Stability of Approximate Quadratic Mappings

Abstract

We investigate the general solution of the quadratic functional equation , in the class of all functions between quasi--normed spaces, and then we prove the generalized Hyers-Ulam stability of the equation by using direct method and fixed point method.

1. Introduction

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 ?

In 1941, the first result concerning the stability of functional equations was presented by Hyers [2]. And then Aoki [3] and Bourgin [4] have investigated the stability theorems of functional equations with unbounded Cauchy differences. In 1978, Th. M. Rassias [5] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. It was shown by Gajda [6] as well as by Th. M. Rassias and Šemrl [7] that one cannot prove the Rassias' type theorem when . Găvruta [8] obtained generalized result of Th. M. Rassias' Theorem which allow the Cauchy difference to be controlled by a general unbounded function. J. M. Rassias [9, 10] established a similar stability theorem linear and nonlinear mappings with the unbounded Cauchy difference.

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:

(1.1)

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 [11, 12] for the details. The Hyers-Ulam stability of the quadratic functional (1.1) was first proved by Skof [13] for functions , where is a normed space and is a Banach space. Cholewa [14] demonstrated that Skof's theorem is also valid if is replaced by an abelian group. Czerwik [15] proved the Hyers-Ulam stability of quadratic functional (1.1) by the similar way to Th. M. Rassias control function [5]. According to the theorem of Borelli and Forti [16], we obtain the following generalization of stability theorem for the quadratic functional (1.1): let be an abelian group and a Banach space; let be a mapping with satisfying the inequality

(1.2)

for all . Assume that one of the following conditions

(1.3)

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

(1.4)

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 [17–23].

In this paper, we consider a new quadratic functional equation

(1.5)

for all vectors in quasi--normed spaces. First, we note that a function is a solution of the functional (1.5) in the class of all functions between vector spaces if and only if the function is quadratic. Further, we investigate the generalized Hyers-Ulam stability of (1.5) by using direct method and fixed point method. As a result of the paper, we have a much better possible estimation of approximate quadratic mappings by quadratic mappings than that of Czerwik [15] and Skof [13].

2. Stability of (1.5)

Now, we consider some basic concepts concerning quasi--normed spaces and some preliminary results. We fix a real number with and let denote either or . Let be a linear space over . A quasi--norm is a real-valued function on satisfying the following.

(1) for all and if and only if .

(2) for all and all .

(3) There is a constant such that for all .

The pair is called a space if is a quasi--norm on . The smallest possible is called the modulus of concavity of . A space is a complete quasi--normed space. A quasi--norm is called a -norm if

(2.1)

for all . In this case, a quasi--Banach space is called a -Banach space. We can refer to [24, 25] for the concept of quasinormed spaces and -Banach spaces. Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz theorem [25] (see also [24]), each quasinorm is equivalent to some -norm. In [26], Tabor has investigated a version of the D. H. Hyers, Th. M. Rassias, and Z. Gajda theorem (see [5, 6]) in quasibanach spaces. Recently, J. M. Rassias and Kim [27] have obtained stability results of general additive equations in quasi--normed spaces.

From now on, let be a quasi--normed space with norm and let be a -Banach space with norm unless we give any specific reference. Now, we are ready to investigate the generalized Hyers-Ulam stability problem for the functional (1.5) using direct method.

Theorem 2.1.

Assume that a function satisfies

(2.2)

for all and that satisfies the following control conditions

(2.3)

for all . Then there exists a unique quadratic function satisfying

(2.4)

for all , where . The function is defined as

(2.5)

for all

Proof.

Putting in (2.2), we get . Replacing by in (2.2), we obtain

(2.6)

for all . Dividing (2.6) by , we get

(2.7)

for all where . Now letting and dividing in (2.7), we have

(2.8)

for all . Therefore we prove from the inequality (2.8) that for any integers with

(2.9)

Since the right-hand side of (2.9) tends to zero as , the sequence is Cauchy for all and thus converges by the completeness of . Define by

(2.10)

Letting in (2.2), respectively, and dividing both sides by and after then taking the limit in the resulting inequality, we have

(2.11)

and so the function is quadratic.

Taking the limit in (2.9) with as , we obtain that

(2.12)

which yields the estimation (2.4).

To prove the uniqueness of the quadratic function subject to (2.4), let us assume that there exists a quadratic function which satisfies (1.5) and the inequality (2.4). Obviously, we obtain that

(2.13)

for all . Hence it follows from (2.4) that

(2.14)

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

Theorem 2.2.

