Theory and Modern Applications

General uniqueness theorem concerning the stability of additive, quadratic, and cubic functional equations

Abstract

We prove a general uniqueness theorem that can easily be applied to the proof of (generalized) Hyers-Ulam stability of the additive, quadratic, cubic, or the cubic-quadratic-additive type functional equation. By using this uniqueness theorem, we can omit the repeated proof for uniqueness of the relevant solutions of those equations.

1 Introduction

In 1940, Ulam [1] posed a problem concerning the stability of functional equations: Give conditions in order for a linear function near an approximately linear function to exist. AÂ year later, Hyers [2] gave an answer to the problem of Ulam for additive functions defined on Banach spaces. After Hyersâ€™ result, many mathematicians have extended Ulamâ€™s problem to other functional equations and generalized Hyersâ€™ result in various directions (see [3â€“7]).

Let V and W be real vector spaces. For a given mapping $$f : V \to W$$, we define

\begin{aligned}& Af(x,y) := f(x+y) - f(x) - f(y), \\& Qf(x,y) := f(x+y) - 2f(x) + f(x-y) - 2f(y), \\& Cf(x,y) := f(x+2y) - 3f(x+y) + 3f(x) - f(x-y) - 6f(y), \\& f_{o}(x) := \frac{f(x)-f(-x)}{2}, \\& f_{o}^{(1)}(x) := \frac{a^{3}f_{o}(x)-f_{o}(ax)}{a^{3}-a}, \\& f_{o}^{(2)}(x) := -\frac{af_{o}(x)-f_{o}(ax)}{a^{3}-a}, \\& f_{e}(x) := \frac{f(x)+f(-x)}{2} \end{aligned}

for all $$x, y \in V$$. A mapping $$f : V \to W$$ is called an additive mapping, a quadratic mapping, and a cubic mapping if f satisfies the functional equation $$Af(x,y) = 0$$, $$Qf(x,y) = 0$$, and $$Cf(x,y) = 0$$ for all $$x, y \in V$$, respectively. We remark that the mappings $$g, h, k : {\mathbf{R}} \to{\mathbf{R}}$$ given by $$g(x) = ax$$, $$h(x) = ax^{2}$$, and $$k(x) = ax^{3}$$ are solutions of $$Ag(x,y) = 0$$, $$Qh(x,y) = 0$$, and $$Ck(x,y) = 0$$, respectively.

A mapping $$f : V \to W$$ is called a cubic-quadratic-additive mapping if and only if f is represented by the sum of an additive mapping, a quadratic mapping, and a cubic mapping. A functional equation is called a cubic-quadratic-additive type functional equation if and only if each of its solutions is a cubic-quadratic-additive mapping. The mapping $$f : {\mathbf{R}} \to{\mathbf{R}}$$ given by $$f(x) = ax^{3} + bx^{2} + cx$$ is a solution of the cubic-quadratic-additive type functional equation.

In the study of the stability problems for cubic-quadratic-additive type functional equations, we frequently encounter the cases where we should prove the uniqueness of the cubic-quadratic-additive mappings (see [8â€“18]). Research in this uniqueness problem still has many untouched possibilities to explore.

In this paper, we prove a general uniqueness theorem that can be easily applied to the stability of the cubic-quadratic-additive type functional equations. Using this uniqueness theorem, we do not need to repeat the proof of uniqueness in studying the stability of functional equations mentioned above.

2 Main results

In this section, let X and Y be real normed spaces and let V and W be real vector spaces. In the following theorem, we prove that if, for any given mapping f, there exists a mapping F (near f) with some properties possessed by cubic-quadratic-additive mappings, then the mapping F must be uniquely determined.

Theorem 2.1

Let $$a > 1$$ be a real constant, let $$\Phi: V \backslash\{ 0 \} \to[0,\infty)$$ be a function satisfying one of the following conditions:

\begin{aligned}& \lim_{n \to\infty} \frac{\Phi ( a^{n}x )}{a^{n}} = 0, \end{aligned}
(1)
\begin{aligned}& \lim_{n \to\infty} a^{n} \Phi \biggl( \frac{x}{a^{n}} \biggr) = \lim_{n \to\infty} \frac{\Phi ( a^{n}x )}{a^{2n}} = 0, \end{aligned}
(2)
\begin{aligned}& \lim_{n \to\infty} a^{2n} \Phi \biggl( \frac{x}{a^{n}} \biggr) = \lim_{n \to\infty} \frac{\Phi ( a^{n}x )}{a^{3n}} = 0, \end{aligned}
(3)
\begin{aligned}& \lim_{n \to\infty} a^{3n} \Phi \biggl( \frac{x}{a^{n}} \biggr) = 0 \end{aligned}
(4)

for all $$x \in V \backslash\{ 0 \}$$, and let $$f : V \to Y$$ be a given mapping. If there exists a mapping $$F : V \to Y$$ such that

$$\bigl\Vert f(x) - F(x) \bigr\Vert \leq\Phi(x)$$
(5)

for all $$x \in V \backslash\{ 0 \}$$ and

$$F_{o}^{(1)}(ax) := aF_{o}^{(1)}(x), \qquad F_{e}(ax) := a^{2}F_{e}(x),\qquad F_{o}^{(2)}(ax) := a^{3}F_{o}^{(2)}(x)$$
(6)

