Ulam Stability of a General Linear Functional Equation in Modular Spaces

Abstract

Using the direct method, we prove the Ulam stability results for the general linear functional equation of the form ${\sum }_{i=1}^m{A}_i(f\left({\phi }_i\left(\overline{x}\right))\right)=D\left(\overline{x}\right)$ for all $\overline{x}\in X^n$, where f is the unknown mapping from a linear space X over a field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ into a linear space Y over field $\mathbb{K}$; n and m are positive integers; ${\phi }_1,\dots ,{\phi }_m$ are linear mappings from $X^n$ to X; $A_1,\dots ,A_m$ are continuous endomorphisms of Y; and $D:X^n\to Y$ is fixed. In this paper, the stability inequality is considered with regard to a convex modular on Y, which is lower semicontinuous and satisfies an additional condition (the ${\Delta }_2$-condition). Our main result generalizes many similar stability outcomes published so far for modular space. It also shows that there is some kind of symmetry between the stability results for equations in modular spaces and those in classical normed spaces.

1. Introduction

In 1940, S. M. Ulam (cf. [1]) posed the stability problem for the functional equation of group homomorphisms. Quite soon, Hyers [2] provided the affirmative answer to this problem in real Banach spaces by using the approach that has subsequently been called the direct method. After that, the problem of the stability of various types of equations (not only functional ones) was extensively studied by many authors (see [3,4,5,6,7,8,9,10] for various types of information; examples; and further references).

For instance, in 2015, Bahyrycz and Olko [11] (see also [12]) published stability results for the following general functional equation:

$$\sum _{i=1}^m{B}_{i}f\left(\sum _{j=1}^n{b}_{ij}{x}_j\right)+B=0,$$

(1)

where f is the unknown mapping from a linear space X over $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ into a Banach space Y over $\mathbb{K}$, $B_i$ and $b_{ij}$ are scalars, and B is a vector from Y. They used the fixed-point approach suggested in [13]. It should be mentioned that the application of fixed-point methods in Ulam stability was initiated in [14,15]. The result reads as follows.

Theorem 1.

Let $B\in Y$ be fixed and assume that either $\left({\sum }_{i=1}^m{B}_i\right)B\ne 0$ or $B=0$. Let $g:X\to Y$ and $\theta :X^n\to [0,\infty )$ satisfy the inequality

$$\parallel \sum _{i=1}^m{B}_{i}g\left(\sum _{j=1}^n{b}_{ij}{x}_j\right)+B\parallel \le \theta (x_1,\dots ,x_n),x_1,\dots ,x_n\in X.$$

(2)

Further, assume that $Ø\ne I\subset \{1,\dots ,m\}$, $I_0:=\{1,\dots ,m\}\backslash I\ne Ø$, $c_1,\dots ,c_n\in \mathbb{K}$ and ${\omega }_1,\dots ,{\omega }_n\in [0,\infty )$ exist such that

(i) 

${\sum }_{j=1}^n{a}_{ij}{c}_j=1$for$i\in I$;

(ii) 

${\sum }_{i\in I_0}|B_i|{\omega }_i<\left|{\sum }_{i\in I}{B}_i\right|$;

(iii) 

$\theta \left(\left({\sum }_{j=1}^n{b}_{ij}{c}_j\right)x_1,\dots ,\left({\sum }_{j=1}^n{b}_{ij}{c}_j\right)x_n\right)\le {\omega }_i\theta (x_1,\dots ,x_n)$for$i\in I_0$and$x_1,\dots ,x_n\in X$.

Then, there is a unique solution $G:X\to Y$ to functional Equation (1) with

$$\parallel g\left(x\right)-G\left(x\right)\parallel \le \frac{\theta (c_{1}x,\dots ,c_{n}x)}{|{\sum }_{i\in I}{B}_i|-{\sum }_{i\in I_0}\left|B_i\right|{\omega }_i},x\in X.$$

Moreover, G is the unique solution to (1) such that

$$\parallel g\left(x\right)-G\left(x\right)\parallel \le \beta \theta (c_{1}x,\dots ,c_{n}x),x\in X,$$

with some constant $\beta \in (0,\infty )$.

