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A generalized Hyers-Ulam stability of a Pexiderized logarithmic functional equation in restricted domains

Abstract

Let + and B be the set of positive real numbers and a Banach space, respectively, f, g, h : +B and ψ: + 2 be a nonnegative function of some special forms. Generalizing the stability theorem for a Jensen-type logarithmic functional equation, we prove the Hyers-Ulam stability of the Pexiderized logarithmic functional inequality

| | f ( x y ) - g ( x ) - h ( y ) | | ψ ( x , y )

in restricted domains of the form {(x, y) : xkysd} for fixed k, s , d > 0. We also discuss an L-version of the Hyers-Ulam stability of the inequality. 2000 MSC: 39B22.

1. Introduction

The Hyers-Ulam stability problems of functional equations go back to 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 [2, 3] under the assumption that the target space of the involved mappings is a Banach space. After the result of Hyers, Aoki [4] and Bourgin [5, 6] treated with this problem, however, there were no other results on this problem until 1978 when Rassias [7] treated again with the inequality of Aoki [4]. Following the Rassias' result a great number of articles on the subject have been published concerning numerous functional equations in various directions [819]. Among the results, the stability problem in a restricted domain was investigated by Skof, who proved the stability problem of the Cauchy functional equation in a restricted domain [20]. Developing this result, Jung, Rassias and Rassias considered the stability problems in restricted domains for the Jensen functional equation [21, 22] and Jensen-type functional equations [23]. We also refer the reader to [2429] for some interesting results on functional equations and their Hyers-Ulam stabilities in restricted conditions. In this article, generalizing the result in [8], we consider the Hyers-Ulam stability of the Pexiderized Jensen functional equation

| | f ( x y ) - g ( x ) - h ( y ) | | ψ ( x , y )
(1.1)

in the restricted domains U k,s,d = {(x, y): x > 0, y > 0, xkysd} for fixed k, s and d > 0, where ψ(x, y) = ϕ (xy), ϕ (x) or ϕ (y). Making use of the result, we prove the asymptotic behavior of f, g and h satisfying

| | f ( x y ) - g ( x ) - h ( y ) | | 0
(1.2)

as xkys→∞. Finally, we discuss the Hyers-Ulam stability of the inequality

| | f ( x y ) - g ( x ) - h ( y ) | | L ( U k , s , d ) ε
(1.3)

and its asymptotic behavior.

2. Stability in classical sense

We call L: +B a logarithmic function provided that

L ( x y ) - L ( x ) - L ( y ) = 0

for all x, y > 0. Let ϕ : + → [0, ∞). We assume that

Φ ( x ) : = k = 1 2 - k ϕ ( x 2 k ) + 2 ϕ ( x 2 k - 1 ) + ϕ ( 1 ) <

for all x > 0. As a direct consequence of Aoki [4] or Bourgin [5, 6], we obtain the generalized Hyers-Ulam stability for the logarithmic functional equation, viewing 〈+, ×〉 as a multiplicative group.

Theorem A. Suppose that f : +B satisfies

| | f ( x y ) - f ( x ) - f ( y ) | | ϕ ( x y ) + ϕ ( x ) + ϕ ( y ) + ϕ ( 1 )

for all x, y > 0. Then, there exists a unique logarithmic function L : +B satisfying

| | f ( x ) - L ( x ) | | Φ ( x )

for all x > 0.

In this section, we first consider the logarithmic functional inequality (1.1) in the restricted domain

U k , s , d = { ( x , y ) : x > 0 , y > 0 , x k y s d }

for fixed k, s and d > 0.

Theorem 2.1. Let d > 0, k, s , ks. Suppose that f, g, h : +B satisfy

| | f ( x y ) - g ( x ) - h ( y ) | | ϕ ( x y )
(2.1)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L1 : +B such that

| | f ( x ) - L 1 ( x ) - f ( 1 ) | | Φ ( x )
(2.2)

for all x +.

Proof. For given x,y +, choosing a z > 0 such that xkyszs-kd, xkzs-kd, yszs-kd and zs-kd, we have

| | f ( x y ) - f ( x ) - f ( y ) + f ( 1 ) | | | | f ( x y ) - g ( x z - 1 ) - h ( y z ) | | + | | - f ( x ) + g ( x z - 1 ) + h ( z ) | | + | | - f ( y ) + g ( z - 1 ) + h ( y z ) | | + | | f ( 1 ) - g ( z - 1 ) - h ( z ) | | ϕ ( x y ) + ϕ ( x ) + ϕ ( y ) + ϕ ( 1 ) .
(2.3)

Now, by Theorem A, we get the result.