for all $$x \in V$$, then F is given by

$$F(x) = \left \{ \textstyle\begin{array}{l@{\quad}l} \lim_{n \to\infty} ( \frac{f_{o}^{(1)} ( a^{n}x )}{a^{n}} + \frac{f_{e} ( a^{n}x )}{a^{2n}} + \frac{f_{o}^{(2)} ( a^{n}x )}{a^{3n}} ) & \textit{if }\Phi \textit{ satisfies }\text{(1)}, \\ \lim_{n \to\infty} ( a^{n} f_{o}^{(1)} ( \frac{x}{a^{n}} ) + \frac{f_{e} ( a^{n}x )}{a^{2n}} + \frac{f_{o}^{(2)} ( a^{n}x )}{a^{3n}} ) & \textit{if }\Phi\textit{ satisfies }\text{(2)}, \\ \lim_{n \to\infty} ( a^{n} f_{o}^{(1)} ( \frac{x}{a^{n}} ) + a^{2n} f_{e} ( \frac{x}{a^{n}} ) + \frac{f_{o}^{(2)} ( a^{n}x )}{a^{3n}} ) & \textit{if }\Phi\textit{ satisfies }\text{(3)}, \\ \lim_{n \to\infty} ( a^{n} f_{o}^{(1)} ( \frac{x}{a^{n}} ) + a^{2n} f_{e} ( \frac{x}{a^{n}} ) + a^{3n} f_{o}^{(2)} ( \frac{x}{a^{n}} ) ) & \textit{if }\Phi\textit{ satisfies }\text{(4)} \end{array}\displaystyle \right .$$
(7)

for all $$x \in V \backslash\{ 0 \}$$. In other words, F is the unique mapping satisfying the conditions (5) andÂ (6).

Proof

Assume that F is a mapping satisfying (5) and (6) for a given mapping $$f : V \to Y$$. First, we consider the mapping $$F_{o}^{(1)}$$. If $$\Phi: V \backslash\{ 0 \} \to[0,\infty)$$ satisfies the condition (1), then it follows from (6) that

\begin{aligned}& \biggl\Vert F_{o}^{(1)}(x) - \frac{f_{o}^{(1)} ( a^{n}x )}{a^{n}} \biggr\Vert \\& \quad = \frac{1}{a^{n}} \bigl\Vert F_{o}^{(1)} \bigl( a^{n}x \bigr) - f_{o}^{(1)} \bigl( a^{n}x \bigr) \bigr\Vert \\& \quad = \frac{1}{2(a^{3}-a)a^{n}} \bigl\Vert a^{3}F \bigl( a^{n}x \bigr) - a^{3}f \bigl( a^{n}x \bigr) - a^{3}F \bigl( -a^{n}x \bigr) + a^{3}f \bigl( -a^{n}x \bigr) \\& \qquad {}-F \bigl( a^{n+1}x \bigr) + f \bigl( a^{n+1}x \bigr) + F \bigl( -a^{n+1}x \bigr) - f \bigl( -a^{n+1}x \bigr) \bigr\Vert \\& \quad \leq \frac{1}{2(a^{3}-a)a^{n}} \bigl( a^{3} \bigl\Vert F \bigl( a^{n}x \bigr) - f \bigl( a^{n}x \bigr) \bigr\Vert + a^{3} \bigl\Vert F \bigl( -a^{n}x \bigr) - f \bigl( -a^{n}x \bigr) \bigr\Vert \\& \qquad {}+ \bigl\Vert F \bigl( a^{n+1}x \bigr) - f \bigl( a^{n+1}x \bigr) \bigr\Vert + \bigl\Vert F \bigl( -a^{n+1}x \bigr) - f \bigl( -a^{n+1}x \bigr) \bigr\Vert \bigr) \\& \quad \leq \frac{a^{3} \Phi( a^{n}x ) + a^{3} \Phi( -a^{n}x ) + \Phi( a^{n+1}x ) + \Phi( -a^{n+1}x )}{2(a^{3}-a) a^{n}} \\& \quad \to 0,\quad \mbox{as } n\to\infty \end{aligned}

for all $$x \in V \backslash\{ 0 \}$$; that is, we see that $$F_{o}^{(1)}(x) = \lim_{n \to\infty} \frac{1}{a^{n}} f_{o}^{(1)}(a^{n}x)$$ for all $$x \in V \backslash\{ 0 \}$$.

If $$\Phi: V \backslash\{ 0 \} \to[0,\infty)$$ satisfies the condition (2), (3), or (4), then it follows from (6) that

\begin{aligned}& \biggl\Vert F_{o}^{(1)}(x) - a^{n} f_{o}^{(1)} \biggl( \frac{x}{a^{n}} \biggr) \biggr\Vert \\& \quad = a^{n} \biggl\Vert F_{o}^{(1)} \biggl( \frac{x}{a^{n}} \biggr) - f_{o}^{(1)} \biggl( \frac{x}{a^{n}} \biggr) \biggr\Vert \\& \quad = \frac{a^{n}}{2(a^{3}-a)} \biggl\Vert a^{3} F \biggl( \frac{x}{a^{n}} \biggr) - a^{3} f \biggl( \frac{x}{a^{n}} \biggr) - a^{3} F \biggl( \frac{-x}{a^{n}} \biggr) + a^{3} f \biggl( \frac{-x}{a^{n}} \biggr) \\& \qquad {}- F \biggl( \frac{x}{a^{n-1}} \biggr) + f \biggl( \frac{x}{a^{n-1}} \biggr) + F \biggl( \frac{-x}{a^{n-1}} \biggr) - f \biggl( \frac{-x}{a^{n-1}} \biggr) \biggr\Vert \\& \quad \leq \frac{a^{n}}{2(a^{3}-a)} \biggl( a^{3} \biggl\Vert F \biggl( \frac{x}{a^{n}} \biggr) - f \biggl( \frac{x}{a^{n}} \biggr) \biggr\Vert + a^{3} \biggl\Vert F \biggl( \frac{-x}{a^{n}} \biggr) - f \biggl( \frac{-x}{a^{n}} \biggr) \biggr\Vert \\& \qquad {}+ \biggl\Vert F \biggl( \frac{x}{a^{n-1}} \biggr) - f \biggl( \frac{x}{a^{n-1}} \biggr) \biggr\Vert + \biggl\Vert F \biggl( \frac{-x}{a^{n-1}} \biggr) - f \biggl( \frac{-x}{a^{n-1}} \biggr) \biggr\Vert \biggr) \\& \quad \leq \frac{1}{2(a^{3}-a)} \biggl( a^{n+3} \Phi \biggl( \frac{x}{a^{n}} \biggr) + a^{n+3} \Phi \biggl( \frac{-x}{a^{n}} \biggr) + a^{n} \Phi \biggl( \frac{x}{a^{n-1}} \biggr) + a^{n} \Phi \biggl( \frac{-x}{a^{n-1}} \biggr)\biggr) \\& \quad \to 0,\quad \mbox{as } n \to\infty \end{aligned}

