- Research Article
- Open Access
Stability of Quadratic Functional Equations via the Fixed Point and Direct Method
© Eunyoung Son et al. 2010
- Received: 13 October 2009
- Accepted: 19 January 2010
- Published: 1 March 2010
Cădariu and Radu applied the fixed point theorem to prove the stability theorem of Cauchy and Jensen functional equations. In this paper, we prove the generalized Hyers-Ulam stability via the fixed point method and investigate new theorems via direct method concerning the stability of a general quadratic functional equation.
- Banach Space
- Functional Equation
- Fixed Point Theorem
- Stability Theorem
- Unbounded Function
In 1940, Ulam  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 . 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 , Bourgin , and Forti  have investigated the stability theorems of functional equations which generalize the Hyers' result. In 1978, Rassias  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  gave an affirmative solution to this question for by following the same approach as in . Rassias  established a similar stability theorem for the unbounded Cauchy difference controlled by a product of unbounded function . G vruţa  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  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  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 , 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
In 1996, Isac and Rassias  applied the stability theory of functional equations to prove fixed point theorems and study some new applications in nonlinear analysis.Radu , 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. ,Jung [33, 34],Jung and Lee ,Jung and Min ,Jung and Rassias  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  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. .
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
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
for all that is, Thus we see that for any and so is strictly contractive with constant on .
for all which implies
which yields inequality (2.4).
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.
for all that is, Thus we see that for any and so is strictly contractive with constant on .
for all which implies
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).
for all .
Letting and then applying Theorem 2.2 with contractive constant , we obtain easily the result.
for all .
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 .
so the function is quadratic.
for all . Therefore letting , one has for all , completing the proof of uniqueness.
for all . Therefore the function is quadratic.
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.
for all .
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.
for all and for all if , where if .
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.
This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0070940).
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