Corollary 2.2. Let ϵ,d > 0, k, s , ks. Suppose that f, g, h : +B satisfy

| | f ( x y ) - g ( x ) - h ( y ) | | ε
(2.4)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L1: +B such that

| | f ( x ) - L 1 ( x ) - f ( 1 ) | | 4 ε
(2.5)

for all x +.

Remark 2.1. Note that the Corollary 2.2 fails if k = s. Indeed, let L: +B be a nonzero logarithmic function. Define g(x) = h(x) = L(x) for all x > 0 and

f ( x ) = L ( x ) , x d 1 / s , 0 , 0 < x < d 1 / s .

Then, it is easy to see that the inequality (2.4) holds for all x, y > 0, with xyd1/s. Assume that there exists a logarithmic function L1 satisfying (2.5). Then, we have

| | L 1 ( x ) | | | f ( 1 ) | + 4 ε = 4 ε
(2.6)

for all 0 < x < d1/s. The inequality (2.6) implies L1 = 0. Indeed, if L1(x0) ≠ 0 for some x0 > 0, then we have L1(1/x0) = -L1(x0) ≠ 0. Thus, we may assume that 0 < x0 < 1. Now, we encounter the contradiction

| n L 1 ( x 0 ) | = | L 1 ( x 0 n ) | 4 ε

for all large integers n. Thus, L1 = 0 and the inequality (2.5) implies

| | L ( x ) | | 4 ε
(2.7)

for all xd1/s. Similarly, using (2.7), we can show that L = 0, which contracts to the choice of L.

As a direct consequence of Corollary 2.2, we have the following.

Corollary 2.3. [8] Let p,q,P,Q be nonzero real numbers and ε , d > 0 , k , s , k p s q . Suppose that f : +B satisfies

| | f ( x p y q ) - P f ( x ) - Q f ( y ) | | ε
(2.8)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L : +B such that

| | f ( x ) - L ( x ) - f ( 1 ) | | 4 ε
(2.9)

for all x +.

Proof. Replacing x by x 1 p , y by y 1 q in (2.8), we have

| | f ( x y ) - P f ( x 1 p ) - Q f ( y 1 q ) | | ε

for all x,y > 0, with x k p y s q d. Letting g ( x ) =Pf ( x 1 p ) ,h ( y ) =Qf ( y 1 q ) , applying Corollary 2.2 and letting L(x) = L1(x), we get the result.

Theorem 2.4. Let d > 0, s ≠ 0. Suppose that f, g, h: +B satisfy

| | f ( x y ) - g ( x ) - h ( y ) | | ϕ ( x )
(2.10)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L2 : +B such that

| | g ( x ) - L 2 ( x ) - g ( 1 ) | | Φ ( x )
(2.11)

for all x +.

Proof. For given x,y +, choosing a z > 0 such that xkykzsd, xkyszsd, ykzsd and yszsd, we have

| | g ( x y ) - g ( x ) - g ( y ) + g ( 1 ) | | | | - f ( x y z ) + g ( x y ) + h ( z ) | | + | | f ( x y z ) - g ( x ) - h ( y z ) | | + | | f ( y z ) - g ( y ) - h ( z ) | | + | | - f ( y z ) + g ( 1 ) + h ( y z ) | | ϕ ( x y ) + ϕ ( x ) + ϕ ( y ) + ϕ ( 1 ) .
(2.12)

Now, by Theorem A, we get the result.

Corollary 2.5. Let ϵ, d > 0, s ≠ 0. Suppose that f, g, h : +B satisfy

| | f ( x y ) - g ( x ) - h ( y ) | | ε
(2.13)

for all x, y U k,s,d . Then, there exists a unique logarithmic function L2: +B such that

| | g ( x ) - L 2 ( x ) - g ( 1 ) | | 4 ε
(2.14)

for all x +.

Remark 2.2. Similarly as in Corollary 2.2, the above result fails if s = 0. Let L : +B be a nonzero logarithmic function. Define f(x) = h(x) = L(x) for all x > 0 and

g ( x ) = L ( x ) , x d 1 / k , 0 , 0 < x < d 1 / s .