for all $$x \in V \backslash\{ 0 \}$$; that is, we see that $$F_{o}^{(1)}(x) = \lim_{n \to\infty} a^{n} f_{o}^{(1)} ( \frac{x}{a^{n}} )$$ for all $$x \in V \backslash\{ 0 \}$$.

Second, we consider the mapping $$F_{e}$$. If $$\Phi: V \backslash\{ 0 \} \to[0,\infty)$$ satisfies the condition (1) or (2), then it follows from (6) that

\begin{aligned}& \biggl\Vert F_{e}(x) - \frac{f_{e} ( a^{n}x )}{a^{2n}} \biggr\Vert \\& \quad = \frac{1}{a^{2n}} \bigl\Vert F_{e} \bigl( a^{n}x \bigr) - f_{e} \bigl( a^{n}x \bigr) \bigr\Vert = \frac{1}{2a^{2n}} \bigl\Vert F \bigl( a^{n}x \bigr) - f \bigl( a^{n}x \bigr) + F \bigl( -a^{n}x \bigr) - f \bigl( -a^{n}x \bigr) \bigr\Vert \\& \quad \leq \frac{1}{2a^{2n}} \bigl\Vert F \bigl( a^{n}x \bigr) - f \bigl( a^{n}x \bigr) \bigr\Vert + \frac{1}{2a^{2n}} \bigl\Vert F \bigl( -a^{n}x \bigr) - f \bigl( -a^{n}x \bigr) \bigr\Vert \\& \quad \leq \frac{\Phi( a^{n}x ) + \Phi( -a^{n}x )}{ 2a^{2n}} \\& \quad \to 0, \quad \mbox{as } n \to\infty \end{aligned}

for all $$x \in V \backslash\{0\}$$; that is, we see that $$F_{e}(x) = \lim_{n \to\infty} \frac{1}{a^{2n}} f_{e}(a^{n}x)$$ for all $$x \in V \backslash\{ 0 \}$$.

If $$\Phi: V \to[0,\infty)$$ satisfies the condition (3) or (4), we get

\begin{aligned}& \biggl\Vert F_{e}(x)- a^{2n} f_{e} \biggl( \frac{x}{a^{n}} \biggr) \biggr\Vert \\& \quad = a^{2n} \biggl\Vert F_{e} \biggl( \frac{x}{a^{n}} \biggr) - f_{e} \biggl( \frac{x}{a^{n}} \biggr) \biggr\Vert \\& \quad = \frac{a^{2n}}{2} \biggl\Vert F \biggl( \frac{x}{a^{n}} \biggr) - f \biggl( \frac{x}{a^{n}} \biggr) + F \biggl( \frac{-x}{a^{n}} \biggr) - f \biggl( \frac{-x}{a^{n}} \biggr) \biggr\Vert \\& \quad \leq \frac{a^{2n}}{2} \biggl\Vert F \biggl( \frac{x}{a^{n}} \biggr) - f \biggl( \frac{x}{a^{n}} \biggr) \biggr\Vert + \frac{a^{2n}}{2} \biggl\Vert F \biggl( \frac{-x}{a^{n}} \biggr) - f \biggl( \frac{-x}{a^{n}} \biggr) \biggr\Vert \\& \quad \leq \frac{a^{2n}}{2} \biggl( \Phi \biggl( \frac{x}{a^{n}} \biggr) + \Phi \biggl( \frac{-x}{a^{n}} \biggr) \biggr) \\& \quad \to 0, \quad \mbox{as } n \to\infty \end{aligned}

for all $$x \in V \backslash\{ 0 \}$$. Then $$F_{e}(x) = \lim_{n \to\infty} a^{2n} f_{e} ( \frac{x}{a^{n}} )$$ for all $$x \in V \backslash\{ 0 \}$$ holds.

Finally, we consider the mapping $$f_{o}^{(2)}$$. If $$\Phi: V\backslash\{ 0 \} \to[0,\infty)$$ satisfies the condition (1), (2), or (3), then it follows from (6) that