The stability of the homogeneous version of Equation (1) (i.e., with $B=0$) was first investigated by Forti [16]. The equation generalizes numerous functional equations that are well known. In particular, the special cases of it are the equations of Cauchy

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

Jensen

$$f\left(\frac{1}{2}(x+y)\right)=\frac{1}{2}\left(f\left(x\right)+f\left(y\right)\right),$$

Jordan–von Neumann

$$f(x+y)+f(x-y)=2f\left(x\right)+2f\left(y\right),$$

but also the equations of Drygas, Fréchet, and Popoviciu; the monomial and polynomial functions (see [17]); the p-Wright affine function; and various others (e.g., cubic, quartic, quintic etc.). For examples of stability results for the mentioned equations, we refer to [18,19,20,21,22,23,24,25,26,27,28,29]. Further stability outcomes concerning (1) can be found, e.g., in [30,31,32,33,34] (see also [35,36] for analogous investigations concerning some particular cases of (1)). For information on the solutions to some of these functional equations, we refer to [17,37].

In [38], the authors introduced the following linear functional equation:

$$\sum _{i=1}^m{A}_i\left(f\left({\phi }_i\left(\overline{x}\right)\right)\right)+b=0,\overline{x}:=(x_1,\cdots ,x_n)\in X^n,$$

(3)

where again m and n are positive integers; $f:X\to Y$ is a mapping from a linear space X into a Banach space Y; and, for every $i\in {\mathbb{N}}_m:=\{1,\cdots ,m\}$, ${\phi }_i$ is a linear mapping from $X^n$ into X and $A_i$ is a continuous endomorphism of Y and $b\in Y$. Using the classical Banach contraction theorem, they proved the stability of (3) in Banach spaces. Notice also that Equation (3) is a generalization of (1). The stability of another very general equation that could be considered a generalization of (1) was studied in [39].

Roughly speaking, the issue of Ulam stability can be formulated as follows: how much the approximate solutions of an equation differ from the exact solutions of this equation. The next definition explains more precisely how this notion could be understood in metric spaces (${\mathbb{R}}_{+}$ denotes the set of non-negative reals and $A^B$ means the family of all mappings from a set $B\ne Ø$ into a set $A\ne Ø$).

Definition 1.

Let $S\ne Ø$ and $U\ne Ø$ be nonempty sets, $(W,d)$ and $(V,\rho )$ be metric spaces, and $\mathcal{D}\subset W^U$ and $\mathcal{C}\subset {\mathbb{R}}_{+}^S$ be nonempty. Let $\mathcal{F},\mathcal{E}:\mathcal{D}\to V^S$ and $\mathcal{G}:\mathcal{C}\to {\mathbb{R}}_{+}^U$ be given. If for every $\psi \in \mathcal{D}$ and $\delta \in \mathcal{C}$ with

$$\rho \left(\left(\mathcal{F}\psi \right)\left(s\right),\left(\mathcal{E}\psi \right)\left(s\right)\right)\le \delta \left(s\right),s\in S,$$

there is $\varphi \in \mathcal{D}$ satisfying the equation

$$\mathcal{F}\psi =\mathcal{E}\psi$$

(4)

and such that

$$d\left(\varphi \left(t\right),\psi \left(t\right)\right)\le \left(\mathcal{G}\delta \right)\left(t\right),t\in U,$$

then we say that Equation (4) is $\mathcal{G}$-stable in the Ulam sense.

However, the notion of an approximate solution and difference between two functions can be defined in different ways (see, e.g., [26,40,41,42,43,44]), depending on the tools that we use to measure distances. One such tool is a modular.

The notion of a modular space was introduced by Nakano [45] and next redefined and generalized by Musielak and Orlicz [46,47]. In the last decade, several authors studied the Ulam stability of functional equations in modular spaces (see [44,48,49,50]). For instance, using a fixed-point method due to Khamsi [51], Sadeghi established, in [49], a stability result for a generalized Jensen functional equation in a convex modular space. Additionally, using the same technique, Wongkum et al. [52] proved a stability result for a quartic functional equation.

