In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

Let be a group and let be a metric group with metric . Given , does there exist a such that if satisfies for all , then a homomorphism exists with for all ?

The concept of stability for functional equations arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus we say that a functional equation is stable if any mapping approximately satisfying the equation is near to a true solution such that and for some function depending on the given function In 1941, the first result concerning the stability of functional equations for the case where and are Banach spaces was presented by Hyers [2]. In fact, he proved that each solution of the inequality for all can be approximated by a unique additive function defined by such that for every . Moreover, if is continuous in for each fixed , then the function is linear. And then Aoki [3], Bourgin [4], and Forti [5] have investigated the stability theorems of functional equations which generalize the Hyers' result. In 1978, Rassias [6] attempted to weaken the condition for the bound of Cauchy difference controlled by a sum of unbounded function and provided a generalization of Hyers' theorem. In 1991, Gajda [7] gave an affirmative solution to this question for by following the same approach as in [6]. Rassias [8] established a similar stability theorem for the unbounded Cauchy difference controlled by a product of unbounded function . Gvruţa [9] provided a further generalization of Rassias' theorem by replacing the bound of Cauchy difference by a general control function. During the last two 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 [10–15]).

Let and be real vector spaces. A function is called a quadratic function if and only if is a solution function of the quadratic functional equation

for all 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 , where the mapping is given by . See [16, 17] for the details.

The Hyers-Ulam stability of the quadratic functional equation (1.1) was first proved by Skof [18] for functions , where is a normed space and is a Banach space. Cholewa noticed that Skof's theorem is also valid if is replaced by an Abelian group. Czerwik [19] proved the generalized Hyers-Ulam stability of quadratic functional equation (1.1) in the spirit of Rassias approach. On the other hand, according to the theorem of Borelli and Forti [20], we know the following generalization of stability theorem for quadratic functional equation. Let be a 2-divisible Abelian group and a Banach space, and let be a mapping with satisfying the inequality

for all . Assume that one of the series

holds for all , then there exists a unique quadratic function such that

for all . 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 [21–27].

In 1996, Isac and Rassias [28] applied the stability theory of functional equations to prove fixed point theorems and study some new applications in nonlinear analysis.Radu [29], Cãdariu and Radu [30, 31] applied the fixed point theorem of alternative to the investigation of Cauchy and Jensen functional equations. Recently, Jung et al. [32],Jung [33, 34],Jung and Lee [35],Jung and Min [36],Jung and Rassias [37] have obtained the generalized Hyers-Ulam stability of functional equations via the fixed point method.

Now, we see that the norm defined by a real inner product space satisfies the following equality:

for all vectors Thus employing the last equality, we introduce to consider the following functional equation

with several variables for any fixed with . It is obvious that if in (1.6), then the solution function is even and thus it reduces to (1.1). Conversely, we observe that the general solution of (1.6) in the class of all functions between vector spaces is exactly a quadratic function. In this paper, we are going to investigate the general solution of (1.6) and then we are to prove the generalized Hyers-Ulam stability of (1.6) for a large class of functions from vector spaces into complete -normed spaces by using fixed point method, and direct method.

For notational convenience, given a mapping , we define the difference operator of (1.6) by

for all , which is called the approximate remainder of the functional equation (1.6) and acts as a perturbation of the equation.

We now introduce a fundamental result of fixed point theory. We refer to [38] for the proof of it. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [39].

Theorem.

Let be a generalized complete metric space (i.e., may assume infinite values). Assume that is a strictly contractive operator, that is, there exists a Lipschitz constant with such that for all Then for a given element one of the following assertions is true:

for all ;

there exists a nonnegative integer such that

for all ;

the sequence converges to a fixed point of ;

is the unique fixed point of in

for all

Throughout this paper, we consider a -Banach space. Let be a real number with and let denote either real field or complex field . Suppose is a vector space over . A function is called a -norm if and only if it satisfies

, if and only if ;

, for all and all ;

, for all

A -Banach space is a -normed space which is complete with respect to the -norm. Now we are ready to investigate the generalized Hyers-Ulam stability problem for the functional equation (1.6) using the fixed point method. From now on, let X be a linear space and let Y be a -Banach space over unless we give any specific reference where is a fixed real number with

Theorem 2.2.

Let be a function with for which there exists a function such that there exists a constant satisfying the inequalities

for all . Then there exists a unique quadratic function defined by such that

for all

Proof.

Let us define to be the set of all functions and introduce a generalized metric on as follows:

Then it is easy to show that is complete (see the proof of Theorem of [35]). Now we define an operator by

for all First, we assert that is strictly contractive with constant on . Given , let be an arbitrary constant with that is, Then it follows from (2.3) that

for all that is, Thus we see that for any and so is strictly contractive with constant on .

Next, if we put in (2.2) and we divide both sides by , then we get

for all which implies

Thus applying Theorem 2.1 to the complete generalized metric space with contractive constant , we see from Theorem 2.1 that there exists a function which is a fixed point of , that is, such that as By mathematical induction we know that

for all Since as there exists a sequence such that as and for every Hence, it follows from the definition of that

for all This implies

for all By Theorem 2.1 we obtain

which yields inequality (2.4).