\begin{aligned}& \biggl\Vert F_{o}^{(2)}(x) - \frac{f_{o}^{(2)} ( a^{n}x )}{a^{3n}} \biggr\Vert \\& \quad = \frac{1}{a^{3n}} \bigl\Vert F_{o}^{(2)} \bigl( a^{n}x \bigr) - f_{o}^{(2)} \bigl( a^{n}x \bigr) \bigr\Vert \\& \quad = \frac{1}{2(a^{3}-a)a^{3n}} \bigl\Vert -aF \bigl( a^{n}x \bigr) + af \bigl( a^{n}x \bigr) + aF \bigl( -a^{n}x \bigr) - af \bigl( -a^{n}x \bigr) \\& \qquad {}+F \bigl( a^{n+1}x \bigr) - f \bigl( a^{n+1}x \bigr) -F \bigl( -a^{n+1}x \bigr) + f \bigl( -a^{n+1}x \bigr) \bigr\Vert \\& \quad \leq \frac{1}{2(a^{3}-a)a^{3n}} \bigl( a \bigl\Vert F \bigl( a^{n}x \bigr) - f \bigl( a^{n}x \bigr) \bigr\Vert + a \bigl\Vert F \bigl( -a^{n}x \bigr) - f \bigl( -a^{n}x \bigr) \bigr\Vert \\& \qquad {}+ \bigl\Vert F \bigl( a^{n+1}x \bigr) - f \bigl( a^{n+1}x \bigr) \bigr\Vert + \bigl\Vert F \bigl( -a^{n+1}x \bigr) - f \bigl( -a^{n+1}x \bigr) \bigr\Vert \bigr) \\& \quad \leq \frac{a\Phi ( a^{n}x ) + a\Phi ( -a^{n}x ) + \Phi ( a^{n+1}x ) + \Phi ( -a^{n+1}x )}{2(a^{3}-a)a^{3n}} \\& \quad \to 0,\quad \mbox{as } n \to\infty \end{aligned}

for all $$x \in V \backslash\{ 0 \}$$; that is, we see that $$F_{o}^{(2)}(x) = \lim_{n \to\infty} \frac{1}{a^{3n}} f_{o}^{(2)}(a^{n}x)$$ for all $$x \in V \backslash\{ 0 \}$$.

If $$\Phi: V \backslash\{ 0 \} \to[0,\infty)$$ satisfies the condition (4), then it follows from (4) and (6) that

\begin{aligned}& \biggl\Vert F_{o}^{(2)}(x) - a^{3n} f_{o}^{(2)} \biggl( \frac{x}{a^{n}} \biggr) \biggr\Vert \\& \quad = a^{3n} \biggl\Vert F_{o}^{(2)} \biggl( \frac{x}{a^{n}} \biggr) - f_{o}^{(2)} \biggl( \frac{x}{a^{n}} \biggr) \biggr\Vert \\& \quad = \frac{a^{3n}}{2(a^{3}-a)} \biggl\Vert -aF \biggl( \frac{x}{a^{n}} \biggr) + af \biggl( \frac{x}{a^{n}} \biggr) + aF \biggl( \frac{-x}{a^{n}} \biggr) - af \biggl( \frac{-x}{a^{n}} \biggr) \\& \qquad {}+F \biggl( \frac{x}{a^{n-1}} \biggr) - f \biggl( \frac{x}{a^{n-1}} \biggr) - F \biggl( \frac{-x}{a^{n-1}} \biggr) + f \biggl( \frac{-x}{a^{n-1}} \biggr) \biggr\Vert \\& \quad \leq \frac{a^{3n}}{2(a^{3}-a)} \biggl( a \biggl\Vert F \biggl( \frac{x}{a^{n}} \biggr) - f \biggl( \frac{x}{a^{n}} \biggr) \biggr\Vert + a \biggl\Vert F \biggl( \frac{-x}{a^{n}} \biggr) - f \biggl( \frac{-x}{a^{n}} \biggr) \biggr\Vert \\& \qquad {}+ \biggl\Vert F \biggl( \frac{x}{a^{n-1}} \biggr) - f \biggl( \frac{x}{a^{n-1}} \biggr) \biggr\Vert + \biggl\Vert F \biggl( \frac{-x}{a^{n-1}} \biggr) - f \biggl( \frac{-x}{a^{n-1}} \biggr) \biggr\Vert \biggr) \\& \quad \leq \frac{a^{3n}}{2(a^{3}-a)} \biggl( a \Phi \biggl( \frac{x}{a^{n}} \biggr) + a \Phi \biggl( \frac{-x}{a^{n}} \biggr) + \Phi \biggl( \frac{x}{a^{n-1}} \biggr) + \Phi \biggl( \frac{-x}{a^{n-1}} \biggr)\biggr) \\& \quad \to 0,\quad \mbox{as } n \to\infty \end{aligned}

for all $$x \in V \backslash\{ 0 \}$$; that is, we see that $$F_{o}^{(2)}(x) = \lim_{n \to\infty} a^{3n} f_{o}^{(2)} ( \frac{x}{a^{n}} )$$ for all $$x \in V \backslash\{ 0 \}$$. Since $$F(x) = F_{o}^{(1)}(x) + F_{e}(x) + F_{o}^{(2)}(x)$$, F is given by the equalities in (7) and F is uniquely determined for any case.â€ƒâ–¡

In general, it is not easy to apply TheoremÂ 2.1 for practical applications. Hence, we introduce a couple of corollaries which are useful for investigating the uniqueness problems in the stability of the cubic-quadratic-additive functional equations.

Corollary 2.2

Let $$a > 1$$ be a real constant and let $$\phi: V \backslash\{ 0 \} \to[0,\infty)$$ be a function satisfying either

$$\Phi(x) := \sum_{i=0}^{\infty}\frac{\phi ( a^{i}x )}{a^{i}} < \infty$$
(8)

or

$$\Phi(x) := \sum_{i=0}^{\infty}a^{3i} \phi \biggl( \frac{x}{a^{i}} \biggr) < \infty$$
(9)

for all $$x \in X \backslash\{ 0 \}$$. For any given mapping $$f : V \to Y$$, if there exists a mapping $$F : V \to Y$$ satisfying the inequality

$$\bigl\Vert f(x) - F(x) \bigr\Vert \leq\Phi(x)$$
(10)

for all $$x \in V \backslash\{ 0 \}$$ and the condition (6) for all $$x \in V$$, then F is a unique mapping satisfying the conditions (6) and (10).