In the present paper, we use the direct method (analogous to [2]) to investigate the stability of the functional equation

$$\sum _{i=1}^m{A}_i(f\left({\phi }_i\left(\overline{x}\right))\right)=D\left(\overline{x}\right),\overline{x}:=(x_1,\cdots ,x_n)\in X^n,$$

(5)

for mappings f from a linear space X into a complete modular space $Y_{\rho }$, where n and m are positive integers, ${\phi }_1,\dots ,{\phi }_m$ are linear mappings from $X^n$ to X; $A_1,\dots ,A_m$ are continuous endomorphisms of Y; and the function $D:X^n\to Y_{\rho }$ is non-constant.

In particular, our results generalize some earlier stability outcomes for the modular spaces in [44,48,49,50,52].

2. Preliminaries

We first recall some basic notions and properties in modular spaces, as in [6,7,44,46,47].

Definition 2.

A functional$\rho :Y\to [0,+\infty ]$is called a modular if, for every$x,y\in Y$,

M1. 

$\rho \left(x\right)=0$if and only if$x=0$;

M2. 

$\rho \left(\alpha x\right)=\rho \left(x\right)$for every$\alpha \in \mathbb{K}$with$\left|\alpha \right|=1$;

M3. 

$\rho (\alpha x+\beta y)\le \rho \left(x\right)+\rho \left(y\right)$for every$\alpha ,\beta \in {\mathbb{R}}_{+}$with$\alpha +\beta =1$.

If we replace condition M3 with the following one:

M4. 

$\rho (\alpha x+\beta y)\le \alpha \rho \left(x\right)+\beta \rho \left(y\right)$for every$\alpha ,\beta \in {\mathbb{R}}_{+}$with$\alpha +\beta =1$,

then the modular ρ is called a convex modular.

If $\rho$ is a modular in Y, then the set

$$Y_{\rho }:=\{x\in Y:\underset{\lambda \to 0}{lim}\rho \left(\lambda x\right)=0\}$$

is called a modular space. Let us note that $Y_{\rho }$ is a linear subspace of Y.

Definition 3.

A modular ρ on Y is said to satisfy the ${\Delta }_2$-condition if there is $k>0$ such that $\rho \left(2y\right)\le k\rho \left(y\right)$ for every $y\in Y_{\rho }$.

It is easily seen that every norm is a convex modular that fulfills the ${\Delta }_2$-condition. If $\rho$ is a norm in Y, then clearly $Y_{\rho }=Y$, which means that our considerations also include the case where Y is a classical normed space.

Remark 1.

(a)

If $\rho$ is a modular on Y and $y\in Y$, then the function ${\mathbb{R}}_{+}\ni t\to \rho \left(ty\right)$ is non-decreasing, i.e., $\rho \left(ay\right)\le \rho \left(by\right)$ for every $a,b\in {\mathbb{R}}_{+}$ with $a<b$ (it is enough to take $y=0$ in M3).

(b) 

For a convex modular $\rho$ on Y, we have $\rho \left(\alpha y\right)\le \left|\alpha \right|\rho \left(y\right)$ for all $y\in Y$ and $\alpha \in \mathbb{K}$ with $\left|\alpha \right|\le 1$ and, moreover,

$$\rho \left(\sum _{j=1}^n{\alpha }_j{y}_j\right)\le \sum _{j=1}^n{\alpha }_j\rho \left(y_j\right)$$

(6)

for all $y_1,\dots ,y_n\in Y$ and ${\alpha }_1,\dots ,{\alpha }_n\in {\mathbb{R}}_{+}$ with ${\sum }_{j=1}^n{\alpha }_j\le 1$.

Definition 4. 

Let ρ be a modular on Y and${\left(y_n\right)}_n$be a sequence in Y. Then,

(i) 

${\left(y_n\right)}_n$is ρ-convergent to a point$y\in Y$(which we denote by$y=\rho -{lim}_n{y}_n$), if$\rho (y_n-y)\to 0$as$n\to +\infty$;

(ii) 

${\left(y_n\right)}_n$is ρ-Cauchy if for any$ϵ>0$, we have$\rho (y_n-y_m)<ϵ$for sufficiently large$m,n\in \mathbb{N}$;

(iii) 

$Y_{\rho }$is said to be ρ-complete if every ρ-Cauchy sequence in$Y_{\rho }$is ρ-convergent.

