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

Abstract

We prove the Hyers-Ulam stability of the logarithmic functional equation of Heuvers and Kannappan 𝑓(𝑥+𝑦)−𝑔(𝑥𝑦)−ℎ(1/𝑥+1/𝑦)=0,𝑥,𝑦>0, in both classical and distributional senses. As a classical sense, the Hyers-Ulam stability of the inequality |𝑓(𝑥+𝑦)−𝑔(𝑥𝑦)−ℎ(1/𝑥+1/𝑦)|≤𝜖,𝑥,𝑦>0 will be proved, where 𝑓,𝑔,ℎ∶ℝ₊→ℂ. As a distributional analogue of the above inequality, the stability of inequality ‖𝑢∘(𝑥+𝑦)−𝑣∘(𝑥𝑦)−𝑤∘(1/𝑥+1/𝑦)‖≤𝜖 will be proved, where 𝑢,𝑣,𝑤∈𝒟^′(ℝ₊) 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 𝐺 is an additive semigroup, 𝐵 is a Banach space, 𝑓∶𝐺→𝐵, 𝜖≥0, and 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜖 (1.1) for all 𝑥,𝑦∈𝐺. Then there exists a unique function 𝐴∶𝐺→𝐵 satisfying 𝐴𝑥+𝑦=𝐴(𝑥)+𝐴𝑦 (1.2) for which 𝑓(𝑥)−𝐴(𝑥)≤𝜖 (1.3) for all 𝑥∈𝐺.

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: 𝐿𝑥𝑦−𝐿(𝑥)−𝐿𝑦=0,𝑥,𝑦>0. (1.4)

Theorem 1.2. Let ℝ₊ be the set of positive real numbers. Suppose that 𝑓∶ℝ₊→𝐶, 𝜖≥0, and 𝑓𝑥𝑦−𝑓(𝑥)−𝑓𝑦≤𝜖 (1.5) for all 𝑥,𝑦>0. Then there exists a unique function 𝐿∶ℝ₊→ℂ satisfying (1.4) for which 𝑓(𝑥)−𝐿(𝑥)≤𝜖 (1.6) for all 𝑥>0.

We call the function 𝐿 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: 𝑓1𝑥+𝑦−𝑔𝑥𝑦−ℎ𝑥+1𝑦=0,𝑥,𝑦>0 (1.7) for 𝑓,𝑔,ℎ∶ℝ₊→ℝ. In particular, it is shown that the general solution of (1.7) has the form 𝑓(𝑥)=𝑐₁+𝑐₂(𝑥),𝑔(𝑥)+𝐿=𝑐₁(𝑥),ℎ(𝑥)+𝐿=𝑐₂+𝐿(𝑥), (1.8) where 𝐿∶ℝ₊→ℝ 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: 𝑢∘𝐴−𝑢∘𝑃₁−𝑢∘𝑃₂≤𝜖, (1.9) 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 𝐸_𝑡(𝑥)=(4𝜋𝑡)^−𝑛/2−|𝑥|exp²4𝑡,𝑥∈ℝ^𝑛,𝑡>0, (1.10) 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 𝑈𝑥+𝑦,𝑡+𝑠−𝑈(𝑥,𝑡)−𝑈𝑦,𝑠≤𝜖 (1.11) for all 𝑥,𝑦∈ℝ^𝑛,  𝑡,𝑠>0, where 𝑈 is an infinitely differentiable function in ℝ^𝑛×(0,∞) given by 𝑈(𝑥,𝑡)=𝑢∗𝐸_𝑡(𝑥). 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 𝛿_𝑡(𝑥)∶=𝑡^−𝑛𝛿(𝑥/𝑡),  𝑥∈ℝ^𝑛,  𝑡>0, where 𝛿(𝑥)=𝑞𝑒^{−(1−|𝑥|2)−1},|𝑥||𝑥|<1,0,≥1,𝑞=_|𝑥|<1𝑒^{−(1−|𝑥|2)−1}𝑑𝑥⁻¹, (1.12)(1.13) 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 1𝑢∘𝑥+𝑦−𝑣∘𝑥𝑦−𝑤∘𝑥+1𝑦≤𝜖, (1.14) where 𝑢,𝑣,𝑤∈𝒟^′(ℝ₊), ∘ denotes the pullback of distributions and the inequality ‖⋅‖≤𝜖 in (1.14) means |⟨⋅,𝜑⟩|≤𝜖‖𝜑‖_𝐿¹ for all test functions 𝜑(𝑥,𝑦) defined on ℝ²₊. Since the tempered distributions are defined in whole real line or whole space ℝ^𝑛, 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 ℝ². As a direct consequence of the result, we obtain the Hyers-Ulam stability of (1.7) in 𝐿^∞-sense, that is, the Hyers-Ulam stability of the inequality 1𝑓(𝑥+𝑦)−𝑔(𝑥𝑦)−ℎ𝑥+1𝑦_𝐿∞≤𝜖 (1.15) 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 𝑓1𝑥+𝑦−𝑔𝑥𝑦−ℎ𝑥+1𝑦≤𝜖,𝑥,𝑦>0, (2.1) where 𝑓,𝑔,ℎ∶ℝ₊→ℝ and 𝜖≥0.

