On the stability of a mixed type functional equation in generalized functions

Abstract

We reformulate the following mixed type quadratic and additive functional equation with n-independent variables

$$2f\left(\sum _{i=1}^n{x}_i\right)+\sum _{\begin{array}{c}\hfill 1\le i,j\le n\hfill \\ \hfill i\ne j\hfill \end{array}}f\left(x_i-x_j\right)=\left(n+1\right)\sum _{i=1}^{n}f\left(x_i\right)+\left(n-1\right)\sum _{i=1}^{n}f\left(-x_i\right)$$

as the equation for the spaces of generalized functions. Using the fundamental solution of the heat equation, we solve the general solution and prove the Hyers-Ulam stability of this equation in the spaces of tempered distributions and Fourier hyperfunctions.

Mathematics Subject Classification 2000: 39B82; 39B52.

1. Introduction

In 1940, Ulam [1] raised a question concerning the stability of group homomorphisms as follows:

Let G₁ be a group and let G₂ be a metric group with the metric d(·,·). Given ε > 0, does there exist a δ > 0 such that if a function h : G₁ → G₂ satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G₁, then there exists a homomorphism H : G₁ → G₂ with d(h(x), H(x)) < ε for all x ∈ G₁?

In 1941, Hyers [2] firstly presented the stability result of functional equations under the assumption that G₁ and G₂ are Banach spaces. In 1978, Rassias [3] generalized Hyers' result to the unbounded Cauchy difference. After that stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [4–7]). Among them, Towanlong and Nakmahachalasint [8] introduced the following functional equation with n-independent variables

$$2f\left(\sum _{i=1}^n{x}_i\right)+\sum _{\begin{array}{c}\hfill 1\le i,j\le n\hfill \\ \hfill i\ne j\hfill \end{array}}f\left(x_i-x_j\right)=\left(n+1\right)\sum _{i=1}^{n}f\left(x_i\right)+\left(n-1\right)\sum _{i=1}^{n}f\left(-x_i\right),$$

(1.1)

where n is a positive integer with n ≥ 2. For real vector spaces X and Y, they proved that a function f : X → Y satisfies (1.1) if and only if there exist a quadratic function q : X → Y satisfying

$$q\left(x+y\right)+q\left(x-y\right)=2q\left(x\right)+2q\left(y\right)$$

and an additive function a : X → Y satisfying

$$a\left(x+y\right)=a\left(x\right)+a\left(y\right)$$

such that

$$f\left(x\right)=q\left(x\right)+a\left(x\right)$$

for all x ∈ X. For this reason, equation (1.1) is called the mixed type quadratic and additive functional equation. We refer to [9–14] for the stability results of other mixed type functional equations.

In this article, we consider equation (1.1) in the spaces of generalized functions such as the space ${\mathcal{S}}^{\prime }\left(ℝ\right)$ of tempered distributions and the space ${\mathcal{F}}^{\prime }\left(ℝ\right)$ of Fourier hyperfunctions. Making use of similar approaches in [15–20], we reformulate equation (1.1) and the related inequality for the spaces of generalized functions as follows:

$$2u\circ A+\sum _{\begin{array}{c}\hfill \mathsf{\text{1}}\le i,j\le n,\hfill \\ \hfill i\ne j\hfill \end{array}}u\circ B_{ij}=\left(n+1\right)\sum _{i=1}^{n}u\circ P_i+\left(n-1\right)\sum _{i=1}^{n}u\circ Q_i,$$

(1.2)

$$∥2u\circ A+\sum _{\begin{array}{c}\hfill \mathsf{\text{1}}\le i,j\le n,\hfill \\ \hfill i\ne j\hfill \end{array}}u\circ B_{ij}-\left(n+1\right)\sum _{i=1}^{n}u\circ P_i-\left(n-1\right)\sum _{i=1}^{n}u\circ Q_i∥\le \epsilon ,$$

(1.3)

where A, B _ij , P _i and Q _i are the functions defined by

$$\begin{array}{c}A\left(x_1,...,x_n\right)=x_1+\cdots +x_n,\\ B_{ij}\left(x_1,...,x_n\right)=x_i-x_j,1\le i,j\le n,i\ne j,\\ P_i\left(x_1,...,x_n\right)=x_i,1\le i\le n,\\ Q_i\left(x_1,...,x_n\right)=-x_i,1\le i\le n.\end{array}$$

Here ○ denotes the pullback of generalized functions and the inequality ||v|| ≤ ε in (1.3) means that $\left|〈v,\phi 〉\right|\le \epsilon {\left|\right|\phi \left|\right|}_{L^1}$ for all test functions φ.