Proof

If Ï• satisfies (8), then we have

$$\lim_{n \to\infty} \frac{\Phi ( a^{n}x )}{a^{n}} = \lim_{n \to\infty} \sum_{i=0}^{\infty}\frac{\phi ( a^{n+i}x )}{a^{n+i}} = \lim _{n \to\infty} \sum_{i=n}^{\infty}\frac{\phi ( a^{i}x )}{a^{i}} = 0,$$

i.e., Î¦ satisfies the condition (1) for all $$x \in V \backslash\{ 0 \}$$.

For the case when Ï• satisfies (9), it holds that

$$\lim_{n \to\infty} a^{3n} \Phi \biggl( \frac{x}{a^{n}} \biggr) = \lim_{n \to\infty} \sum_{i=0}^{\infty}a^{3n+3i} \phi \biggl( \frac{x}{a^{n+i}} \biggr) = \lim _{n \to\infty} \sum_{i=n}^{\infty}a^{3i} \phi \biggl( \frac{x}{a^{i}} \biggr) = 0,$$

i.e., Î¦ satisfies the condition (4) for all $$x \in V \backslash\{ 0 \}$$. Hence, our assertion is true in view of TheoremÂ 2.1.â€ƒâ–¡

Corollary 2.3

Let $$a > 1$$ be a real constant, let $$\phi, \psi: V \backslash\{ 0 \} \to[0,\infty)$$ be functions satisfying each of the following conditions:

\begin{aligned} &\sum_{i=0}^{\infty}a^{i} \psi \biggl( \frac{x}{a^{i}} \biggr) < \infty,\qquad \sum _{i=0}^{\infty}\frac{\phi ( a^{i}x )}{a^{2i}} < \infty, \\ &\tilde{\Phi}(x) := \sum_{i=0}^{\infty}a^{i} \phi \biggl( \frac{x}{a^{i}} \biggr) < \infty, \qquad \tilde{ \Psi}(x) := \sum_{i=0}^{\infty}\frac{\psi ( a^{i}x )}{a^{2i}} < \infty \end{aligned}
(11)

for all $$x \in V \backslash\{ 0 \}$$, and let $$f : V \to Y$$ be an arbitrarily given mapping. If there exists a mapping $$F : V \to Y$$ satisfying the inequality

$$\bigl\Vert f(x) - F(x) \bigr\Vert \leq\tilde{\Phi}(x) + \tilde{\Psi}(x)$$
(12)

for all $$x \in V \backslash\{ 0 \}$$ and the condition (6) for all $$x \in V$$, then F is a unique mapping satisfying the conditions (6) for all $$x \in V$$ and the inequality (12) for all $$x \in V \backslash\{ 0 \}$$.

Proof

If we put $$\Phi(x) = \tilde{\Phi}(x) + \tilde{\Psi}(x)$$, then it follows from (12) that

$$\frac{1}{a^{4n}} \Phi \bigl( a^{2n}x \bigr) = \sum _{i=0}^{\infty}\frac{1}{a^{4n-i}} \phi \bigl( a^{2n-i}x \bigr) + \sum_{i=0}^{\infty}\frac{1}{a^{4n+2i}} \psi \bigl( a^{2n+i}x \bigr)$$

for all $$x \in V\backslash\{ 0 \}$$. We make a change of the summation indices in the preceding equality with $$j = i-2n$$ and $$k = 2n+i$$ to get

\begin{aligned}& \frac{1}{a^{4n}} \Phi \bigl( a^{2n}x \bigr) \\& \quad = \frac{1}{a^{2n}} \sum_{j=-2n}^{\infty}a^{j} \phi \biggl( \frac{x}{a^{j}} \biggr) + \sum _{k=2n}^{\infty}\frac{1}{a^{2k}} \psi \bigl( a^{k}x \bigr) \\& \quad = \frac{1}{a^{2n}} \sum_{i=1}^{2n} \frac{1}{a^{i}} \phi \bigl( a^{i}x \bigr) + \frac{1}{a^{2n}} \sum _{i=0}^{\infty}a^{i} \phi \biggl( \frac{x}{a^{i}} \biggr) + \sum_{i=2n}^{\infty}\frac{1}{a^{2i}} \psi \bigl( a^{i}x \bigr) \\& \quad = \frac{1}{a^{n}} \sum_{i=1}^{n-1} \frac{a^{i}}{a^{n}} \frac{1}{a^{2i}} \phi \bigl( a^{i}x \bigr) + \sum _{i=n}^{2n} \frac{a^{i}}{a^{2n}} \frac{1}{a^{2i}} \phi \bigl( a^{i}x \bigr) + \frac{1}{a^{2n}} \tilde{\Phi}(x) + \sum_{i=2n}^{\infty}\frac{1}{a^{2i}} \psi \bigl( a^{i}x \bigr) \\& \quad \leq \frac{1}{a^{n}} \sum_{i=1}^{\infty}\frac{1}{a^{2i}} \phi \bigl( a^{i}x \bigr) + \sum _{i=n}^{\infty}\frac{1}{a^{2i}} \phi \bigl( a^{i}x \bigr) + \frac{1}{a^{2n}} \tilde{\Phi}(x) + \sum _{i=2n}^{\infty}\frac{1}{a^{2i}} \psi \bigl( a^{i}x \bigr) \end{aligned}

for any $$x \in V \backslash\{ 0 \}$$. Hence, it follows from (11) that

$$\lim_{n \to\infty} \frac{1}{a^{4n}} \Phi \bigl( a^{2n}x \bigr) = 0$$

for all $$x \in V \backslash\{ 0 \}$$. On the other hand, we use the above equality to get

$$\lim_{n \to\infty} \frac{1}{a^{4n+2}} \Phi \bigl( a^{2n+1}x \bigr) = \frac{1}{a^{2}} \lim_{n \to\infty} \frac{1}{a^{4n}} \Phi \bigl( a^{2n}ax \bigr) = 0$$

for all $$x \in V \backslash\{ 0 \}$$. From the above two equalities, we conclude that

$$\lim_{n \to\infty} \frac{1}{a^{2n}} \Phi \bigl( a^{n}x \bigr) = 0$$

for all $$x \in V \backslash\{ 0 \}$$.

