Stability in Generalized Functions

Abstract

We consider the following additive functional equation with 𝑛-independent variables: 𝑓(∑^𝑛𝑖=1𝑥_𝑖)=∑^𝑛𝑖=1𝑓(𝑥_𝑖)+∑^𝑛𝑖=1𝑓(𝑥_𝑖−𝑥_𝑖−1) in the spaces of generalized functions. Making use of the heat kernels, we solve the general solutions and the stability problems of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the mollifiers, we extend these results to the space of distributions.

1. Introduction

The most famous functional equation is the Cauchy equation𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦),(1.1) any solution of which is called additive. It is well known that every measurable solution of (1.1) is of the form 𝑓(𝑥)=𝑎𝑥 for some constant 𝑎. In 1941, Hyers proved the stability theorem for (1.1) as follows.

Theorem 1.1 (see [1]). Let 𝐸₁ be a normed vector space, 𝐸₂ a Banach space. Suppose that 𝑓∶𝐸₁→𝐸₂ satisfies the inequality ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜖,(1.2) for all 𝑥,𝑦∈𝐸₁, then there exists the unique additive mapping 𝑔∶E₁→𝐸₂ such that ‖𝑓(𝑥)−𝑔(𝑥)‖≤𝜖,(1.3) for all 𝑥∈𝐸₁.

The above stability theorem was motivated by Ulam [2]. As noted in the above theorem, the stability problem of the functional equations means how the solution of the inequality differs from the solution of the original equation. Forti [3] noticed that the theorem of Hyers is still true if 𝐸₁ is replaced by an arbitrary semigroup. In 1950 Aoki [4] and in 1978 Rassias [5] generalized Hyers’ result to the unbounded Cauchy difference. Thereafter, many authors studied the stability problems for (1.1) in various settings (see [6, 7]).

During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [8–17]). Among them, the following additive functional equation with 𝑛-independent variables:𝑓^𝑛_𝑖=1𝑥_𝑖=^𝑛_𝑖=1𝑓𝑥_𝑖+^𝑛_𝑖=1𝑓𝑥_𝑖−𝑥_𝑖−1(1.4) was proposed by Nakmahachalasint [18], where 𝑛 is a positive integer with 𝑛>1 and 𝑥₀≡𝑥_𝑛. He solved the general solutions and the stability problems for the above equation. Actually, he proved that (1.4) is equivalent to (1.1).

In this paper, in a similar manner as in [19–23], we solve the general solutions and the stability problems for (1.4) in the spaces of generalized functions such as the space 𝒮^(ℝ^𝑚) of tempered distributions, the space ℱ^(ℝ^𝑚) of Fourier hyperfunctions, and the space 𝒟^(ℝ^𝑚) of distributions. Making use of the pullbacks, we first reformulate (1.4) and the related inequality in the spaces of generalized functions as follows:𝑢∘𝐴=^𝑛_𝑖=1𝑢∘𝑃_𝑖+^𝑛_𝑖=1‖‖‖‖𝑢∘𝐴−_𝑛𝑢∘𝐵_𝑖,(1.5)_𝑖=1𝑢∘𝑃_𝑖−^𝑛_𝑖=1𝑢∘𝐵_𝑖‖‖‖‖≤𝜖,(1.6) where 𝐴, 𝑃_𝑖, and 𝐵_𝑖 are the functions defined by 𝑃𝐴_𝑖𝑥₁,…,𝑥_𝑛=𝑥₁+⋯+𝑥_𝑛,𝐵𝑥₁,…,𝑥_𝑛=𝑥_𝑖,1≤𝑖≤𝑛,_𝑖𝑥₁,…,𝑥_𝑛=𝑥_𝑖−𝑥_𝑖−1,1≤𝑖≤𝑛.(1.7) Here ∘ denotes the pullback of generalized functions, and the inequality ‖𝑣‖≤𝜖 in (1.6) means that |⟨𝑣,𝜑⟩|≤𝜖‖𝜑‖_𝐿¹ for all test functions 𝜑.

In Section 2, we will prove that every solution 𝑢 in 𝒮^(ℝ^𝑚) or ℱ^(ℝ^𝑚) of (1.5) has the form 𝑢=𝑎⋅𝑥,(1.8) where 𝑎∈ℂ^𝑚. Also, we shall figure out that every solution 𝑢 in 𝒮^(ℝ^𝑚) or ℱ^(ℝ^𝑚) of the inequality (1.6) can be written uniquely in the form 𝑢=𝑎⋅𝑥+𝜇(𝑥),(1.9) where 𝜇 is a bounded measurable function such that ‖𝜇‖_𝐿^∞≤((10𝑛−3)/(2(2𝑛−1)))𝜖. Subsequently, in Section 3, these results are extended to the space 𝒟^(ℝ^𝑚).

