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Theory and Modern Applications

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 \((\mathbb{R},|\cdot|)\). □

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Acknowledgements

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

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Kim, S.O., Senthil Kumar, B.V. & Bodaghi, A. Approximation on the reciprocal-cubic and reciprocal-quartic functional equations in non-Archimedean fields. Adv Differ Equ 2017, 77 (2017). https://doi.org/10.1186/s13662-017-1128-z

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