(iv) 

A subset$C\subset Y_{\rho }$is called ρ-closed if C contains every$x\in Y_{\rho }$such that there is a sequence${\left(x_n\right)}_n$in C which is ρ-convergent to x.

Notice that if ${\left(x_n\right)}_n$ is $\rho$-convergent to x, then ${\left(\alpha x_n\right)}_n$ is $\rho$-convergent to $\alpha x$, for $\alpha \in {\mathbb{R}}_{+}$, $\alpha \le 1$. This does not need to hold if $\left|\alpha \right|>1$, unless $\rho$ satisfies ${\Delta }_2$.

Definition 5.

A modular ρ on Y is said to be lower semi-continuous if every sequence ${\left(x_n\right)}_n$ in $Y_{\rho }$ that is ρ-convergent to some $x\in Y_{\rho }$, satisfies the inequality

$$\rho \left(x\right)\le \underset{n\to +\infty }{lim\; inf}\rho \left(x_n\right).$$

3. Stability of Equation (5)

In the sequel, X and Y are linear spaces over the same field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ and $\rho$ denotes a convex, lower semi-continuous modular on Y that satisfies the ${\Delta }_2$-condition, with a constant $k>0$. Moreover, we always assume that $Y_{\rho }$ is $\rho$-complete.

Let $m>1$ and n be positive integers, ${\phi }_i:X^n\to X$ for $i\in {\mathbb{N}}_m:=\{1,2,\cdots ,m\}$, and $A_1,\dots ,A_m$ be endomorphisms of $Y_{\rho }$ that commute (i.e., $A_i\circ A_j=A_j\circ A_i$ for $i,j\in {\mathbb{N}}_m$). Moreover, we assume that each $A_i$ is continuous with respect to the topology of the modular space $Y_{\rho }$ (as in [53]).

An arbitrary element $(x_1,\cdots ,x_n)$ of $X^n$ will be denoted by $\overline{x}$, and, for every non empty $I\subset {\mathbb{N}}_m$, we define $A_I:Y_{\rho }\to Y_{\rho }$ by $A_I\left(x\right):={\sum }_{i\in I}{A}_i\left(x\right)$ for $x\in Y_{\rho }$. If $I=\{1,2,\cdots ,m\}$, then we simply write A instead of $A_{\{1,2,\cdots ,m\}}$. Next, given $I\subset {\mathbb{N}}_m$, by $i\notin I$ we mean that $i\in {\mathbb{N}}_m\backslash I$.

Our main result concerns the stability of (5) in modular spaces.

Theorem 2.

Let $D:X^n\to Y_{\rho }$ and $\psi :X\to X^n$ be such that, for every $\overline{x}\in X^n$,

$$\sum _{i=1}^m{A}_i\left(D(\psi \circ {\phi }_i\left(\overline{x}\right))\right)=\sum _{i=1}^m{A}_i\left(D({\phi }_i\circ \psi \left(x_1\right),\cdots ,{\phi }_i\circ \psi \left(x_n\right))\right).$$

(7)

Suppose the following exist: $\theta :X^n\to {\mathbb{R}}_{+}$, a proper subset I of ${\mathbb{N}}_m$ and positive real numbers ${\omega }_i$ and ${\alpha }_i$, for $i\notin I$, such that $A_I$ possesses an eigenvalue $M\ne 0$ with eigenspace $Y_{\rho }^M$ and $f\left(X\right)\cup D\left(X^n\right)\subset Y_{\rho }^M$. Assume that