Then, the inequality (2.13) holds for all x, y > 0, with xkd but (2.14) does not hold for any logarithmic function L2.

As a direct consequence of Corollary 2.5, we have the following.

Corollary 2.6. [8] Let p, q, P, Q be nonzero real numbers and ϵ, d > 0, k, s with s ≠ 0. Suppose that f : +B satisfies

| | f ( x p y q ) - P f ( x ) - Q f ( y ) | | ε
(2.15)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L : +B such that

| | f ( x ) - L ( x ) - f ( 1 ) | | 4 ε | P |
(2.16)

for all x +.

Proof. Replacing x by x 1 p , y by y 1 q in (2.15), we have

| | f ( x y ) - P f ( x 1 p ) - Q f ( y 1 q ) | | ε

for all x,y > 0, with x k p y s q d. Letting g ( x ) =Pf ( x 1 p ) ,h ( y ) =Qf ( y 1 q ) , applying Corollary 2.5 and dividing the result by |P|, we get the result with L ( x ) = 1 P L 2 ( x p ) .

Theorem 2.7. Let d > 0, k ≠ 0. Suppose that f, g, h : +B satisfy

| | f ( x y ) - g ( x ) - h ( y ) | | ϕ ( y )
(2.17)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L3 : +B such that

| | h ( x ) - L 3 ( x ) - h ( 1 ) | | Φ ( x )
(2.18)

for all x +.

Proof. For given x,y +, choosing a z > 0 such that xsyszkd, xkyszkd, xszkd and xkzkd, we have

| | h ( x y ) - h ( x ) - h ( y ) + h ( 1 ) | | | | - f ( x y z ) + g ( z ) + h ( x y ) | | + | | f ( x y z ) - g ( x z ) - h ( y ) | | + | | f ( z x ) - g ( z ) - h ( x ) | | + | | - f ( x z ) + g ( x z ) + h ( 1 ) | | ϕ ( x y ) + ϕ ( x ) + ϕ ( y ) + ϕ ( 1 ) .
(2.19)

Now, by Theorem A, we get the result.

Corollary 2.8. Let ϵ, d > 0, k ≠ 0. Suppose that f, g, h: +B satisfy

| | f ( x y ) - g ( x ) - h ( y ) | | ε
(2.20)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L3 : +B such that

| | h ( x ) - L 3 ( x ) - h ( 1 ) | | 4 ε
(2.21)

for all x +.

Remark 2.3. Similarly, as in Remark 2.2, we can show that the above result fails if k = 0. Also, as a direct consequence of the result, we have the following.

Corollary 2.9. [8] Let p, q, P, Q be nonzero real numbers and ϵ, d > 0, k, s with k ≠ 0. Suppose that f : +B satisfies

| | f ( x p y q ) - P f ( x ) - Q f ( y ) | | ε
(2.22)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L: +B such that

| | f ( x ) - L ( x ) - f ( 1 ) | | 4 ε | Q |
(2.23)

for all x +.

Theorem 2.10. Let ϵ, d > 0, k, s ≠ 0, ks. Suppose that f, g, h : +B satisfy

| | f ( x y ) - g ( x ) - h ( y ) | | ε
(2.24)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L : +B such that

| | f ( x ) - L ( x ) - f ( 1 ) | | 4 ε , | | g ( x ) - L ( x ) - g ( 1 ) | | 4 ε , | | h ( x ) - L ( x ) - h ( 1 ) | | 4 ε

for all x +.

Proof. In view of Corollaries 2.2, 2.5 and 2.8, it suffices to prove that L1 = L2 = L3. For given x,y > 0, choose a z > 0 such that xkyszs-kd, zs-kd. Then, in view of (2.24), we have

| | f ( x y ) - g ( x z - 1 ) - h ( y z ) | | ε ,
(2.25)
| | - f ( 1 ) + g ( z - 1 ) + h ( z ) | | ε .
(2.26)

Using the inequalities (2.10) and (2.15), we have

| | g ( x z - 1 ) - g ( x ) - g ( z - 1 ) + g ( 1 ) | | 4 ε ,
(2.27)
| | h ( y z ) - h ( z ) - h ( y ) + h ( 1 ) | | 4 ε
(2.28)

for all x,y,z > 0. From (2.25)-(2.28), using the triangle inequality, we have

| | f ( x y ) - g ( x ) - h ( y ) - f ( 1 ) + g ( 1 ) + h ( 1 ) | | 10 ε
(2.29)

for all x,y > 0. From the inequalities (2.5), (2.14), (2.21), (2.29) using the triangle inequality, we have

| | L 1 ( x y ) - L 2 ( x ) - L 3 ( y ) | | 22 ε .
(2.30)

Putting y = 1 and x = 1 in (2.30) separately, and using the fact that for all x > 0, n , L j (xn) = nL j (x), j = 1,2,3, we can show that L1 = L2 and L1 = L3. This completes the proof.

