Stability of a Jensen Type Logarithmic Functional Equation on Restricted Domains and Its Asymptotic Behaviors

Abstract

Let be the set of positive real numbers, a Banach space, and , with . We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality in restricted domains of the form for fixed with or and . As consequences of the results we obtain asymptotic behaviors of the inequality as .

1. Introduction

The stability problems of functional equations have been originated by 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 satisfies the inequality

(1.1)

for all . Then there exists a unique function satisfying

(1.2)

for which

(1.3)

for all .

In 1950-1951 this result was generalized by the authors Aoki [3] and Bourgin [4, 5]. Unfortunately, no results appeared until 1978 when Th. M. Rassias generalized the Hyers' result to a new approximately linear mappings [6]. Following the Rassias' result, a great number of the papers on the subject have been published concerning numerous functional equations in various directions [6–16]. For more precise descriptions of the Hyers-Ulam stability and related results, we refer the reader to the paper of Moszner [17]. Among the results, the stability problem in a restricted domain was investigated by Skof, who proved the stability problem of the inequality (1.1) in a restricted domain [16]. Developing this result, Jung considered the stability problems in restricted domains for the Jensen functional equation [11] and Jensen type functional equations [14]. The results can be summarized as follows: let and be a real normed space and a real Banach space, respectively. For fixed , if satisfies the functional inequalities (such as that of Cauchy, Jensen and Jensen type, etc.) for all with , the inequalities hold for all . We also refer the reader to [18–26] for some interesting results on functional equations and their Hyers-Ulam stabilities in restricted conditions.

Throughout this paper, we denote by the set of positive real numbers, a Banach space, , and with . We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality

(1.4)

in the restricted domains of the form for fixed with or , and . As a result, we prove that if the inequality (1.4) holds for all , there exists a unique function satisfying

(1.5)

for which

(1.6)

for all if ,

(1.7)

for all if , and

(1.8)

for all if . As a consequence of the result we obtain the stability of the inequality

(1.9)

in the restricted domains of the form for fixed with or , and . Also we obtain asymptotic behaviors of the inequalities (1.4) and (1.9) as and , respectively.

2. Hyers-Ulam Stability in Restricted Domains

We call the functions satisfying (1.5)logarithmic functions. As a direct consequence of Theorem 1.1, we obtain the stability of the logarithmic functional equation, viewing as a multiplicative group (see also the result of Forti [9]).

Theorem A.

Suppose that , , and

(2.1)

for all . Then there exists a unique logarithmic function satisfying

(2.2)

for all .

We first consider the usual logarithmic functional inequality (2.1) in the restricted domains .

Theorem 2.1.

Let , with or . Suppose that satisfies

(2.3)

for all , with . Then there exists a unique logarithmic function such that

(2.4)

for all .

Proof.

From the symmetry of the inequality we may assume that . For given , choose a such that , , and . Then we have

(2.5)

This completes the proof.

Now we consider the Hyers-Ulam stability of the Jensen type logarithmic functional inequality (1.4) in the restricted domains .

Theorem 2.2.

Let . Suppose that satisfies

(2.6)

for all , with . Then there exists a unique logarithmic function such that

(2.7)

for all .

Proof.

Replacing by , by in (2.6) we have

(2.8)

for all , with .

For given , choose a such that , , , and . Replacing by , by ; by , by ; by , by ; by , by in (2.8) we have

(2.9)

Now by Theorem A, there exists a unique logarithmic function such that

(2.10)

for all . This completes the proof.

As a matter of fact, we obtain that in Theorem 2.2 provided that and or is a rational number, or and or is a rational number.

Theorem 2.3.

Let , , . Suppose that and or is a rational number, or and or is a rational number, and satisfies

(2.11)

for all , with . Then one has

(2.12)

for all .

Proof.

We prove (2.12) only for the case that and or is a rational number since the other case is similarly proved. From (2.7) and (2.11), using the triangle inequality we have

(2.13)

for all , with , where . If , putting in (2.13) we have

(2.14)

for all , with . It is easy to see that for all and all rational numbers . Thus if is a rational number, it follows from (2.14) that

(2.15)

for all , with . If there exists such that , we can choose a rational number such that and (it is realized when is large if , and when is large if ). Now we have

(2.16)

Thus it follows that . If is a rational number, it follows from (2.14) that

(2.17)

for all , with , which implies

(2.18)

for all , with . Similarly, using (2.18) we can show that . If , choosing such that , putting in (2.13) and using the triangle inequality we have

(2.19)

for all . Similarly, using (2.19) we can show that . Thus the inequality (2.12) follows from (2.7). This completes the proof.