Theorem 2.1. Suppose that 𝑓,𝑔,ℎ∶ℝ₊→ℂ satisfy (2.1). Then there exists a logarithmic function 𝐿∶ℝ₊→ℝ such that 𝑓(𝑥)(𝑥)(1𝑔(𝑥)(𝑥)(1|ℎ(𝑥)(𝑥)(1−𝐿−𝑓)≤4𝜖,−𝐿−𝑔)≤4𝜖,−𝐿−ℎ)|≤4𝜖 (2.2) for all 𝑥>0.

Proof. Let 𝑥𝑦=𝑡,  1/𝑥+1/𝑦=𝑠. Then we have 𝑓(𝑡𝑠)−𝑔(𝑡)−ℎ(𝑠)≤𝜖 (2.3) for all 𝑡>0,  𝑠>0 such that 𝑡𝑠²≥4. For given 𝑡,𝑠>0, choose a large 𝑢>0 so that 𝑡𝑠²𝑢²≥4,  𝑡𝑠𝑢²≥4,  𝑠𝑢²≥4,  𝑠²𝑢²≥2. Then in view of (2.3), we have 𝑓()(𝑡)(𝑓()()(𝑢𝑓()(𝑠)(𝑢𝑓()(1)(𝑡𝑠𝑢−𝑔−ℎ𝑠𝑢)≤𝜖,𝑡𝑠𝑢−𝑔𝑡𝑠−ℎ)≤𝜖,𝑠𝑢−𝑔−ℎ)≤𝜖,𝑠𝑢−𝑔−ℎ𝑠𝑢)≤𝜖. (2.4)(2.5)(2.6)(2.7) From (2.4)–(2.7), using the triangle inequality we have 𝑔(𝑡𝑠)−𝑔(𝑡)−𝑔(𝑠)+𝑔(1)≤4𝜖 (2.8) for all 𝑡,𝑠>0. Changing the roles of 𝑔 and ℎ in (2.3), we can show that |ℎ(𝑡𝑠)−ℎ(𝑡)−ℎ(𝑠)+ℎ(1)|≤4𝜖 (2.9) for all 𝑡,𝑠>0. Now we prove that 𝑓(𝑡𝑠)−𝑓(𝑡)−𝑓(𝑠)+𝑓(1)≤4𝜖 (2.10) for all 𝑡,𝑠>0. Replacing 𝑡 by 𝑢 and 𝑠 by 𝑠/𝑢 in (2.3), we have 𝑓(𝑠)(𝑢)𝑠−𝑔−ℎ𝑢≤𝜖,4𝑢≤𝑠². (2.11) Similarly, we have 𝑓(𝑡𝑠)−𝑔(𝑢)−ℎ𝑡𝑠𝑢≤𝜖,4𝑢≤𝑡²𝑠²,𝑓(𝑡)𝑢−𝑔𝑠−ℎ𝑡𝑠𝑢≤𝜖,4𝑢≤𝑡²𝑓(1)𝑢𝑠,−𝑔𝑠𝑠−ℎ𝑢≤𝜖,4𝑢≤𝑠. (2.12)(2.13)(2.14)
For given 𝑡,𝑠>0, let 𝑢=1/4min{𝑠²,𝑡²𝑠²,𝑡²𝑠,𝑠}. Then, from (2.11)–(2.14), using the triangle inequality we get the inequality (2.10).
Now by Theorem 1.2, there exist functions 𝐿_𝑗∶ℝ₊→ℝ,  𝑗=1,2,3, satisfying the logarithmic functional equation 𝐿_𝑗(𝑡𝑠)=𝐿_𝑗(𝑡)+𝐿_𝑗(𝑠),𝑗=1,2,3, (2.15) for which 𝑓(𝑡)−𝐿₁(𝑡)(1𝑔(𝑡)−𝑓)≤4𝜖,−𝐿₂(𝑡)(1ℎ(𝑡)−𝑔)≤4𝜖,−𝐿₃(𝑡)−ℎ(1)≤4𝜖. (2.16)(2.17)(2.18)
Now we show that 𝐿₁=𝐿₂=𝐿₃. From (2.3), we have 𝑓(𝑡)(𝑡)(1𝑓(𝑡)(𝑡)(1−𝑔−ℎ)≤𝜖,𝑡≥2,−ℎ−𝑔)≤𝜖,𝑡≥4. (2.19)(2.20)
From (2.16), (2.17), and (2.19), using the triangle inequality we have 𝐿₁(𝑡)−𝐿₂(𝑡)≤9𝜖+𝑓(1)−𝑔(1)−ℎ(1)∶=𝑀,𝑡≥2. (2.21)
In view of (2.15) and (2.21), we have 𝐿₁(𝑡)−𝐿₂(𝑡=1)|𝑛|𝐿₁𝑡^𝑛−𝐿₂𝑡^𝑛≤1|𝑛|𝑀 (2.22) for all 𝑡>0,  𝑡≠1 and 𝑡^𝑛≥2. Letting 𝑛→∞ for 𝑡>1 and letting 𝑛→−∞ for 0<𝑡<1, we have 𝐿₁(𝑡)=𝐿₂(𝑡) for 𝑡≠1. Since 𝐿₁(1)=𝐿₂(1)=0, we have 𝐿₁(𝑡)=𝐿₂(𝑡) for all 𝑡>0. Similarly we can show that 𝐿₁=𝐿₃. This completes the proof.

