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

Abstract

Let ℝ₊ and B be the set of positive real numbers and a Banach space, respectively, f, g, h : ℝ₊ → B and $\psi :{ℝ}_{+}^2\to ℝ$ 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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \psi \left(x,y\right)$$

in restricted domains of the form {(x, y) : x^ky^s≥ d} for fixed k, s ∈ ℝ, 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 [8–19]. 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 [24–29] 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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \psi \left(x,y\right)$$

(1.1)

in the restricted domains U _k,s,d = {(x, y): x > 0, y > 0, x^ky^s≥ d} for fixed k, s ∈ ℝ 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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\to 0$$

(1.2)

as x^ky^s→∞. Finally, we discuss the Hyers-Ulam stability of the inequality

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)|{|}_{L^{\infty }\left(U_{k,s,d}\right)}\le \epsilon$$

(1.3)

and its asymptotic behavior.

2. Stability in classical sense

We call L: ℝ₊ → B a logarithmic function provided that

$$L\left(xy\right)-L\left(x\right)-L\left(y\right)=0$$

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

$$\Phi \left(x\right):=\sum _{k=1}^{\infty }{2}^{-k}\left(\varphi \left(x^{2^k}\right)+2\varphi \left(x^{2^{k-1}}\right)+\varphi \left(1\right)\right)<\infty$$

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 〈ℝ₊, ×〉 as a multiplicative group.

Theorem A. Suppose that f : ℝ₊ → B satisfies

$$\left|\right|f\left(xy\right)-f\left(x\right)-f\left(y\right)\left|\right|\le \varphi \left(xy\right)+\varphi \left(x\right)+\varphi \left(y\right)+\varphi \left(1\right)$$

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

$$\left|\right|f\left(x\right)-L\left(x\right)\left|\right|\le \Phi \left(x\right)$$

for all x > 0.

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

$$U_{k,s,d}=\left\{\left(x,y\right):x>0,y>0,x^k{y}^s\ge d\right\}$$

for fixed k, s ∈ ℝ and d > 0.

Theorem 2.1. Let d > 0, k, s ∈ ℝ, k ≠ s. Suppose that f, g, h : ℝ₊ → B satisfy

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \varphi \left(xy\right)$$

(2.1)

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

$$\left|\right|f\left(x\right)-L_1\left(x\right)-f\left(1\right)\left|\right|\le \Phi \left(x\right)$$

(2.2)

for all x ∈ ℝ₊.

Proof. For given x,y ∈ ℝ₊, choosing a z > 0 such that x^ky^sz^s-k≥ d, x^kz^s-k≥ d, y^sz^s-k≥ d and z^s-k≥ d, we have

$$\begin{array}{ll}\hfill \left|\right|f\left(xy\right)-f\left(x\right)-f\left(y\right)+f\left(1\right)\left|\right|& \le \left|\right|f\left(xy\right)-g\left(x{z}^{-1}\right)-h\left(yz\right)\left|\right|\\ +\left|\right|-f\left(x\right)+g\left(x{z}^{-1}\right)+h\left(z\right)\left|\right|\\ +\left|\right|-f\left(y\right)+g\left(z^{-1}\right)+h\left(yz\right)\left|\right|\\ +\left|\right|f\left(1\right)-g\left(z^{-1}\right)-h\left(z\right)\left|\right|\\ \le \varphi \left(xy\right)+\varphi \left(x\right)+\varphi \left(y\right)+\varphi \left(1\right).\end{array}$$

(2.3)

Now, by Theorem A, we get the result.

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \epsilon$$

(2.4)

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

$$\left|\right|f\left(x\right)-L_1\left(x\right)-f\left(1\right)\left|\right|\le 4\epsilon$$

(2.5)

for all x ∈ ℝ₊.

