Abstract and Applied Analysis

Volume 2012 (2012), Article ID 435310, 15 pages

doi:10.1155/2012/435310

Research Article

On a Stability of Logarithmic-Type Functional Equation in Schwartz Distributions

Jae-Young Chung

Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea

Received 29 August 2012; Accepted 18 October 2012

Academic Editor: Abdelghani Bellouquid

Copyright © 2012 Jae-Young Chung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove the Hyers-Ulam stability of the logarithmic functional equation of Heuvers and Kannappan , in both classical and distributional senses. As a classical sense, the Hyers-Ulam stability of the inequality will be proved, where . As a distributional analogue of the above inequality, the stability of inequality will be proved, where and denotes the pullback of distributions.

1. Introduction

The stability problems of functional equations have originated with Ulam in 1940 (see [1]). One of the first assertions to be obtained is the following result, essentially due to Hyers [2], that gives an answer for the question of Ulam.

Theorem 1.1. Suppose that is an additive semigroup, is a Banach space, , , and

for all . Then there exists a unique function satisfying

for which

for all .

As a direct consequence of the result, we obtain the stability of the logarithmic functional equation (see also the result of Forti [3]) as follows:

Theorem 1.2. Let be the set of positive real numbers. Suppose that , , and

for all . Then there exists a unique function satisfying (1.4) for which

for all .

We call the function satisfying (1.4) logarithmic function. The logarithmic functional equation has been modified in various forms [4, 5] and Heuvers and Kannappan introduced the functional equation [6] as follows:

for . In particular, it is shown that the general solution of (1.7) has the form

where is a logarithmic function.

In 1950, Schwartz introduced the theory of distributions in his monograph Théorie des distributions [7]. In this book Schwartz systematizes the theory of generalized functions, basing it on the theory of linear topological spaces, relating all the earlier approaches, and obtaining many important results. After his elegant theory appeared, many important concepts and results on the classical spaces of functions have been generalized to the space of distributions.

Making use of differentiation of distributions, several authors have dealt with functional equations in the spaces of Schwartz distributions, converting given functional equations to differential equations, and finding the solutions in the space of distributions (see [8–11]). However, when we try to consider the Hyers-Ulam stability problems of functional equations, the differentiation is not available for solving them in both the space of infinitely differentiable functions and the space of distributions. In the paper [12], using convolutional approach we initiated the following distributional version of the well-known Hyers-Ulam stability problem for the Cauchy functional equation:

where is the pullback. See Section 3 for the pullback and see below the definition of the norm in (1.9). Using the heat kernel

we proved the stability problems (1.9) in the space of tempered distributions [7] by converting the inequality (1.9) to the classical stability problems

for all ,  , where is an infinitely differentiable function in given by . We also refer the reader to [13] for the stability of Pexider equations in the space of tempered distributions. In [14] we extend the stability problems in the space of tempered distributions to the space of distributions. Instead of the heat kernel, using the regularizing function ,  ,  , where

we prove that the unknown distributions in the functional inequalities are tempered distributions and then use the same method as in [12, 13].

In this paper, developing the previous method in [12–14], we consider a distributional version of the Hyers-Ulam stability of (1.7) in the space of distributions as

where , denotes the pullback of distributions and the inequality in (1.14) means for all test functions defined on . Since the tempered distributions are defined in whole real line or whole space , the methods used in [14] are not available for the inequality (1.14). For the proof of the above problem, we need some technical method than those employed in [12–14]. Indeed, we will show a method to control a functional inequality satisfied in a subset of . As a direct consequence of the result, we obtain the Hyers-Ulam stability of (1.7) in -sense, that is, the Hyers-Ulam stability of the inequality

will be obtained. Finally, we also find locally integrable solutions of (1.7) as a consequence of the stability of the inequality (1.15).

2. Stability in Classical Sense

In this section, we prove the Hyers-Ulam stability of the functional inequality

where and .

Theorem 2.1. Suppose that satisfy (2.1). Then there exists a logarithmic function such that

for all .

Proof. Let ,  . Then we have

for all ,   such that . For given , choose a large so that ,  ,  ,  . Then in view of (2.3), we have

From (2.4)–(2.7), using the triangle inequality we have

for all . Changing the roles of and in (2.3), we can show that

for all . Now we prove that

for all . Replacing by and by in (2.3), we have

Similarly, we have

For given , let . Then, from (2.11)–(2.14), using the triangle inequality we get the inequality (2.10).

Now by Theorem 1.2, there exist functions ,  , satisfying the logarithmic functional equation

for which

Now we show that . From (2.3), we have

From (2.16), (2.17), and (2.19), using the triangle inequality we have

In view of (2.15) and (2.21), we have

for all ,   and . Letting for and letting for , we have for . Since , we have for all . Similarly we can show that . This completes the proof.

Letting in Theorem 2.1, in view of the inequalities (2.4), (2.5), and (2.6), using the triangle inequality we have

for all . Thus by Theorem 1.2 we have the following.

Theorem 2.2. Let satisfy the inequality

for all . Then there exists a logarithmic function such that

for all .

3. Schwartz Distributions

Let be an open subset of . We briefly introduce the space of distributions. We denote by , where is the set of nonnegative integers, and ,  , .

