Approximation on the reciprocal-cubic and reciprocal-quartic functional equations in non-Archimedean fields

Abstract

The aim of this paper is to study the generalized Hyers-Ulam stability of a form of reciprocal-cubic and reciprocal-quartic functional equations in non-Archimedean fields. Some related examples for the singular cases of these new functional equations on an Archimedean field are indicated.

1 Introduction

The study of the stability of functional equations was instigated by the famous question of Ulam [1] during a Mathematical Colloquium at the University of Wiskonsin in the year 1940. In the successive year, Hyers [2] provided a partial answer to the question of Ulam. Later, Hyers’s result was extended and generalized for a Cauchy functional equation by Bourgin [3], Th.M. Rassias [4], Gruber [5], Aoki [6], J.M. Rassias [7] and Găvruta [8] in various adaptations. After that several stability articles, many textbooks and research monographs have investigated the result for various functional equations, also for mappings with more general domains and ranges; for instance, see [9–16] and [17].

In 2010, Ravi and Senthil Kumar [18] obtained Ulam-Găvruta-Rassias stability for the Rassias reciprocal functional equation

$$ r(x+y)=\frac{r(x)r(y)}{r(x)+r(y)}, $$

(1.1)

where $r:X\longrightarrow\mathbb{R}$ is a mapping with X as the space of non-zero real numbers. The reciprocal function $r(x)=\frac{c}{x}$ is a solution of the functional equation (1.1). The functional equation (1.1) holds good for the ‘reciprocal formula’ of any electric circuit with two resistors connected in parallel [19]. Ravi et al. [20] obtained the solution of a new generalized reciprocal-type functional equation in two variables of the form

$$ r(x+y)=\frac{kr(x+(k-1)y)r((k-1)x+y)}{r(x+(k-1)y)+r((k-1)x+y)}, $$

(1.2)

where $k>2$ is a positive integer, and investigated its generalized Hyers-Ulam stability in non-Archimedean fields. Then Senthil Kumar et al. [21] found a general solution of a reciprocal-type functional equation

$$ f(x+y)=\frac{f (\frac{k_{1}x+k_{2}y}{k} )f (\frac {k_{2}x+k_{1}y}{k} )}{f (\frac{k_{1}x+k_{2}y}{k} )+f (\frac {k_{2}x+k_{1}y}{k} )} $$

(1.3)

and investigated its generalized Hyers-Ulam-Rassias stability in non-Archimedean fields, where $k>2$, $k_{1}$ and $k_{2}$ are positive integers with $k=k_{1}+k_{2}$ and $k_{1}\neq k_{2}$. The other results pertaining to the stability of different reciprocal-type functional equations can be found in [22–24] and [25].

For the first time, Kim and Bodaghi [26] introduced and studied the Ulam-Găvruta-Rassias stability for the quadratic reciprocal functional equation

$$ f(2x+y)+f(2x-y) = \frac{2f(x)f(y)[4f(y)+f(x)]}{(4f(y)-f(x))^{2}}. $$

(1.4)

Then the functional equation (1.4) was generalized in [27] as

$$ f \bigl((a+1)x+ay \bigr)+f \bigl((a+1)x-ay \bigr) = \frac{2f(x)f(y) [(a+1)^{2}f(y)+a^{2}f(x) ]}{ ((a+1)^{2}f(y)-a^{2}f(x) )^{2}}, $$

(1.5)

where $a\in\mathbb{Z}$ with $a\neq0,-1$. In [27], the authors established the generalized Hyers-Ulam-Rassias stability for the functional equation (1.5) in non-Archimedean fields. Since then Ravi et al. [28] investigated the generalized Hyers-Ulam-Rassias stability of a reciprocal-quadratic functional equation of the form

$$ r(x+2y)+r(2x+y)=\frac{r(x)r(y) [5r(x)+5r(y)+8\sqrt{r(x)r(y)} ]}{ [2r(x)+2r(y)+5\sqrt{r(x)r(y)} ]^{2}} $$

(1.6)

in intuitionistic fuzzy normed spaces; for another form of a reciprocal-quadratic functional equation, see [29].

In this paper, we introduce the reciprocal-cubic functional equation

$$ c(2x+y)+c(x+2y)=\frac{9c(x)c(y) [c(x)+c(y)+2c(x)^{\frac {1}{3}}c(y)^{\frac{1}{3}} (c(x)^{\frac{1}{3}}+c(y)^{\frac {1}{3}} ) ]}{ [2c(x)^{\frac{2}{3}}+2c(y)^{\frac {2}{3}}+5c(x)^{\frac{1}{3}}c(y)^{\frac{1}{3}} ]^{3}} $$

(1.7)

and the reciprocal-quartic functional equation

$$ q(2x+y)+q(2x-y)=\frac{2q(x)q(y) [q(x)+16q(y)+24\sqrt{q(x)q(y)} ]}{ [4\sqrt{q(y)}-\sqrt{q(x)} ]^{4}}. $$

(1.8)

It can be verified that the reciprocal-cubic function $c(x)=\frac {1}{x^{3}}$ and the reciprocal-quartic function $q(x)=\frac{1}{x^{4}}$ are solutions of the functional equations (1.7) and (1.8), respectively. Then we investigate the generalized Hyers-Ulam stability of these new functional equations in the framework of non-Archimedean fields. We extend the results concerning Hyers-Ulam stability, Hyers-Ulam-Rassias stability and Ulam-Găvruta-Rassias stability controlled by the mixed product-sum of powers of norms for equations (1.7) and (1.8). We also provide related examples that the functional equations (1.7) and (1.8) are not stable for the singular cases.