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

$$f\left(x\right)=\left\{\begin{array}{cc}L\left(x\right),\hfill & x\ge d^{1/s},\hfill \\ 0,\hfill & 0<x<d^{1/s}.\hfill \end{array}\right.$$

Then, it is easy to see that the inequality (2.4) holds for all x, y > 0, with xy ≥ d^1/s. Assume that there exists a logarithmic function L₁ satisfying (2.5). Then, we have

$$\left|\right|L_1\left(x\right)\left|\right|\le \left|f\left(1\right)\right|+4\epsilon =4\epsilon$$

(2.6)

for all 0 < x < d^1/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

$$\left|n{L}_1\left(x_0\right)\right|=\left|L_1\left(x_0^n\right)\right|\le 4\epsilon$$

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

$$\left|\right|L\left(x\right)\left|\right|\le 4\epsilon$$

(2.7)

for all x ≥ d^1/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 $\epsilon ,d>0,k,s\in ℝ,\frac{k}{p}\ne \frac{s}{q}$. Suppose that f : ℝ₊ → B satisfies

$$\left|\right|f\left(x^p{y}^q\right)-Pf\left(x\right)-Qf\left(y\right)\left|\right|\le \epsilon$$

(2.8)

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

$$\left|\right|f\left(x\right)-L\left(x\right)-f\left(1\right)\left|\right|\le 4\epsilon$$

(2.9)

for all x ∈ ℝ₊.

Proof. Replacing x by $x^{\frac{1}{p}}$, y by $y^{\frac{1}{q}}$ in (2.8), we have

$$\left|\right|f\left(xy\right)-Pf\left(x^{\frac{1}{p}}\right)-Qf\left(y^{\frac{1}{q}}\right)\left|\right|\le \epsilon$$

for all x,y > 0, with $x^{\frac{k}{p}}{y}^{\frac{s}{q}}\ge d$. Letting $g\left(x\right)=Pf\left(x^{\frac{1}{p}}\right),h\left(y\right)=Qf\left(y^{\frac{1}{q}}\right)$, 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: ℝ₊ → B satisfy

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \varphi \left(x\right)$$

(2.10)

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

$$\left|\right|g\left(x\right)-L_2\left(x\right)-g\left(1\right)\left|\right|\le \Phi \left(x\right)$$

(2.11)

for all x ∈ ℝ₊.

Proof. For given x,y ∈ ℝ₊, choosing a z > 0 such that x^ky^kz^s≥ d, x^ky^sz^s≥ d, y^kz^s≥ d and y^sz^s≥ d, we have

$$\begin{array}{ll}\hfill \left|\right|g\left(xy\right)-g\left(x\right)-g\left(y\right)+g\left(1\right)\left|\right|& \le \left|\right|-f\left(xyz\right)+g\left(xy\right)+h\left(z\right)\left|\right|\\ +\left|\right|f\left(xyz\right)-g\left(x\right)-h\left(yz\right)\left|\right|\\ +\left|\right|f\left(yz\right)-g\left(y\right)-h\left(z\right)\left|\right|\\ +\left|\right|-f\left(yz\right)+g\left(1\right)+h\left(yz\right)\left|\right|\\ \le \varphi \left(xy\right)+\varphi \left(x\right)+\varphi \left(y\right)+\varphi \left(1\right).\end{array}$$

(2.12)

Now, by Theorem A, we get the result.

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \epsilon$$

(2.13)

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

$$\left|\right|g\left(x\right)-L_2\left(x\right)-g\left(1\right)\left|\right|\le 4\epsilon$$

(2.14)

for all x ∈ ℝ₊.

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

$$g\left(x\right)=\left\{\begin{array}{cc}L\left(x\right),\hfill & x\ge d^{1/k},\hfill \\ 0,\hfill & 0<x<d^{1/s}.\hfill \end{array}\right.$$

Then, the inequality (2.13) holds for all x, y > 0, with x^k≥ d 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 ∈ ℝ with s ≠ 0. Suppose that f : ℝ₊ → B satisfies

$$\left|\right|f\left(x^p{y}^q\right)-Pf\left(x\right)-Qf\left(y\right)\left|\right|\le \epsilon$$

(2.15)

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

$$\left|\right|f\left(x\right)-L\left(x\right)-f\left(1\right)\left|\right|\le \frac{4\epsilon }{\left|P\right|}$$

(2.16)

for all x ∈ ℝ₊.

