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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 751680, 9 pages
http://dx.doi.org/10.1155/2013/751680
Research Article

Stability of a Logarithmic Functional Equation in Distributions on a Restricted Domain

1Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea
2Department of Mathematics, University of Louisville, Louisville, KY 40292, USA

Received 25 May 2013; Accepted 2 August 2013

Academic Editor: Soon-Mo Jung

Copyright © 2013 Jaeyoung Chung and Prasanna K. Sahoo. 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

Let be the set of real numbers, , , and . As classical and versions of the Hyers-Ulam stability of the logarithmic type functional equation in a restricted domain, we consider the following inequalities: , and in the sectors . As consequences of the results, we obtain asymptotic behaviors of the previous inequalities. We also consider its distributional version , where , , , , , and the inequality means that for all test functions .

1. Introduction

The Hyers-Ulam stability problem of functional equations was originated in 1940 when Ulam proposed a question concerning the approximate homomorphisms from a group to a metric group (see [1]). A partial answer was given by Hyers et al. [2, 3] under the assumption that the target space of the involved mappings is a Banach space. It is possible to prove stability results similar to Hyers for functions that do not have bounded Cauchy difference. In 1950, Aoki [4] first proved such a result for additive functions. Bourgin [5, 6] and Aoki [4] studied the Ulam problem from 1949 to 1951. The area rested there for a while until 1978 when Rassias [7] published a generalized version of Hyers’ result on linear mappings, where the Cauchy difference was allowed to be unbounded. Rassias’ work provided an impetus for the study on the stability of functional equations (see [2, 731]).

Let be the set of real numbers, the set of positive real numbers, and the set of complex numbers. The subset, for fixed real number , of the plane, , will be referred to as a sector. A function is said to be logarithmic if and only if it satisfies the logarithmic functional equation: for all . There are several variants of logarithmic functional equations (see [1416]). It was shown by Heuvers and Kannappan [16] that the logarithmic functional equation is equivalent to the following functional equation: They have also studied the following pexiderized version of (3): The general solution of the functional equation (4) has the form (see [16]) where is a logarithmic function and are arbitrary constants.

In this paper, we study Hyers-Ulam stability of the functional equation (4). In Section 2, we treat the Hyers-Ulam stability of the functional equation (4) in the classical sense and present its asymptotic behavior. In Section 3, we consider the stability of (4) in -sense and its asymptotic behavior. Finally, in Section 4 we present the stability of (4) in Schwartz distributions.

2. Stability of (4) in Classical Sense and Its Asymptotic Behavior

In this section, we consider the classical Hyers-Ulam stability of the functional equation (4) on the sector and then study its asymptotic behavior.

The following theorem is a direct consequence of the Hyers’ result [3] (see also result of Forti [32]).

Theorem 1. Let be a nonnegative real number. Suppose that satisfies for all . Then there exists a unique logarithmic function such that

Next, we establish the Hyers-Ulam stability of the functional equation (4) on the restricted domain .

Theorem 2. Suppose that , , and , , satisfy the functional inequality for all . Then there exists a unique logarithmic function such that for all .

Proof. For given , choose a real number such that and let Then it is easy to check that , for all . Replacing , by , in (8), respectively, for we have From (12)–(15), using the triangle inequality we have for all . Similarly, for given , choose such that and let Then it is easy to check that , for all . Next, replacing , by , in (8), respectively, for , we have From (19), using the triangle inequality, we have for all . Now we prove that for all . For given , choose such that and let Then , for all . Replacing , by , in (8), respectively, for , we have From (24), using the triangle inequality we get (21).
Now by Theorem 1, there exist for satisfying the logarithmic functional equation for which Now we show that . Putting and in (12) separately, we have From (26), (27), and (29), using the triangle inequality we have Let . Then we can choose a positive integer such that for all integers . In view of (25), and (31) we have for all integer . Letting in (32), we have for all . For , we have . Thus, we have for all . Similarly, using (26), (28), and (30) we can show that . The uniqueness of the logarithmic function is obvious. This completes the proof of the theorem.

Letting in Theorem 2 and using the inequalities (12)–(14) together with the triangle inequality, we obtain for all . Thus, by Theorem 1 we have the following theorem.

Theorem 3. Let . Suppose that satisfies the functional inequality for all . Then there exists a unique logarithmic function such that

Now we prove the following asymptotic result concerning (8).

Theorem 4. Suppose that satisfy the asymptotic condition as . Then there exists a logarithmic function and such that for all .

Proof. By the condition (36), for any positive integer , there exists such that for all with . By Theorem 1, there exists a logarithmic function such that for all . Replacing by in (39) and using the triangle inequality, we have for all . Thus, we obtain for all and . Letting in (43), we have for all . Finally, letting in (39), (40), and (41), we have for all . Finally, substituting (44) in (36) we get . Letting and we obtain the asserted result.

3. Stability of (4) in -Sense and Its Asymptotic Behavior

In this section, we consider the Hyers-Ulam stability of the functional equation (4) in -sense on the sector and then examine its asymptotic behavior. Consider the functional inequality where and is fixed, where denotes the essential supremum norm of on the set . We employ the function on defined by where It is easy to see that is an infinitely differentiable function with support . Let be a locally integrable function and ,  . Then for each , is a smooth function of and for almost every as .

