Stability of Quadratic Functional Equations via the Fixed Point and Direct Method

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Abstract

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.

Keywords

Banach Space Functional Equation Fixed Point Theorem Stability Theorem Unbounded Function 

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 Open image in new window be a group and let Open image in new window be a metric group with metric Open image in new window . Given Open image in new window , does there exist a Open image in new window such that if Open image in new window satisfies Open image in new window for all Open image in new window , then a homomorphism Open image in new window exists with Open image in new window for all Open image in new window ?

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 Open image in new window is stable if any mapping Open image in new window approximately satisfying the equation Open image in new window is near to a true solution Open image in new window such that Open image in new window and Open image in new window for some function Open image in new window depending on the given function Open image in new window In 1941, the first result concerning the stability of functional equations for the case where Open image in new window and Open image in new window are Banach spaces was presented by Hyers [2]. In fact, he proved that each solution Open image in new window of the inequality Open image in new window for all Open image in new window can be approximated by a unique additive function Open image in new window defined by Open image in new window such that Open image in new window for every Open image in new window . Moreover, if Open image in new window is continuous in Open image in new window for each fixed Open image in new window , then the function Open image in new window 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 Open image in new window and provided a generalization of Hyers' theorem. In 1991, Gajda [7] gave an affirmative solution to this question for Open image in new window 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 Open image in new window . G Open image in new window vruţa [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, 11, 12, 13, 14, 15]).

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

for all Open image in new window It is well known that a function Open image in new window between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function Open image in new window such that Open image in new window for all Open image in new window , where the mapping Open image in new window is given by Open image in new window . 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 Open image in new window , where Open image in new window is a normed space and Open image in new window is a Banach space. Cholewa noticed that Skof's theorem is also valid if Open image in new window 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 Open image in new window be a 2-divisible Abelian group and Open image in new window a Banach space, and let Open image in new window be a mapping with Open image in new window satisfying the inequality

for all Open image in new window . Assume that one of the series

holds for all Open image in new window , then there exists a unique quadratic function Open image in new window such that

for all Open image in new window . 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, 22, 23, 24, 25, 26, 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 Open image in new window Thus employing the last equality, we introduce to consider the following functional equation

with several variables for any fixed Open image in new window with Open image in new window . It is obvious that if Open image in new window 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 Open image in new window -normed spaces by using fixed point method, and direct method.

2. Stability of (1.6) by Fixed Point Method

For notational convenience, given a mapping Open image in new window , we define the difference operator Open image in new window of (1.6) by

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

Open image in new window for all Open image in new window ;

there exists a nonnegative integer Open image in new window such that

Open image in new window for all Open image in new window ;

the sequence Open image in new window converges to a fixed point Open image in new window of Open image in new window ;

Open image in new window is the unique fixed point of Open image in new window in Open image in new window

Open image in new window for all Open image in new window

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

Open image in new window , if and only if Open image in new window ;

Open image in new window , for all Open image in new window and all Open image in new window ;

Open image in new window , for all Open image in new window

A Open image in new window -Banach space is a Open image in new window -normed space which is complete with respect to the Open image in new window -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 Open image in new window -Banach space over Open image in new window unless we give any specific reference where Open image in new window is a fixed real number with Open image in new window

Theorem 2.2.

Let Open image in new window be a function with Open image in new window for which there exists a function Open image in new window such that there exists a constant Open image in new window satisfying the inequalities
for all Open image in new window . Then there exists a unique quadratic function Open image in new window defined by Open image in new window such that

for all Open image in new window

Proof.

Let us define Open image in new window to be the set of all functions Open image in new window and introduce a generalized metric Open image in new window on Open image in new window as follows:
Then it is easy to show that Open image in new window is complete (see the proof of Theorem Open image in new window of [35]). Now we define an operator Open image in new window by
for all Open image in new window First, we assert that Open image in new window is strictly contractive with constant Open image in new window on Open image in new window . Given Open image in new window , let Open image in new window be an arbitrary constant with Open image in new window that is, Open image in new window Then it follows from (2.3) that

for all Open image in new window that is, Open image in new window Thus we see that Open image in new window for any Open image in new window and so Open image in new window is strictly contractive with constant Open image in new window on Open image in new window .

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

for all Open image in new window which implies Open image in new window

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

which yields inequality (2.4).

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

for all Open image in new window , which implies that Open image in new window is a solution of (1.6) and so the mapping Open image in new window is quadratic.

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

The following theorem is an alternative result of Theorem 2.2.

Theorem.

Let Open image in new window be a function with Open image in new window for which there exists a function Open image in new window such that there exists a constant Open image in new window satisfying the inequalities
for all Open image in new window . Then there exists a unique quadratic function Open image in new window defined by Open image in new window such that

for all Open image in new window

Proof.

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

for all Open image in new window

Then it follows from (2.15) that

for all Open image in new window that is, Open image in new window Thus we see that Open image in new window for any Open image in new window and so Open image in new window is strictly contractive with constant Open image in new window on Open image in new window .