2 Preliminaries

In this section, we recall the basic concepts of a non-Archimedean field.

Definition 2.1

By a non-Archimedean field, we mean a field $\mathbb{K}$ equipped with a function (valuation) $|\cdot|$ from $\mathbb{K}$ into $[0,\infty)$ such that $|p|=0$ if and only if $p=0$, $|pq|=|p||q|$ and $|p+q|\leq \max\{|p|, |q|\}$ for all $p, q\in\mathbb{K}$.

Clearly, $|1|=|{-}1|=1$ and $|n|\leq1$ for all $n\in\mathbb{N}$. We always assume, in addition, that $|\cdot|$ is non-trivial, i.e., there exists $a_{0}\in\mathbb {K}$ such that $|a_{0}|\neq{0,1}$. Due to the fact that

$$\vert p_{n}-p_{m}\vert \leq\max \bigl\{ \vert p_{j+1}-p_{j}\vert :m\leq j\leq n-1 \bigr\} \quad(n>m), $$

a sequence $\{p_{n}\}$ is Cauchy if and only if $\{p_{n+1}-p_{n}\}$ converges to zero in a non-Archimedean field. By a complete non-Archimedean field, we mean that every Cauchy sequence is convergent in the field.

An example of a non-Archimedean valuation is the mapping $|\cdot|$ taking everything but 0 into 1 and $|0|=0$. This valuation is called trivial. Another example of a non-Archimedean valuation on a field $\mathbb{A}$ is the mapping

$$ \vert k\vert = \textstyle\begin{cases} 0 &\text{if } k=0,\\ \frac{1}{k} &\text{if } k>0,\\ -\frac{1}{k} &\text{if } k< 0 \end{cases} $$

for any $k\in\mathbb{A}$.

Let p be a prime number. For any non-zero rational number $x=p^{r}\frac{m}{n}$ in which m and n are co-prime to the prime number p, consider the p-adic absolute value $|x|_{p}=p^{-r}$ on $\mathbb{Q}$. It is easy to check that $|\cdot|_{p}$ is a non-Archimedean norm on $\mathbb{Q}$. The completion of $\mathbb{Q}$ with respect to $|\cdot|_{p}$, which is denoted by $\mathbb{Q}_{p}$, is said to be the p-adic number field. Note that if $p>2$, then $\vert 2^{n}\vert _{p}=1$ for all integers n.

Throughout this paper, we consider that $\mathbb{X}$ and $\mathbb {Y}$ are a non-Archimedean field and a complete non-Archimedean field, respectively. From now on, for a non-Archimedean field $\mathbb{X}$, we put $\mathbb{X}^{*}=\mathbb{X}\setminus\{0\}$. For the purpose of simplification, let us define the difference operators $\Delta_{1}c, \Delta_{2}q:\mathbb{X}^{*}\times\mathbb{X}^{*}\longrightarrow\mathbb{Y}$ by

$$ \Delta_{1}c(x,y) =c(2x+y)+c(x+2y)-\frac{9c(x)c(y) [c(x)+c(y)+2c(x)^{\frac{1}{3}}c(y)^{\frac{1}{3}} (c(x)^{\frac {1}{3}}+c(y)^{\frac{1}{3}} ) ]}{ [2c(x)^{\frac {2}{3}}+2c(y)^{\frac{2}{3}}+5c(x)^{\frac{1}{3}}c(y)^{\frac{1}{3}} ]^{3}} $$

and

$$ \Delta_{2}q(x,y)=q(2x+y)+q(2x-y)-\frac{2q(x)q(y) [q(x)+16q(y)+24\sqrt {q(x)q(y)} ]}{ [4\sqrt{q(y)}-\sqrt{q(x)} ]^{4}} $$

for all $x,y\in\mathbb{X^{*}}$.

Definition 2.2

A mapping $c:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ is called a reciprocal-cubic mapping if c satisfies equation (1.7). Also, a mapping $q:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ is called a reciprocal-quartic mapping if q satisfies equation (1.8).

3 Hyers-Ulam stability for equations (1.7) and (1.8)

In this section, we investigate the generalized Hyers-Ulam stability of equations (1.7) and (1.8) in non-Archimedean fields. We also establish the results pertaining to Hyers-Ulam stability, Hyers-Ulam-Rassias stability and Ulam-Găvruta-Rassias stability controlled by product-sum of powers of norms.