Letting 𝑔=ℎ=𝑓 in Theorem 2.1, in view of the inequalities (2.4), (2.5), and (2.6), using the triangle inequality we have 𝑓(𝑡𝑠)−𝑓(𝑡)−𝑓(𝑠)≤3𝜖 (2.23) for all 𝑡,𝑠>0. Thus by Theorem 1.2 we have the following.

Theorem 2.2. Let 𝑓∶ℝ₊→ℝ satisfy the inequality 𝑓1𝑥+𝑦−𝑓𝑥𝑦−𝑓𝑥+1𝑦≤𝜖 (2.24) for all 𝑥,𝑦>0. Then there exists a logarithmic function 𝐿∶ℝ₊→ℝ such that 𝑓(𝑥)−𝐿(𝑥)≤3𝜖 (2.25) for all 𝑥>0.

3. Schwartz Distributions

Let Ω be an open subset of ℝ^𝑛. We briefly introduce the space 𝒟^′(Ω) of distributions. We denote by 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛₀, where ℕ₀ is the set of nonnegative integers, and |𝛼|=𝛼₁+⋯+𝛼_𝑛,  𝜕^𝛼=𝜕^𝛼1₁⋯𝜕^𝛼𝑛_𝑛, 𝜕_𝑗=𝜕/𝜕𝑥_𝑗,𝑗=1,2,…,𝑛.

Definition 3.1. Let 𝐶^∞_𝑐(Ω) be the set of all infinitely differentiable functions on Ω with compact supports. A distribution 𝑢 is a linear form on 𝐶^∞_𝑐(Ω) such that for every compact set 𝐾⊂Ω there exist constants 𝐶>0 and 𝑘∈ℕ₀ for which 𝑢,𝜑≤𝐶_{|𝛼|≤𝑘}sup𝜕^𝛼𝜑 (3.1) holds for all 𝜑∈𝐶^∞_𝑐(Ω) with supports contained in 𝐾. The set of all distributions is denoted by 𝒟^′(Ω).

Let Ω_𝑗 be open subsets of ℝ^𝑛𝑗 for 𝑗=1,2, with 𝑛₁≥𝑛₂.

Definition 3.2. Let 𝑢_𝑗∈𝒟^′(Ω_𝑗) and 𝜆∶Ω₁→Ω₂ be a smooth function such that for each 𝑥∈Ω₁ the derivative 𝜆^′(𝑥) is surjective, that is, the Jacobian matrix ∇𝜆 of 𝜆 has rank 𝑛₂. Then there exists a unique continuous linear map 𝜆^∗∶𝒟^′(Ω₂)→𝒟^′(Ω₁) such that 𝜆^∗𝑢=𝑢∘𝜆 when 𝑢 is a continuous function. We call 𝜆^∗𝑢 the pullback of 𝑢 by 𝜆 and is usually denoted by 𝑢∘𝜆.

In particular, if 𝑆(𝑥,𝑦)=𝑥+𝑦,𝑃₁(𝑥,𝑦)=𝑥,𝑃₂(𝑥,𝑦)=𝑦, the pullbacks 𝑢∘𝑆,  𝑢∘𝑃₁,  𝑢∘𝑃₂ can be written as 𝑢∘𝑆,𝜑𝑥,𝑦=𝑢,𝜑𝑥−𝑦,𝑦𝑑𝑦,𝑢∘𝑃₁,𝜑𝑥,𝑦=𝑢,𝜑𝑥,𝑦𝑑𝑦,𝑢∘𝑃₂,𝜑𝑥,𝑦=𝑢,𝜑𝑥,𝑦𝑑𝑥 (3.2)(3.3)(3.4) for all test functions 𝜑∈𝐶^∞_𝑐(Ω).