Now we prove the Hyers-Ulam stability of the functional equation (4) in -sense on the sector .

Theorem 5. Let , , be locally integrable functions satisfying (45). Then there exist constants such that

Proof. We will use the diffeomorphism Let , and . Then, we have Thus, we have . Consequently, (45) is converted to Now, let Then, we have
For each and , we have We also have Similarly, we have On the other hand, let and , . Then, we have Let . Then it follows from (54)~(58) that Thus, we have the functional inequality for all and , . From now on, we assume that , . From (60), we have for , for , for , for .
For given , choose . Then, using the triangle inequality with (61)~(64), we have for all . Replacing by , by in (60) and changing the roles of and , we have for all . Now we prove that for all . From (60), we have for all , , such that , , , and . For given , choose . Using the triangle inequality with (68), we have Letting in (69), we get (67).
Applying Hyers’ stability theorem from [3] for (65), (66), and (67), we obtain that for each there exist functions ,  , satisfying for which for all .
Now we prove that . From (60), using the triangle inequality we have for all . Since as , in view of (74) it is easy to see that exists for all . Similarly, we can show that exists for all . Putting in (60) and letting so that we have for all . Similarly, we have for all . Using (71), (72), (77), and the triangle inequality, we have for all . From (71) and (80), we have for all ,  , and all integers with . Letting if and letting if in (80), we have for , which implies since . Similarly, using (71), (73), and (78) we obtain that .
Finally, we prove that is independent of . Fixing and letting so that in (60), we have for all . From (81), using the same substitutions as in (61)~(64) we have By Hyers’ stability theorem [3], there exists a unique function satisfying the Cauchy functional equation for which Now we show that for all and . Putting in (81), we have for all . From (71), (84), and (85), using the triangle inequality we have for all . From (86), using the method of proving we can show that for all and . Thus, we have .
Letting in (72) so that , we have Similarly, letting in (73) so that , we have Now we prove the inequality For given , choosing such that replacing by and by in (81), and using the triangle inequality, we have From (90), it is easy to see that exists for all . Letting in (71) so that , we get (89). Replacing by in (87), (88), and (89), we have Finally, we show that the solution of the Cauchy equation (83) has the form for some . Since is the supremum limit of a collection of continuous functions ,  ,   is a Lebesgue measurable function. Also, as we see in the proof of Hyers-Ulam stability theorem (see [3]), the function is given by Thus, is a Lebesgue measurable function since it is the limit of a sequence of Lebesgue measurable functions. It is well known that every Lebesgue measurable solution of the Cauchy functional equation (83) has the form for some . Letting ,  ,   we get the asserted result.

Now we discuss an asymptotic behavior of the inequality (45).

Theorem 6. Let ,  , be locally integrable functions satisfying as . Then there exist constants such that

Proof. By the condition (94), for any positive integer there exists such that for all with . Now by Theorem 5, there exist constants (which are independent of ) such that Letting in (97), we obtain the asserted result.

As a direct consequence of the previous result we have found the solution of functional equation (4) in the -sense.

Corollary 7. Let be locally integrable functions satisfying for all . Then there exist such that

Finally, we discuss the locally integrable solution of the functional equation (c.f. [16]) for all . The following result is a direct consequence of Theorem 2. However, we introduce an alternative proof using Corollary 7. The following method of proof will be useful when we know only regular solution in -sense.

Corollary 8. Every locally integrable solution of the functional equation (100) has the form for some constants .

Proof. It follows from Corollary 7 that (101), (102), and (103) hold in almost everywhere sense; that is, there exists a subset with Lebesgue measure such that (101), (102), and (103) hold for all . For given , let by ,  . Since , we can choose . Let . Then and ,  . Thus, we can write which gives (101). For given , let by . Then, we have . Choose and let . Then , . Thus, using (101) we can write which gives (102). Finally, (103) follows from (100), (101), and (102). This completes the proof of the corollary.

4. Stability of (4) in Schwartz Distributions

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

Definition 9. 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 10. Let and let 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 it is usually denoted by .

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 [29, 33].

In this section, we consider the Hyers-Ulam stability of the functional equation of (4) in Schwartz distributions, that is, the functional inequality where ,  ,  and are defined by and the inequality in (108) means that for all test functions . For each , is a smooth function of and as in the sense that for all .

Theorem 11. Let satisfy (108). Then there exist constants such that

Proof. The idea of the following proof is essentially the same as that of Theorem 5, only with different terminologies. For the reader we give a sketch of proof. Let and be the set and mapping in the proof of Theorem 5, respectively. Then, is given by Taking pullback by in (108), we have where ,   are given by Thus, instead of (54) we have the inequality where ,  ,  and . Using the same approach as in the proof of Theorem 5, we have for some . Taking pullback by in (116), we have for some constants . This completes the proof of the theorem.

Acknowledgment

This work was done during the first author’s visit to the University of Louisville from Kunsan National University during 2012-13. This work was supported by the 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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