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# Fixed Points and Stability of a Generalized Quadratic Functional Equation

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 193035 (2009)

## Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the generalized quadratic functional equation in Banach modules, where are nonzero rational numbers with .

## 1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms: *let* *be a group and let* *be a metric group with the metric* . *Given* , *does there exist* *such that if a mapping* *satisfies the inequality*

for all , then there is a homomorphism with

for all

Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let and be Banach spaces. *Assume that* *satisfies*

*for some* *and all* . *Then there exists a unique additive mapping* *such that*

*for all* .

Aoki [3] and Th. M. Rassias [4] provided a generalization of the Hyers' theorem for additive and linear mappings, respectively, by allowing the Cauchy difference to be unbounded.

Theorem 1.1 (Th. M. Rassias [4]).

Let be a mapping from a normed vector space into a Banach space subject to the inequality

for all , where and are constants with and . Then the limit

exists for all and is the unique additive mapping which satisfies

for all . If , then the inequality (1.5) holds for and (1.7) for . Also, if for each the mapping is continuous in , then is - linear.

Theorem 1.2 (J. M. Rassias [5–7]).

Let be a real normed linear space and let be a real Banach space. Assume that is a mapping for which there exist constants and such that and satisfies the functional inequality

for all . Then there exists a unique additive mapping satisfying

for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed then is linear.

In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by Găvruţa [8], who replaced the bounds and by a general control function .

The functional equation

is called a *quadratic functional equation.* Quadratic functional equations were used to characterize inner product spaces [9–11]. In particular, every solution of the quadratic equation (1.10) is said to be a *quadratic mapping.* It is well known that a mapping between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive mapping such that for all (see [9, 12]). The biadditive mapping is given by

The generalized Hyers-Ulam stability problem for the quadratic functional equation (1.10) was proved by Skof for mappings where is a normed space and is a Banach space (see [13]). Cholewa [14] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. J. M. Rassias [15] and Czerwik [16], proved the stability of the quadratic functional equation (1.10). Grabiec [17] has generalized these results mentioned above. J. M. Rassias [18] introduced and investigated the stability problem of Ulam for the Euler-Lagrange quadratic mappings:

In addition, J. M. Rassias [19] generalized the Euler-Lagrange quadratic mapping (1.12) and investigated its stability problem. The Euler-Lagrange quadratic mapping (1.12) has provided a lot of influence in the development of general Euler-Lagrange quadratic equations (mappings) which is now known as Euler-Lagrange-Rassias quadratic functional equations (mappings).

Jun and Lee [20] proved the generalized Hyers-Ulam stability of a pexiderized quadratic equation. 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 [8, 20–47]). We also refer the readers to the books [48–51].

Let be a set. A function is called a *generalized metric* on if satisfies

(i) if and only if ,

(ii) for all ,

(iii) for all

We recall the following theorem by Margolis and Diaz.

Theorem 1.3 (see [52]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either

for all nonnegative integers or there exists a nonnegative 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 .

Throughout this paper, we assume that are nonzero rational numbers with and that is a unital Banach algebra with unit , norm , and . Assume that is a normed left -module and is a (unit linked) Banach left -module. A quadratic mapping is called -*quadratic* if for all and all .

In this paper, we investigate an -quadratic mapping associated with the generalized quadratic functional equation

and using the fixed point method (see [24, 25, 38, 53–55]), we prove the generalized Hyers-Ulam stability of -quadratic mappings in Banach -modules associated with the functional equation (1.14). In 1996, Isac and Th. M. Rassias [56] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.

For convenience, we use the following abbreviation for a given and a mapping :

for all .

## 2. Fixed Points and Stability of the Generalized Quadratic Functional Equation (1.14)

Proposition 2.1.

A mapping satisfies

for all if and only if is quadratic.

Proof.