Also, if 𝜆 is a diffeomorphism (a bijection with 𝜆,  𝜆⁻¹ smooth functions), the pullback 𝑢∘𝜆 can be written as 𝑢∘𝜆,𝜑=𝑢,𝜑∘𝜆⁻¹(𝑥)∇𝜆⁻¹(𝑥). (3.5)

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

4. Stability in Schwartz Distributions

We employ a function 𝛿 on ℝ^𝑛 defined by 𝛿(𝑥)=𝑞𝑒^{−(1−|𝑥|2)−1},|𝑥||𝑥|<1,0,≥1, (4.1) where 𝑞=_|𝑥|<1𝑒^{−(1−|𝑥|2)−1}𝑑𝑥⁻¹. (4.2)

It is easy to see that 𝛿(𝑥) is an infinitely differentiable function with support {𝑥∶|𝑥|≤1}. Let 𝛿_𝑡(𝑥)∶=𝑡^−𝑛𝛿(𝑥/𝑡),𝑡>0 and 𝑢∈𝒟^′(ℝ^𝑛). Then for each 𝑡>0,𝑢∗𝛿_𝑡(𝑥)=⟨𝑢_𝑦,𝛿_𝑡(𝑥−𝑦)⟩ is a smooth function of 𝑥∈ℝ^𝑛 and 𝑢∗𝛿_𝑡(𝑥)→𝑢 as 𝑡→0⁺ in the sense that lim_𝑡→0+𝑢∗𝛿_𝑡(𝑥)𝜑(𝑥)𝑑𝑥=⟨𝑢,𝜑⟩ (4.3) for all 𝜑∈𝐶^∞_𝑐(ℝ^𝑛). Here after we denote by 𝑆,𝑃,𝑃₁,𝑃₂∶ℝ²→ℝ,𝑅∶ℝ²₊→ℝ,𝐸∶ℝ→ℝ⁺ by 𝑆𝑥,𝑦=𝑥+𝑦,𝑃𝑥,𝑦=𝑥𝑦,𝑃₁𝑥,𝑦=𝑥,𝑃₂𝑅=1𝑥,𝑦=𝑦,𝑥,𝑦𝑥+1𝑦,𝐸(𝑥)=2^𝑥. (4.4) Now we are in a position to prove the Hyers-Ulam stability of the inequality ‖𝑢∘𝑆−𝑣∘𝑃−𝑤∘𝑅‖≤𝜖. (4.5) Recall that the inequality ‖⋅‖≤𝜖 in (4.5) means that |⟨⋅,𝜑⟩|≤𝜖‖𝜑‖_𝐿¹ for all test functions 𝜑(𝑥,𝑦) defined on ℝ²₊.

Theorem 4.1. Let 𝑢,𝑣,𝑤∈𝒟^′(ℝ₊) satisfy (4.5). Then there exist 𝑎,𝑐₁,𝑐₂,𝑐₃∈ℂ such that 𝑢−𝑐₁−𝑎ln𝑥≤4𝜖,𝑣−𝑐₂−𝑎ln𝑥≤4𝜖,𝑤−𝑐₃−𝑎ln𝑥≤4𝜖. (4.6)