$$\begin{array}{}\mathrm{(8)}& \hfill {\phi }_j\circ \psi \left(x\right)& =& x,\hfill \mathrm{(9)}& \hfill \theta ({\phi }_i\circ \psi \left(x_1\right),\cdots ,{\phi }_i\circ \psi \left(x_n\right))& \le & {\omega }_i\theta \left(\overline{x}\right),\hfill \mathrm{(10)}& \hfill {\phi }_p({\phi }_i\circ \psi \left(x_1\right),\cdots ,{\phi }_i\circ \psi \left(x_n\right))& =& {\phi }_i\circ \psi \circ {\phi }_p\left(\overline{x}\right),\hfill \mathrm{(11)}& \hfill \theta \circ \psi \circ {\phi }_i\circ \psi \left(x\right)& \le & {\omega }_i\theta \circ \psi \left(x\right),\hfill \mathrm{(12)}& \hfill \gamma :=\frac{1}{\left|M\right|}\sum _{i\notin I}{\alpha }_i{\omega }_i& <& \min \left(1,\frac{2}{k}\right),\hfill \mathrm{(13)}& \hfill A_i\left(y\right)={\alpha }_{i}y,& and& \sum _{i\notin I}{\alpha }_i\le \left|M\right|\hfill \end{array}$$

for all $j\in I$, $i\notin I$, $p\in {\mathbb{N}}_m$, $\overline{x}\in X^n$, $x\in X$ and $y\in Y_{\rho }$.

If a function $f:X\to Y_{\rho }$ satisfies

$$\rho \left(\sum _{i=1}^m{A}_i\left(f\left({\phi }_i\left(\overline{x}\right)\right)\right)-D\left(\overline{x}\right)\right)\le \theta \left(\overline{x}\right),\overline{x}\in X^n,$$

(14)

then there is a unique solution$G:X\to Y_{\rho }^M$of (5) such that

$$\rho \left(MG\left(x\right)-Mf\left(x\right)\right)\le \frac{k\theta \left(\psi \right(x\left)\right)}{2-k\gamma },x\in X.$$

(15)

Proof. 

Taking $\overline{x}:=\psi \left(x\right)$ in (14), for each $x\in X$ we obtain

$$\rho \left(\sum _{i\in I}{A}_i\left(f\left({\phi }_i\left(\psi \left(x\right)\right)\right)\right)+\sum _{i\notin I}{A}_i\left(f\left({\phi }_i\left(\psi \left(x\right)\right)\right)\right)-D\left(\psi \left(x\right)\right)\right)\le \theta \left(\psi \left(x\right)\right),$$

whence, by (8), we come to

$$\rho \left(M\left[f\left(x\right)-\frac{-1}{M}\left(\sum _{i\notin I}{A}_i\left(f\left({\phi }_i\left(\psi \left(x\right)\right)\right)\right)-D\left(\psi \left(x\right)\right)\right)\right]\right)\le \theta \left(\psi \left(x\right)\right).$$

(16)

Let $\mathcal{M}$ denote the family of all $g:X\to Y_{\rho }^M$. The family $\mathcal{M}$ is nonempty since $f\in \mathcal{M}$. Now, for an arbitrary $g\in \mathcal{M}$, define the mapping $Tg:X\to Y_{\rho }$ by

$$Tg\left(t\right):=\frac{-1}{M}\left(\sum _{i\notin I}{A}_i\left(g\left({\phi }_i\left(\psi \left(t\right)\right)\right)\right)-D\left(\psi \left(t\right)\right)\right),t\in X.$$

Then,

$$\begin{array}{cc}\hfill A_I\left(Tg\left(t\right)\right)& =A_I\left(\frac{-1}{M}\left[\sum _{i\notin I}{A}_i\left(g({\phi }_i\circ \psi \left(t\right))\right)-D\left(\psi \left(t\right)\right)\right]\right)\hfill \\ \hfill & =\frac{-1}{M}\left(\sum _{i\notin I}{A}_i\circ A_I\left(g({\phi }_i\circ \psi \left(t\right))\right)+A_I\left(D\left(\psi \left(t\right)\right)\right)\right)\hfill \\ \hfill & =\frac{-1}{M}\left(\sum _{i\notin I}{A}_i\left(Mg({\phi }_i\circ \psi \left(t\right))\right)-MD(\psi \left(t\right)\right)\hfill \\ \hfill & =MTg\left(t\right),t\in X,g\in \mathcal{M}.\hfill \end{array}$$

This shows that, for every $t\in X$ and $g\in \mathcal{M}$, $Tg\left(t\right)\in Y_{\rho }^M$ and consequently $Tg\in \mathcal{M}$.