Assume that a function satisfies

(2.15)

for all and that satisfies conditions

(2.16)

for all . Then there exists a unique quadratic function satisfying

(2.17)

for all . The function is given by

(2.18)

for all

Proof.

In this case, since and so by assumption.

Replacing by in (2.6), we obtain

(2.19)

for . Therefore we prove from inequality (2.19) that for any integers with

(2.20)

for all . Since the right-hand side of (2.20) tends to zero as , the sequence is Cauchy for all and thus converges by the completeness of . Define by

(2.21)

for all .

Thereafter, applying the same argument as in the proof of Theorem 2.1, we obtain the desired result.

We now introduce a fundamental result of fixed point theory. We refer to [28] for the proof of it, and the reader is referred to papers [29–31].

Theorem 2.3.

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

(A₁) for all ;

(A₂) there exists a nonnegative integer such that

(A_2.1) for all ;

(A_2.2) the sequence converges to a fixed point of ;

(A_2.3) is the unique fixed point of in the set

(A_2.4) for all

For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [32]. In 1996, Isac and Th. M. Rassias [33] applied the stability theory of functional equations to prove fixed point theorems and study some new applications in nonlinear analysis. Cădariu and Radu [29, 31] and Radu [34] applied the fixed point theorem of alternative to the investigation of Cauchy and Jensen functional equations. Recently, Jung et al. [35–40] and Jung and Rassias [41] have obtained the generalized Hyers-Ulam stability of functional equations via the fixed point method.

Now we are ready to investigate the generalized Hyers-Ulam stability problem for the functional (1.5) using the fixed point method.

Theorem 2.4.

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

(2.22)

(2.23)

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

(2.24)

for all

Proof.

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

(2.25)

Then it is easy to show that is complete (see [37, Proof of Theorem ]). Now we define an operator by

(2.26)

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 (2.23) that

(2.27)

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

Next, if we put in (2.22) and we divide both sides by , then we get

(2.28)

for all which implies

Thus applying Theorem 2.3 to the complete generalized metric space with contractive constant , we see from of Theorem 2.3 that there exists a function which is a fixed point of , that is, such that as By mathematical induction we know that

(2.29)

for all

Since as by of Theorem 2.3, there exists a sequence such that as and for every Hence, it follows from the definition of that

(2.30)

for all This implies

(2.31)

for all

In turn, it follows from (2.22) and (2.23) that

(2.32)

for all , which implies that is a solution of (1.5) and so the mapping is quadratic.

By of Theorem 2.3, we obtain

(2.33)

which yields the inequality (2.24).

To prove the uniqueness of , assume now that is another quadratic mapping satisfying the inequality (2.24). Then is a fixed point of with in view of the inequality (2.24). This implies that and so by of Theorem 2.3. The proof is complete.

By a similar way, one can prove the following theorem using the fixed point method.

Theorem 2.5.

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

(2.34)

(2.35)

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

(2.36)

for all

Proof.

We use the same notations for and as in the proof of Theorem 2.4. Thus is a complete generalized metric space. Let us define an operator by

(2.37)

for all Then it follows from (2.35) that

(2.38)

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

Next, if we put in (2.34) and we divide both sides by , then we get by virtue of (2.35)

(2.39)

for all which implies Thereafter, applying the same argument as in the proof of Theorem 2.4, we obtain the desired results.

3. Applications of Main Results

In the following corollary, we have a stability result of (1.5) in the sense of Th. M. Rassias.

Corollary 3.1.

Let and be real numbers such that and for . Assume that a function satisfies the inequality

(3.1)

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

(3.2)

for all and for all if . The function is given by

(3.3)

for all , where if .

Proof.

If , then we get by putting in (3.1). Letting for all and then applying Theorem 2.1 we obtain easily the desired results.

Corollary 3.2.

Let and be real numbers such that and for . Assume that a function satisfies the inequality

(3.4)

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

(3.5)

for all . The function is given by

(3.6)

for all .

In the following corollary, we have a stability result of (1.5) in the sense of Hyers.

Corollary 3.3.

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

(3.7)

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

(3.8)

for all .

In the next corollary, we get a stability result of (1.5) in the sense of J. M. Rassias.

Corollary 3.4.

Let be real numbers such that and , where . Suppose that a function satisfies

(3.9)

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

(3.10)

for all and all if , where if .

Proof.

Letting and applying Theorems 2.1 and 2.2, we get the results.