Theorem 3.1

Let $l\in\{1,-1\}$ be fixed, and let $F:\mathbb{X^{*}}\times\mathbb {X^{*}}\longrightarrow[0,\infty)$ be a mapping such that

$$ \lim_{n\rightarrow\infty} \biggl\vert \frac{1}{27} \biggr\vert ^{ln} F \biggl(\frac{x}{3^{ln+\frac{l+1}{2}}},\frac{y}{3^{ln+\frac{l+1}{2}}} \biggr)=0 $$

(3.1)

for all $x,y\in\mathbb{X^{*}}$. Suppose that $c:\mathbb {X^{*}}\longrightarrow\mathbb{Y}$ is a mapping satisfying the inequality

$$ \bigl\vert \Delta_{1}c(x,y) \bigr\vert \leq F(x,y) $$

(3.2)

for all $x,y\in\mathbb{X^{*}}$. Then there exists a unique reciprocal-cubic mapping $C:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ such that

$$ \bigl\vert c(x)-C(x) \bigr\vert \leq\sup \biggl\{ \biggl\vert \frac{1}{27} \biggr\vert ^{jl+\frac{l-1}{2}}F \biggl( \frac{x}{3^{jl+\frac{l+1}{2}}},\frac {x}{3^{jl+\frac{l+1}{2}}} \biggr): j\in\mathbb{N}\cup\{0\} \biggr\} $$

(3.3)

for all $x\in\mathbb{X^{*}}$.

Proof

Interchanging $(x,y)$ into $(x,x)$ in (3.2), we obtain

$$ \biggl\vert c(x)-\frac{1}{27^{l}}c \biggl(\frac{x}{3^{l}} \biggr) \biggr\vert \leq |27|^{\frac{|l-1|}{2}}F \biggl(\frac{x}{3^{\frac{l+1}{2}}}, \frac {x}{3^{\frac{l+1}{2}}} \biggr) $$

(3.4)

for all $x\in\mathbb{X^{*}}$. Replacing x by $\frac{x}{3^{ln}}$ in (3.4) and multiplying by $\vert \frac{1}{27}\vert ^{ln}$, we have

$$ \biggl\vert \frac{1}{27^{ln}}c \biggl(\frac{x}{3^{ln}} \biggr)-\frac {1}{27^{(n+1)l}}c \biggl(\frac{x}{3^{(n+1)l}} \biggr) \biggr\vert \leq \biggl\vert \frac{1}{27} \biggr\vert ^{ln+\frac{l-1}{2}}F \biggl( \frac{x}{3^{ln+\frac {l+1}{2}}},\frac{x}{3^{ln+\frac{l+1}{2}}} \biggr) $$

(3.5)

for all $x\in\mathbb{X^{*}}$. It follows from relations (3.1) and (3.5) that the sequence $\{\frac{1}{27^{ln}}c (\frac {x}{3^{ln}} ) \}$ is Cauchy. Since $\mathbb{Y}$ is complete, this sequence converges to a mapping $C:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ defined by

$$ C(x) = \lim_{n\rightarrow\infty}\frac{1}{27^{ln}}c \biggl( \frac {x}{3^{ln}} \biggr). $$

(3.6)

On the other hand, for each $x\in\mathbb{X^{*}}$ and non-negative integers n, we have

$$\begin{aligned} \biggl\vert \frac{1}{27^{ln}}c \biggl(\frac{x}{3^{ln}} \biggr)-c(x) \biggr\vert & = \Biggl\vert \sum_{j=0}^{n-1} \biggl\{ \frac{1}{27^{(j+1)l}}c \biggl(\frac {x}{3^{(j+1)l}} \biggr)-\frac{1}{27^{jl}}c \biggl(\frac{x}{3^{jl}} \biggr) \biggr\} \Biggr\vert \\ & \leq\max \biggl\{ \biggl\vert \frac{1}{27^{(j+1)l}}c \biggl( \frac {x}{3^{(j+1)l}} \biggr)-\frac{1}{27^{jl}}c \biggl(\frac{x}{3^{jl}} \biggr) \biggr\vert :0\leq i< n \biggr\} \\ & \leq\max \biggl\{ \biggl\vert \frac{1}{27} \biggr\vert ^{jl+\frac {l-1}{2}}F \biggl(\frac{x}{3^{jl+\frac{l+1}{2}}},\frac{x}{3^{jl+\frac {l+1}{2}}} \biggr):0\leq j< n \biggr\} . \end{aligned}$$

(3.7)

Applying (3.6) and letting $n\rightarrow\infty$ in inequality (3.7), we find that inequality (3.3) holds. Using (3.1), (3.2) and (3.6), for all $x,y\in\mathbb{X^{*}}$, we have

$$\begin{aligned} \bigl\vert \Delta_{1}C(x,y) \bigr\vert & = \lim _{n\rightarrow\infty} \biggl\vert \frac{1}{27} \biggr\vert ^{ln} \biggl\vert \Delta_{1}c \biggl(\frac{x}{3^{ln}}, \frac {y}{3^{ln}} \biggr) \biggr\vert \leq\lim_{n\rightarrow\infty} \biggl\vert \frac{1}{27} \biggr\vert ^{ln}F \biggl( \frac{x}{3^{ln}},\frac{y}{3^{ln}} \biggr)=0. \end{aligned}$$