Proof. Replacing x by $x^{\frac{1}{p}}$, y by $y^{\frac{1}{q}}$ in (2.15), we have

$$\left|\right|f\left(xy\right)-Pf\left(x^{\frac{1}{p}}\right)-Qf\left(y^{\frac{1}{q}}\right)\left|\right|\le \epsilon$$

for all x,y > 0, with $x^{\frac{k}{p}}{y}^{\frac{s}{q}}\ge d$. Letting $g\left(x\right)=Pf\left(x^{\frac{1}{p}}\right),h\left(y\right)=Qf\left(y^{\frac{1}{q}}\right)$, applying Corollary 2.5 and dividing the result by |P|, we get the result with $L\left(x\right)=\frac{1}{P}{L}_2\left(x^p\right)$.

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \varphi \left(y\right)$$

(2.17)

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

$$\left|\right|h\left(x\right)-L_3\left(x\right)-h\left(1\right)\left|\right|\le \Phi \left(x\right)$$

(2.18)

for all x ∈ ℝ₊.

Proof. For given x,y ∈ ℝ₊, choosing a z > 0 such that x^sy^sz^k≥ d, x^ky^sz^k≥ d, x^sz^k≥ d and x^kz^k≥ d, we have

$$\begin{array}{ll}\hfill \left|\right|h\left(xy\right)-h\left(x\right)-h\left(y\right)+h\left(1\right)\left|\right|& \le \left|\right|-f\left(xyz\right)+g\left(z\right)+h\left(xy\right)\left|\right|\\ +\left|\right|f\left(xyz\right)-g\left(xz\right)-h\left(y\right)\left|\right|\\ +\left|\right|f\left(zx\right)-g\left(z\right)-h\left(x\right)\left|\right|\\ +\left|\right|-f\left(xz\right)+g\left(xz\right)+h\left(1\right)\left|\right|\\ \le \varphi \left(xy\right)+\varphi \left(x\right)+\varphi \left(y\right)+\varphi \left(1\right).\end{array}$$

(2.19)

Now, by Theorem A, we get the result.

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \epsilon$$

(2.20)

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

$$\left|\right|h\left(x\right)-L_3\left(x\right)-h\left(1\right)\left|\right|\le 4\epsilon$$

(2.21)

for all x ∈ ℝ₊.

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 ∈ ℝ with k ≠ 0. Suppose that f : ℝ₊ → B satisfies

$$\left|\right|f\left(x^p{y}^q\right)-Pf\left(x\right)-Qf\left(y\right)\left|\right|\le \epsilon$$

(2.22)

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

$$\left|\right|f\left(x\right)-L\left(x\right)-f\left(1\right)\left|\right|\le \frac{4\epsilon }{\left|Q\right|}$$

(2.23)

for all x ∈ ℝ₊.

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \epsilon$$

(2.24)

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

$$\begin{array}{c}\left|\right|f\left(x\right)-L\left(x\right)-f\left(1\right)\left|\right|\le 4\epsilon ,\\ \left|\right|g\left(x\right)-L\left(x\right)-g\left(1\right)\left|\right|\le 4\epsilon ,\\ \left|\right|h\left(x\right)-L\left(x\right)-h\left(1\right)\left|\right|\le 4\epsilon \end{array}$$

for all x ∈ ℝ₊.

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 x^ky^sz^s-k≥ d, z^s-k≥ d. Then, in view of (2.24), we have

$$\left|\right|f\left(xy\right)-g\left(x{z}^{-1}\right)-h\left(yz\right)\left|\right|\le \epsilon ,$$

(2.25)

$$\left|\right|-f\left(1\right)+g\left(z^{-1}\right)+h\left(z\right)\left|\right|\le \epsilon .$$

(2.26)

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

$$\left|\right|g\left(x{z}^{-1}\right)-g\left(x\right)-g\left(z^{-1}\right)+g\left(1\right)\left|\right|\le 4\epsilon ,$$

(2.27)

$$\left|\right|h\left(yz\right)-h\left(z\right)-h\left(y\right)+h\left(1\right)\left|\right|\le 4\epsilon$$

