On the quotient stability of a family of functional equations

Abstract

We show some stability results for a family of functional equations that contains the exponential and the Goła̧b–Schinzel functional equations.

Introduction

Let $\mathbb{R}$, $\mathbb{Z}$ and $\mathbb{N}$ stand, as usual, for the sets of reals, integers and positive integers, respectively; moreover, we write ${\mathbb{R}}_0≔\mathbb{R}\setminus \left\{0\right\}$, $D+E≔\{d+e:d\in D,e\in E\}$ and $cD≔\{cd:d\in D\}$ for $c\in \mathbb{R}$, $D,E\subset \mathbb{R}$.

In what follows $\lambda \in {\mathbb{R}}_0$ and $M:\mathbb{R}\to \mathbb{R}$ denotes a continuous function that is multiplicative (i.e., $M\left(xy\right)=M\left(x\right)M\left(y\right)$ for $x,y\in \mathbb{R}$) and $M\left(\mathbb{R}\right)\ne \left\{0\right\}$. Several particular forms of the functional equation

$$f(x+M\left(f\left(x\right)\right)y)=\lambda f\left(x\right)f\left(y\right)$$

(with the unknown function $f$ mapping, e.g., a real linear space into $\mathbb{R}$) have already found nontrivial applications in meteorology and fluid mechanics, theories of near rings and quasialgebras, finding algebraic substructures, theory of algebraic objects, and description of associative operations; for suitable references see the monograph [1] and the survey paper [2]. Note that, in the particular case: $\lambda =1$ and $M\left(x\right)=1$ for $x\in \mathbb{R}$, Eq. (1) becomes the well-known exponential equation

$$f(x+y)=f\left(x\right)f\left(y\right)$$

and, if $\lambda =1$ and $M\left(x\right)=x$ for $x\in \mathbb{R}$, (1) has the form

$$f(x+f\left(x\right)y)=f\left(x\right)f\left(y\right)\text{,}$$

which is now called the Goła̧b–Schinzel functional equation. Some recent results concerning solutions of Eq. (3) and its generalizations can be found in [3], [4], [5], [6], [7].

The investigation of stability of functional equations started with a problem posed by S.M. Ulam and a solution of it given by Hyers [8]. Among other results, the superstability of the exponential equation (2) has been discovered and in connection with that phenomenon, a new approach has been suggested (see [9]) that is now called: stability in the sense of Ger (for more information on stability and superstability of functional equations we refer the reader to, e.g., [10], [11], [12], [13]; for some recent results on these subjects see, e.g., [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [37]). Motivated by this approach, Chudziak [28] (see also [29]) has proved that, in the class of functions $f$, mapping a linear space $Y$ over a field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ into $\mathbb{K}$, such that the set $A_f≔\{x\in Y;f\left(x\right)\ne 0\}$ has an algebraically interior point, the functional equation

$$f(x+f{\left(x\right)}^{n}y)=\lambda f\left(x\right)f\left(y\right)$$

(with fixed $n\in \mathbb{N}$) is superstable in the sense of Ger. Namely he has shown that every function $f:Y\to \mathbb{K}$ such that the set $A_f$ possesses an algebraically interior point and

$$|\frac{f(x+f{\left(x\right)}^{n}y)}{\lambda f\left(x\right)f\left(y\right)}-1|\le \epsilon \text{whenever }f\left(x\right)f\left(y\right)\ne 0\text{,}$$$$|\frac{\lambda f\left(x\right)f\left(y\right)}{f(x+f{\left(x\right)}^{n}y)}-1|\le \epsilon \text{whenever }f(x+f{\left(x\right)}^{n}y)\ne 0\text{,}$$

with some $\epsilon \in (0,1)$, is a solution of Eq. (4) or it is bounded. Some further results related to or concerning the stability of (3) or (4) can be found in [30], [31], [32], [33].

The result in [28] (and in [29]) does not remain valid if we replace inequalities (5), (6) by the following weaker condition

$$|\frac{f(x+f{\left(x\right)}^{n}y)}{\lambda f\left(x\right)f\left(y\right)}-1|\le \epsilon \text{and}|\frac{\lambda f\left(x\right)f\left(y\right)}{f(x+f{\left(x\right)}^{n}y)}-1|\le \epsilon$$$$\text{whenever }f(x+f{\left(x\right)}^{n}y)f\left(x\right)f\left(y\right)\ne 0\text{.}$$

For instance, for $\lambda =1$, the function $f:\mathbb{R}\to \mathbb{R}$, given by the formulas $f\left(x\right)=\sqrt[n]{x+1}$ for $x>0$ and $f\left(x\right)=0$ for $x\le 0$, satisfies (7) (with $\epsilon =0$ and $\lambda =1$), but it is not a solution of (4) (take $x>0$ and $y=0$). However, it is easy to check that $f$ satisfies (with $\lambda =1$) the functional equation

$$f(x+f{\left(x\right)}^{n}y)f\left(x\right)f\left(y\right)[f(x+f{\left(x\right)}^{n}y)-\lambda f\left(x\right)f\left(y\right)]=0\text{.}$$

In this paper we complete (and generalize to some extent) the results in [29], [28] by showing that every continuous solution $f:\mathbb{R}\to \mathbb{R}$ of (7) satisfies Eq. (8) and, moreover, in the case where $f(x_0+f{\left(x_0\right)}^n{y}_0)f\left(x_0\right)f\left(y_0\right)\ne 0$ for some $x_0,y_0\in \mathbb{R}$, it must be a solution of (4). We also extend these results to the case of Eq. (1); in particular, we prove analogous results for the exponential equation (2). In view of Remark 3, for $M\left(\mathbb{R}\right)\ne \left\{1\right\}$, we should not expect to obtain such results under assumptions weaker than continuity of $f$ (at each point of the domain); in the case $M\left(\mathbb{R}\right)=\left\{1\right\}$ the situation is somewhat different, as it is shown in Proposition 1 and Corollary 1, Corollary 2.

The present paper gives a starting point for investigations of analogous problems in more general classes of functions, e.g., mapping a linear topological space $Y$ over a field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ into $\mathbb{K}$. However, since some additional longer arguments are necessary in such a case, they will be studied in next publications.

At the end of the paper (see Proposition 3) we show that a description of the continuous solutions $f:\mathbb{R}\to \mathbb{R}$ of the equation

$$f(x+M\left(f\left(x\right)\right)y)f\left(x\right)f\left(y\right)[f(x+M\left(f\left(x\right)\right)y)-\lambda f\left(x\right)f\left(y\right)]=0$$

can be easily derived from our main theorem (Theorem 2). In this way we generalize the main results in [34], where the continuous solutions $f:\mathbb{R}\to \mathbb{R}$ of the functional equation

$$f(x+f\left(x\right)y)f\left(x\right)f\left(y\right)[f(x+f\left(x\right)y)-f\left(x\right)f\left(y\right)]=0$$

have been determined.