A Fixed Point Approach to the Stability of a Quadratic Functional Equation in -Algebras

Abstract

We use a fixed point method to investigate the stability problem of the quadratic functional equation in -algebras.

1. Introduction and Preliminaries

In 1940, the following question concerning the stability of group homomorphisms was proposed by Ulam [1]: Under what conditions does there exist a group homomorphism near an approximately group homomorphism? In 1941, Hyers [2] considered the case of approximately additive functions , where and are Banach spaces and satisfies Hyers inequality

(1.1)

for all . Aoki [3] and Th. M. Rassias [4] provided a generalization of the Hyers' theorem for additive mappings and for linear mappings, respectively, by allowing the Cauchy difference to be unbounded (see also [5]).

Theorem 1.1 (Th. M. Rassias).

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

(1.2)

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

(1.3)

exists for all and is the unique additive mapping which satisfies

(1.4)

for all . If then inequality (1.2) holds for and (1.4) for . Also, if for each the mapping is continuous in , then is -linear.

The result of the Th. M. Rassias theorem has been generalized by Gvruţa [6] who permitted the Cauchy difference to be bounded by a general control function. During the last three decades a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [7–20]). We also refer the readers to the books [21–25]. A quadratic functional equation is a functional equation of the following form:

(1.5)

In particular, every solution of the quadratic equation (1.5) 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 [16, 21, 26, 27]. The biadditive mapping is given by

(1.6)

The Hyers-Ulam stability problem for the quadratic functional equation (1.5) was studied by Skof [28] for mappings where is a normed space and is a Banach space. Cholewa [8] noticed that the theorem of Skof is still true if we replace by an Abelian group. Czerwik [9] proved the generalized Hyers-Ulam stability of the quadratic functional equation (1.5). Grabiec [11] has generalized these results mentioned above. Jun and Lee [14] proved the generalized Hyers-Ulam stability of a Pexiderized quadratic functional equation.

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 1.2 (see [29]).

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

(1.7)

for all nonnegative integers or there exists a non-negative 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 will be a -algebra. We denote by the unique positive element such that for each positive element . Also, we denote by and the set of real, complex, and rational numbers, respectively. In this paper, we use a fixed point method (see [7, 15, 17]) to investigate the stability problem of the quadratic functional equation

(1.8)

in -algebras. A systematic study of fixed point theorems in nonlinear analysis is due to Hyers et al. [30] and Isac and Rassias [13].

2. Solutions of (1.8)

Theorem 2.1.

Let be a linear space. If a mapping satisfies and the functional equation (1.8), then is quadratic.

Proof.

Letting and in (1.8) respectively, we get

(2.1)

for all It follows from (1.8) and (2.1) that

(2.2)

for all Letting in (2.2), we get

(2.3)

for all Thus (2.2) implies that

(2.4)

for all Hence is quadratic.

Remark 2.2.

A quadratic mapping does not satisfy (1.8) in general. Let be the mapping defined by for all It is clear that is quadratic and that does not satisfy (1.8).

Corollary 2.3.

Let be a linear space. If a mapping satisfies the functional equation (1.8), then there exists a symmetric biadditive mapping such that for all

3. Generalized Hyers-Ulam Stability of (1.8) in -Algebras

In this section, we use a fixed point method (see [7, 15, 17]) to investigate the stability problem of the functional equation (1.8) in -algebras.

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

(3.1)

for all where is a linear space.

Theorem 3.1.

Let be a linear space and let be a mapping with for which there exists a function such that

(3.2)

for all . If there exists a constant such that

(3.3)

for all , then there exists a unique quadratic mapping such that

(3.4)

for all where

(3.5)

Moreover, if is continuous in for each fixed , then is -quadratic, that is, for all and all

Proof.

Replacing and by and in (3.2), respectively, we get

(3.6)

for all Replacing and by and in (3.2), respectively, we get

(3.7)

for all It follows from (3.6) and (3.7) that

(3.8)

for all Letting in (3.8), we get

(3.9)

for all By (3.3) we have for all Let be the set of all mappings with . We can define a generalized metric on as follows:

(3.10)

is a generalized complete metric space [7].

Let be the mapping defined by

(3.11)

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

(3.12)

for all . Hence

(3.13)

for all . So

(3.14)

for any . It follows from (3.9) that . According to Theorem 1.2, the sequence converges to a fixed point of , that is,

(3.15)

and for all . Also,

(3.16)

and is the unique fixed point of in the set . Thus the inequality (3.4) holds true for all . It follows from the definition of , (3.2), and (3.3) that

(3.17)

for all By Theorem 2.1, the function is quadratic.

Moreover, if is continuous in for each fixed , then by the same reasoning as in the proof of [4] is -quadratic.

Corollary 3.2.

Let and be non-negative real numbers and let be a mapping with such that

(3.18)

for all Then there exists a unique quadratic mapping such that

(3.19)

for all . Moreover, if is continuous in for each fixed , then is -quadratic.

The following theorem is an alternative result of Theorem 3.1 and we will omit the proof.

Theorem 3.3.

Let be a mapping with for which there exists a function satisfying (3.2) for all If there exists a constant such that

(3.20)

for all , then there exists a unique quadratic mapping such that

(3.21)

for all , where is defined as in Theorem 3.1. Moreover, if is continuous in for each fixed , then is -quadratic.

Corollary 3.4.

Let and be non-negative real numbers and let be a mapping with such that

(3.22)

for all . Then there exists a unique quadratic mapping such that

(3.23)

for all . Moreover, if is continuous in for each fixed , then is -quadratic.

For the case we use the Gajda's example [31] to give the following counterexample (see also [9]).

Example 3.5.

Let be defined by

(3.24)

Consider the function by the formula

(3.25)

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

(3.26)

for all To see this, if or then

(3.27)

Now suppose that Then there exists a positive integer such that

(3.28)

Thus

(3.29)

Hence

(3.30)

for all It follows from the definition of and (3.28) that

(3.31)

Thus satisfies (3.26). Let be a quadratic function such that

(3.32)

for all where is a positive constant. Then there exists a constant such that for all . So we have

(3.33)

for all Let with If , then for all So

(3.34)

which contradicts (3.33).