In order to solve the general solution of (1.2) and prove the Hyers-Ulam stability of (1.3), we employ the heat kernel method stated in section 2. In section 3, we prove that every solution u in ${\mathcal{F}}^{\prime }\left(ℝ\right)$ (or ${\mathcal{S}}^{\prime }\left(ℝ\right)$, resp.) of equation (1.2) is of the form

$$u=a{x}^2+bx$$

for some a, b ∈ ℂ. Subsequently, in section 4, we prove that every solution u in ${\mathcal{F}}^{\prime }\left(ℝ\right)$ (or ${\mathcal{S}}^{\prime }\left(ℝ\right)$, resp.) of the inequality (1.3) can be written uniquely in the form

$$u=a{x}^2+bx+\mu \left(x\right),$$

where µ is a bounded measurable function such that ${∥\mu ∥}_{L^{\mathrm{\infty }}}\le \frac{n^2+n-3}{n^2+n-2}\epsilon .$

2. Preliminaries

In this section, we introduce the spaces of tempered distributions and Fourier hyperfunctions. We first consider the space of rapidly decreasing functions which is a test function space of tempered distributions.

Definition 2.1. [21] The space $\mathcal{S}\left(ℝ\right)$ denotes the set of all infinitely differentiable functions φ : ℝ → ℂ such that

$${∥\phi ∥}_{\alpha ,\beta }=\underset{x}{\text{sup}}\left|x^{\alpha }{D}^{\beta }\phi \left(x\right)\right|<\mathrm{\infty }$$

for all nonnegative integers α, β.

In other words, φ(x) as well as its derivatives of all orders vanish at infinity faster than the reciprocal of any polynomial. For that reason, we call the element of $\mathcal{S}\left(ℝ\right)$ as the rapidly decreasing function. It can be easily shown that the function φ(x) = exp(−ax²), a > 0, belongs to $\mathcal{S}\left(ℝ\right)$, but ψ(x) = (1 + x²)⁻¹ is not a member of $\mathcal{S}\left(ℝ\right)$. Next we consider the space of tempered distributions which is a dual space of $\mathcal{S}\left(ℝ\right)$.

Definition 2.2. [21] A linear functional u on $\mathcal{S}\left(ℝ\right)$ is said to be a tempered distribution if there exists constant C ≥ 0 and nonnegative integer N such that

$$\left|〈u,\phi 〉\right|\le C\sum _{\alpha ,\beta \le N}\underset{x}{\text{sup}}\left|x^{\alpha }{D}^{\beta }\phi \right|$$

(2.1)

for all $\phi \in \mathcal{S}\left(ℝ\right)$. The set of all tempered distributions is denoted by ${\mathcal{S}}^{\prime }\left(ℝ\right)$.

For example, every f ∈ L^p(ℝ), 1 ≤ p < ∞, defines a tempered distribution by virtue of the relation

$$〈f,\phi 〉=\int f\left(x\right)\phi \left(x\right)dx,\phi \in \mathcal{S}\left(ℝ\right).$$

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, but not all distributions have one. Imposing the growth condition on || · ||_α,βin (2.1) a new space of test functions has emerged as follows.

Definition 2.3. [22] We denote by $\mathcal{F}\left(ℝ\right)$ the set of all infinitely differentiable functions φ in ℝ such that

$${∥\phi ∥}_{A,B}=\underset{x,\alpha ,\beta }{\text{sup}}\frac{\left|x^{\alpha }{D}^{\beta }\phi \left(x\right)\right|}{{A^{|}\alpha }^{|}{B^{|}\beta }^{|}\alpha !\beta !}<\mathrm{\infty }$$

(2.2)

for some positive constants A, B depending only on φ.