Proof. Let 𝑈={(𝑥,𝑦)∶𝑥+2𝑦>2}, 𝑉={(𝑥,𝑦)∶𝑥>𝑦>0} and define 𝐽∶𝑈→𝑉 by 𝐽=2𝑥,𝑦^𝑥+𝑦+4^𝑥+𝑦−2^𝑥+22,2^𝑥+𝑦−4^𝑥+𝑦−2^𝑥+22. (4.7) Then 𝐽 is a diffeomorphism with 𝐽⁻¹∶𝑉→𝑈,  𝐽⁻¹(𝑥,𝑦)=(log₂𝑥𝑦,log₂(𝑥+𝑦)/𝑥𝑦). Taking pullback by 𝐽 in (4.5) and using (3.5), we have 𝑢∘𝐸∘𝑆−𝑣∘𝐸∘𝑃₁−𝑤∘𝐸∘𝑃₂≤𝜖 (4.8) in 𝑈. Thus it follows that 𝑢∘𝑆−𝑣∘𝑃₁−𝑤∘𝑃₂≤𝜖 (4.9) in 𝑈, where 𝑢=𝑢∘𝐸,𝑣=𝑣∘𝐸,𝑤=𝑤∘𝐸. Denoting by (𝛿_𝑡⊗𝛿_𝑠)(𝑥,𝑦)=𝛿_𝑡(𝑥)𝛿_𝑠(𝑦) and convolving 𝛿_𝑡⊗𝛿_𝑠 in the left hand side of (4.9) we have, in view of (3.2), (𝑢∘𝑆)∗𝛿_𝑡⊗𝛿_𝑠𝑥,𝑦=𝑢∘𝜉+𝜂,𝛿_𝑡(𝑥−𝜉)𝛿_𝑠=𝑢𝑦−𝜂_𝜉,𝛿_𝑡𝑥−𝜉+𝜂𝛿_𝑠=𝑢𝑦−𝜂𝑑𝜂_𝜉,𝛿_𝑡∗𝛿_𝑠=𝑥+𝑦−𝜉𝑢∗𝛿_𝑡∗𝛿_𝑠𝑥+𝑦. (4.10) Similarly we have, in view of (3.3) and (3.4), 𝑣∘𝑃₁∗𝛿_𝑡⊗𝛿_𝑠𝑥,𝑦=𝑣∗𝛿_𝑡(𝑥),𝑤∘𝑃₂∗𝛿_𝑡⊗𝛿_𝑠𝑥,𝑦=𝑤∗𝛿_𝑠𝑦. (4.11) Thus the inequality (4.9) is converted to the classical stability problem 𝑢∗𝛿_𝑡∗𝛿_𝑠𝑥+𝑦−𝑣∗𝛿_𝑡(𝑥)−𝑤∗𝛿_𝑠𝑦≤𝜖𝛿_𝑡⊗𝛿_𝑠𝐿1=𝜖 (4.12) for all 𝑥+2𝑦≥5 and 0<𝑡<1,  0<𝑠<1. From now on, we assume that 0<𝑡<1,  0<𝑠<1. From the inequality (4.12), we have 𝑢∗𝛿_𝑡∗𝛿_𝑠𝑥+𝑦+𝑧−𝑣∗𝛿_𝑡𝑥+𝑦−𝑤∗𝛿_𝑠(𝑧)≤𝜖 (4.13) for 𝑥+𝑦+2𝑧≥5, 𝑢∗𝛿_𝑡∗𝛿_𝑠𝑥+𝑦+𝑧−𝑣∗𝛿_𝑡(𝑥)−𝑤∗𝛿_𝑠𝑦+𝑧≤𝜖 (4.14) for 𝑥+2𝑦+2𝑧≥5, 𝑢∗𝛿_𝑡∗𝛿_𝑠𝑦+𝑧−𝑣∗𝛿_𝑡𝑦−𝑤∗𝛿_𝑠(𝑧)≤𝜖 (4.15) for 𝑦+2𝑧≥5, and 𝑢∗𝛿_𝑡∗𝛿_𝑠𝑦+𝑧−𝑣∗𝛿_𝑡(0)−𝑤∗𝛿_𝑠𝑦+𝑧≤𝜖 (4.16) for  2𝑦+2𝑧≥5.
For given 𝑥,𝑦∈ℝ, choose 𝑧≥(1/2)(5+|𝑥|+2|𝑦|). Then in view of (4.13)–(4.16), using triangle inequality, we have 𝑣∗𝛿_𝑡𝑥+𝑦−𝑣∗𝛿_𝑡(𝑥)−𝑣∗𝛿_𝑡𝑦+𝑣∗𝛿_𝑡(0)≤4𝜖 (4.17) for all 𝑥,𝑦∈ℝ. Replacing (𝑥,𝑡) by (𝑦,𝑠), (𝑦,𝑠) by (𝑥,𝑡) in (4.12) and changing the positions of 𝑣 and 𝑤, we have 𝑤∗𝛿_𝑡𝑥+𝑦−𝑤∗𝛿_𝑡(𝑥)−𝑤∗𝛿_𝑡𝑦+𝑤∗𝛿_𝑡(0)≤4𝜖 (4.18) for all 𝑥,𝑦∈ℝ. Now we prove that 𝑢∗𝛿_𝑡𝑥+𝑦−𝑢∗𝛿_𝑡(𝑥)−𝑢∗𝛿_𝑡𝑦+𝑢∗𝛿_𝑡(0)≤4𝜖 (4.19) for all 𝑥,𝑦∈ℝ. From the inequality (4.12), we have 𝑢∗𝛿_𝑡∗𝛿_𝑠𝑥+𝑦−𝑣∗𝛿_𝑡(𝑧)−𝑤∗𝛿_𝑠𝑥+𝑦−𝑧≤𝜖,𝑢∗𝛿_𝑡∗𝛿_𝑠(𝑥)−𝑣∗𝛿_𝑡𝑧−𝑦−𝑤∗𝛿_𝑠𝑥+𝑦−𝑧≤𝜖,𝑢∗𝛿_𝑡∗𝛿_𝑠𝑦−𝑣∗𝛿_𝑡(𝑧)−𝑤∗𝛿_𝑠𝑦−𝑧≤𝜖,𝑢∗𝛿_𝑡∗𝛿_𝑠(0)−𝑣∗𝛿_𝑡𝑧−𝑦−𝑤∗𝛿_𝑠𝑦−𝑧≤𝜖 (4.20) for all 𝑥,𝑦,𝑧 such that 2𝑥+2𝑦−𝑧≥5, 2𝑥+𝑦−𝑧≥5, 2𝑦−𝑧≥5, and 𝑦−𝑧≥5. For