2. Stability in ℱ^(ℝ^𝑚)

We first introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the 𝑚-dimensional notations, |𝛼|=𝛼₁+⋯+𝛼_𝑚, 𝛼!=𝛼₁!⋯𝛼_𝑚!, 𝜁^𝛼=𝜁^𝛼1₁⋯𝜁^𝛼𝑚_𝑚, and 𝜕^𝛼=𝜕^𝛼1₁⋯𝜕^𝛼𝑚_𝑚, for 𝜁=(𝜁₁,…,𝜁_𝑚)∈ℝ^𝑚, 𝛼=(𝛼₁,…,𝛼_𝑚)∈ℕ^𝑚0, where ℕ₀ is the set of nonnegative integers and 𝜕_𝑗=𝜕/𝜕𝜁_𝑗.

Definition 2.1 (see [24, 25]). We denote by 𝒮(ℝ^𝑚) the Schwartz space of all infinitely differentiable functions 𝜑 in ℝ^𝑚 satisfying ‖𝜑‖_𝛼,𝛽=sup_{𝑥∈ℝ𝑚}||𝑥^𝛼𝜕^𝛽𝜑(𝑥)||<∞,(2.1) for all 𝛼,𝛽∈ℕ^𝑚0. A linear functional 𝑢 on 𝒮(ℝ^𝑚) is said to be tempered distribution if there exist a constant 𝐶≥0 and a nonnegative integer 𝑁 such that ||⟨𝑢,𝜑⟩||≤𝐶_{||𝛽|||𝛼|,≤𝑁}sup_{𝑥∈ℝ𝑚}||𝑥^𝛼𝜕^𝛽𝜑||,(2.2) for all 𝜑∈𝒮(ℝ^𝑚). The set of all tempered distributions is denoted by 𝒮^(ℝ^𝑚).

Note that tempered distributions are generalizations of 𝐿^𝑝-functions. These are very useful for the study of Fourier transforms in generality, since all tempered distributions have a Fourier transform. Imposing the growth condition on ‖⋅‖_𝛼,𝛽 in (2.1), a new space of test functions has emerged as follows.

Definition 2.2 (see [26]). We denote by ℱ(ℝ^𝑚) the set of all infinitely differentiable functions 𝜑 in ℝ^𝑚 such that ‖𝜑‖_𝐴,𝐵=sup_{𝑥,𝛼,𝛽}||𝑥^𝛼𝜕^𝛽𝜑(𝑥)||𝐴^|𝛼|𝐵^|𝛽|𝛼!𝛽!<∞,(2.3) for some positive constants 𝐴,𝐵 depending only on 𝜑. The strong dual of ℱ(ℝ^𝑚), denoted by ℱ^(ℝ^𝑚), is called the Fourier hyperfunction.

It can be verified that the seminorm (2.3) is equivalent to ‖𝜑‖_ℎ,𝑘=sup_𝑥,𝛼||𝜕^𝛼𝜑(𝑥)||exp𝑘|𝑥|ℎ^|𝛼|𝛼!<∞(2.4) for some constants ℎ,𝑘>0. It is easy to see the following topological inclusions:ℱ(ℝ)↪𝒮(ℝ),𝒮^(ℝ)↪ℱ^(ℝ).(2.5) Taking the inclusions (2.5) into account, it suffices to consider the space ℱ^(ℝ^𝑚). In order to solve the general solutions and the stability problems for (1.4) in the spaces ℱ^(ℝ^𝑚) and 𝒮^(ℝ^𝑚), we employ the 𝑚-dimensional heat kernel, fundamental solution of the heat equation, 𝐸_𝑡(𝑥)=𝐸(𝑥,𝑡)=⎧⎪⎨⎪⎩(4𝜋𝑡)^−𝑚/2exp−|𝑥|²4𝑡,𝑥∈ℝ^𝑚0,𝑥∈ℝ_𝑚,𝑡>0,,𝑡≤0.(2.6) Since for each 𝑡>0, 𝐸(⋅,𝑡) belongs to the space ℱ(ℝ^𝑚), the convolution 𝑢̃(𝑥,𝑡)=(𝑢∗𝐸)(𝑥,𝑡)=𝑢_𝑦,𝐸_𝑡(𝑥−𝑦),𝑥∈ℝ^𝑚,𝑡>0(2.7) is well defined for all 𝑢 in ℱ^(ℝ^𝑚), which is called the Gauss transform of 𝑢. Subsequently, the semigroup property 𝐸_𝑡∗𝐸_𝑠(𝑥)=𝐸_𝑡+𝑠(𝑥),𝑥∈ℝ^𝑚,𝑡,𝑠>0(2.8) of the heat kernel is very useful to convert (1.5) into the classical functional equation defined on upper-half plane. We also use the following famous result, so-called heat kernel method, which states as follows.

Theorem 2.3 (see [27]). Let 𝑢∈𝒮^(ℝ^𝑚), then its Gauss transform 𝑢̃ is a 𝐶^∞-solution of the heat equation 𝜕𝜕𝑡−Δ𝑢̃(𝑥,𝑡)=0,(2.9) satisfying the following:(i) there exist positive constants 𝐶, 𝑀, and 𝑁 such that ||𝑢̃(𝑥,𝑡)||≤𝐶𝑡^−𝑀(1+|𝑥|)^𝑁inℝ^𝑚×(0,𝛿),(2.10)(ii)𝑢(̃𝑥,𝑡)→𝑢 as 𝑡→0⁺ in the sense that for every 𝜑∈𝒮(ℝ^𝑚), ⟨𝑢,𝜑⟩=lim_𝑡→0+𝑢̃(𝑥,𝑡)𝜑(𝑥)𝑑𝑥.(2.11)Conversely, every 𝐶^∞-solution 𝑈(𝑥,𝑡) of the heat equation satisfying the growth condition (2.10) can be uniquely expressed as 𝑈(𝑥,𝑡)=𝑢(̃𝑥,𝑡) for some 𝑢∈𝒮′(ℝ^𝑚).