Thus, the mapping C satisfies (1.7) and hence it is a reciprocal-cubic mapping. In order to prove the uniqueness of C, let us consider another reciprocal-cubic mapping $C^{\prime}:\mathbb {X^{*}}\longrightarrow\mathbb{Y}$ satisfying (3.3). Then

$$\begin{aligned} & \bigl\vert C(x)-C^{\prime}(x) \bigr\vert \\ &\quad= \lim_{m\rightarrow\infty} \biggl\vert \frac{1}{27} \biggr\vert ^{lm} \biggl\vert C \biggl(\frac{x}{3^{lm}}x \biggr)-C^{\prime} \biggl(\frac{x}{3^{lm}} \biggr) \biggr\vert \\ & \quad\leq\lim_{m\rightarrow\infty} \biggl\vert \frac{1}{27} \biggr\vert ^{lm} \max \biggl\{ \biggl\vert C \biggl( \frac{x}{3^{lm}} \biggr)-c \biggl(\frac {x}{3^{lm}} \biggr) \biggr\vert , \biggl\vert c \biggl( \frac{x}{3^{lm}} \biggr)-C^{\prime} \biggl( \frac{x}{3^{lm}} \biggr) \biggr\vert \biggr\} \\ &\quad\leq\lim_{m\rightarrow\infty}\lim_{n\rightarrow \infty}\max \biggl\{ \max \biggl\{ \biggl\vert \frac{1}{27} \biggr\vert ^{(j+m)l+\frac{l-1}{2}}F \biggl(\frac{x}{3^{(j+m)l+\frac{l+1}{2}}},\frac {x}{3^{(j+m)l+\frac{l+1}{2}}} \biggr):m\leq j\leq n+m \biggr\} \biggr\} \\ &\quad =0 \end{aligned}$$

for all $x\in\mathbb{X^{*}}$, which shows that C is unique. This finishes the proof. □

From now on, we assume that $|2|<1$. The following corollaries are immediate consequences of Theorem 3.1 concerning the stability of (1.7).

Corollary 3.2

Let $\epsilon>0$ be a constant. If $c:\mathbb{X^{*}}\longrightarrow \mathbb{Y}$ satisfies $\vert \Delta_{1}c(x,y)\vert \leq\epsilon$ for all $x,y\in\mathbb {X^{*}}$, then there exists a unique reciprocal-cubic mapping $C:\mathbb {X^{*}}\longrightarrow\mathbb{Y}$ satisfying (1.7) and $\vert c(x)-C(x)\vert \leq\epsilon$ for all $x\in\mathbb{X^{*}}$.

Proof

Defining $F(x,y)=\epsilon$ and applying Theorem 3.1 for the case $l=-1$, we get the desired result. □

Corollary 3.3

Let $\epsilon\geq0$ and $r\neq-3$ be fixed constants. If $c:\mathbb {X^{*}}\longrightarrow\mathbb{Y}$ satisfies $\vert \Delta_{1}c(x,y)\vert \leq\epsilon (\vert x\vert ^{r}+\vert y\vert ^{r} )$ for all $x,y\in\mathbb{X^{*}}$, then there exists a unique reciprocal-cubic mapping $C:\mathbb{X^{*}}\longrightarrow \mathbb{Y}$ satisfying (1.7) and

$$ \bigl\vert c(x)-C(x) \bigr\vert \leq \textstyle\begin{cases} \frac{2\epsilon}{|3|^{r}}\vert x\vert ^{r}, & r>-3,\\ 2\epsilon|3|^{3}\vert x\vert ^{r}, & r< -3 \end{cases} $$

for all $x\in\mathbb{X^{*}}$.

Proof

The result follows immediately from Theorem 3.1 by taking $F(x,y)=\epsilon (\vert x\vert ^{r}+\vert y\vert ^{r} )$. □

Corollary 3.4

Let $c:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ be a mapping, and let there exist real numbers p, q, ${r =p+q \neq-3}$ and $\epsilon\geq0$ such that $\vert \Delta_{1}c(x,y)\vert \leq\epsilon \vert x\vert ^{p}\vert y\vert ^{q}$ for all $x,y\in\mathbb{X^{*}}$. Then there exists a unique reciprocal-cubic mapping $C:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ satisfying (1.7) and

$$ \bigl\vert c(x)-C(x) \bigr\vert \leq \textstyle\begin{cases} \frac{\epsilon}{|3|^{r}}\vert x\vert ^{r}, & r>-3,\\ \epsilon|3|^{3}|\vert x\vert ^{r}, & r< -3 \end{cases} $$

for all $x\in\mathbb{X^{*}}$.

Proof

The required result is obtained by choosing $F(x,y)=\epsilon \vert x\vert ^{p}\vert y\vert ^{q}$ for all $x,y\in\mathbb{X^{*}}$ in Theorem 3.1. □

Corollary 3.5

Let $\epsilon\geq0$ and $r\neq-3$ be real numbers and $c:\mathbb {X^{*}}\longrightarrow\mathbb{Y}$ be a mapping satisfying the functional inequality

$$ \bigl\vert \Delta_{1}c(x,y) \bigr\vert \leq\epsilon \bigl(\vert x\vert ^{\frac {r}{2}}\vert y\vert ^{\frac{r}{2}}+ \bigl(\vert x\vert ^{r}+\vert y\vert ^{r} \bigr) \bigr) $$

for all $x,y\in\mathbb{X^{*}}$. Then there exists a unique reciprocal-cubic mapping $C:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ satisfying (1.7) and

$$ \bigl\vert c(x)-C(x) \bigr\vert \leq \textstyle\begin{cases} \frac{3\epsilon}{|3|^{r}}\vert x\vert ^{r}, & r>-3,\\ 3\epsilon|3|^{3}\vert x\vert ^{r}, & r< -3 \end{cases} $$

for all $x\in\mathbb{X^{*}}$.