Now, we show that for every $n\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$,

$$\rho \left(M\left(T^{n+1}f-T^{n}f\right)\left(t\right)\right)\le {\gamma }^n\theta \circ \psi \left(t\right),t\in X.$$

(17)

The case $n=0$ coincides with (16). So assume that (17) holds for a non-negative integer n. Then, by (6), (11), the definitions of T and $\gamma$ (see (12)), and (13),

$$\begin{array}{cc}\hfill \rho \left(M\right(T^{n+2}f\left(t\right)& -T^{n+1}f\left(t\right)\left)\right)\hfill \\ \hfill & =\rho \left(\sum _{i\notin I}{A}_i\left(T^{n+1}f({\phi }_i\circ \psi \left(t\right))-T^{n}f({\phi }_i\circ \psi \left(t\right))\right)\right)\hfill \\ \hfill & \le \frac{1}{\left|M\right|}\sum _{i\notin I}{\alpha }_i\rho \left(M\left(T^{n+1}f({\phi }_i\circ \psi \left(t\right))-T^{n}f({\phi }_i\circ \psi \left(t\right))\right)\right)\hfill \\ \hfill & \le \frac{1}{\left|M\right|}\sum _{i\notin I}{\alpha }_i{\gamma }^n\theta \circ \psi \circ {\phi }_i\circ \psi \left(t\right)\hfill \\ \hfill & \le \frac{1}{\left|M\right|}\sum _{i\notin I}{\alpha }_i{\omega }_i{\gamma }^n\theta \circ \psi \left(t\right)\hfill \\ \hfill & \le {\gamma }^{n+1}\theta \circ \psi \left(t\right).\hfill \end{array}$$

Thus, we have shown that (17) holds for every $n\in {\mathbb{N}}_0$.

Next, by (6), (12), and the ${\Delta }_2$ property, for all fixed $m,n\in {\mathbb{N}}_0$, and $t\in X$, one can obtain

$$\begin{array}{cc}\hfill \rho \left(M\left(T^{n}f-T^{n+m}f\right)\left(t\right)\right)& =\rho \left(M\sum _{i=0}^{m-1}{\left(\frac{1}{2}\right)}^{i+1}{2}^{i+1}\left(T^{n+i}f-T^{n+i+1}f\right)\left(t\right)\right)\hfill \\ \hfill & \le \sum _{i=0}^{m-1}{\left(\frac{k}{2}\right)}^{i+1}\rho \left(M\left(T^{n+i}f-T^{n+i+1}f\right)\left(t\right)\right)\hfill \\ \hfill & \le \frac{k{\gamma }^n}{2}\sum _{i=0}^{m-1}{\left(\frac{k\gamma }{2}\right)}^i\theta \left(\psi \left(t\right)\right)\hfill \\ \hfill & \le k{\gamma }^n\frac{\theta \left(\psi \right(t\left)\right)}{2-k\gamma }.\hfill \end{array}$$

Since $\gamma <1$, we conclude that ${\left(M{T}^{n}f\left(t\right)\right)}_n$ is a $\rho$-Cauchy sequence in $Y_{\rho }$ for every $t\in X$. Since $Y_{\rho }$ is $\rho$-complete and $Y_{\rho }^M$ is $\rho$-closed, so ${\left(M{T}^{n}f\left(t\right)\right)}_n$ is $\rho$-convergent in $Y_{\rho }^M$. This allows us to define a function $G:X\to Y_{\rho }^M$ by

$$G\left(t\right):=\frac{1}{M}\rho -\underset{n\to +\infty }{lim}M{T}^{n}f\left(t\right),t\in X.$$

Since $\rho$ is lower semi-continuous, one has

$$\begin{array}{cc}\hfill \rho \left(MG\left(t\right)-Mf\left(t\right)\right)& \le \underset{n\to +\infty }{lim\; inf}\rho \left(M{T}^{m}f\left(t\right)-Mf\left(t\right)\right)\hfill \\ \hfill & \le \frac{k\theta \left(\psi \right(t\left)\right)}{2-k\gamma },t\in X,\hfill \end{array}$$

whereby we have (15).