Similarly, we have

$$a^{2n} \Phi \biggl( \frac{x}{a^{2n}} \biggr) = \sum _{i=0}^{\infty}a^{2n+i} \phi \biggl( \frac{x}{a^{2n+i}} \biggr) + \sum_{i=0}^{\infty}\frac{1}{a^{2i-2n}} \psi \bigl( a^{i-2n}x \bigr)$$

for all $$x \in V \backslash\{ 0 \}$$. If we make a change of the summation indices in the last equality with $$j = i+2n$$ and $$k = i-2n$$, then we get

\begin{aligned}& a^{2n} \Phi \biggl( \frac{x}{a^{2n}} \biggr) \\& \quad = \sum_{j=2n}^{\infty}a^{j} \phi \biggl( \frac{x}{a^{j}} \biggr) + \frac{1}{a^{2n}} \sum _{k=-2n}^{\infty}\frac{1}{a^{2k}} \psi \bigl( a^{k}x \bigr) \\& \quad = \sum_{i=2n}^{\infty}a^{i} \phi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{2n}} \sum _{i=1}^{2n} a^{2i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{2n}} \sum_{i=0}^{\infty}\frac{1}{a^{2i}} \psi \bigl( a^{i}x \bigr) \\& \quad = \sum_{i=2n}^{\infty}a^{i} \phi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{n}} \sum _{i=1}^{n-1} \frac{a^{i}}{a^{n}} a^{i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \sum_{i=n}^{2n} \frac{a^{i}}{a^{2n}} a^{i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{2n}} \tilde{\Psi}(x) \\& \quad \leq \sum_{i=2n}^{\infty}a^{i} \phi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{n}} \sum _{i=1}^{\infty}a^{i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \sum_{i=n}^{\infty}a^{i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{2n}} \tilde{\Psi}(x) \end{aligned}

for any $$x \in V \backslash\{ 0 \}$$. Thus, it follows from (11) that

\begin{aligned}& \lim_{n \to\infty} a^{2n} \Phi \biggl( \frac{x}{a^{2n}} \biggr) = 0, \\& \lim_{n \to\infty} a^{2n+1} \Phi \biggl( \frac{x}{a^{2n+1}} \biggr) = a \lim_{n \to\infty} a^{2n} \Phi \biggl( \frac{1}{a^{2n}} \frac{x}{a} \biggr) = 0 \end{aligned}

for each $$x \in V \backslash\{ 0 \}$$. Thus, we see that

$$\lim_{n \to\infty} a^{n} \Phi \biggl( \frac{x}{a^{n}} \biggr) = 0$$

for each $$x \in V \backslash\{ 0 \}$$.

Altogether, Î¦ satisfies (2) for all $$x \in V \backslash\{ 0 \}$$. Hence, TheoremÂ 2.1 implies that our conclusion of this corollary is true.â€ƒâ–¡

Corollary 2.4

Let $$a > 1$$ be a real constant, let $$\phi, \psi: V \backslash\{ 0 \} \to[0,\infty)$$ be functions satisfying each of the following conditions:

\begin{aligned} &\sum_{i=0}^{\infty}a^{2i} \psi \biggl( \frac{x}{a^{i}} \biggr) < \infty,\qquad \sum _{i=0}^{\infty}\frac{\phi ( a^{i}x )}{a^{3i}} < \infty, \\ &\tilde{\Phi}(x) := \sum_{i=0}^{\infty}a^{2i} \phi \biggl( \frac{x}{a^{i}} \biggr) < \infty,\qquad \tilde{ \Psi}(x) := \sum_{i=0}^{\infty}\frac{\psi ( a^{i}x )}{a^{3i}} < \infty \end{aligned}
(13)

for all $$x \in V \backslash\{ 0 \}$$, and let $$f : V \to Y$$ be an arbitrarily given mapping. If there exists a mapping $$F : V \to Y$$ satisfying the inequality

$$\bigl\Vert f(x) - F(x) \bigr\Vert \leq\tilde{\Phi}(x) + \tilde{\Psi}(x)$$
(14)

for all $$x \in V \backslash\{ 0 \}$$ and the condition (6) for all $$x \in V$$, then F is a unique mapping satisfying the conditions (6) for all $$x \in V$$ and (14) for all $$x \in V \backslash\{ 0 \}$$.