It can be verified that the seminorm (2.2) is equivalent to

$${∥\phi ∥}_{h,k}=\underset{x,\alpha }{\text{sup}}\frac{\left|D^{\alpha }\phi \left(x\right)\right|\text{exp}k\left|x\right|}{{h^{|}\alpha }^{|}\alpha !}<\mathrm{\infty }$$

for some constants h, k > 0.

Definition 2.4. [22] The strong dual space of $\mathcal{F}\left(ℝ\right)$ is called the Fourier hyperfunctions. We denote the Fourier hyperfunctions by ${\mathcal{F}}^{\prime }\left(ℝ\right)$.

It is easy to see the following topological inclusions:

$$\mathcal{F}\left(ℝ\right)\hookrightarrow \mathcal{S}\left(ℝ\right),\mathcal{S}\prime \left(ℝ\right)\hookrightarrow {\mathcal{F}}^{\prime }\left(ℝ\right).$$

(2.3)

Taking the relations (2.3) into account, it suffices to consider the space ${\mathcal{F}}^{\prime }\left(ℝ\right)$. In order to solve the general solution and the stability problem of (1.2) in the space ${\mathcal{F}}^{\prime }\left(ℝ\right)$, we employ the fundamental solution of the heat equation called the heat kernel,

$$E_t\left(x\right)=E\left(x,t\right)=\left\{\begin{array}{cccc}\hfill {\left(4\pi t\right)}^{-1/2}\text{exp}\left(-x^2/4t\right)\hfill & \hfill ,\hfill & \hfill x\in ℝ,\hfill & \hfill t>0,\hfill \\ \hfill 0\hfill & \hfill ,\hfill & \hfill x\in ℝ,\hfill & \hfill t\le 0.\hfill \end{array}\right.$$

Since for each t > 0, E(·, t) belongs to the space $\mathcal{F}\left(ℝ\right)$, the convolution

$$ũ\left(x,t\right)=\left(u*E\right)\left(x,t\right)=〈u_y,E_t\left(x-y\right)〉,x\in ℝ,t>0$$

is well defined for all $u\in {\mathcal{F}}^{\prime }\left(ℝ\right)$. We call $ũ$ as the Gauss transform of u. Semigroup property of the heat kernel

$$\left(E_t*E_s\right)\left(x\right)=E_{t+s}\left(x\right)$$

holds for convolution. It is useful to convert equation (1.2) into the classical functional equation defined on upper-half plane. We also use the following famous result called heat kernel method, which states as follows.

Theorem 2.5. [23] Let $u\in {\mathcal{S}}^{\prime }\left(ℝ\right)$. Then its Gauss transform $ũ$ is a C^∞-solution of the heat equation

$$\left(\partial /\partial t-\mathrm{\Delta }\right)ũ\left(x,t\right)=0$$

satisfying

(i) There exist positive constants C, M and N such that

$$\left|ũ\left(x,t\right)\right|\le C{t}^{-M}{\left(1+\left|x\right|\right)}^{N}inℝ\times \left(0,\delta \right).$$

(2.4)

(ii) $ũ\left(x,t\right)\to u$ as t → 0⁺ in the sense that for every $\phi \in \mathcal{S}\left(ℝ\right)$,

$$〈u,\phi 〉=\underset{t\to 0^{+}}{\text{lim}}\int ũ\left(x,t\right)\phi \left(x\right)dx.$$

Conversely, every C^∞-solution U(x, t) of the heat equation satisfying the growth condition (2.4) can be uniquely expressed as $U\left(x,t\right)=ũ\left(x,t\right)$ for some $u\in {\mathcal{S}}^{\prime }\left(ℝ\right)$.