Proof

Considering $F(x,y)=\epsilon (\vert x\vert ^{\frac{r}{2}}\vert y\vert ^{\frac{r}{2}}+ (\vert x\vert ^{r}+\vert y\vert ^{r} ) )$ in Theorem 3.1, one can find the result. □

We have the following result which is analogous to Theorem 3.1 for the functional equation (1.8). We include the proof for the sake of completeness.

Theorem 3.6

Let $l\in\{1,-1\}$ be fixed, and let $G:\mathbb{X^{*}}\times\mathbb {X^{*}}\longrightarrow[0,\infty)$ be a mapping such that

$$ \lim_{n\rightarrow\infty} \biggl\vert \frac{1}{81} \biggr\vert ^{ln} G \biggl(\frac{x}{3^{ln+\frac{l+1}{2}}},\frac{y}{3^{ln+\frac{l+1}{2}}} \biggr)=0 $$

(3.8)

for all $x,y\in\mathbb{X^{*}}$. Suppose that $q:\mathbb {X^{*}}\longrightarrow\mathbb{Y}$ is a mapping satisfying the inequality

$$ \bigl\vert \Delta_{2}q(x,y) \bigr\vert \leq G(x,y) $$

(3.9)

for all $x,y\in\mathbb{X^{*}}$. Then there exists a unique reciprocal-quartic mapping $Q:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ such that

$$ \bigl\vert q(x)-Q(x) \bigr\vert \leq\sup \biggl\{ \biggl\vert \frac{1}{81} \biggr\vert ^{jl+\frac{l-1}{2}}F \biggl( \frac{x}{3^{jl+\frac{l+1}{2}}},\frac {x}{3^{jl+\frac{l+1}{2}}} \biggr): j\in\mathbb{N}\cup\{0\} \biggr\} $$

(3.10)

for all $x\in\mathbb{X^{*}}$.

Proof

Replacing $(x,y)$ by $(x,x)$ in (3.9), we get

$$ \biggl\vert q(x)-\frac{1}{81^{l}}q \biggl(\frac{x}{3^{l}} \biggr) \biggr\vert \leq |81|^{\frac{|l-1|}{2}}G \biggl(\frac{x}{3^{\frac{l+1}{2}}}, \frac {x}{3^{\frac{l+1}{2}}} \biggr) $$

(3.11)

for all $x\in\mathbb{X^{*}}$. Switching x into $\frac{x}{3^{ln}}$ in (3.11) and multiplying by $\vert \frac{1}{81}\vert ^{ln}$, we arrive at

$$ \biggl\vert \frac{1}{81^{ln}}c \biggl(\frac{x}{3^{ln}} \biggr)-\frac {1}{81^{(n+1)l}}c \biggl(\frac{x}{3^{(n+1)l}} \biggr) \biggr\vert \leq \biggl\vert \frac{1}{81} \biggr\vert ^{ln+\frac{l-1}{2}}G \biggl( \frac{x}{3^{ln+\frac {l+1}{2}}},\frac{x}{3^{ln+\frac{l+1}{2}}} \biggr) $$

(3.12)

for all $x\in\mathbb{X^{*}}$. Relations (3.8) and (3.12) imply that $\{\frac{1}{81^{ln}}q (\frac{x}{3^{ln}} ) \}$ is a Cauchy sequence. Due to the completeness of $\mathbb{Y}$, there is a mapping $Q:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ so that

$$ Q(x) = \lim_{n\rightarrow\infty}\frac{1}{81^{ln}}q \biggl( \frac {x}{3^{ln}} \biggr) $$

(3.13)

for all $x\in\mathbb{X^{*}}$. The rest of the proof is similar to the proof of Theorem 3.1. □

Here, we bring some corollaries regarding the stability of functional equation (1.8) which are a direct consequence of Theorem 3.6.

Corollary 3.7

Let $\delta>0$ be a constant, and let $q:\mathbb{X^{*}}\longrightarrow \mathbb{Y}$ satisfy $\vert \Delta_{2}q(x,y)\vert \leq\delta$ for all $x,y\in\mathbb {X^{*}}$. Then there exists a unique reciprocal-quartic mapping $Q:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ satisfying (1.8) and $\vert q(x)-Q(x)\vert \leq\delta$ for all $x\in\mathbb{X^{*}}$.