Similarly, we can represent Fourier hyperfunctions as a special case of the results as in [28]. In this case, the estimate (2.10) is replaced by the following.

For every 𝜖>0, there exists a positive constant 𝐶_𝜖 such that

||𝑢̃(𝑥,𝑡)||≤𝐶_𝜖exp𝜖|𝑥|𝑡+1inℝ×(0,𝛿).(2.12)

We need the following lemma in order to solve the general solutions for the additive functional equation in the spaces of ℱ^(ℝ^𝑚) and 𝒮^(ℝ^𝑚). In what follows, we denote 𝑥₀≡𝑥_𝑛 and 𝑡₀≡𝑡_𝑛.

Lemma 2.4. Suppose that 𝑓∶ℝ^𝑚×(0,∞)→ℂ is a continuous function satisfying 𝑓^𝑛_𝑖=1𝑥_𝑖,^𝑛_𝑖=1𝑡_𝑖=^𝑛_𝑖=1𝑓𝑥_𝑖,𝑡_𝑖+^𝑛_𝑖=1𝑓𝑥_𝑖−𝑥_𝑖−1,𝑡_𝑖+𝑡_𝑖−1,(2.13) for all 𝑥₁,…,𝑥_𝑛∈ℝ^𝑚, 𝑡₁,…,𝑡_𝑛>0, then the solution 𝑓 has the form 𝑓(𝑥,𝑡)=𝑎⋅𝑥,(2.14) for some 𝑎∈ℂ^𝑚.

Proof. Putting (𝑥₁,…,𝑥_𝑛)=(0,…,0) in (2.13) yields 𝑓0,^𝑛_𝑖=1𝑡_𝑖=^𝑛_𝑖=1𝑓0,𝑡_𝑖+^𝑛_𝑖=1𝑓0,𝑡_𝑖+𝑡_𝑖−1,(2.15) for all 𝑡₁,…,𝑡_𝑛>0. In view of (2.15), we see that 𝑐∶=lim_𝑡→0+𝑓(0,𝑡)(2.16) exists. Letting 𝑡₁=⋯=𝑡_𝑛→0⁺ in (2.15) gives 𝑐=0. Setting (𝑥₁,𝑥₂,…,𝑥_𝑛)=(𝑥,0,…,0) and letting 𝑡₁=𝑡, 𝑡₂=⋯=𝑡_𝑛→0⁺ in (2.13), we have 𝑓(−𝑥,𝑡)=−𝑓(𝑥,𝑡),(2.17) for all 𝑥∈ℝ^𝑚, 𝑡>0. Substituting (𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛) with (𝑥,𝑦,0,…,0) and letting 𝑡₁=𝑡, 𝑡₂=𝑠, 𝑡₃=⋯=𝑡_𝑛→0⁺ in (2.13), we obtain from (2.17) that 𝑓(𝑥+𝑦,𝑡+𝑠)+𝑓(𝑥−𝑦,𝑡+𝑠)=2𝑓(𝑥,𝑡),(2.18) for all 𝑥,𝑦∈ℝ^𝑚, 𝑡,𝑠>0. Putting 𝑦=0 in (2.18) yields 𝑓(𝑥,𝑡+𝑠)=𝑓(𝑥,𝑡),(2.19) which shows that 𝑓(𝑥,𝑡) is independent with respect to 𝑡>0. For that reason, we see from (2.18) that 𝐹(𝑥)∶=𝑓(𝑥,𝑡) satisfies 𝐹(𝑥+𝑦)+𝐹(𝑥−𝑦)=2𝐹(𝑥),(2.20) for all 𝑥,𝑦∈ℝ^𝑚. Replacing 𝑥 by (𝑥+𝑦)/2 and 𝑦 by (𝑥−𝑦)/2 in (2.20), we have 𝐹(𝑥+𝑦)=𝐹(𝑥)+𝐹(𝑦),(2.21) for all 𝑥,𝑦∈ℝ^𝑚. Given the continuity, we obtain 𝑓(𝑥,𝑡)=𝐹(𝑥)=𝑎⋅𝑥,(2.22) for some 𝑎∈ℂ^𝑚.

From the above lemma, we can solve the general solutions for the additive functional equation in the spaces of ℱ^(ℝ^𝑚) and 𝒮^(ℝ^𝑚).

Theorem 2.5. Every solution 𝑢 in ℱ′(ℝ^𝑚) (or 𝒮′(ℝ^𝑚), resp.) of (1.5) has the form 𝑢=𝑎⋅𝑥,(2.23) for some 𝑎∈ℂ^𝑚.