Proof

If we put $$\Phi(x) = \tilde{\Phi}(x) + \tilde{\Psi}(x)$$, then it follows from (13) that

$$\frac{1}{a^{6n}} \Phi\bigl(a^{2n}x\bigr) = \sum _{i=0}^{\infty}\frac{1}{a^{6n-2i}} \phi \bigl( a^{2n-i}x \bigr) + \sum_{i=0}^{\infty}\frac{1}{a^{6n+3i}} \psi \bigl( a^{2n+i}x \bigr)$$

for all $$x \in V \backslash\{ 0 \}$$. We make a change of the summation indices in the preceding equality with $$j = i-2n$$ and $$k = 2n+i$$ to get

\begin{aligned}& \frac{1}{a^{6n}} \Phi\bigl(a^{2n}x\bigr) \\& \quad = \frac{1}{a^{2n}} \sum_{j=-2n}^{\infty}a^{2j} \phi \biggl( \frac{x}{a^{j}} \biggr) + \sum _{k=2n}^{\infty}\frac{1}{a^{3k}} \psi \bigl( a^{k}x \bigr) \\& \quad = \frac{1}{a^{2n}} \sum_{i=1}^{2n} \frac{1}{a^{2i}} \phi \bigl( a^{i}x \bigr) + \frac{1}{a^{2n}} \sum _{i=0}^{\infty}a^{2i} \phi \biggl( \frac{x}{a^{i}} \biggr) + \sum_{i=2n}^{\infty}\frac{1}{a^{3i}} \psi \bigl( a^{i}x \bigr) \\& \quad = \frac{1}{a^{n}} \sum_{i=1}^{n-1} \frac{1}{a^{2i+n}} \phi \bigl( a^{i}x \bigr) + \sum _{i=n}^{2n} \frac{1}{a^{2n+2i}} \phi \bigl( a^{i}x \bigr) + \frac{1}{a^{2n}} \tilde{\Phi}(x) + \sum _{i=2n}^{\infty}\frac{1}{a^{3i}} \psi \bigl( a^{i}x \bigr) \\& \quad \leq \frac{1}{a^{n}} \sum_{i=1}^{\infty}\frac{1}{a^{3i}} \phi \bigl( a^{i}x \bigr) + \sum _{i=n}^{\infty}\frac{1}{a^{3i}} \phi \bigl( a^{i}x \bigr) + \frac{1}{a^{2n}} \tilde{\Phi}(x) + \sum _{i=2n}^{\infty}\frac{1}{a^{3i}} \psi \bigl( a^{i}x \bigr) \end{aligned}

for any $$x \in V \backslash\{ 0 \}$$. Hence, by (13), we get

$$\lim_{n \to\infty} \frac{1}{a^{6n}} \Phi \bigl( a^{2n}x \bigr) = 0$$

for all $$x \in V \backslash\{ 0 \}$$. On the other hand, we use the above equality to get

$$\lim_{n \to\infty} \frac{1}{a^{6n+3}} \Phi \bigl( a^{2n+1}x \bigr) = \frac{1}{a^{3}} \lim_{n \to\infty} \frac{1}{a^{6n}} \Phi \bigl( a^{2n}ax \bigr) = 0$$

for all $$x \in V \backslash\{ 0 \}$$.

From the above two equalities, we conclude that

$$\lim_{n \to\infty} \frac{1}{a^{3n}} \Phi \bigl( a^{n}x \bigr) = 0$$

for all $$x \in V \backslash\{ 0 \}$$.

Similarly, we have

$$a^{4n} \Phi \biggl( \frac{x}{a^{2n}} \biggr) = \sum _{i=0}^{\infty}a^{4n+2i} \phi \biggl( \frac{x}{a^{2n+i}} \biggr) + \sum_{i=0}^{\infty}\frac{1}{a^{3i-4n}} \psi \bigl( a^{i-2n}x \bigr)$$

for all $$x \in V \backslash\{ 0 \}$$. If we make a change of the summation indices in the last equality with $$j = i+2n$$ and $$k = i-2n$$, then we get

\begin{aligned}& a^{4n} \Phi \biggl( \frac{x}{a^{2n}} \biggr) \\& \quad = \sum_{j=2n}^{\infty}a^{2j} \phi \biggl( \frac{x}{a^{j}} \biggr) + \frac{1}{a^{2n}} \sum _{k=-2n}^{\infty}\frac{1}{a^{3k}} \psi \bigl( a^{k}x \bigr) \\& \quad = \sum_{i=2n}^{\infty}a^{2i} \phi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{2n}} \sum _{i=1}^{2n} a^{3i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{2n}} \sum_{i=0}^{\infty}\frac{1}{a^{3i}} \psi \bigl( a^{i}x \bigr) \\& \quad = \sum_{i=2n}^{\infty}a^{2i} \phi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{n}} \sum _{i=1}^{n-1} \frac{a^{i}}{a^{n}} a^{2i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \sum_{i=n}^{2n} \frac{a^{i}}{a^{2n}} a^{2i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{2n}} \tilde{\Psi}(x) \\& \quad \leq \sum_{i=2n}^{\infty}a^{2i} \phi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{n}} \sum _{i=1}^{\infty}a^{2i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \sum_{i=n}^{\infty}a^{2i} \psi \biggl( \frac{x}{a^{i}} \biggr) + \frac{1}{a^{2n}} \tilde{\Psi}(x) \end{aligned}

for any $$x \in V \backslash\{ 0 \}$$. Thus, we obtain

\begin{aligned}& \lim_{n \to\infty} a^{4n} \Phi \biggl( \frac{x}{a^{2n}} \biggr) = 0, \\& \lim_{n \to\infty} a^{4n+2} \Phi \biggl( \frac{x}{a^{2n+1}} \biggr) = a^{2} \lim_{n \to\infty} a^{4n} \Phi \biggl( \frac{1}{a^{2n}} \frac{x}{a} \biggr) = 0 \end{aligned}

for each $$x \in V \backslash\{ 0 \}$$. Thus, we see that

$$\lim_{n \to\infty} a^{2n} \Phi \biggl( \frac{x}{a^{n}} \biggr) = 0$$

for each $$x \in V \backslash\{ 0 \}$$.

