Open Access

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

  • Sang Og Kim1,
  • Beri Venkatachalapathy Senthil Kumar2 and
  • Abasalt Bodaghi3Email author
Advances in Difference Equations20172017:77

DOI: 10.1186/s13662-017-1128-z

Received: 30 January 2017

Accepted: 23 February 2017

Published: 9 March 2017

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

MSC

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 [916] 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 [2224] 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 \((\mathbb{R},|\cdot|)\). □

Declarations

Acknowledgements

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

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Hallym University
(2)
C. Abdul Hakeem College of Engineering and Technology
(3)
Young Researchers and Elite Club, Islamshahr Branch, Islamic Azad University

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