Proof. Convolving the tensor product 𝐸_𝑡1(𝑥₁)⋯𝐸_𝑡𝑛(𝑥_𝑛) of the heat kernels on both sides of (1.5), we have (𝑢∘=𝐴)𝑢∗∘𝐴𝐸,_𝑡1𝐸𝑥_1𝑡1⋯𝐸_𝑡𝑛𝑥_𝑛𝜉₁,…,𝜉_𝑛=𝑢,⋯𝐸𝜉_1𝑡1−𝑥₁⋯𝐸_𝑡𝑛𝜉_𝑛−𝑥_𝑛=𝑢,⋯𝐸_𝑡1𝜉₁−𝑥₁+𝑥₂+⋯+𝑥_𝑛𝐸_𝑡2𝜉₂−𝑥₂⋯𝐸_𝑡𝑛𝜉_𝑛−𝑥_𝑛𝑑𝑥₂⋯𝑑𝑥_𝑛=𝑢,𝐸_𝑡1𝜉₁+⋯+𝜉_𝑛−𝑥₁−⋯−𝑥_𝑛𝐸_𝑡2𝑥₂⋯𝐸_𝑡𝑛𝑥_𝑛𝑑𝑥₂⋯𝑑𝑥_𝑛=𝑢,𝐸∗⋯_{𝑡1+⋯+𝑡𝑛}∗𝐸_𝑡𝑛𝜉₁+⋯+𝜉_𝑛−𝑥₁=𝑢̃𝜉₁𝜉₁+⋯+𝜉_𝑛+⋯𝑢+∘𝜉𝑃_𝑛,_𝑖𝑡₁+⋯+𝑡_𝑛,𝑢∘𝐵_𝑖∗𝐸_𝑡1𝑥₁⋯𝐸_𝑡𝑛𝑥_𝑛𝜉₁,…,𝜉_𝑛=𝑢̃𝜉_𝑖,𝑡_𝑖,∗𝐸_𝑡1𝑥₁⋯𝐸_𝑡𝑛𝑥_𝑛𝜉₁,…,𝜉_𝑛=𝑢̃𝜉_𝑖−𝜉_𝑖−1,𝑡_𝑖+𝑡_𝑖−1,(2.24) where 𝑢̃ is the Gauss transform of 𝑢. Thus, (1.5) is converted into the following classical functional equation: 𝑢̃^𝑛_𝑖=1𝑥_𝑖,^𝑛_𝑖=1𝑡_𝑖=^𝑛_𝑖=1𝑢̃𝑥_𝑖,𝑡_𝑖+^𝑛_𝑖=1𝑢̃𝑥_𝑖−𝑥_𝑖−1,𝑡_𝑖+𝑡_𝑖−1,(2.25) for all 𝑥₁,…,𝑥_𝑛∈ℝ^𝑚, 𝑡₁,…,𝑡_𝑛>0. It follows from Lemma 2.4 that the solution 𝑢̃ of (2.25) has the form 𝑢̃(𝑥,𝑡)=𝑎⋅𝑥,(2.26) for some 𝑎∈ℂ^𝑚. Letting 𝑡→0⁺ in (2.26), we finally obtain the general solution for (1.5).

We are going to solve the stability problems for the additive functional equation in the spaces of ℱ^(ℝ^𝑚) and 𝒮^(ℝ^𝑚).

Lemma 2.6. Suppose that 𝑓∶ℝ^𝑚×(0,∞)→ℂ is a continuous function satisfying |||||𝑓_𝑛_𝑖=1𝑥_𝑖,^𝑛_𝑖=1𝑡_𝑖−^𝑛_𝑖=1𝑓𝑥_𝑖,𝑡_𝑖−^𝑛_𝑖=1𝑓𝑥_𝑖−𝑥_𝑖−1,𝑡_𝑖+𝑡_𝑖−1|||||≤𝜖,(2.27) for all 𝑥₁,…,𝑥_𝑛∈ℝ^𝑚, 𝑡₁,…,𝑡_𝑛>0, then there exists a unique 𝑎∈ℂ^𝑚 such that ||𝑓(𝑥,𝑡)−𝑎⋅𝑥||≤10𝑛−32(2𝑛−1)𝜖,(2.28) for all 𝑥∈ℝ^𝑚, 𝑡>0.

