a generalization of Hyers’ theorem by proving the existence of unique linear mappings near approximate additive mappings. It was shown by Gajda [6], as well as by Rassias and Šemrl [7] that one cannot prove a stability theorem of the additive equation for a specific function. Găvruţa [8] obtained generalized result of Rassias’ theorem which allows the Cauchy difference to be controlled by a general unbounded function. Isac and Rassias [9] generalized the Hyers’ theorem by introducing a mapping subject to the conditions:, ; ,(3); .

These stability results can be applied in stochastic analysis [10], financial, and actuarial mathematics, as well as in psychology and sociology.

In 1987 Gajda and Ger [11] showed that one can get analogous stability results for subadditive multifunctions. In 1978 Gruber [12] remarked that Ulam’s problem is of particular interest in probability theory and in the case of functional equations of different types. We refer the readers to [2, 532] and references therein for more detailed results on the stability problems of various functional equations.

We recall some basic facts concerning quasi-Banach space. A quasi-norm is a real-valued function on satisfying the following. for all and if and only if . for all and all . There is a constant such that for all , . The pair is called a quasi-normed space if is a quasi-norm on . A quasi-Banach space is a complete quasi-normed space. A quasi-norm is called a -norm if for all , . In this case, a quasi-Banach space is called a -Banach space. Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz theorem (see [33]), each quasi-norm is equivalent to some -norm. Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms. In this paper, we consider the generalized Hyers-Ulam stability of the following functional equation: for a fixed positive integer with in quasi-Banach spaces.

Throughout this paper, assume that is a quasi-normed space with quasi-norm and that is a -Banach space with -norm .

2. Stability of Functional Equation (1.4) in Quasi-Banach Spaces

For simplicity, we use the following abbreviation for a given mapping : for all .

We start our work with the following theorem which can be regard as a general solution of functional equation (1.4).

Theorem 2.1. Let and be real vector spaces. A mapping satisfies in (1.4) if and only if is additive.

Proof. Setting in (1.4) , we obtain Since , we have Setting in (1.4), we obtain Putting , we get Putting , we get Let in (2.6), we obtain So, (2.6) turns to the following: From (2.5) and (2.8), we have Replacing by and by in (2.8) and comparing it with (2.9), we get Letting in (2.5), (2.8), and (2.10), respectively, we obtain From (2.11) we have Replacing and by their equivalents by using (2.12) in (2.10), we get Replacing by in (2.13), we get Replacing by in (2.13), we get Similarly, replacing by in (2.13), we obtain Replacing by and by in (2.15) and (2.16), respectively, we obtain Adding both sides of (2.17) and using (2.14), we get Comparing (2.18) and (2.13), we obtain for all . So, if a mapping satisfying (1.4) it must be additive. Conversely, let be additive, it is clear that satisfying (1.4), and the proof is complete.

Now, we investigate the generalized Hyers-Ulam stability of functional equation (1.4) in quasi-Banach spaces.

Theorem 2.2. Let be a function satisfying for all , and for all . Suppose that a function with satisfies the inequality: for all . Then there exists a unique additive mapping defined by for all and the mapping satisfies the inequality: for all .

Proof. Putting and in (2.22) and using , we obtain for all . By a simple induction we can prove that for all and . Thus for all and all . (2.20) and (2.27) show the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges in for all . Hence we can define the mapping by for all . Letting in (2.26), we obtain (2.24). Now we show that the mapping is additive. We conclude from (2.21), (2.22), and (2.28) for all . So Hence, by Theorem 2.1, the mapping is additive. Now we prove the uniqueness assertion of , by this mean let be another mapping satisfies (2.24). It follows from (2.24) for all . The right-hand side tends to zero as , hence for all . This show the uniqueness of .

Corollary 2.3. Let be nonnegative real numbers such that . Suppose that a mapping with satisfies the inequality: for all . Then there exists a unique additive mapping such that for all .
Proof. This is a simple consequence of Theorem 2.2.

The following corollary is Hyers-Ulam-type stability for the functional equation (1.4).

Corollary 2.4. Let be nonnegative real number. Suppose that a mapping with satisfies the inequality: for all . Then there exists a unique additive mapping such that for all .

Proof. In Theorem 2.2, let for all .

The following corollary is Isac-Rassias-type stability for the functional equation (1.4).

Corollary 2.5. Let be a mapping such that Let be nonnegative real numbers. Suppose that a mapping with satisfies the inequality for all . Then there exists a unique additive mapping such that for all , where .
Proof. The proof follows from Theorem 2.2 by taking for all .

Remark 2.6. In Theorem 2.2, If we replace control function by , then . Therefore in this case, is superstable.

Theorem 2.7. Let be a mapping such that for all and for all . Suppose that a function with satisfies the inequality: for all . Then there exists a unique additive mapping defined by for all and the mapping satisfies the inequality: for all .

Proof. Putting and in (2.43) we obtain for all . Replacing by in (2.46) and multiplying both sides of (2.46) to , we get for all and all . Since is a -Banach space, we have for all and all nonnegative and with . Therefore, we have from (2.42) and (2.48) that the sequence is a Cauchy in for all . Because of is complete, the sequence converges for all . Hence, we can define the mapping by for all . Putting and passing the limit in (2.48), we obtain (2.45). Showing the additivity and uniqueness of is similar to Theorem 2.2, and the proof is complete.

Corollary 2.8. Let be nonnegative real numbers such that . Suppose that a mapping with satisfies the inequality: for all . Then there exists a unique additive mapping such that for all .
Proof. This is a simple consequence of Theorem 2.7.

Remark 2.9. We can formulate similar statement to Corollaries 2.4 and 2.5 for Theorem 2.7. Moreover, In Theorem 2.7, If we replace control function by , then . Therefore in this case, is superstable.

Now, we apply a fixed point method and prove the generalized Hyers-Ulam stability of functional equation (1.4).

We recall a fundamental result in fixed point theory.

Theorem 2.10 (see [34]). Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschits constant . Then, for a given element , exactly one of the following assertions is true:
either for all orthere exists such that for all .Actually, if holds, then the sequence is a convergent to a fixed point of and is the unique fixed point of in ; for all .

Theorem 2.11. Let with be a mapping for which there exists a function such that for all . If there exists an such that , then there exists a unique additive mapping satisfying for all .

Proof. Putting and in (2.52) and using , we obtain for all . Hence, for all .
Let . We introduce a generalized metric on as follows: It is easy to show that is a generalized complete metric space.
Now we consider the mapping defined by for all and all . Let and let ba an arbitrary constant with . From the definition of , we have for all . By the assumption and last inequality, we have for all . So for all . It follows from (2.56) that . Therefore, according to Theorem 2.10, the sequence converges to a fixed point of , that is, and for all . Also is the unique fixed point of in the set and that is, inequality (2.54) holds true for all . It follows from the definition of , and (2.52) and (2.53) that Hence, by Theorem 2.1, the mapping is additive.

Corollary 2.12. Let and be real numbers. Let with such that for all . Then there exists a unique additive mapping satisfies the inequality: for all .

Proof. Setting for all in Theorem 2.11. Then by , we get the desired result.

Remark 2.13. We can formulate similar statement to Theorem 2.11 in which we can define the sequence under suitable conditions on the function and then obtain similar result to Corollary 2.12.

Remark 2.14. We can formulate similar statements for stability of (1.4) on Banach spaces.

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