(2.28)

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)-f\left(1\right)+g\left(1\right)+h\left(1\right)\left|\right|\le 10\epsilon$$

(2.29)

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

$$\left|\right|L_1\left(xy\right)-L_2\left(x\right)-L_3\left(y\right)\left|\right|\le 22\epsilon .$$

(2.30)

Putting y = 1 and x = 1 in (2.30) separately, and using the fact that for all x > 0, n ∈ 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 ∈ ℝ with k ≠ 0, s ≠ 0 and k ≠ s. Suppose that f : ℝ₊ → B satisfies

$$\left|\right|f\left(x^p{y}^q\right)-Pf\left(x\right)-Qf\left(y\right)\left|\right|\le \epsilon$$

(2.31)

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

$$\left|\right|f\left(x\right)-L\left(x\right)-f\left(1\right)\left|\right|\le \text{min}\left\{4\epsilon ,\frac{4\epsilon }{\left|P\right|},\frac{4\epsilon }{\left|Q\right|}\right\}$$

(2.32)

for all x ∈ ℝ₊.

3. Asymptotic behaviors

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

Theorem 3.1. Let k, s ∈ ℝ, k ≠ s. Suppose that f, g, h : ℝ₊ → B satisfy the asymptotic condition

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\to 0$$

(3.1)

as x^ky^s→ ∞. Then, there exists a unique logarithmic function L : ℝ₊ → B such that

$$f\left(x\right)=L\left(x\right)+f\left(1\right)$$

(3.2)

for all x ∈ ℝ₊.

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \frac{1}{n}$$

(3.3)

for all x, y > 0, with x^ky^s≥ d _n . By Corollary 2.2, there exists a unique logarithmic function L _n : ℝ₊ → B such that

$$\left|\right|f\left(x\right)-L_n\left(x\right)-f\left(1\right)\left|\right|\le \frac{4}{n}$$

(3.4)

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

$$\left|\right|L_n\left(x\right)-L_m\left(x\right)\left|\right|\le \frac{4}{n}+\frac{4}{m}\le 8$$

(3.5)

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

$$\left|\right|L_n\left(x\right)-L_m\left(x\right)\left|\right|=\frac{1}{r}\left|\right|L_n\left(x^r\right)-L_m\left(x^r\right)\left|\right|\le \frac{8}{r}.$$

(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 : ℝ₊ → B satisfy the asymptotic condition

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\to 0$$

(3.7)

as x^ky^s→ ∞. Then, there exists a unique logarithmic function L: ℝ₊ → B such that

$$g\left(x\right)=L\left(x\right)+g\left(1\right)$$

(3.8)

for all x ∈ ℝ₊.

Using Corollary 2.8, we obtain the following.

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\to 0$$

(3.9)

as x^ky^s→ ∞. Then, there exists a unique logarithmic function L : ℝ₊ → B such that

$$h\left(x\right)=L\left(x\right)+h\left(1\right)$$

(3.10)

for all x ∈ ℝ₊.

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\to 0$$

(3.11)

as x^ky^s→ ∞. Then, there exists a unique logarithmic function L : ℝ₊ → B and c₁, c₂ ∈ B such that

$$\begin{array}{c}f\left(x\right)=L\left(x\right)+c_1+c_2,\\ g\left(x\right)=L\left(x\right)+c_1,\\ h\left(x\right)=L\left(x\right)+c_2\end{array}$$

for all x ∈ ℝ₊.

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)\left|\right|\le \frac{1}{n}$$

(3.12)

for all x, y > 0, with x^ky^s≥ d _n . By Theorem 2.10, there exists a unique logarithmic function L _n : ℝ₊ → B such that

$$\left|\right|f\left(x\right)-L_n\left(x\right)-f\left(1\right)\left|\right|\le \frac{4}{n},$$

(3.13)

$$\left|\right|g\left(x\right)-L_n\left(x\right)-g\left(1\right)\left|\right|\le \frac{4}{n},$$

(3.14)

$$\left|\right|h\left(x\right)-L_n\left(x\right)-h\left(1\right)\left|\right|\le \frac{4}{n}$$