Now, we prove that G satisfies Equation (5). First, we show by induction that, for every $k\in {\mathbb{N}}_0$,

$$\rho \left(\sum _{p=1}^m{A}_p\circ \left(T^{k}f\right)\circ {\phi }_p\left(\overline{x}\right)-D\left(\overline{x}\right)\right)\le {\gamma }^k\theta \left(\overline{x}\right),\overline{x}\in X^n.$$

(18)

The case $k=0$ is just (14). Next, assume that (18) holds for $k\in {\mathbb{N}}_0$. Putting

$$\overline{{\phi }_i\circ \psi (\overline{x}}):=\left({\phi }_i\circ \psi \left(x_1\right),\cdots ,{\phi }_i\circ \psi \left(x_n\right))\right),$$

by the assumption (7), for every $i\notin I$ and every $\overline{x}\in X^n$, we obtain

$$\begin{array}{cc}\hfill \sum _{p=1}^m& A_p\circ \left(T^{k+1}f\right)\circ {\phi }_p\left(\overline{x}\right)=\sum _{p=1}^m{A}_p\circ \left(T\left(T^{k}f\right)\right)\circ {\phi }_p\left(\overline{x}\right)\hfill \\ \hfill & =\frac{-1}{M}\sum _{i\notin I}{A}_i\left(\sum _{p=1}^m{A}_p\circ \left(T^{k}f\right)\circ {\phi }_p\left(\overline{{\phi }_i\circ \psi \left(\overline{x}\right)}\right)\right)\hfill \\ \hfill & +\frac{1}{M}\sum _{p=1}^m{A}_p\left(D\circ \psi \circ {\phi }_p\left(\overline{x}\right))\right)\hfill \\ \hfill & =\frac{-1}{M}\sum _{i\notin I}{A}_i\left(\sum _{p=1}^m{A}_p\left(\left(T^{k}f\right)({\phi }_p\left(\overline{{\phi }_i\circ \psi (\overline{x}}\right)\right)\right)\hfill \\ \hfill & +\frac{1}{M}\sum _{p=1}^m{A}_p\left(D({\phi }_p\circ \psi \left(x_1\right),\cdots ,{\phi }_p\circ \psi \left(x_n\right))\right)\hfill \\ \hfill & =\frac{-1}{M}\sum _{i\notin I}{A}_i\left(\sum _{p=1}^m{A}_p\left(\left(T^{k}f\right)({\phi }_p\left(\overline{{\phi }_i\circ \psi (\overline{x}}\right)\right)\right)\hfill \\ \hfill & +\frac{1}{M}\left[\sum _{i\notin I}{A}_i\left(D\left(\overline{{\phi }_i\circ \psi (\overline{x}}\right))\right)+\sum _{i\in I}{A}_i\left(D\left(\overline{x}\right)\right)\right]\hfill \\ \hfill & =\frac{-1}{M}\sum _{i\notin I}{A}_i\left(\sum _{p=1}^m{A}_p\left(\left(T^{k}f\right)\left({\phi }_p\right(\overline{{\phi }_i\circ \psi \left(\overline{x}\right)}\right)-D\left(\overline{{\phi }_i\circ \psi \left(\overline{x}\right)}\right)\right)+D\left(\overline{x}\right),\hfill \end{array}$$

whence

$$\begin{array}{cc}\hfill \rho & \left(\sum _{p=1}^m{A}_p\circ \left(T^{k+1}f\right)\circ {\phi }_p\left(\overline{x}\right)-D\left(\overline{x}\right)\right)\hfill \\ \hfill & =\rho \left(\frac{-1}{M}\sum _{i\notin I}{A}_i\circ \left[\sum _{p=1}^m{A}_p(\left(T^{k}f\right)\left({\phi }_p\left(\overline{{\phi }_i\circ \psi (\overline{x}}\right)\right)-D\left(\overline{{\phi }_i\circ \psi (\overline{x}}\right)\right]\right)\hfill \\ \hfill & \le \rho \left(\frac{-1}{M}\sum _{i\notin I}{\alpha }_i\left[\sum _{p=1}^m{A}_p(\left(T^{k}f\right)\left({\phi }_p\left(\overline{{\phi }_i\circ \psi (\overline{x}}\right)\right)-D\left(\overline{{\phi }_i\circ \psi (\overline{x}}\right)\right]\right)\hfill \\ \hfill & \le \frac{1}{\left|M\right|}\sum _{i\notin I}{\alpha }_i\rho \left(\sum _{p=1}^m{A}_p(\left(T^{k}f\right)\left({\phi }_p\left(\overline{{\phi }_i\circ \psi (\overline{x}}\right)\right)-D\left(\overline{{\phi }_i\circ \psi (\overline{x}}\right)\right)\hfill \\ \hfill & \le {\gamma }^k\frac{1}{\left|M\right|}\sum _{i\notin I}{\alpha }_i\theta \left(\overline{{\phi }_i\circ \psi (\overline{x}})\right)\hfill \\ \hfill & \le {\gamma }^k\frac{1}{\left|M\right|}\sum _{i\notin I}{\alpha }_i{\omega }_i\theta \left(\overline{x}\right)\hfill \\ \hfill & \le {\gamma }^{k+1}\theta \left(\overline{x}\right).\hfill \end{array}$$