Proof. Putting (𝑥₁,…,𝑥_𝑛)=(0,…,0) in (2.27) yields |||||𝑓0,_𝑛_𝑖=1𝑡_𝑖−^𝑛_𝑖=1𝑓0,𝑡_𝑖−^𝑛_𝑖=1𝑓0,𝑡_𝑖+𝑡_𝑖−1|||||≤𝜖,(2.29) for all 𝑡₁,…,𝑡_𝑛>0. In view of (2.29), we see that 𝑐∶=limsup_𝑡→0+𝑓(0,𝑡)(2.30) exists. Letting 𝑡₁=⋯=𝑡_𝑛→0⁺ in (2.29) gives |𝑐|≤𝜖2𝑛−1.(2.31) Setting (𝑥₁,𝑥₂,…,𝑥_𝑛)=(𝑥,0,…,0) and letting 𝑡₁=𝑡, 𝑡₂=⋯=𝑡_𝑛→0⁺ in (2.27), we have ||𝑓(𝑥,𝑡)+𝑓(−𝑥,𝑡)+(2𝑛−3)𝑐||≤𝜖,(2.32) for all 𝑥∈ℝ^𝑚, 𝑡>0. Substituting (𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛)=(𝑥,𝑥,0,…,0) and letting 𝑡₁=𝑡₂=𝑡, 𝑡₃=⋯=𝑡_𝑛→0⁺ in (2.13), we obtain ||𝑓(2𝑥,2𝑡)−3𝑓(𝑥,𝑡)−𝑓(−𝑥,𝑡)−𝑓(0,2𝑡)−(2𝑛−5)𝑐||≤𝜖,(2.33) for all 𝑥∈ℝ^𝑚, 𝑡>0. Adding (2.33) to (2.32) yields ||𝑓(2𝑥,2𝑡)−2𝑓(𝑥,𝑡)−𝑓(0,2𝑡)+2𝑐||≤2𝜖,(2.34) for all 𝑥∈ℝ^𝑚, 𝑡>0. Letting 𝑡₁=𝑡₂=𝑡, 𝑡₃=⋯=𝑡_𝑛→0⁺ in (2.29) gives ||4𝑓(0,𝑡)+(2𝑛−5)𝑐||≤𝜖,(2.35) for all 𝑡>0. Combining (2.31), (2.34), and (2.35), we have |||𝑓(2𝑥,2𝑡)2−𝑓(𝑥,𝑡)|||≤41(02𝑛𝑛−−31)𝜖,(2.36) for all 𝑥∈ℝ^𝑚, 𝑡>0. Making use of induction argument in (2.36), we obtain ||||𝑓2^𝑘𝑥,2^𝑘𝑡2^𝑘−𝑓(𝑥,𝑡)||||≤10𝑛−32(2𝑛−1)𝜖,(2.37) for all 𝑘∈ℕ, 𝑥∈ℝ^𝑚, 𝑡>0. Replacing 𝑥, 𝑡 by 2^𝑙𝑥, 2^𝑙𝑡 in (2.37), respectively, and dividing the result by 2^𝑙, we see that for 𝑘≥𝑙>0, ||||𝑓2^𝑘+𝑙𝑥,2^𝑘+𝑙𝑡2^𝑘+𝑙−𝑓2^𝑙𝑥,2^𝑙𝑡2^𝑙||||≤10𝑛−32^𝑙+1(2𝑛−1)𝜖.(2.38) Since the right-hand side of (2.38) tends to 0 as 𝑙→∞, the sequence {2^−𝑘𝑓(2^𝑘𝑥,2^𝑘𝑡)} is a Cauchy sequence which converges uniformly. Thus, we may define 𝐴(𝑥,𝑡)∶=lim_𝑘→∞𝑓2^𝑘𝑥,2^𝑘𝑡2^𝑘,(2.39) for all 𝑥∈ℝ^𝑚, 𝑡>0. Now, we verify from (2.27) that the function 𝐴 satisfies 𝐴^𝑛_𝑖=1𝑥_𝑖,^𝑛_𝑖=1𝑡_𝑖=^𝑛_𝑖=1𝐴𝑥_𝑖,𝑡_𝑖+^𝑛_𝑖=1𝐴𝑥_𝑖−𝑥_𝑖−1,𝑡_𝑖+𝑡_𝑖−1,(2.40) for all 𝑥₁,…,𝑥_𝑛∈ℝ^𝑚, 𝑡₁,…,𝑡_𝑛>0. As observed in Lemma 2.4, the continuous solution of (2.40) has the form 𝐴(𝑥,𝑡)=𝑎⋅𝑥,(2.41) for some 𝑎∈ℂ^𝑚. It follows from (2.37) that 𝐴 is the unique function in ℝ^𝑚×(0,∞) satisfying ||𝑓(𝑥,𝑡)−𝐴(𝑥,𝑡)||≤10𝑛−32(2𝑛−1)𝜖,(2.42) for all 𝑥∈ℝ^𝑚, 𝑡>0.

From the above lemma, we have the following stability theorem for the additive functional equation in the spaces of ℱ^(ℝ^𝑚) and 𝒮^(ℝ^𝑚).

Theorem 2.7. Suppose that 𝑢 in ℱ^(ℝ^𝑚) (or 𝒮^(ℝ^𝑚), resp.) satisfies the inequality (1.6), then there exists a unique 𝑎∈ℂ^𝑚 such that ‖𝑢−𝑎⋅𝑥‖≤10𝑛−32(2𝑛−1)𝜖.(2.43)

Proof. Convolving the tensor product 𝐸_𝑡1(𝑥₁)⋯𝐸_𝑡𝑛(𝑥_𝑛) of the heat kernels on both sides of (1.6), we have |||||𝑢̃_𝑛_𝑖=1𝑥_𝑖,^𝑛_𝑖=1𝑡_𝑖−^𝑛_𝑖=1𝑢̃𝑥_𝑖,𝑡_𝑖−^𝑛_𝑖=1𝑢̃𝑥_𝑖−𝑥_𝑖−1,𝑡_𝑖+𝑡_𝑖−1|||||≤𝜖,(2.44) for all 𝑥₁,…,𝑥_𝑛∈ℝ^𝑚, 𝑡₁,…,𝑡_𝑛>0, where 𝑢̃ is the Gauss transform of 𝑢. By Lemma 2.6, we have ||𝑢̃(𝑥,𝑡)−𝑎⋅𝑥||≤10𝑛−32(2𝑛−1)𝜖,(2.45) for all 𝑥∈ℝ^𝑚, 𝑡>0. Letting 𝑡→0⁺ in (2.45), we obtain the conclusion.