Theorem 2.4.

Let with or . Suppose that satisfies

(2.20)

for all , with . Then there exists a unique logarithmic function such that

(2.21)

for all if , and

(2.22)

for all if .

Proof.

Assume that . For given , choose a such that , , and . Replacing by , by ; by , by ; by , by ; by 1, by in (2.20) we have

(2.23)

Dividing (2.23) by and using Theorem A, we obtain that there exists a unique logarithmic function such that

(2.24)

for all . Assume that . For given , choose a such that and . Replacing by , by ; by , by ; by , by ; by 1, by in (2.20) we have

(2.25)

Dividing (2.25) by and using Theorem A, we obtain that there exists a unique logarithmic function such that

(2.26)

for all . This completes the proof.

From Theorem 2.4, using the same approach as in the proof of Theorem 2.3 we have the following.

Theorem 2.5.

Let , with or . Suppose that and or is a rational number, or and or is a rational number, and satisfies

(2.27)

for all , with . Then one has

(2.28)

for all if , and

(2.29)

for all if .

We call an additive function provided that

(2.30)

for all . Using Theorem 2.2 we have the following.

Corollary 2.6 (see [22]).

Let , with . Suppose that satisfies

(2.31)

for all , with . Then there exists a unique additive function such that

(2.32)

for all .

Proof.

Replacing by , by in (2.31) and setting we have

(2.33)

for all , with . Using Theorem 2.2, we have

(2.34)

for all , which implies

(2.35)

for all . Letting we get the result.

Using Theorem 2.3, we have the following.

Corollary 2.7.

Let , with . Suppose that and or is a rational number, or and or is a rational number, and satisfies

(2.36)

for all , with . Then one has

(2.37)

for all .

Using Theorem 2.4, we have the following.

Corollary 2.8.

Let , with or . Suppose that satisfies

(2.38)

for all , with . Then there exists a unique additive function such that

(2.39)

for all if , and

(2.40)

for all if .

Using Theorem 2.5, we have the following.

Corollary 2.9.

Let , with or . Suppose that and or is a rational number, or and or is a rational number, and satisfies

(2.41)

for all , with . Then one has

(2.42)

for all if , and

(2.43)

for all if .

3. Asymptotic Behavior of the Inequality

In this section, we consider asymptotic behaviors of the inequalities (1.4) and (2.1).

Theorem 3.1.

Let satisfy one of the conditions; , . Suppose that satisfies the asymptotic condition

(3.1)

as . Then is a logarithmic function.

Proof.

By the condition (3.1), for each , there exists such that

(3.2)

for all , with . By Theorem 2.1, there exists a unique logarithmic function such that

(3.3)

for all . From (3.4) we have

(3.4)

for all and all positive integers . Now, the inequality (3.4) implies . Indeed, for all and rational numbers we have

(3.5)

Letting in (3.5), we have . Thus, letting in (3.3), we get the result.

Theorem 3.2.

Let satisfy one of the conditions; , , . Suppose that satisfies the asymptotic condition

(3.6)

as . Then there exists a unique logarithmic function such that

(3.7)

for all .

Proof.

By the condition (3.6), for each , there exists such that

(3.8)

for all , with . By Theorems 2.2 and 2.4, there exists a unique logarithmic function such that

(3.9)

if ,

(3.10)

if , and

(3.11)

if . For all cases (3.9), (3.10), and (3.11), there exists such that

(3.12)

for all and all positive integers . Now as in the proof of Theorem 3.1, it follows from (3.12) that for all . Letting in (3.9), (3.10), and (3.11) we get the result.

Similarly using Theorems 2.3 and 2.5, we have the following.

Theorem 3.3.

Let satisfy one of the conditions; , , . Suppose that and or is a rational number, or and or is a rational number, and satisfies the asymptotic condition

(3.13)

as . Then is a constant function.

Using Corollaries 2.6 and 2.8 we have the following.

Corollary 3.4.

Let , satisfy one of the conditions , , or . Suppose that satisfies

(3.14)

as . Then there exists a unique additive function such that

(3.15)

for all .

Using Corollaries 2.7 and 2.9 we have the following.

Corollary 3.5.

Let , satisfy one of the conditions , , or . Suppose that and or is a rational number, or and or is a rational number, and satisfies

(3.16)

as . Then is a constant function.