, 2017:77

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

• Sang Og Kim
• Beri Venkatachalapathy Senthil Kumar
• Abasalt Bodaghi
Open Access
Research

## 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.

## Keywords

reciprocal functional equation reciprocal-cubic functional equation reciprocal-quartic functional equation generalized Hyers-Ulam stability non-Archimedean field

39B82 39B72

## 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, 10, 11, 12, 13, 14, 15, 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, 23, 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 $$(\mathbb{R},|\cdot|)$$. □

## Notes

### Acknowledgements

The authors would like to thank the referee for careful reading of the paper and giving some useful suggestions.

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## Authors and Affiliations

• Sang Og Kim
• 1
• Beri Venkatachalapathy Senthil Kumar
• 2
• Abasalt Bodaghi
• 3
1. 1.Department of MathematicsHallym UniversityChuncheonRepublic of Korea
2. 2.C. Abdul Hakeem College of Engineering and TechnologyMelvisharamIndia
3. 3.Young Researchers and Elite Club, Islamshahr BranchIslamic Azad UniversityIslamshahrIran