3. Stability in 𝒟^(ℝ^𝑚)

In this section, we shall extend the previous results to the space of distributions. Recall that a distribution 𝑢 is a linear functional on 𝐶^∞𝑐(ℝ^𝑚) of infinitely differentiable functions on ℝ^𝑚 with compact supports such that for every compact set 𝐾⊂ℝ^𝑚, there exist constants 𝐶>0 and 𝑁∈ℕ₀ satisfying ||⟨𝑢,𝜑⟩||≤𝐶_{|𝛼|≤𝑁}sup||𝜕^𝛼𝜑||,(3.1) for all 𝜑∈𝐶^∞𝑐(ℝ^𝑚) with supports contained in 𝐾. The set of all distributions is denoted by 𝒟^(ℝ^𝑚). It is well known that the following topological inclusions hold: 𝐶^∞𝑐(ℝ^𝑚)↪𝒮(ℝ^𝑚),𝒮^(ℝ^𝑚)↪𝒟^(ℝ^𝑚).(3.2) As we see in [19, 20, 23], by the semigroup property of the heat kernel, (1.5) can be controlled easily in the spaces ℱ^(ℝ^𝑚) and 𝒮^(ℝ^𝑚). But we cannot employ the heat kernel in the space 𝒟^(ℝ^𝑚). For that reason, instead of the heat kernel, we use the regularizing functions. We denote by 𝜓 the function on ℝ^𝑚 satisfying 𝜓(𝑥)=𝐴0,exp−1−|𝑥|²⁻¹,||𝑥𝑥||≥<11,,(3.3) where 𝐴=_|𝑥|<1exp−1−|𝑥|²⁻¹𝑑𝑥⁻¹.(3.4) It is easy to see that 𝜓 is an infinitely differentiable function supported in the set {𝑥∶|𝑥|≤1} with ∫𝜓(𝑥)𝑑𝑥=1. For each 𝑡>0, we define 𝜓_𝑡(𝑥)∶=𝑡^−𝑚𝜓(𝑥/𝑡), then 𝜓_𝑡 has all the properties of 𝜓 except that the support of 𝜓_𝑡 is contained in the ball of radius 𝑡 with center at 0. If 𝑢∈𝒟^(ℝ^𝑚), then for each 𝑡>0, (𝑢∗𝜓_𝑡)(𝑥)=⟨𝑢_𝑦,𝜓_𝑡(𝑥−𝑦)⟩ is a smooth function in ℝ^𝑚 and (𝑢∗𝜓_𝑡)(𝑥)→𝑢as𝑡→0⁺ in the sense of distributions, that is, for every 𝜑∈𝐶^∞𝑐(ℝ^𝑚),⟨𝑢∗𝜓_𝑡,𝜑⟩=𝑢∗𝜓_𝑡(𝑥)𝜑(𝑥)𝑑𝑥⟶⟨𝑢,𝜑⟩as𝑡⟶0⁺.(3.5) For each 𝑡>0, the function 𝑢∗𝜓_𝑡 is called a regularization of 𝑢, and the transform which maps 𝑢 to 𝑢∗𝜓_𝑡 is called a mollifier. Making use of the mollifiers, we can solve the general solution for the additive functional equation in the space 𝒟^(ℝ^𝑚) as follows.

Theorem 3.1. Every solution 𝑢 in 𝒟^(ℝ^𝑚) of (1.5) has the form 𝑢=𝑎⋅𝑥,(3.6) for some 𝑎∈ℂ^𝑚.