Similarly, we can represent Fourier hyperfunctions as initial values of solutions of the heat equation as a special case of the results as in [24]. In this case, the condition (i) in the above theorem is replaced by the following:

For every ε > 0 there exists a positive constant C _ε such that

$$\left|ũ\left(x,t\right)\right|\le C_{\epsilon }\text{exp}\left(\epsilon \left(\left|x\right|+1/t\right)\right)\mathsf{\text{in}}ℝ\times \left(0,\delta \right).$$

3. General solution in ${\mathcal{F}}^{\prime }\left(ℝ\right)$

We are now going to solve the general solution of (1.2) in the space of ${\mathcal{F}}^{\prime }\left(ℝ\right)$ (or ${\mathcal{S}}^{\prime }\left(ℝ\right)$, resp.). In order to do so, we employ the heat kernel mentioned in the previous section. Convolving the tensor product $E_{t_1}\left(x_1\right)...E_{t_n}\left(x_n\right)$ of the heat kernels on both sides of (1.2) we have

$$\begin{array}{c}\left[\left(u\circ A\right)*\left(E_{t_1}\left(x_1\right)...E_{t_n}\left(x_n\right)\right)\right]\left({\xi }_1,...,{\xi }_n\right)\\ =〈u\circ A,E_{t_1}\left({\xi }_1-x_1\right)...E_{t_n}\left({\xi }_n-x_n\right)〉\\ =〈u,\int \cdots \int E_{t_1}({\xi }_1-x_1+x_2+\cdots +x_n\mathsf{\text{)}}{E}_{t_2}\left({\xi }_2-x_2\right)...E_{t_n}\left({\xi }_n-x_n\right)d{x}_2...d{x}_n〉\\ =〈u,\int \cdots \int E_{t_1}({\xi }_1+\cdots +{\xi }_n-x_1-\cdots -x_n\mathsf{\text{)}}{E}_{t_2}\left(x_2\right)...E_{t_n}\left(x_n\right)d{x}_2...d{x}_n〉\\ =〈u,\left(E_{t_1}*...*E_{t_n}\right)\left({\xi }_1+\cdots +{\xi }_n-x_1\right)〉\\ =〈u,E_{t_1+\cdots +t_n}\left({\xi }_1+\cdots +{\xi }_n\right)〉\\ =ũ\left({\xi }_1+\cdots +{\xi }_n,t_1+\cdots +t_n\right),\\ \left[\left(u\circ B_{ij}\right)*\left(E_{t_1}\left(x_1\right)...E_{t_n}\left(x_n\right)\right)\right]\left({\xi }_1,...,{\xi }_n\right)=ũ\left({\xi }_i-{\xi }_j,t_i+t_j\right),\\ \left[\left(u\circ P_i\right)*\left(E_{t_1}\left(x_1\right)...E_{t_n}\left(x_n\right)\right)\right]\left({\xi }_1,...,{\xi }_n\right)=ũ\left({\xi }_i,t_i\right),\\ \left[\left(u\circ Q_i\right)*\left(E_{t_1}\left(x_1\right)...E_{t_n}\left(x_n\right)\right)\right]\left({\xi }_1,...,{\xi }_n\right)=ũ\left(-{\xi }_i,t_i\right),\end{array}$$

where $ũ$ is the Gauss transform of u. Thus, (1.2) is converted into the following classical functional equation

$$\begin{array}{c}2ũ\left(\sum _{i=1}^n{x}_i,\sum _{i=1}^n{t}_i\right)+\sum _{\begin{array}{c}\hfill 1\le i,j\le n,\hfill \\ \hfill i\ne j\hfill \end{array}}ũ\left(x_i-x_j,t_i+t_j\right)\\ =\left(n+1\right)\sum _{i=1}^nũ\left(x_i,t_i\right)+\left(n-1\right)\sum _{i=1}^nũ\left(-x_i,t_i\right)\end{array}$$

for all x₁, . . . , x _n ∈ ℝ, t₁, . . . , t _n > 0. We here need the following lemma which will be crucial role in the proof of main theorem.

