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

(1.1)

for all , then there is a homomorphism with

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

exists for all and is the unique additive mapping which satisfies

(1.7)

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

(1.8)

for all . Then there exists a unique additive mapping satisfying

(1.9)

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

(1.10)

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

(1.11)

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:

(1.12)

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

(1.13)

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

(1.14)

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 :

(1.15)

for all .

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

Proposition 2.1.

A mapping satisfies

(2.1)

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

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)
(2.6)

for all Letting in (2.5), we get

(2.7)

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

(2.8)

for all Therefore,

(2.9)

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

(2.10)

for all . So

(2.11)

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

(2.12)

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

(2.13)

for all (see [9, 12]). Hence

(2.14)

for all . Hence satisfies (2.1).

Corollary 2.2.

Let be a mapping satisfying

(2.15)

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

(2.16)

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

(2.17)

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

(2.18)

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

(2.19)

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

(2.20)

for all .

Proof.

It follows from that

(2.21)

for all .

Letting in (2.19), we get

(2.22)

for all and all . Hence

(2.23)

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

(2.24)

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

Now we consider the mapping defined by

(2.25)

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

(2.26)

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

(2.27)

for all . So

(2.28)

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,

(2.29)

and for all . Also is the unique fixed point of in the set and

(2.30)

that is, the inequality (2.20) holds true for all . It follows from the definition of , (2.19), and (2.21) that

(2.31)

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

(2.32)

Since is quadratic and is linear,

(2.33)

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

(2.34)

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

(2.35)

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

(2.36)

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

(2.37)

for all .

Proof.

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

(2.38)

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

(2.39)

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

(2.40)

for all .

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

(2.41)

for all .

Corollary 2.6.

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.

Theorem 2.7.

Let be an even 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

(2.44)

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

(2.45)

for all .

Proof.

Since it follows from (2.19) that and

(2.46)

for all and all . Therefore,

(2.47)

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

(2.48)

for all , where

(2.49)

Letting in (2.48), we get

(2.50)

for all Hence

(2.51)

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

(2.52)

Now we consider the mapping defined by

(2.53)

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

(2.54)

for all .

Proof.

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

(2.55)

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

(2.56)

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

(2.57)

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

(2.58)

for all .

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

(2.59)

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

(2.60)

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

(2.61)

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

(2.62)

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

(2.63)

for all . We get from (2.62) that

(2.64)

for all and all , where

(2.65)

Hence

(2.66)

for all . Letting in (2.66), we get

(2.67)

for all . Therefore,

(2.68)

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

(2.69)

Now we consider the mapping defined by

(2.70)

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

(2.71)

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

(2.72)

for all .

Proof.

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

(2.73)

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

(2.74)

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

(2.75)

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

(2.76)

for all . Hence

(2.77)

for all .

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

(2.78)

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

(2.79)

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

(2.80)

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

(2.81)

for all To see this, if or then

(2.82)

Now suppose that Then there exists a nonnegative integer such that

(2.83)

Therefore,

(2.84)

Hence

(2.85)

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

(2.86)

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

(2.87)

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

(2.88)

for all Let with If , then for all So

(2.89)

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.