Let satisfy (2.1). Since letting in (2.1), we get . Letting in (2.1), we get

for all . It follows from (2.1) that for all Hence

for all We decompose into the even part and the odd part by putting

for all It is clear that for all It is easy to show that the mappings and satisfy (2.2) and (2.3). Thus we have

for all Letting in (2.5), we get

for all . It follows from (2.2), (2.5), and (2.7) that

for all Therefore,

for all . So is quadratic. We claim that For this, it follows from (2.2) and (2.6) that

for all . So

for all . Letting in (2.11), we get for all . So it follows from (2.11) that

for all . Replacing by and by in (2.12), we infer that is additive. To complete the proof we have two cases.

Case 1 ().

Since is additive and satisfies (2.1), letting and replacing by in (2.1), we get for all . Since , we get

Case 2 ().

Since is additive and satisfies (2.2), we have for all . Since , we get

Hence and this proves that is quadratic.

Conversely, let be quadratic. Then there exists a unique symmetric biadditive mapping such that for all and

for all . Hence satisfies (2.1).

Corollary 2.2.

Let be a mapping satisfying

for all and all If for each the mapping is continuous in , then is -quadratic.

Proof.

Let By Proposition 2.1, is quadratic. Thus is -quadratic. Let and let be a sequence of rational numbers such that Since is -quadratic and the mapping is continuous in for each , we have

for all . So is -quadratic. Letting in (2.15), we get

for all and all It is clear that (2.17) is also true for For each element Since is -quadratic and for all and all we have

for all and all So the -quadratic mapping is also -quadratic. This completes the proof.

Now we prove the generalized Hyers-Ulam stability of -quadratic mappings in Banach -modules.

Theorem 2.3.

Let be a mapping with for which there exists a function such that

for all and all . Let be a constant such that for all If for each the mapping is continuous in , then there exists a unique -quadratic mapping satisfying

for all .

Proof.

It follows from that

for all .

Letting in (2.19), we get

for all and all . Hence

for all and all . Let We introduce a generalized metric on as follows:

It is easy to show that is a generalized complete metric space [24].

Now we consider the mapping defined by

Let and let be an arbitrary constant with . From the definition of , we have

for all . By the assumption and the last inequality, we have

for all . So

for any . It follows from (2.23) (by letting ) that . According to Theorem 1.3, 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, the inequality (2.20) holds true for all . It follows from the definition of , (2.19), and (2.21) that

for all and all . By Proposition 2.1 (by letting ), the mapping is quadratic. Let be a continuous linear functional. For any , we consider the mapping defined by

Since is quadratic and is linear,

for all So is quadratic. Also is measurable since it is the pointwise limit of the sequence

It follows from [48, Corollary 10.2] that for all Then

for all Hence for all and all By Corollary 2.2, the mapping is -quadratic.

Corollary 2.4.

Let and be nonnegative real numbers such that and let be a mapping satisfying the inequality

for all and all . If for each the mapping is continuous in , then there exists a unique -quadratic mapping such that

for all .

Proof.

Letting and in (2.36), we get Now, the proof follows from Theorem 2.3 by taking

for all . Then we can choose and we get the desired result.

Remark 2.5.

Let be a mapping with for which there exists a function such that

for all and all . Let be a constant such that for all . By a similar method to the proof of Theorem 2.3, one can show that if for each the mapping is continuous in , then there exists a unique -quadratic mapping satisfying

for all .

For the case (where are nonnegative real numbers and with , there exists a unique -quadratic mapping satisfying

for all .

Corollary 2.6.

Let and let be nonnegative real numbers such that and let be a mapping satisfying the inequality

for all and all . If for each the mapping is continuous in , then is -quadratic.

Theorem 2.7.

Let be an even mapping for which there exists a function satisfying (2.19) and

for all and all . Let be a constant such that the mapping

satisfying for all If for each the mapping is continuous in , then there exists a unique -quadratic mapping satisfying

for all .

Proof.

Since it follows from (2.19) that and

for all and all . Therefore,

for all and all . Letting and replacing by and by in (2.47), we get

for all , where

Letting in (2.48), we get

for all Hence

for all Let We introduce a generalized metric on as follows:

Now we consider the mapping defined by

Similar to the proof of Theorem 2.3, we deduce that the sequence converges to a fixed point of which is -quadratic. Also is the unique fixed point of in the set and satisfies (2.45).