Proof

It is enough to put $G(x,y)=\delta$ in Theorem 3.6 when $l=-1$. □

Corollary 3.8

Let $\delta\geq0$ and $\alpha\neq-4$ be fixed constants. If $q:\mathbb {X^{*}}\longrightarrow\mathbb{Y}$ satisfies $\vert \Delta_{2}q(x,y)\vert \leq\delta (\vert x\vert ^{\alpha }+\vert y\vert ^{\alpha} )$ for all $x,y\in\mathbb{X^{*}}$, then there exists a unique reciprocal-quartic mapping $Q:\mathbb {X^{*}}\longrightarrow\mathbb{Y}$ satisfying (1.8) and

$$ \bigl\vert q(x)-Q(x) \bigr\vert \leq \textstyle\begin{cases} \frac{2\delta}{|3|^{\alpha}} \vert x\vert ^{\alpha}, & \alpha >-4,\\ 2\delta|3|^{4}\vert x\vert ^{\alpha}, & \alpha< -4 \end{cases} $$

for all $x\in\mathbb{X^{*}}$.

Proof

Considering $G(x,y)=\delta (\vert x\vert ^{\alpha}+\vert y\vert ^{\alpha} )$ for all $x,y\in\mathbb{X^{*}}$ in Theorem 3.6, we reach the result. □

Corollary 3.9

Let $q:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ be a mapping, and let there exist real numbers a, b, ${\alpha=a+b \neq-4}$ and $\delta\geq0$ such that

$$ \bigl\vert D_{2}q(x,y) \bigr\vert \leq\delta \vert x\vert ^{a}\vert y\vert ^{b} $$

for all $x,y\in\mathbb{X^{*}}$. Then there exists a unique reciprocal-quartic mapping $Q:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ satisfying (1.8) and

$$ \bigl\vert q(x)-Q(x) \bigr\vert \leq \textstyle\begin{cases} \frac{\delta}{|3|^{\alpha}} \vert x\vert ^{\alpha}, & \alpha >-4,\\ \delta|3|^{4}\vert x\vert ^{\alpha}, & \alpha< -4 \end{cases} $$

for all $x\in\mathbb{X^{*}}$.

Proof

Choosing $G(x,y)=\delta \vert x\vert ^{\alpha} \vert y\vert ^{\alpha}$ in Theorem 3.6, one can derive the desired result. □

Corollary 3.10

Let $\delta\geq0$ and $\alpha\neq-4$ be real numbers and $q:\mathbb {X^{*}}\longrightarrow\mathbb{Y}$ be a mapping satisfying the functional inequality

$$ \bigl\vert D_{2}q(x,y) \bigr\vert \leq\delta \bigl(\vert x \vert ^{\frac{\alpha }{2}}\vert y\vert ^{\frac{\alpha}{2}}+ \bigl(\vert x\vert ^{\alpha }+\vert y\vert ^{\alpha} \bigr) \bigr) $$

for all $x,y\in\mathbb{X^{*}}$. Then there exists a unique reciprocal-quartic mapping $Q:\mathbb{X^{*}}\longrightarrow\mathbb{Y}$ satisfying (1.8) and

$$ \bigl\vert q(x)-Q(x) \bigr\vert \leq \textstyle\begin{cases} \frac{3\delta}{|3|^{\alpha}} \vert x\vert ^{\alpha}, & \alpha >-4,\\ 3\delta|3|^{4}\vert x\vert ^{\alpha}, & \alpha< -4 \end{cases} $$

for all $x\in\mathbb{X^{*}}$.

Proof

The proof follows immediately by taking $G(x,y)=\delta (\vert x\vert ^{\frac{\alpha}{2}}\vert y\vert ^{\frac{\alpha}{2}}+ (\vert x\vert ^{\alpha}+\vert y\vert ^{\alpha} ) )$ in Theorem 3.6. □

4 Related examples

In this section, applying the idea of the well-known counter-example provided by Gajda [30], we show that Corollary 3.3 for $r=-3$ and Corollary 3.8 for $\alpha=-4$ do not hold in $\mathbb {R}$ with usual $|\cdot|$. Note that $(\mathbb {R},|\cdot|)$ is an Archimedean field.

Consider the function

$$ \varphi(x)= \textstyle\begin{cases} \frac{\delta}{x^{3}} & \text{for }x\in(0,\infty),\\ \delta, & \text{otherwise}, \end{cases} $$

(4.1)

where $\varphi:\mathbb{R^{*}}\longrightarrow\mathbb{R}$. Let $f:\mathbb {R^{*}}\longrightarrow\mathbb{R}$ be defined by

$$ f(x)=\sum_{n=0}^{\infty}27^{-n} \varphi \bigl(3^{-n}x \bigr) $$

(4.2)

for all $x\in\mathbb {R}^{*}$.

Theorem 4.1

If the function $f:\mathbb{R^{*}}\longrightarrow\mathbb{R}$ defined in (4.2) satisfies the functional inequality

$$ \bigl\vert \Delta_{1}f(x,y) \bigr\vert \leq \frac{28\delta}{13} \bigl(\vert x\vert ^{-3}+\vert y\vert ^{-3} \bigr) $$

(4.3)

for all $x,y\in X$, then there do not exist a reciprocal-cubic mapping $c:\mathbb{R^{*}}\longrightarrow\mathbb{R}$ and a constant $\mu>0$ such that

$$ \bigl\vert f(x)-c(x) \bigr\vert \leq\mu \vert x\vert ^{-3} $$

(4.4)

for all $x\in\mathbb{R^{*}}$.