As a direct consequence of Theorem 2.10, we have the following.

Corollary 2.11. [8] Let p, q, P ,Q be nonzero real numbers and ϵ, d > 0, k, s with k ≠ 0, s ≠ 0 and ks. Suppose that f : +B satisfies

| | f ( x p y q ) - P f ( x ) - Q f ( y ) | | ε
(2.31)

for all x,y U k,s,d . Then, there exists a unique logarithmic function L: +B such that

| | f ( x ) - L ( x ) - f ( 1 ) | | min 4 ε , 4 ε | P | , 4 ε | Q |
(2.32)

for all x +.

3. Asymptotic behaviors

In this section, we consider asymptotic behaviors of f,g, h satisfying (1.2).

Theorem 3.1. Let k, s , ks. Suppose that f, g, h : +B satisfy the asymptotic condition

| | f ( x y ) - g ( x ) - h ( y ) | | 0
(3.1)

as xkys→ ∞. Then, there exists a unique logarithmic function L : +B such that

f ( x ) = L ( x ) + f ( 1 )
(3.2)

for all x +.

Proof. By the condition (3.1), for each n , there exists d n > 0 such that

| | f ( x y ) - g ( x ) - h ( y ) | | 1 n
(3.3)

for all x, y > 0, with xkysd n . By Corollary 2.2, there exists a unique logarithmic function L n : +B such that

| | f ( x ) - L n ( x ) - f ( 1 ) | | 4 n
(3.4)

for all x +. Replacing n by m in (3.4) and using the triangle inequality we have

| | L n ( x ) - L m ( x ) | | 4 n + 4 m 8
(3.5)

for all x +. Now, for all x > 0 and all rational numbers r > 0, we have

| | L n ( x ) - L m ( x ) | | = 1 r | | L n ( x r ) - L m ( x r ) | | 8 r .
(3.6)

Letting r → ∞ in (3.6), we have L n = L m . Letting n → ∞ in (3.4), we get the result.

Using Corollary 2.5, we obtain the following.

Theorem 3.2. Let s ≠ 0. Suppose that f, g, h : +B satisfy the asymptotic condition

| | f ( x y ) - g ( x ) - h ( y ) | | 0
(3.7)

as xkys→ ∞. Then, there exists a unique logarithmic function L: +B such that

g ( x ) = L ( x ) + g ( 1 )
(3.8)

for all x +.

Using Corollary 2.8, we obtain the following.

Theorem 3.3. Let k ≠ 0. Suppose that f, g, h : +B satisfies the asymptotic condition

| | f ( x y ) - g ( x ) - h ( y ) | | 0
(3.9)

as xkys→ ∞. Then, there exists a unique logarithmic function L : +B such that

h ( x ) = L ( x ) + h ( 1 )
(3.10)

for all x +.

Theorem 3.4. Let k, s ≠ 0 and ks. Suppose that f, g, h : +B satisfy the asymptotic condition

| | f ( x y ) - g ( x ) - h ( y ) | | 0
(3.11)

as xkys→ ∞. Then, there exists a unique logarithmic function L : +B and c1, c2 B such that

f ( x ) = L ( x ) + c 1 + c 2 , g ( x ) = L ( x ) + c 1 , h ( x ) = L ( x ) + c 2

for all x +.

Proof. By the condition (3.11), for each n , there exists d n > 0 such that

| | f ( x y ) - g ( x ) - h ( y ) | | 1 n
(3.12)

for all x, y > 0, with xkysd n . By Theorem 2.10, there exists a unique logarithmic function L n : +B such that

| | f ( x ) - L n ( x ) - f ( 1 ) | | 4 n ,
(3.13)
| | g ( x ) - L n ( x ) - g ( 1 ) | | 4 n ,
(3.14)
| | h ( x ) - L n ( x ) - h ( 1 ) | | 4 n
(3.15)

for all x +. Similarly, as in the proof of Theorem 3.1, we have L n = L m for all n, m . Letting n → ∞ in (3.13)-(3.15), and using (3.11), we get the result.