Definition 3.1. Let be the set of all infinitely differentiable functions on with compact supports. A distribution is a linear form on such that for every compact set there exist constants and for which

holds for all with supports contained in . The set of all distributions is denoted by .

Let be open subsets of for , with .

Definition 3.2. Let and be a smooth function such that for each the derivative is surjective, that is, the Jacobian matrix of has rank . Then there exists a unique continuous linear map such that when is a continuous function. We call the pullback of by and is usually denoted by .

In particular, if , the pullbacks ,  ,  can be written as

for all test functions .

Also, if is a diffeomorphism (a bijection with ,   smooth functions), the pullback can be written as

For more details of distributions we refer the reader to [7, 15].

4. Stability in Schwartz Distributions

We employ a function on defined by

where

It is easy to see that is an infinitely differentiable function with support . Let and . Then for each is a smooth function of and as in the sense that

for all . Here after we denote by by

Now we are in a position to prove the Hyers-Ulam stability of the inequality

Recall that the inequality in (4.5) means that for all test functions defined on .

Theorem 4.1. Let satisfy (4.5). Then there exist such that

Proof. Let , and define by

Then is a diffeomorphism with ,  . Taking pullback by in (4.5) and using (3.5), we have

in . Thus it follows that

in , where . Denoting by and convolving in the left hand side of (4.9) we have, in view of (3.2),

Similarly we have, in view of (3.3) and (3.4),

Thus the inequality (4.9) is converted to the classical stability problem

for all and ,  . From now on, we assume that ,  . From the inequality (4.12), we have

for ,

for ,

for , and

for  .

For given , choose . Then in view of (4.13)–(4.16), using triangle inequality, we have

for all . Replacing by , by in (4.12) and changing the positions of and , we have

for all . Now we prove that

for all . From the inequality (4.12), we have

for all such that , , , and . For given , choose . Then in view of (4.20), using triangle inequality, we have

Letting in (4.21), we get the inequality (4.19).

Now in view of (4.17), (4.18), and (4.19), it follows from Theorem 1.1 that for each , there exist functions , satisfying

for which

for all .

Now we prove that . From (4.12), using the triangle inequality, we have

for all . Since as , in view of (4.26) it is easy to see that

exists for all . Similarly, we can show that

exists for all . Putting in (4.12) and letting so that , we have

for all . Similarly, we have

for all . Using (4.23), (4.24), (4.29), and the triangle inequality, we have

for all . From (4.22) and (4.31), we have

for all and all integers with . Letting if and letting if in (4.32) we have for , which implies since . Similarly, using (4.23), (4.25), and (4.30) we can show that .

Finally we prove that is independent of . Fixing and letting so that in (4.12), we have

for all . The same substitution as the inequalities (4.13)–(4.16) gives

Using the stability Theorem [2], we obtain that there exists a unique function satisfying the Cauchy functional equation

for which

Now we show that for all and . Putting in (4.33), we have

for all . From (4.23), (4.36), and (4.37), using the triangle inequality, we have

for all . From (4.38), using the method of proving , we can show that for all and . Thus we have .

Letting in (4.24) so that , we have

for some . Similarly, letting in (4.25) so that , we have

for some . Now we prove the inequality

for some . Putting ,   in (4.33) and using the triangle inequality, we have

From (4.42), there exists a sequence such that converges. Letting in (4.23), we get (4.41). Taking pullback by in (4.39), (4.40), and (4.41), we have

Finally we show that the solution of the Cauchy equation (4.35) has the form for some . Recall that is the supremum limit of a collection of continuous functions . Thus, if we let , then both and are Lebesgue measurable functions. Now, as we see in the proof of Hyers-Ulam stability Theorem [2], the function is given by

Thus, let . Then and are Lebesgue measurable functions as limits of sequences of Lebesgue measurable functions. It is well known that every Lebesgue measurable solution of the Cauchy functional equation (4.35) has the form for some . This completes the proof.

As a consequence of the above result we have the following.

Corollary 4.2. Let , be locally integrable functions satisfying

Then there exist such that

Proof. Every locally integrable function defines a distribution via the equation

Viewing as distributions, the inequality (4.45) implies

By Theorem 4.1, we have

for all . Viewing as a subspace of   (dense subspace) and using the Hahn-Banach theorem we obtain that the inequalities (4.49) hold for all . Now since we get the inequalities (4.46). This completes the proof.

As a direct consequence of the above result we solve the functional equation

in -sense, that is, we obtain the following.

Corollary 4.3. Let , be locally integrable functions satisfying

Then there exist such that

Finally, we discuss the locally integrable solution of (4.50) (c.f. [6]).

Corollary 4.4. Every locally integrable solution of (4.50) has the form

for some .

Proof. It follows from Corollary 4.3 that the equalities (4.53), (4.54), and (4.55) hold in almost everywhere sense, that is, there exists a subset with Lebesgue measure such that the equalities (4.53), (4.54), and (4.55) hold for all . For given , let by . Since , we can choose . Let . Then and , . Thus we can write

which gives (4.53). For given , let by . Then we have . Choose and let . Then . Thus, using (4.53), we can write

which gives (4.54). Finally, the equality (4.55) follows from (4.50), (4.53), and (4.54). This completes the proof.

Acknowledgment

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (no. 2012008507).

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