given 𝑥,𝑦∈ℝ, choose 𝑧≤−5−2|𝑥|−2|𝑦|. Then in view of (4.20), using triangle inequality, we have 𝑢∗𝛿_𝑡∗𝛿_𝑠𝑥+𝑦−𝑢∗𝛿_𝑡∗𝛿_𝑠(𝑥)−𝑢∗𝛿_𝑡∗𝛿_𝑠𝑦+𝑢∗𝛿_𝑡∗𝛿_𝑠(0)≤4𝜖. (4.21) Letting 𝑠→0⁺ 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 0<𝑡<1, there exist functions 𝐴_𝑗(⋅,𝑡),𝑗=1,2,3, satisfying 𝐴_𝑗𝑥+𝑦,𝑡=𝐴_𝑗(𝑥,𝑡)+𝐴_𝑗𝑦,𝑡,𝑥,𝑦∈ℝ, (4.22) for which 𝑢∗𝛿_𝑡(𝑥)−𝐴₁(𝑥,𝑡)−𝑢∗𝛿_𝑡(0)≤4𝜖,𝑣∗𝛿_𝑡(𝑥)−𝐴₂(𝑥,𝑡)−𝑣∗𝛿_𝑡(0)≤4𝜖,𝑤∗𝛿_𝑡(𝑥)−𝐴₃(𝑥,𝑡)−𝑤∗𝛿_𝑡(0)≤4𝜖 (4.23)(4.24)(4.25) for all 𝑥∈ℝ.
Now we prove that 𝐴₁=𝐴₂=𝐴₃. From (4.12), using the triangle inequality, we have 𝑣∗𝛿_𝑡(𝑥)≤𝜖+𝑢∗𝛿_𝑡∗𝛿_𝑠𝑥+𝑦+𝑤∗𝛿_𝑠𝑦 (4.26) for all 𝑥+2𝑦≥5. Since (𝑢∗𝛿_𝑡∗𝛿_𝑠)(𝑥)→(𝑢∗𝛿_𝑠)(𝑥) as 𝑡→0⁺, in view of (4.26) it is easy to see that 𝑔(𝑥)∶=limsup_𝑡→0+𝑣∗𝛿_𝑡(𝑥) (4.27) exists for all 𝑥∈ℝ. Similarly, we can show that ℎ(𝑥)∶=limsup_𝑠→0+𝑤∗𝛿_𝑠(𝑥) (4.28) exists for all 𝑥∈ℝ. Putting 𝑦=0 in (4.12) and letting 𝑠→0⁺ so that 𝑤∗𝛿_𝑠(0)→ℎ(0), we have 𝑢∗𝛿_𝑡(𝑥)−𝑣∗𝛿_𝑡(𝑥)−ℎ(0)≤𝜖 (4.29) for all 𝑥≥5. Similarly, we have 𝑢∗𝛿_𝑡(𝑥)−𝑤∗𝛿_𝑡(𝑥)−𝑔(0)≤𝜖 (4.30) for all 𝑥≥5/2. Using (4.23), (4.24), (4.29), and the triangle inequality, we have 𝐴₁(𝑥,𝑡)−𝐴₂(𝑥,𝑡)≤9𝜖+𝑢∗𝛿_𝑡(0)−𝑣∗𝛿_𝑡(0)−ℎ(0)∶=𝑀(𝑡) (4.31) for all 𝑥≥5. From (4.22) and (4.31), we have 𝐴₁(𝑥,𝑡)−𝐴₂(=1𝑥,𝑡)|𝑘|𝐴₁(𝑘𝑥,𝑡)−𝐴₂(≤1𝑘𝑥,𝑡)|𝑘|𝑀(𝑡) (4.32) for all 𝑥∈ℝ,𝑥≠0 and all integers 𝑘 with 𝑘𝑥≥5. Letting 𝑘→∞ if 𝑥>0 and letting 𝑘→−∞ if 𝑥<0 in (4.32) we have 𝐴₁(𝑥,𝑡)=𝐴₂(𝑥,𝑡) for 𝑥≠0, which implies 𝐴₁=𝐴₂ since 𝐴₁(0,𝑡)=𝐴₂(0,𝑡)=0. Similarly, using (4.23), (4.25), and (4.30) we can show that 𝐴₁=𝐴₃.
Finally we prove that 𝐴₁ is independent of 𝑡. Fixing 𝑥∈ℝ and letting 𝑡→0⁺ so that 𝑣∗𝛿_𝑡(𝑥)→𝑔(𝑥) in (4.12), we have 𝑢∗𝛿_𝑠𝑥+𝑦−𝑔(𝑥)−𝑤∗𝛿_𝑠𝑦≤𝜖 (4.33) for all 𝑥+2𝑦≥5. The same substitution as the inequalities (4.13)–(4.16) gives 𝑔𝑥+𝑦−𝑔(𝑥)−𝑔𝑦+𝑔(0)≤4𝜖. (4.34) Using the stability Theorem [2], we obtain that there exists a unique