Proof

First, we are going to show that f satisfies (4.3). By computation, we have

$$\bigl\vert f(x) \bigr\vert = \Biggl\vert \sum _{n=0}^{\infty}27^{-n} \varphi \bigl(3^{-n}x \bigr) \Biggr\vert \leq\sum _{n=0}^{\infty}\frac{\delta}{27^{n}}=\frac{27\delta}{26}. $$

Therefore, we see that f is bounded by $\frac{27\delta}{26}$ on $\mathbb{R}$. If $\vert x\vert ^{-3}+\vert y\vert ^{-3}\geq1$, then the left-hand side of (4.3) is less than $\frac{28\delta}{13}$. Now, suppose that $0<\vert x\vert ^{-3}+\vert y\vert ^{-3}<1$. Hence, there exists a positive integer k such that

$$ \frac{1}{27^{k+1}}\leq \vert x\vert ^{-3}+\vert y \vert ^{-3}< \frac{1}{27^{k}}. $$

(4.5)

Thus, relation (4.5) requires $27^{k} (\vert x\vert ^{-3}+\vert y\vert ^{-3} )<1$ or, equivalently, $27^{k}x^{-3}<1$, $27^{k}y^{-3}<1$. So, $\frac{x^{3}}{27^{k}}>1$, $\frac{y^{3}}{27^{k}}>1$. The last inequalities imply that $\frac{x^{3}}{27^{k-1}}>27>1$, $\frac{y^{3}}{27^{k-1}}>27>1$; and consequently,

$$\frac{1}{3^{k-1}}(x)>1,\quad\quad \frac{1}{3^{k-1}}(y)>1, \quad\quad\frac {1}{3^{k-1}}(2x+y)>1,\quad\quad \frac{1}{3^{k-1}}(x+2y)>1. $$

Therefore, for each value of $n=0,1,2,\dots,k-1$, we obtain

$$\frac{1}{3^{n}}(x)>1, \quad\quad\frac{1}{3^{n}}(y)>1, \qquad\frac{1}{3^{n}}(2x+y)>1,\qquad \frac {1}{3^{n}}(x+2y)>1 $$

and $\Delta_{1}\varphi(3^{-n}x,3^{-n}y)=0$ for $n=0,1,2,\dots,k-1$. Using (4.1) and the definition of f, we obtain

$$\begin{aligned} \bigl\vert \Delta_{1}f(x,y) \bigr\vert & \leq\sum _{n=k}^{\infty}\frac{\delta }{27^{n}}+\sum _{n=k}^{\infty}\frac{\delta}{27^{n}}+\frac{54}{729}\sum _{n=k}^{\infty}\frac{\delta}{27^{n}}\leq2\delta\sum _{n=k}^{\infty}\frac {1}{27^{n}}+ \frac{2\delta}{27}\sum_{n=k}^{\infty} \frac{1}{27^{n}} \\ &\leq\frac{56\delta}{27}\frac{1}{27^{k}} \biggl(1-\frac{1}{27} \biggr)^{-1}\leq\frac{28\delta}{13}\frac{1}{27^{k}}\leq \frac{28\delta }{13}\frac{1}{27^{k+1}}\leq\frac{28\delta}{13} \bigl(\vert x\vert ^{-3}+\vert y\vert ^{-3} \bigr) \end{aligned}$$

for all $x,y \in\mathbb{R^{*}}$. Therefore, inequality (4.3) holds. We claim that the reciprocal-cubic functional equation (1.7) is not stable for $r=-3$ in Corollary 3.3. Assume that there exists a reciprocal-cubic mapping $c:\mathbb {R^{*}}\longrightarrow\mathbb{R}$ satisfying (4.4). Therefore,

$$ \bigl|f(x)\bigr|\leq(\mu+1)|x|^{-3}. $$

(4.6)

However, we can choose a positive integer m with $m\delta>\mu+1$. If $x\in (1,3^{m-1} )$, then $3^{-n}x\in(1,\infty)$ for all $n=0,1,2,\dots,m-1$, and thus

$$ \bigl|f(x)\bigr| =\sum_{n=0}^{\infty}\frac{\varphi(3^{-n}x)}{27^{n}} \geq\sum_{n=0}^{m-1}\frac{\frac{27^{n}\delta}{x^{3}}}{27^{n}}= \frac{m\delta }{x^{3}}>(\mu+1)x^{-3}, $$

which contradicts (4.6). This completes the proof. □

Now, we consider the function $\phi:\mathbb {R^{*}}\longrightarrow\mathbb{R}$ defined via

$$ \phi(x)= \textstyle\begin{cases} \frac{\lambda}{x^{4}} & \text{for }x\in(0,\infty),\\ \lambda,& \text{otherwise}. \end{cases} $$

(4.7)

Also, let $g:\mathbb{R^{*}}\longrightarrow\mathbb{R}$ be defined by

$$ g(x)=\sum_{n=0}^{\infty}81^{-n} \phi \bigl(3^{-n}x \bigr) $$

(4.8)

for all $x\in\mathbb {R}^{*}$. In analogy with Theorem 4.1, we show that Corollary 3.8 does not hold for $\alpha=-4$ in $\mathbb{R}$ with usual $|\cdot|$.