Lemma 3.1. A continuous function f : ℝ × (0, ∞) → ℂ satisfies the functional equation

$$\begin{array}{c}2f\left(\sum _{i=1}^n{x}_i,\sum _{i=1}^n{t}_i\right)+\sum _{\begin{array}{c}\hfill 1\le i,j\le n\hfill \\ \hfill i\ne j\hfill \end{array}}f\left(x_i-x_j,t_i+t_j\right)\\ =\left(n+1\right)\sum _{i=1}^{n}f\left(x_i,t_i\right)+\left(n-1\right)\sum _{i=1}^{n}f\left(-x_i,t_i\right)\end{array}$$

(3.1)

for all x₁, . . . , x _n ∈ ℝ, t₁, . . . , t _n > 0 if and only if there exist constants a, b, c ∈ ℂ such that

$$f\left(x,t\right)=a{x}^2+bx+ct$$

for all x ∈ ℝ, t > 0.

Proof. Putting (x₁, . . . , x _n ) = (0, . . . , 0) in (3.1) yields

$$f\left(0,\sum _{{i=1}^n}{t}_i\right)+\sum _{1\le i<j\le n}f\left(0,t_i+t_j\right)=n\sum _{i=1}^{n}f\left(0,t_i\right)$$

(3.2)

for all t₁, . . . , t _n > 0. In view of (3.2) we see that

$$c:=\underset{t\to 0^{+}}{\text{lim}}f\left(0,t\right)$$

exists. Letting t₁ = · · · = t _n → 0⁺ in (3.2) gives c = 0. Setting (x₁, x₂, x₃, . . . , x _n ) = (x, y, 0, . . . , 0) and letting t₁ = t, t₂ = s, t₃ = · · · = t _n → 0⁺ in (3.1) we have

$$\begin{array}{c}2f\left(x+y,t+s\right)+f\left(x-y,t+s\right)+f\left(-x+y,t+s\right)\\ =3f\left(x,t\right)+3f\left(y,s\right)+f\left(-x,t\right)+f\left(-y,s\right)\end{array}$$

(3.3)

for all x, y ∈ ℝ, t, s > 0. Replacing x and y with −x and −y in (3.3) yields

$$\begin{array}{c}2f\left(-x-y,t+s\right)+f\left(-x+y,t+s\right)+f\left(x-y,t+s\right)\\ =3f\left(-x,t\right)+3f\left(-y,s\right)+f\left(x,t\right)+f\left(y,s\right)\end{array}$$

(3.4)

for all x, y ∈ ℝ, t, s > 0. We now define the even part and the odd part of the function f by

$$f_e\left(x,t\right)=\frac{f\left(x,t\right)+f\left(-x,t\right)}{2},f_o\left(x,t\right)=\frac{f\left(x,t\right)-f\left(-x,t\right)}{2}$$

for all x ∈ ℝ, t > 0. Adding (3.3) to (3.4) we verify that f _e satisfies

$$f_e\left(x+y,t+s\right)+f_e\left(x-y,t+s\right)=2f_e\left(x,t\right)+2f_e\left(y,s\right)$$

(3.5)

for all x, y ∈ ℝ, t, s > 0. Similarly, taking the difference of (3.3) and (3.4) we see that f _o satisfies

$$f_o\left(x+y,t+s\right)=f_o\left(x,t\right)+f_o\left(y,s\right)$$

(3.6)

for all x, y ∈ ℝ, t, s > 0. It follows from (3.5), (3.6) and given the continuity that f _e and f _o are of the forms

$$f_e\left(x,t\right)=a{x}^2+c_{1}t,f_o\left(x,t\right)=bx+c_{2}t$$

for some constants a, b, c₁, c₂ ∈ ℂ. Finally we have

$$f\left(x,t\right)=f_e\left(x,t\right)+f_o\left(x,t\right)=a{x}^2+bx+ct,$$

where c = c₁ + c₂.

Conversely, if f (x, t) = ax² + bx + c for some a, b, c ∈ ℂ, then it is obvious that f satisfies equation (3.1).   □

According to the above lemma, we solve the general solution of (1.2) in the space of ${\mathcal{F}}^{\prime }\left(ℝ\right)$ (or ${\mathcal{S}}^{\prime }\left(ℝ\right)$, resp.) as follows.