function 𝐴 satisfying the Cauchy functional equation 𝐴𝑥+𝑦−𝐴(𝑥)−𝐴𝑦=0 (4.35) for which 𝑔(𝑥)−𝐴(𝑥)−𝑔(0)≤4𝜖. (4.36) Now we show that 𝐴₁(𝑥,𝑡)=𝐴(𝑥) for all 𝑥∈ℝ and 0<𝑡<1. Putting 𝑦=0 in (4.33), we have 𝑢∗𝛿_𝑠(𝑥)−𝑔(𝑥)−𝑤∗𝛿_𝑠(0)≤𝜖 (4.37) for all 𝑥≥5. From (4.23), (4.36), and (4.37), using the triangle inequality, we have 𝐴₁(𝑥,𝑡)−𝐴(𝑥)≤9𝜖+𝑢∗𝛿_𝑡(0)−𝑤∗𝛿_𝑡(0)−𝑔(0) (4.38) for all 𝑥≥5. From (4.38), using the method of proving 𝐴₁=𝐴₂, we can show that 𝐴₁(𝑥,𝑡)=𝐴(𝑥) for all 𝑥∈ℝ and 𝑡. Thus we have 𝐴₁=𝐴₂=𝐴₃=𝐴.
Letting 𝑡→0⁺ in (4.24) so that 𝑣∗𝛿_𝑡(0)→𝑔(0), we have 𝑣−𝐴(𝑥)−𝑐₂≤4𝜖 (4.39) for some 𝑐₂∈ℂ. Similarly, letting 𝑡→0⁺ in (4.25) so that 𝑤∗𝛿_𝑡(0)→ℎ(0), we have 𝑤−𝐴(𝑥)−𝑐₃≤4𝜖 (4.40) for some 𝑐₃∈ℂ. Now we prove the inequality 𝑢−𝐴(𝑥)−𝑐₁≤4𝜖 (4.41) for some 𝑐₁∈ℂ. Putting 𝑥=−5,  𝑦=5 in (4.33) and using the triangle inequality, we have 𝑢∗𝛿_𝑠(0)≤𝜖+𝑔(−5)+𝑤∗𝛿_𝑠(5). (4.42) From (4.42), there exists a sequence 𝑠_𝑛→0⁺ such that 𝑢∗𝛿_𝑠𝑛(0) converges. Letting 𝑡=𝑠_𝑛→0⁺ in (4.23), we get (4.41). Taking pullback by 𝐸⁻¹(𝑥)=log₂𝑥 in (4.39), (4.40), and (4.41), we have 𝑢−𝐴log₂𝑥−𝑐₁≤4𝜖,𝑣−𝐴log₂𝑥−𝑐₂≤4𝜖,𝑤−𝐴log₂𝑥−𝑐₃≤4𝜖. (4.43) Finally we show that the solution 𝐴 of the Cauchy equation (4.35) has the form 𝐴(𝑥)=𝑎𝑥 for some 𝑎∈ℂ. Recall that 𝑔 is the supremum limit of a collection of continuous functions 𝑣∗𝛿_𝑡,0<𝑡<1. Thus, if we let 𝑔=𝑔₁+𝑖𝑔₂, then both 𝑔₁ and 𝑔₂ are Lebesgue measurable functions. Now, as we see in the proof of Hyers-Ulam stability Theorem [2], the function 𝐴 is given by 𝐴(𝑥)=lim_𝑛→∞2^−𝑛𝑔2^𝑛𝑥. (4.44) Thus, let 𝐴=𝐴₁+𝑖𝐴₂. Then 𝐴₁ and 𝐴₂ are Lebesgue measurable functions as limits of sequences of Lebesgue measurable functions. It is well known that every Lebesgue measurable solution 𝐴 of the Cauchy functional equation (4.35) has the form 𝐴(𝑥)=𝑐𝑥 for some 𝑐∈ℂ. This completes the proof.