Theorem 4.2

If the function $g:\mathbb{R^{*}}\longrightarrow\mathbb{R}$ defined in (4.8) satisfies the functional inequality

$$ \bigl\vert \Delta_{2}g(x,y) \bigr\vert \leq \frac{61\lambda}{20} \bigl(\vert x\vert ^{-4}+\vert y\vert ^{-4} \bigr) $$

(4.9)

for all $x,y\in X$, then there do not exist a reciprocal-quartic mapping $q:\mathbb{R^{*}}\longrightarrow\mathbb{R}$ and a constant $\beta>0$ such that

$$ \bigl\vert g(x)-q(x) \bigr\vert \leq\beta \vert x\vert ^{-4} $$

(4.10)

for all $x\in\mathbb{R^{*}}$.

Proof

Let us first prove that g satisfies (4.9).

$$\bigl\vert g(x) \bigr\vert = \Biggl\vert \sum _{n=0}^{\infty}81^{-n} \phi \bigl(3^{-n}x \bigr) \Biggr\vert \leq\sum _{n=0}^{\infty}\frac{\lambda}{81^{n}}=\frac{81\lambda}{80}. $$

Hence, we find that g is bounded by $\frac{81\lambda}{80}$ on $\mathbb {R}$. If $\vert x\vert ^{-4}+\vert y\vert ^{-4}\geq1$, then the left-hand side of (4.9) is less than $\frac{61\lambda}{20}$. Now, suppose that $0<\vert x\vert ^{-4}+\vert y\vert ^{-4}<1$. Then there exists a positive integer m such that

$$ \frac{1}{81^{m+1}}\leq \vert x\vert ^{-4}+\vert y\vert ^{-4}< \frac{1}{81^{m}}. $$

By arguments similar to those in Theorem 4.1, the relation $\vert x\vert ^{-4}+\vert y\vert ^{-4}<\frac{1}{81^{m}}$ implies

$$\frac{1}{3^{m-1}}(x)>1,\qquad \frac{1}{3^{m-1}}(y)>1, \qquad\frac {1}{3^{m-1}}(2x+y)>1,\qquad \frac{1}{3^{m-1}}(2x-y)>1. $$

Therefore, for any $n=0,1,2,\dots,m-1$, we get

$$\frac{1}{3^{n}}(x)>1, \qquad\frac{1}{3^{n}}(y)>1, \qquad\frac{1}{3^{n}}(2x+y)>1,\qquad \frac {1}{3^{n}}(2x-y)>1 $$

and $\Delta_{2}\phi(3^{-n}x,3^{-n}y)=0$ for $n=0,1,2,\dots,m-1$. Using (4.7) and the definition of g, we find

$$\begin{aligned} \bigl\vert \Delta_{2}g(x,y) \bigr\vert & \leq\sum _{n=m}^{\infty}\frac{\lambda }{81^{n}}+\sum _{n=m}^{\infty}\frac{\lambda}{81^{n}}+\frac{82}{81}\sum _{n=k}^{\infty}\frac{\lambda}{81^{n}}\leq2\lambda\sum _{n=m}^{\infty}\frac {1}{81^{n}}+ \frac{82\lambda}{81}\sum_{n=m}^{\infty} \frac{1}{81^{n}} \\ &\leq\frac{244\lambda}{81}\frac{1}{81^{m}} \biggl(1-\frac{1}{81} \biggr)^{-1}\leq\frac{244\lambda}{80}\frac{1}{81^{m}}\leq \frac{244\lambda }{80}\frac{1}{81^{k+1}} \\ &\leq\frac{61\lambda}{20} \bigl(\vert x\vert ^{-4}+\vert y\vert ^{-4} \bigr) \end{aligned}$$

for all $x,y \in\mathbb{R^{*}}$. This shows that inequality (4.9) holds. Here, we prove that the reciprocal-quartic functional equation (1.8) is not stable for $\alpha=-4$ in Corollary 3.8. Assume that there exists a reciprocal-quartic mapping $q:\mathbb{R^{*}}\longrightarrow\mathbb{R}$ satisfying (4.10). Hence

$$ \bigl|g(x)\bigr|\leq(\beta+1)|x|^{-4}. $$

(4.11)

On the other hand, we can choose a positive integer k with $k\lambda >\beta+1$. If $x\in (1,3^{k-1} )$, then $3^{-n}x\in(1,\infty )$ for all $n=0,1,2,\dots,k-1$, and so

$$ \bigl|g(x)\bigr| =\sum_{n=0}^{\infty}\frac{\phi(3^{-n}x)}{81^{n}} \geq\sum_{n=0}^{k-1}\frac{\frac{81^{n}\lambda}{x^{4}}}{81^{n}}= \frac{k\lambda }{x^{4}}>(\beta+1)x^{-4}, $$

which contradicts (4.11). Therefore, the reciprocal-quartic functional equation (1.8) is not stable in the case $\alpha=-4$ in Corollary 3.8 for $(\mathbb{R},|\cdot|)$. □