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

for all Open image in new window which implies Open image in new window

Thus according to Open image in new window of Theorem 2.1, there exists a function Open image in new window which is a fixed point of Open image in new window , that is, Open image in new window such that
By Theorem 2.1 Open image in new window we obtain

which yields the inequality (2.16).

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

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

Corollary.

Let Open image in new window be a real number with Open image in new window . Assume that a function Open image in new window with Open image in new window satisfies the inequality
for all Open image in new window . Then there exists a unique quadratic function Open image in new window given by Open image in new window which satisfies the inequality

for all Open image in new window .

Proof.

Letting Open image in new window and then applying Theorem 2.2 with contractive constant Open image in new window , we obtain easily the result.

Corollary.

for all Open image in new window and Open image in new window if Open image in new window . Then there exists a unique quadratic function Open image in new window which satisfies the inequality

for all Open image in new window .

Proof.

Letting Open image in new window for all Open image in new window and then applying Theorem 2.2 with contractive constant Open image in new window and Theorem 2.3 with contractive constant Open image in new window , we obtain easily the results.

3. Stability of (1.6) by Direct Method

In the next two theorems, let Open image in new window be a mapping satisfying one of the conditions

for all Open image in new window .

Theorem.

Assume that a function Open image in new window satisfies
for all Open image in new window and Open image in new window satisfies the condition (3.1). Then there exists a unique quadratic function Open image in new window satisfying

for all Open image in new window

Proof.

Putting Open image in new window in (3.3), we get Open image in new window . Putting Open image in new window in (3.3), we obtain
where Open image in new window for any Open image in new window . Thus it follows from formula (3.7) and triangle inequality that
for all Open image in new window and all Open image in new window which is verified by induction. Therefore we prove from inequality (3.8) that for any integers Open image in new window with Open image in new window
for all Open image in new window . Since the right-hand side of (3.9) tends to zero as Open image in new window , the sequence Open image in new window is a Cauchy sequence for all Open image in new window and thus converges by the completeness of Open image in new window . Define Open image in new window by
Taking the limit in (3.8) as Open image in new window , we obtain that
for all Open image in new window . Letting Open image in new window for all Open image in new window in (3.3), respectively, and dividing both sides by Open image in new window and after then taking the limit in the resulting inequality, we have

so the function Open image in new window is quadratic.

To prove the uniqueness of the quadratic function Open image in new window subject to (3.4), let us assume that there exists a quadratic function Open image in new window which satisfies (1.6) and inequality (3.4). Obviously, we obtain that
for all Open image in new window . Hence it follows from (3.4) that

for all Open image in new window . Therefore letting Open image in new window , one has Open image in new window for all Open image in new window , completing the proof of uniqueness.

Theorem.

Assume that a function Open image in new window satisfies
for all Open image in new window and Open image in new window satisfies condition (3.2). Then there exists a unique quadratic function Open image in new window satisfying
for all Open image in new window . The function Q is given by

for all Open image in new window

Proof.

for Open image in new window .

Therefore we prove from inequality (3.18) that for any integers Open image in new window with Open image in new window
for all Open image in new window . Since the right-hand side of (3.19) tends to zero as Open image in new window , the sequence Open image in new window is a Cauchy sequence for all Open image in new window , and thus converges by the completeness of Open image in new window . Define Open image in new window by
for all Open image in new window . Taking the limit in (3.19) with Open image in new window as Open image in new window , we obtain that
Replacing Open image in new window in (3.3) by Open image in new window , multiplying both sides by Open image in new window and then taking the limit as Open image in new window in the resulting inequality, we have

for all Open image in new window . Therefore the function Open image in new window is quadratic.

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

for all Open image in new window and all Open image in new window . Therefore letting Open image in new window , one has Open image in new window for all Open image in new window . This completes the proof.

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

Corollary.

Let Open image in new window be an Open image in new window -normed space with Open image in new window and Open image in new window a Open image in new window -Banach space, respectively. Let Open image in new window be real numbers with Open image in new window for all Open image in new window , and let Open image in new window be real numbers such that either Open image in new window or Open image in new window . Assume that a function Open image in new window satisfies the inequality
for all Open image in new window and Open image in new window if Open image in new window . Then there exists a unique quadratic function Open image in new window which satisfies the inequality

for all Open image in new window .

Proof.

Letting Open image in new window for all Open image in new window and then applying Theorems 3.1 and 3.2, we obtain easily the results.

We observe that if Open image in new window and Open image in new window 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 Open image in new window bounded by the product of powers of Open image in new window -norms.

Corollary.

for all Open image in new window and Open image in new window if Open image in new window . Then there exists a unique quadratic function Open image in new window which satisfies the inequality

for all Open image in new window and for all Open image in new window if Open image in new window , where Open image in new window if Open image in new window .

Proof.

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

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

Notes

Acknowledgment

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0070940).

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© Eunyoung Son et al. 2010

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Authors and Affiliations

  1. 1.Department of MathematicsChungnam National UniversityDaejeonSouth Korea

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