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

Corollary 4.2. Let 𝑓,𝑔,ℎ∶ℝ₊→ℂ,𝑗=1,2,3, be locally integrable functions satisfying 1𝑓(𝑥+𝑦)−𝑔(𝑥𝑦)−ℎ𝑥+1𝑦_𝐿∞≤𝜖. (4.45) Then there exist 𝑎,𝑐₁,𝑐₂,𝑐₃∈ℂ such that 𝑓(𝑥)−𝑐₁−𝑎ln𝑥_𝐿∞≤4𝜖,𝑔(𝑥)−𝑐₂−𝑎ln𝑥_𝐿∞≤4𝜖,ℎ(𝑥)−𝑐₃−𝑎ln𝑥_𝐿∞≤4𝜖. (4.46)

Proof. Every locally integrable function 𝑓 defines a distribution via the equation ⟨𝑓,𝜑⟩=𝑓(𝑥)𝜑(𝑥)𝑑𝑥,𝜑∈𝐶^∞_𝑐ℝ₊. (4.47) Viewing 𝑓,𝑔,ℎ as distributions, the inequality (4.45) implies 𝑓∘𝑆−𝑔∘𝑃−ℎ∘𝑅≤𝜖. (4.48) By Theorem 4.1, we have 𝑓(𝑥)−𝑐₁−𝑎ln𝑥𝜑(𝑥)𝑑𝑥≤4𝜖𝜑_𝐿1,𝑔(𝑥)−𝑐₂−𝑎ln𝑥𝜑(𝑥)𝑑𝑥≤4𝜖𝜑_𝐿1,ℎ(𝑥)−𝑐₃−𝑎ln𝑥𝜑(𝑥)𝑑𝑥≤4𝜖𝜑_𝐿1 (4.49) for all 𝜑∈𝐶^∞_𝑐(ℝ₊). Viewing 𝐶^∞_𝑐 as a subspace of 𝐿¹  (dense subspace) and using the Hahn-Banach theorem we obtain that the inequalities (4.49) hold for all 𝜑∈𝐿¹. Now since 𝐿^∞=(𝐿¹)^′ we get the inequalities (4.46). This completes the proof.

As a direct consequence of the above result we solve the functional equation 𝑓1𝑥+𝑦−𝑔𝑥𝑦−ℎ𝑥+1𝑦=0 (4.50) in 𝐿^∞-sense, that is, we obtain the following.

Corollary 4.3. Let 𝑓,𝑔,ℎ∶ℝ₊→ℂ,𝑗=1,2,3, be locally integrable functions satisfying 1𝑓(𝑥+𝑦)−𝑔(𝑥𝑦)−ℎ𝑥+1𝑦_𝐿∞=0. (4.51) Then there exist 𝑎,𝑐₁,𝑐₂,𝑐₃∈ℂ such that 𝑓(𝑥)−𝑐₁−𝑎ln𝑥_𝐿∞=0,𝑔(𝑥)−𝑐₂−𝑎ln𝑥_𝐿∞=0,ℎ(𝑥)−𝑐₃−𝑎ln𝑥_𝐿∞=0. (4.52)

Finally, we discuss the locally integrable solution 𝑓,𝑔,ℎ∶ℝ₊→ℂ of (4.50) (c.f. [6]).

Corollary 4.4. Every locally integrable solution 𝑓,𝑔,ℎ∶ℝ₊→ℂ of (4.50) has the form 𝑓(𝑥)=𝑐₁+𝑐₂𝑔(𝑥)+𝑎ln𝑥,=𝑐₁ℎ(𝑥)+𝑎ln𝑥,=𝑐₂+𝑎ln𝑥 (4.53)(4.54)(4.55) for some 𝑎,𝑐₁,𝑐₂∈ℂ.

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 Ω⊂ℝ₊ with Lebesgue measure 𝑚(Ω^𝑐)=0 such that the equalities (4.53), (4.54), and (4.55) hold for all 𝑥∈Ω. For given 𝑥>0, let 𝑝,𝑞∶(0,𝑥)→ℝ by 𝑝(𝑡)=1/𝑡+(1/(𝑥−𝑡)),𝑞(𝑡)=𝑡(𝑥−𝑡). Since 𝑚[(𝑝⁻¹(Ω)∩𝑞⁻¹(Ω))^𝑐]=𝑚[𝑝⁻¹(Ω^𝑐)∪𝑞⁻¹(Ω^𝑐)]=0, we can choose 𝑦∈𝑝⁻¹(Ω)∩𝑞⁻¹(Ω). Let 𝑧=𝑥−𝑦. Then 𝑦+𝑧=𝑥 and 𝑦𝑧=𝑦(𝑥−𝑦)=𝑞(𝑦)∈Ω, 1/𝑦+1/𝑧=𝑝(𝑦)∈Ω. Thus we can write 𝑓(𝑥)1=𝑔𝑦𝑧+ℎ𝑦+1𝑧=𝑐₁+𝑎ln𝑦𝑧+𝑐₂1+𝑎ln𝑦+1𝑧=𝑐₁+𝑐₂+𝑎ln𝑦+𝑧=𝑐₁+𝑐₂+𝑎ln𝑥, (4.56) which gives (4.53). For given 𝑥>0, let 𝑝∶ℝ₊→ℝ by 𝑝(𝑡)=1/𝑡+𝑡/𝑥. Then we have 𝑝⁻¹(Ω)≠∅. Choose 𝑦∈𝑝⁻¹(Ω) and let 𝑧=𝑥/𝑦. Then 𝑦𝑧=𝑥,1/𝑦+1/𝑧∈Ω. Thus, using (4.53), we can write 𝑔(𝑥)1=𝑓𝑦+𝑧−ℎ𝑦+1𝑧=𝑐₁+𝑐₂+𝑎ln𝑦+𝑧−𝑐₂1−𝑎ln𝑦+1𝑧=𝑐₁+𝑎ln𝑦𝑧=𝑐₁+𝑎ln𝑥, (4.57) 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).