Corollary 2.8.

Let and let be nonnegative real numbers and let be an even mapping satisfying the inequality (2.36) for all and all . If for each the mapping is continuous in , then there exists a unique -quadratic mapping such that

for all .

Proof.

Letting and in (2.36), we get Now the proof follows from Theorem 2.7 by taking

for all . Then we can choose and we get the desired result.

Remark 2.9.

Let be an even mapping with for which there exists a function such that

for all and all . Let be a constant such that the mapping

satisfying for all By a similar method to the proof of Theorem 2.7, one can show that if for each the mapping is continuous in , then there exists a unique -quadratic mapping satisfying

for all .

For the case (where are nonnegative real numbers and , there exists a unique -quadratic mapping satisfying

for all .

Corollary 2.10.

Let and let be nonnegative real numbers such that and let be an even mapping satisfying the inequality (2.42) for all and all . If for each the mapping is continuous in , then is -quadratic.

We may omit the evenness of the mapping in Theorem 2.7.

Theorem 2.11.

Let be a mapping for which there exists a function satisfying (2.19) and (2.43) for all and all . Let be a constant such that the mapping

satisfying for all If for each the mapping is continuous in , then there exists a unique -quadratic mapping satisfying

for all .

Proof.

Since it follows from (2.19) that We decompose into the even part and the odd part It follows from (2.19) that

for all and all . By Theorem 2.7, there exists a unique -quadratic mapping satisfying

for all . We get from (2.62) that

for all and all , where

Hence

for all . Letting in (2.66), we get

for all . Therefore,

for all . Let We introduce a generalized metric on as follows:

Now we consider the mapping defined by

Similar to the proof of Theorem 2.3, we deduce that the sequence converges to a fixed point of which is quadratic and

Also is odd since is odd. Therefore, since is quadratic too. Now (2.61) follows from (2.63) and (2.71).

Corollary 2.12.

Let and let be nonnegative real numbers and let be a mapping satisfying the inequality (2.36) for all and all . If for each the mapping is continuous in , then there exists a unique -quadratic mapping such that

for all .

Proof.

Letting and in (2.36), we get Now the proof follows from Theorem 2.11 by taking

for all . Then we can choose and we get the desired result.

Remark 2.13.

Let be a mapping with for which there exists a function such that

for all and all . Let be a constant such that the mapping

satisfying for all By a similar method to the proof of Theorem 2.11, one can show that if for each the mapping is continuous in , then there exists a unique -quadratic mapping satisfying

for all . Hence

for all .

For the case (where are nonnegative real numbers and , there exists a unique -quadratic mapping satisfying

for all .

For the case , we have the following counterexample which is a modification of the example of Czerwik [16].

Example 2.14.

Let be defined by

where is a positive real number. Consider the function by the formula

where It is clear that is continuous and bounded by on . We prove that

for all To see this, if or then

Now suppose that Then there exists a nonnegative integer such that

Therefore,

Hence

for all From the definition of and (2.83), we have

Therefore, satisfies (2.81). Let be a quadratic function such that

for all Then there exists a constant such that for all (see [57]). So we have

for all Let with If , then for all So

which contradicts (2.88).

Corollary 2.15.

Let and let be nonnegative real numbers such that and let be a mapping satisfying the inequality (2.42) for all and all . If for each the mapping is continuous in , then is -quadratic.

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## Acknowledgments

The authors would like to thank the referees for bringing some useful references to their attention. The second author was supported by Korea Research Foundation Grant KRF-2008-313-C00041.

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Najati, A., Park, C. Fixed Points and Stability of a Generalized Quadratic Functional Equation.
*J Inequal Appl* **2009, **193035 (2009). https://doi.org/10.1155/2009/193035

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### Keywords

- Banach Space
- Functional Equation
- Stability Problem
- Normed Linear Space
- Unique Fixed Point