Proof. Convolving the tensor product 𝜓_𝑡1(𝑥₁)⋯𝜓_𝑡𝑛(𝑥_𝑛) of the regularizing functions on both sides of (1.5), we have (𝑢∘=𝐴⟨)𝑢∗∘𝐴𝜓,𝜓_𝑡1_𝑡1𝑥₁⋯𝜓_𝑡𝑛𝑥_𝑛𝜉₁,…,𝜉_𝑛=𝑢,⋯𝜓𝜉_1𝑡1−𝑥₁⋯𝜓_𝑡𝑛𝜉_𝑛−𝑥_𝑛⟩=𝑢,⋯𝜓_𝑡1𝜉₁−𝑥₁+𝑥₂+⋯+𝑥_𝑛𝜓_𝑡2𝜉₂−𝑥₂⋯𝜓_𝑡𝑛𝜉_𝑛−𝑥_𝑛𝑑𝑥₂⋯𝑑𝑥_𝑛=⟨𝑢,𝜓_𝑡1𝜉₁+⋯+𝜉_𝑛−𝑥₁−⋯−𝑥_𝑛𝜓_𝑡2𝑥₂⋯𝜓_𝑡𝑛𝑥_𝑛𝑑𝑥₂⋯𝑑𝑥_𝑛=𝑢∗𝜓_𝑡1∗⋯∗𝜓_𝑡𝑛𝜉₁+⋯+𝜉_𝑛−𝑥₁⟩𝑢∗∘⋯𝑃_𝑖∗𝜓_𝑡𝑛𝜉₁+⋯+𝜉_𝑛,𝑢∘𝐵_𝑖∗𝜓_𝑡1𝑥₁⋯𝜓_𝑡𝑛𝑥_𝑛𝜉₁,…,𝜉_𝑛=𝑢∗𝜓_𝑡𝑖𝜉_𝑖,∗𝜓_𝑡1𝑥₁⋯𝜓_𝑡𝑛𝑥_𝑛𝜉₁,…,𝜉_𝑛=𝑢∗𝜓_𝑡𝑖∗𝜓_𝑡𝑖−1𝜉_𝑖−𝜉_𝑖−1.(3.7) Thus, (1.5) is converted into the following functional equation: 𝑢∗𝜓_𝑡1∗⋯∗𝜓_𝑡𝑛𝑥₁+⋯+𝑥_𝑛=^𝑛_𝑖=1𝑢∗𝜓_𝑡𝑖𝑥_𝑖+^𝑛_𝑖=1𝑢∗𝜓_𝑡𝑖∗𝜓_𝑡𝑖−1𝑥_𝑖−𝑥_𝑖−1,(3.8) for all 𝑥₁,…,𝑥_𝑛∈ℝ^𝑚, 𝑡₁,…,𝑡_𝑛>0. In view of (3.8), it is easy to see that, for each fixed 𝑥∈ℝ^𝑚, 𝑓(𝑥)∶=lim_𝑡→0+𝑢∗𝜓_𝑡(𝑥)(3.9) exists. Putting (𝑥₁,…,𝑥_𝑛)=(0,…,0) and letting 𝑡₁=⋯=𝑡_𝑛→0⁺ in (3.8) yield 𝑓(0)=0. Setting (𝑥₁,𝑥₂,…,𝑥_𝑛)=(𝑥,0,…,0) and letting 𝑡₁=𝑡, 𝑡₂=⋯=𝑡_𝑛→0⁺ in (3.8) give 𝑢∗𝜓_𝑡(−𝑥)=−𝑢∗𝜓_𝑡(𝑥),(3.10) for all 𝑥∈ℝ^𝑚, 𝑡>0. Substituting (𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛) with (𝑥,𝑦,0,…,0) and letting 𝑡₁=𝑡, 𝑡₂=𝑠, 𝑡₃=⋯=𝑡_𝑛→0⁺ in (3.8), we obtain from (3.10) that 𝑢∗𝜓_𝑡∗𝜓_𝑠(𝑥+𝑦)+𝑢∗𝜓_𝑡∗𝜓_𝑠(𝑥−𝑦)=2𝑢∗𝜓_𝑡(𝑥),(3.11) for all 𝑥,𝑦∈ℝ^𝑚, 𝑡,𝑠>0. Letting 𝑡→0⁺ in (3.11) yields 𝑢∗𝜓_𝑠(𝑥+𝑦)+𝑢∗𝜓_𝑠(𝑥−𝑦)=2𝑓(𝑥),(3.12) for all 𝑥,𝑦∈ℝ^𝑚, 𝑠>0. Putting 𝑦=0 in (3.12) gives 𝑓(𝑥)=𝑢∗𝜓_𝑠(𝑥),(3.13) for all 𝑥∈ℝ^𝑚, 𝑠>0. Applying (3.13) to (3.11), we see that 𝑓 satisfies 𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)=2𝑓(𝑥),(3.14) which is equivalent to the Cauchy equation (1.1) for all 𝑥,𝑦∈ℝ^𝑚. Since 𝑓 is a smooth function in view of (3.13), it follows that 𝑓(𝑥)=𝑎⋅𝑥 for some 𝑎∈ℂ^𝑚. Letting 𝑠→0⁺ in (3.13), we finally obtain the general solution for (1.5).

Now, we shall extend the stability theorem for the additive equation mentioned in the previous section to the space 𝒟^(ℝ^𝑚).