Theorem 3.2. Every solution u in ${\mathcal{F}}^{\prime }\left(ℝ\right)$ (or ${\mathcal{S}}^{\prime }\left(ℝ\right)$, resp.) of equation (1.2) has the form

$$u=a{x}^2+bx,$$

for some a, b ∈ ℂ.

Proof. Convolving the tensor product $E_{t_1}\left(x_1\right)...E_{t_n}\left(x_n\right)$ of the heat kernels on both sides of (1.2) we have

$$\begin{array}{c}2ũ\left(\sum _{i=1}^n{x}_i,\sum _{i=1}^n{t}_i\right)+\sum _{\begin{array}{c}\hfill 1\le i,j\le n,\hfill \\ \hfill i\ne j\hfill \end{array}}ũ\left(x_i-x_j,t_i+t_j\right)\\ =\left(n+1\right)\sum _{i=1}^nũ\left(x_i,t_i\right)+\left(n-1\right)\sum _{i=1}^nũ\left(-x_i,t_i\right)\end{array}$$

(3.7)

for all x₁, . . . , x _n ∈ ℝ, t₁, . . . , t _n > 0. It follows from Lemma 3.1 that the solution $ũ$ of equation (3.7) has the form

$$ũ\left(x,t\right)=a{x}^2+bx+ct$$

(3.8)

for some a, b, c ∈ ℂ. Letting t → 0⁺ in (3.8), we finally obtain the general solution of (1.2).   □

4. Stability in ${\mathcal{F}}^{\prime }\left(ℝ\right)$

In this section, we are going to state and prove the Hyers-Ulam stability of (1.3) in the space of ${\mathcal{F}}^{\prime }\left(ℝ\right)$ (or ${\mathcal{S}}^{\prime }\left(ℝ\right)$, resp.).

Lemma 4.1. Suppose that f : ℝ × (0, ∞) → ℂ is a continuous function satisfying

$$\begin{array}{c}\left|2f\left(\sum _{i=1}^n{x}_i,\sum _{i=1}^n{t}_i\right)+\sum _{\begin{array}{c}\hfill 1\le i,j\le n\hfill \\ \hfill i\ne j\hfill \end{array}}f\left(x_i-x_j,t_i+t_j\right)\right.\\ \left.-\left(n+1\right)\sum _{i=1}^{n}f\left(x_i,t_i\right)-\left(n-1\right)\sum _{i=1}^{n}f\left(-x_i,t_i\right)\right|\le \epsilon \end{array}$$

(4.1)

for all x₁, . . . , x _n ∈ ℝ, t₁, . . . , t _n > 0, then there exists the unique function g : ℝ × (0, ∞) → ℂ satisfying equation (3.1) such that

$$|f\left(x,t\right)-g\left(x,t\right)|\le \frac{n^2+n-3}{n^2+n-2}\epsilon$$

for all x ∈ ℝ, t > 0.

Proof. Putting (x₁, . . . , x _n ) = (0, . . . , 0) in (4.1) yields

$$\left|f\left(0,\sum _{i=1}^n{t}_i\right)+\sum _{1\le i<j\le n}f\left(0,t_i+t_j\right)-n\sum _{i=1}^{n}f\left(0,t_i\right)\right|\le \frac{\epsilon }{2}$$

(4.2)

for all t₁, . . . , t _n > 0. In view of (4.2) we see that

$$c:\underset{t\to 0^{+}}{\text{lim}\text{sup}}f\left(0,t\right)$$

exists. Letting t₁ = · · · = t _n → 0⁺ in (4.2) gives

$$\left|c\right|\le \frac{\epsilon }{n^2+n-2}.$$

(4.3)

Setting (x₁, x₂, x₃, . . . , x _n ) = (x, x, 0, . . . , 0) and letting t₁ = t₂ = t, t₃ = · · · = t _n → 0⁺ in (4.1) we have

$$\left|f\left(2x,2t\right)+f\left(0,2t\right)-3f\left(x,t\right)-f\left(-x,t\right)-\frac{c\left(n^2+n-6\right)}{2}\right|\le \frac{\epsilon }{2}$$