4. Stability in L-sense and its asymptotic behavior

Let f, g, h be locally integrable functions on +. In this section, we consider the L-version of Hyers-Ulam stability of the inequality

| | f ( x y ) - g ( x ) - h ( y ) | | L ( U k , s , d ) ε ,
(4.1)

where k ≠ 0, s ≠ 0, ks, d > 0 are fixed and U k,s,d = {(x, y): xkysd}. Let ω on be a nonnegative infinitely differentiable function satisfying the conditions

supp ω { x : | x | 1 }

and

ω ( x ) d x = 1 .

Let ω t (x): = t-1ω(x/t), t > 0 and f be a locally integrable function. Then, for each t > 0,f * ω t (x) = ∫ f(y)ω t (x - y) dy is a smooth function of x and f * ω t (x) → f(x) for almost every x as t → 0+. Now, we are in a position to prove the Hyers-Ulam stability of the inequality (3.1).

Theorem 4.1. Let f, g, h be locally integrable functions satisfying (3.1). Then, there exist c1, c2, c3, a such that

| | f ( x ) - c 1 - a ln x | | L ( + ) 4 ε , | | g ( x ) - c 2 - a ln x | | L ( + ) 4 ε , | | h ( x ) - c 3 - a ln x | | L ( + ) 4 ε .

Proof. Using the change of variables x by 2xand y by 2yin (4.1), we have

| | f ( 2 x + y ) - g ( 2 x ) - h ( 2 y ) | | L ( U d ) ε ,
(4.2)

where U d = { ( x , y ) : k x + s y log 2 d : = d 1 } . Now, let

F ( x ) = f ( 2 x ) , G ( x ) = g ( 2 x ) , H ( x ) = h ( 2 x ) .
(4.3)

Then, we have

| | F ( x + y ) - G ( x ) - H ( y ) | | L ( U d ) ε .
(4.4)

Convolving ω t (x)ω s (y) in (4.4) as in the proof of [8, Theorem 3.1], we have

| F * ω t * ω s ( x + y ) - G * ω t ( x ) - H * ω s ( y ) | ε
(4.5)

holds for all kx+sy d 2 := d 1 + k 2 + s 2 and 0 < t < 1, 0 < s < 1. Using the same method as in [9, Theorem 4.3], we get the result.

Now, we discuss an asymptotic behavior of the inequality (4.1).

Theorem 4.2. Let f, g, h : +, j = 1, 2, 3, be locally integrable functions satisfying

| | f ( x y ) - g ( x ) - h ( y ) | | L ( U k , s , d ) 0
(4.6)

as d → ∞. Then, there exist a, c1, c2, c3 such that

| | f ( x ) - c 1 - a ln x | | L ( + ) = 0 , | | g ( x ) - c 2 - a ln x | | L ( + ) = 0 , | | h ( x ) - c 3 - a ln x | | L ( + ) = 0 .

Proof. By the condition (4.6), for any positive integer n, there exists d n > 0 such that

| | f ( x y ) - g ( x ) - h ( y ) | | L ( U k , s , d n ) 1 n
(4.7)

for all x,y U k , s , d n . Now, by Theorem 4.1, there exist a, c1, c2, c3 (which are independent of n) such that

| | f ( x ) - c 1 - a ln x | | L ( + ) 4 n ,
(4.8)
| | g ( x ) - c 2 - a ln x | | L ( + ) 4 n ,
(4.9)
| | h ( x ) - c 3 - a ln x | | L ( + ) 4 n .
(4.10)

Letting n → ∞ in (4.8)-(4.10), we get the result.

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Acknowledgements

This study 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. 2010-0016963).

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Correspondence to Jae-Young Chung.

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Chung, JY. A generalized Hyers-Ulam stability of a Pexiderized logarithmic functional equation in restricted domains. J Inequal Appl 2012, 15 (2012). https://doi.org/10.1186/1029-242X-2012-15

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Keywords

  • logarithmic functional equation
  • Hyers-Ulam stability
  • asymptotic behavior