Theorem 3.2. Suppose that 𝑢 in 𝒟^(ℝ^𝑚) satisfies the inequality (1.6), then there exists a unique 𝑎∈ℂ^𝑚 such that ‖𝑢−𝑎⋅𝑥‖≤10𝑛−32(2𝑛−1)𝜖.(3.15)

Proof. It suffices to show that every distribution satisfying (1.6) belongs to the space 𝒮^(ℝ^𝑚). Convolving the tensor product 𝜓_𝑡1(𝑥₁)⋯𝜓_𝑡𝑛(𝑥_𝑛) on both sides of (1.6), we have |||||𝑢∗𝜓_𝑡1∗⋯∗𝜓_𝑡𝑛𝑥₁+⋯+𝑥_𝑛−^𝑛_𝑖=1𝑢∗𝜓_𝑡𝑖𝑥_𝑖−^𝑛_𝑖=1𝑢∗𝜓_𝑡i∗𝜓_𝑡𝑖−1𝑥_𝑖−𝑥_𝑖−1|||||≤𝜖,(3.16) for all 𝑥₁,…,𝑥_𝑛∈ℝ^𝑚, 𝑡₁,…,𝑡_𝑛>0. In view of (3.16), it is easy to see that for each fixed 𝑥, 𝑓(𝑥)∶=limsup_𝑡→0+𝑢∗𝜓_𝑡(𝑥)(3.17) exists. Putting (𝑥₁,…,𝑥_𝑛)=(0,…,0) and letting 𝑡₁=⋯=𝑡_𝑛→0⁺ in (3.16) yield ||𝑓(0)||≤𝜖2𝑛−1.(3.18) Setting (𝑥₁,𝑥₂,…,𝑥_𝑛)=(𝑥,0,…,0) and letting 𝑡₁=𝑡, 𝑡₂=⋯=𝑡_𝑛→0⁺ in (3.16), we have ||𝑢∗𝜓_𝑡(𝑥)+𝑢∗𝜓_𝑡(−𝑥)+(2𝑛−3)𝑓(0)||≤𝜖,(3.19) for all 𝑥∈ℝ^𝑚, 𝑡>0. Substituting (𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛) with (𝑥,𝑦,0,…,0) and letting 𝑡₁=𝑡, 𝑡₂=𝑠, 𝑡₃=⋯=𝑡_𝑛→0⁺ in (3.16), we have ||𝑢−∗2𝜓𝑢_𝑡∗∗𝜓𝜓_𝑠𝑡(𝑥+𝑦)−𝑢∗𝜓_𝑡∗𝜓_𝑠(𝑦−𝑥)(𝑥)−𝑢∗𝜓_𝑠(𝑦)−𝑢∗𝜓_𝑠(−𝑦)−(2𝑛−5)𝑓(0)||≤𝜖,(3.20) for all 𝑥,𝑦∈ℝ^𝑚, 𝑡,𝑠>0. It follows from (3.19) that the inequality (3.20) can be rewritten as ||𝑢∗𝜓_𝑡∗𝜓_𝑠(𝑥+𝑦)−𝑢∗𝜓_𝑡∗𝜓_𝑠(𝑦−𝑥)−2𝑢∗𝜓_𝑡(𝑥)+2𝑓(0)||≤2𝜖,(3.21) for all 𝑥,𝑦∈ℝ^𝑚, 𝑡,𝑠>0. Letting 𝑡→0⁺ in (3.21) yields ||𝑢∗𝜓_𝑠(𝑥+𝑦)−𝑢∗𝜓_𝑠(𝑦−𝑥)−2𝑓(𝑥)+2𝑓(0)||≤2𝜖,(3.22) for all 𝑥,𝑦∈ℝ^𝑚, 𝑠>0. Using (3.19) we may write the inequality (3.22) as ||𝑢∗𝜓_𝑠(𝑥+𝑦)+𝑢∗𝜓_𝑠(𝑥−𝑦)−2𝑓(𝑥)+(2𝑛−1)𝑓(0)||≤3𝜖,(3.23) for all 𝑥,𝑦∈ℝ^𝑚, 𝑡,𝑠>0. Putting 𝑦=0 in (3.23) and dividing the result by 2 give |||𝑢∗𝜓_𝑠(𝑥)−𝑓(𝑥)+(2𝑛2−1)𝑓(0)|||≤23𝜖,(3.24) for all 𝑥∈ℝ^𝑚,𝑠>0. From (3.23) and (3.24), we have ||𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)−2𝑓(𝑥)||≤6𝜖,(3.25) which is equivalent to |||2𝑓𝑥+𝑦2−𝑓(𝑥)−𝑓(𝑦)|||≤6𝜖,(3.26) for all 𝑥,𝑦∈ℝ^𝑚. Thus, by virtue of the result as in [29], there exists a unique function 𝑔∶ℝ^𝑚→ℂ satisfying 𝑔(𝑥+𝑦)=𝑔(𝑥)+𝑔(𝑦)(3.27) such that ||𝑓(𝑥)−𝑔(𝑥)||≤6𝜖+||𝑔(0)||,(3.28) for all 𝑥∈ℝ^𝑚. It follows from (3.18), (3.24), and (3.28) that ||𝑢∗𝜓_𝑠(𝑥)−𝑔(𝑥)||≤8𝜖+||𝑔(0)||,(3.29) for all 𝑥∈ℝ^𝑚, 𝑠>0. Letting 𝑠→0⁺ in (3.29), we obtain ‖𝑢−𝑔(𝑥)‖≤8𝜖+||𝑔(0)||.(3.30) Inequality (3.30) implies that ℎ(𝑥)∶=𝑢−𝑔(𝑥) belongs to (𝐿¹)^=𝐿^∞. Thus, we conclude that 𝑢=𝑔(𝑥)+ℎ(𝑥)∈𝒮^(ℝ^𝑚).