(3.15)

for all x ∈ ℝ₊. Similarly, as in the proof of Theorem 3.1, we have L _n = L _m for all n, m ∈ ℕ. 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 ℝ⁺. In this section, we consider the L^∞-version of Hyers-Ulam stability of the inequality

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)|{|}_{L^{\infty }\left(U_{k,s,d}\right)}\le \epsilon ,$$

(4.1)

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

$$\text{supp}\omega \subset \text{{}x:\left|x\right|\le 1\}$$

and

$$\int \omega \left(x\right)dx=1.$$

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 ∈ ℂ and f * ω _t (x) → f(x) for almost every x ∈ ℂ 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 ∈ ℂ such that

$$\begin{array}{c}\left|\right|f\left(x\right)-c_1-a\text{ln}x|{|}_{L^{\infty }\left({ℝ}_{+}\right)}\le 4\epsilon ,\\ \left|\right|g\left(x\right)-c_2-a\text{ln}x|{|}_{L^{\infty }\left({ℝ}_{+}\right)}\le 4\epsilon ,\\ \left|\right|h\left(x\right)-c_3-a\text{ln}x|{|}_{L^{\infty }\left({ℝ}_{+}\right)}\le 4\epsilon .\end{array}$$

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

$$\left|\right|f\left(2^{x+y}\right)-g\left(2^x\right)-h\left(2^y\right)|{|}_{L^{\infty }\left(U_d\right)}\le \epsilon ,$$

(4.2)

where $U_d=\left\{\left(x,y\right):kx+sy\ge {\text{log}}_2^d:=d_1\right\}$. Now, let

$$F\left(x\right)=f\left(2^x\right),G\left(x\right)=g\left(2^x\right),H\left(x\right)=h\left(2^x\right).$$

(4.3)

Then, we have

$$\left|\right|F\left(x+y\right)-G\left(x\right)-H\left(y\right)|{|}_{L^{\infty }\left(U_d\right)}\le \epsilon .$$

(4.4)

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

$$|F*{\omega }_t*{\omega }_s\left(x+y\right)-G*{\omega }_t\left(x\right)-H*{\omega }_s\left(y\right)|\le \epsilon$$

(4.5)

holds for all $kx+sy\ge d_2:=d_1+\sqrt{k^2+s^2}$ 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 : ℝ₊ → ℂ, j = 1, 2, 3, be locally integrable functions satisfying

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)|{|}_{L^{\infty }\left(U_{k,s,d}\right)}\to 0$$

(4.6)

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

$$\begin{array}{c}\left|\right|f\left(x\right)-c_1-a\text{ln}x|{|}_{L^{\infty }\left({ℝ}_{+}\right)}=0,\\ \left|\right|g\left(x\right)-c_2-a\text{ln}x|{|}_{L^{\infty }\left({ℝ}_{+}\right)}=0,\\ \left|\right|h\left(x\right)-c_3-a\text{ln}x|{|}_{L^{\infty }\left({ℝ}_{+}\right)}=0.\end{array}$$

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

$$\left|\right|f\left(xy\right)-g\left(x\right)-h\left(y\right)|{|}_{L^{\infty }\left(U_{k,s,d_n}\right)}\le \frac{1}{n}$$

(4.7)

for all $x,y\in U_{k,s,d_n}$. Now, by Theorem 4.1, there exist a, c₁, c₂, c₃ ∈ ℂ (which are independent of n) such that

$$\left|\right|f\left(x\right)-c_1-a\text{ln}x|{|}_{L^{\infty }\left({ℝ}_{+}\right)}\le \frac{4}{n},$$

(4.8)

$$\left|\right|g\left(x\right)-c_2-a\text{ln}x\text{|}{\text{|}}_{L^{\infty }\left({ℝ}_{+}\right)}\le \frac{4}{n},$$

(4.9)

$$\left|\right|h\left(x\right)-c_3-a\text{ln}x|{|}_{L^{\infty }\left({ℝ}_{+}\right)}\le \frac{4}{n}.$$

(4.10)

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