(4.4)

for all x ∈ ℝ, t > 0. Replacing x by - x in (4.4) yields

$$\left|f\left(-2x,2t\right)+f\left(0,2t\right)-3f\left(-x,t\right)-f\left(x,t\right)-\frac{c\left(n^2+n-6\right)}{2}\right|\le \frac{\epsilon }{2}$$

(4.5)

for all x ∈ ℝ, t > 0. Let f _e and f _o be even and odd part of f defined in Lemma 3.1, respectively. Using the triangle inequality in (4.4) and (4.5) we get the inequalities

$$\left|\frac{g_e\left(2x,2t\right)}{4}-g_e\left(x,t\right)+\frac{g_e\left(0,2t\right)}{4}\right|\le \frac{\epsilon }{8},$$

(4.6)

$$\left|\frac{f_o\left(2x,2t\right)}{2}-f_o\left(x,t\right)\right|\le \frac{\epsilon }{4}$$

(4.7)

for all x ∈ ℝ, t > 0, where $g_e\left(x,t\right):=f_e\left(x,t\right)+\frac{c\left(n^2+n-6\right)}{4}.$

We first consider the even case. Using the iterative method in (4.6) we obtain

$$\left|\frac{g_e\left(2^{k}x,2^{k}t\right)}{4^k}-g_e\left(x,t\right)+\sum _{j=1}^k\frac{g_e\left(0,2^{j}t\right)}{4^j}\right|\le \frac{\epsilon }{6}$$

(4.8)

for all k ∈ ℕ, x ∈ ℝ, t > 0. Letting t₁ = t, t₂ = s, t₃ = · · · = t _n → 0⁺ in (4.2) we have

$$\left|g_e\left(0,t+s\right)-g_e\left(0,t\right)-g_e\left(0,s\right)\right|\le \frac{\epsilon }{4}$$

(4.9)

for all t, s > 0. We verify from (4.9) that

$$h\left(t\right):=\underset{k\to \mathrm{\infty }}{\text{lim}}\frac{g_e\left(0,2^{k}t\right)}{2^k}$$

converges and is the unique function satisfying

$$h\left(t+s\right)=h\left(t\right)+h\left(s\right),$$

(4.10)

$$|h\left(t\right)-g_e\left(0,t\right)|\le \frac{\epsilon }{4}$$

(4.11)

for all t, s > 0. Combining (4.10) and (4.11) we get

$$\left|\left(1-2^{-k}\right)h\left(t\right)-\sum _{i=1}^k\frac{g_e\left(0,2^{k}t\right)}{4^k}\right|\le \frac{\epsilon }{12}$$

(4.12)

for all k ∈ ℕ, t > 0. Adding (4.8) to (4.12) we have

$$\left|{\tilde{g}}_e\left(x,t\right)-\frac{{\tilde{g}}_e\left(2^{k}x,2^{k}t\right)}{4^k}\right|\le \frac{\epsilon }{4}$$

(4.13)

for all k ∈ ℕ, x ∈ ℝ, t > 0, where ${\tilde{g}}_e\left(x,t\right):=g_e\left(x,t\right)-h\left(t\right)$. From (4.1) and (4.13) we verify that

$$G_e\left(x,t\right):=\underset{k\to \mathrm{\infty }}{\text{lim}}\frac{{\tilde{g}}_e\left(2^{k}x,2^{k}t\right)}{4^k}$$

is the unique function satisfying equation (3.1) and the inequality

$$|{\tilde{g}}_e\left(x,t\right)-G_e\left(x,t\right)|\le \frac{\epsilon }{4}$$

(4.14)

for all x ∈ ℝ, t > 0. If we define a function q(x, t) := G _e (x, t) + h(t), then q also satisfies (3.1). By Lemma 3.1 and evenness of q we have

$$q\left(x,t\right)=a{x}^2+c_{1}t$$

for some a, c₁ ∈ ℂ. It follows from (4.3) and (4.14) that

$$|f_e\left(x,t\right)-a{x}^2-c_{1}t|\le \frac{n^2+n-4}{2\left(n^2+n-2\right)}\epsilon$$

(4.15)

for all x ∈ ℝ, t > 0.