Altogether, Î¦ satisfies (3) for all $$x \in V \backslash\{ 0 \}$$. Hence, TheoremÂ 2.1 implies that our conclusion of this corollary is true.â€ƒâ–¡

3 Applications

In this section, we apply the theorem and corollaries of the last section to show that if for any given mapping f, there exists an additive, a quadratic, a cubic, a quadratic-additive, a cubic-additive, a cubic-quadratic, or a cubic-quadratic-additive mapping F near f, then the mapping F is uniquely determined.

The proofs of the first three corollaries immediately follow from Corollaries 2.2, 2.3, and 2.4, respectively, because each cubic-quadratic-additive mapping satisfies the conditions in (6) provided a is a rational number.

Corollary 3.1

Let $$a > 1$$ be a rational number and let $$\phi: V \backslash\{ 0 \} \to[0,\infty)$$ be a function satisfying the condition (8) or (9) for all $$x \in V \backslash\{ 0 \}$$. Let $$f : V \to Y$$ be a given mapping. If there exists a cubic-quadratic-additive mapping $$F : V \to Y$$ satisfying the inequality (10), then F is uniquely determined.

Corollary 3.2

Let $$a > 1$$ be a rational number and let $$\phi, \psi: V \backslash\{ 0 \} \to[0,\infty)$$ be functions satisfying the conditions in (11) for all $$x \in V \backslash\{ 0 \}$$. Let $$f : V \to Y$$ be a given mapping. If there exists a cubic-quadratic-additive mapping $$F : V \to Y$$ satisfying the inequality (12), then F is uniquely determined.

Corollary 3.3

Let $$a > 1$$ be a rational number and let $$\phi, \psi: V \backslash\{ 0 \} \to[0,\infty)$$ be functions satisfying the conditions in (13) for all $$x \in V \backslash\{ 0 \}$$. Let $$f : V \to Y$$ be a given mapping. If there exists a cubic-quadratic-additive mapping $$F : X \to Y$$ satisfying the inequality (14), then F is uniquely determined.

If $$p < 1$$ then $$\Phi(x) := K \| x \|^{p}$$ satisfies (1); if $$1 < p < 2$$ then $$\Phi(x)$$ satisfies (2); if $$2 < p < 3$$ then $$\Phi(x)$$ satisfies (3); and if $$p > 3$$ then $$\Phi(x)$$ satisfies (4). Hence, by TheoremÂ 2.1, we get the following corollaries concerning the Hyers-Ulam-Rassias stability. For the detailed concept of the Hyers-Ulam-Rassias stability, we refer to [1, 2, 4, 6, 17].

When we prove the Hyers-Ulam-Rassias stability, Y is usually assumed to be a Banach space. In this paper, however, we only need to assume that Y is a real normed space provided the validity of inequality (5), (10), (12), (14), or (15) is already guaranteed.

Corollary 3.4

Let $$p \notin\{ 1, 2, 3 \}$$ and $$\theta> 0$$ be real constants, let X, Y be real normed spaces, and let $$f : X \to Y$$ be an arbitrarily given mapping. If there exists a mapping $$F : X \to Y$$ satisfying the inequality

$$\bigl\Vert f(x) - F(x) \bigr\Vert \leq\theta \Vert x \Vert ^{p}$$
(15)

for all $$x \in X \backslash\{ 0 \}$$ and the conditions in (6) for all $$x \in X$$, then F is a unique mapping satisfying the conditions in (6) for all $$x \in X$$ and the inequality (15) for all $$x \in X \backslash\{ 0 \}$$.

Since each of the cubic, additive, and cubic-additive mappings satisfies the conditions in (6), using CorollaryÂ 3.2, we can easily prove the following corollary.

Corollary 3.5

Let $$p \notin\{ 1, 2, 3 \}$$ and $$\theta> 0$$ be real constants, let X, Y be real normed spaces, and let $$f : X \to Y$$ be an arbitrarily given mapping. If there exists an additive, a quadratic, a cubic, a quadratic-additive, a cubic-additive, a cubic-quadratic, or a cubic-quadratic-additive mapping $$F : X \to Y$$ satisfying the inequality (15) for all $$x \in X \backslash\{ 0 \}$$, then F is uniquely determined.

If we set $$\phi(x) = \varepsilon$$ in CorollaryÂ 3.1, then Ï• satisfies the condition (8). Hence, CorollaryÂ 3.1 implies the following result.

Corollary 3.6

Let V be a real vector space, let Y be a real normed space, and let $$f : V \to Y$$ be an arbitrarily given mapping. If there exists an additive, a quadratic, a cubic, a quadratic-additive, a cubic-additive, a cubic-quadratic, or a cubic-quadratic-additive mapping $$F : X \to Y$$ satisfying the inequality

$$\bigl\Vert f(x) - F(x) \bigr\Vert \leq\varepsilon$$

for all $$x \in V \backslash\{ 0 \}$$ and for some $$\varepsilon> 0$$, then F is uniquely determined.

Remark 3.7

In 2005, Baker [3] proved the Hyers-Ulam stability of a large class of functional equations of the form

$$\sum_{k=0}^{m} f_{k} ( \alpha_{k} x + \beta_{k} y ) = 0,$$
(16)

which includes the additive, the quadratic, the cubic, the quadratic-additive, the cubic-additive, the cubic-quadratic, and the cubic-quadratic-additive type functional equations; in fact, he proved the Hyers-Ulam stability of equation (16) without addressing the uniqueness of the relevant solution of that equation, while the main aim of this paper is to prove a general uniqueness theorem for those equations. From this viewpoint, we can say that this paper complements the results of Baker.

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Acknowledgements

Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No.Â 2013R1A1A2005557).

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Lee, YH., Jung, SM. General uniqueness theorem concerning the stability of additive, quadratic, and cubic functional equations. Adv Differ Equ 2016, 75 (2016). https://doi.org/10.1186/s13662-016-0803-9