In turn, it follows from (2.2) and (2.3) that

for all , which implies that is a solution of (1.6) and so the mapping is quadratic.

To prove the uniqueness of , assume now that is another quadratic mapping satisfying inequality (2.4). Then is a fixed point of and Since the mapping is a unique fixed point of in the set we see that by Theorem 2.1 The proof is complete.

The following theorem is an alternative result of Theorem 2.2.

Theorem.

Let be a function with for which there exists a function such that there exists a constant satisfying the inequalities

for all . Then there exists a unique quadratic function defined by such that

for all

Proof.

We use the same notations for and as in the proof of Theorem 2.2. Thus is a complete generalized metric space. Let us define an operator by

for all

Then it follows from (2.15) that

for all that is, Thus we see that for any and so is strictly contractive with constant on .

Next, if we put in (2.14) and we multiply both sides by , then we get by virtue of (2.15)

for all which implies

Thus according to of Theorem 2.1, there exists a function which is a fixed point of , that is, such that

By Theorem 2.1 we obtain

which yields the inequality (2.16).

Replacing instead of in the last part of Theorem 2.2, we can prove that is a unique quadratic function satisfying (2.16) for all

As applications, one has the following corollaries concerning the stability of (1.6).

Corollary.

Let be a real number with . Assume that a function with satisfies the inequality

for all . Then there exists a unique quadratic function given by which satisfies the inequality

for all .

Proof.

Letting and then applying Theorem 2.2 with contractive constant , we obtain easily the result.

Corollary.

Let be an -normed space with and a -Banach space, respectively. Let be real numbers such that for all and let be real numbers such that either or . Assume that a function with satisfies the inequality

for all and if . Then there exists a unique quadratic function which satisfies the inequality

for all and if . The function is given by

for all .

Proof.

Letting for all and then applying Theorem 2.2 with contractive constant and Theorem 2.3 with contractive constant , we obtain easily the results.

In the next two theorems, let be a mapping satisfying one of the conditions

for all .

Theorem.

Assume that a function satisfies

for all and satisfies the condition (3.1). Then there exists a unique quadratic function satisfying

for all , where . The function is given by

for all

Proof.

Putting in (3.3), we get . Putting in (3.3), we obtain

for . Dividing (3.6) by , we get

where for any . Thus it follows from formula (3.7) and triangle inequality that

for all and all which is verified by induction. Therefore we prove from inequality (3.8) that for any integers with

for all . Since the right-hand side of (3.9) tends to zero as , the sequence is a Cauchy sequence for all and thus converges by the completeness of . Define by

Taking the limit in (3.8) as , we obtain that

for all . Letting for all in (3.3), respectively, and dividing both sides by and after then taking the limit in the resulting inequality, we have

so the function is quadratic.

To prove the uniqueness of the quadratic function subject to (3.4), let us assume that there exists a quadratic function which satisfies (1.6) and inequality (3.4). Obviously, we obtain that

for all . Hence it follows from (3.4) that

for all . Therefore letting , one has for all , completing the proof of uniqueness.

Theorem.

Assume that a function satisfies

for all and satisfies condition (3.2). Then there exists a unique quadratic function satisfying

for all . The function Q is given by

for all

Proof.

In this case, since and so by assumption. Replacing by in (3.6), we obtain

for .

Therefore we prove from inequality (3.18) that for any integers with

for all . Since the right-hand side of (3.19) tends to zero as , the sequence is a Cauchy sequence for all , and thus converges by the completeness of . Define by

for all . Taking the limit in (3.19) with as , we obtain that

Replacing in (3.3) by , multiplying both sides by and then taking the limit as in the resulting inequality, we have

for all . Therefore the function is quadratic.

To prove the uniqueness, let be another quadratic function satisfying (3.16). Then it is easy to see that the following identities and hold for all . Thus we have

for all and all . Therefore letting , one has for all . This completes the proof.

In the following corollary, we have a stability result of (1.6) with difference operator bounded by the sum of powers of -norms.

Corollary.

Let be an -normed space with and a -Banach space, respectively. Let be real numbers with for all , and let be real numbers such that either or . Assume that a function satisfies the inequality

for all and if . Then there exists a unique quadratic function which satisfies the inequality

for all and if . The function is given by

for all .

Proof.

Letting for all and then applying Theorems 3.1 and 3.2, we obtain easily the results.

We observe that if and in Corollary 3.3, then the stability result obtained by the fixed point method in Corollary 2.5 is somewhat different from the stability result obtained by direct method in Corollary 3.3. The stability result in Corollary 3.3 is sharper than that of Corollary 2.5.

In the next corollary, we get a stability result of (1.6) with difference operator bounded by the product of powers of -norms.

Corollary.

Let be an -normed space with and a -Banach space, respectively, and let be real numbers such that and , where . Suppose that a function satisfies

for all and if . Then there exists a unique quadratic function which satisfies the inequality

for all and for all if , where if .

Proof.

We remark that satisfies condition (3.1) for the case or condition (3.2) for the case . By Theorems 3.1 and 3.2, we get the results.

We observe that if in Corollary 3.4, then the stability result obtained by the fixed point method with contractive constants respectively, coincides with the stability result (3.28) obtained by direct method.