Next, we consider the odd case. From (4.7), in the similar manner, we verify that

$$F_o\left(x,t\right):=\underset{k\to \mathrm{\infty }}{\text{lim}}\frac{f_o\left(2^{k}x,2^{k}t\right)}{2^k}$$

is the unique function satisfying equation (3.1) and the inequality

$$\left|F_o\left(x,t\right)-f_o\left(x,t\right)\right|\le \frac{\epsilon }{2}$$

(4.16)

for all x ∈ ℝ, t > 0. By Lemma 3.1 and oddness of F _o we have

$$F_o\left(x,t\right)=bx+c_{2}t$$

for some b, c₂ ∈ ℂ.

Therefore, from (4.15) and (4.16), we obtain

$$\begin{array}{ll}\hfill |f\left(x,t\right)& -\left(a{x}^2+bx+ct\right)|\\ \le |f_e\left(x,t\right)-\left(a{x}^2+c_{1}t\right)|+|f_0\left(x,t\right)-\left(bx+c_{2}t\right)|\\ \le \frac{n^2+n-3}{n^2+n-2}\epsilon \end{array}$$

for all x ∈ ℝ, t > 0, where c = c₁ + c₂.   □

From the above lemma we immediately prove the Hyers-Ulam stability of (1.3) in the space of ${\mathcal{F}}^{\prime }\left(ℝ\right)$ (or ${\mathcal{S}}^{\prime }\left(ℝ\right)$, resp.) as follows.

Theorem 4.2. Suppose that u in ${\mathcal{F}}^{\prime }\left(ℝ\right)$ ( or ${\mathcal{S}}^{\prime }\left(ℝ\right)$, resp.) satisfies the inequality (1.3), then there exists the unique quadratic additive function q(x) = ax² + bx such that

$$∥u-q\left(x\right)∥\le \frac{n^2+n-3}{n^2+n-2}\epsilon .$$

(4.17)

Proof. Convolving the tensor product $E_{t_1}\left(x_1\right)...E_{t_n}\left(x_n\right)$ of the heat kernels on both sides of (1.3) we verify that the inequality (1.3) is converted into

$$\begin{array}{c}\left|2ũ\left(\sum _{i=1}^n{x}_i,\sum _{i=1}^n{t}_i\right)+\sum _{\begin{array}{c}\hfill 1\le i,j\le n\hfill \\ \hfill i\ne j\hfill \end{array}}ũ\left(x_i-x_j,t_i+t_j\right)\right.\\ \left.-\left(n+1\right)\sum _{i=1}^nũ\left(x_i,t_i\right)-\left(n-1\right)\sum _{i=1}^nũ\left(-x_i,t_i\right)\right|\le \epsilon \end{array}$$

for all x₁, . . . , x _n ∈ ℝ, t₁, . . . , t _n > 0. According to Lemma 4.1, there exists the unique function g(x, t) = ax² + bx + ct such that

$$|ũ\left(x,t\right)-g\left(x,t\right)|\le \frac{n^2+n-3}{n^2+n-2}\epsilon$$

(4.18)

for all x ∈ ℝ, t > 0. Letting t → 0⁺ in (4.18) finally we have the stability result (4.17).   □

Remark 4.3. The above norm inequality $∥u-q\left(x\right)∥\le \frac{n^2+n-3}{n^2+n-2}\epsilon$ implies that u − q(x) belongs to (L¹)' = L^∞. Thus, every solution u of the inequality (4.17) in ${\mathcal{F}}^{\prime }\left(ℝ\right)$ (or ${\mathcal{S}}^{\prime }\left(ℝ\right)$, resp.) can be rewritten uniquely in the form

$$u=q\left(x\right)+\mu \left(x\right),$$

where µ is a bounded measurable function such that ${∥u∥}_{L^{\mathrm{\infty }}}\le \frac{n^2+n-3}{n^2+n-2}\epsilon$.