This means that (18) holds for every $k\in {\mathbb{N}}_0$. Now, since the topology of the modular space $Y_{\rho }$ is a linear topology, for each $\overline{x}\in X^n$, we obtain

$$\sum _{i=1}^m{A}_i\circ G\circ {\phi }_i\left(\overline{x}\right)-D\left(\overline{x}\right)=\rho -\underset{k\to +\infty }{lim}\left(\sum _{i=1}^m{A}_i\left(\left(T^{k}f\right)\left({\phi }_i\left(\overline{x}\right)\right)\right)-D\left(\overline{x}\right)\right).$$

and consequently

$$\begin{array}{cc}\hfill \rho \left(\sum _{i=1}^m{A}_i\circ G\circ {\phi }_i\left(\overline{x}\right))-D\left(\overline{x}\right)\right)& \le \underset{k\to +\infty }{lim\; inf}\rho \left(\sum _{i=1}^m{A}_i\left(\left(T^{k}f\right)\left({\phi }_i\left(\overline{x}\right)\right)\right)-D\left(\overline{x}\right)\right)\hfill \\ \hfill & \le \underset{k\to +\infty }{lim\; inf}{\gamma }^k\theta \left(\overline{x}\right),\hfill \end{array}$$

because $\rho$ is lower semi-continuous. As $\gamma <1$, this implies that

$$\sum _{p=1}^m{A}_p\circ G\circ {\phi }_p\left(\overline{x}\right)=D\left(\overline{x}\right),\overline{x}\in X^n.$$

Finally, to show the uniqueness of G, assume that $G_1:X\to Y_{\rho }^M$ also is a solution of (5) that satisfies (15). First, we prove that G and $G_1$ are both fixed points of T. Since G is a solution of (5), we obtain

$$\sum _{i\in I}{A}_i\circ G\circ {\phi }_i\circ \psi \left(t\right)+\sum _{i\notin I}{A}_i\circ G\circ {\phi }_i\circ \psi \left(t\right)=D\circ \psi \left(t\right),t\in X.$$

Using (8), we obtain

$$\sum _{i\in I}{A}_i\circ G\left(t\right)+\sum _{i\notin I}{A}_i\circ G\circ {\phi }_i\left(\psi \left(t\right)\right)=D\left(\psi \left(t\right)\right),t\in X.$$

Moreover, $A_I\circ G\left(t\right)=MG\left(t\right)$ for every $t\in X$ and therefore $TG=G$. Using the same argument, we obtain $T{G}_1=G_1$.

Now, we prove by induction that, for every $n\in {\mathbb{N}}_0$,

$$\rho \left(M{T}^{n}G\left(t\right)-M{T}^n{G}_1\left(t\right)\right)\le \frac{{\gamma }^n{k}^2\theta \left(\psi \left(t\right)\right)}{2-k\gamma },t\in X.$$

(19)

We have

$$\begin{array}{cc}\hfill \rho \left(MG\left(t\right)-M{G}_1\left(t\right)\right)& \le \frac{1}{2}\rho \left(2M\left(G\right(t)-f(t\left)\right)\right)+\frac{1}{2}\rho \left(2M(G_1\left(t\right)-f\left(t\right))\right)\hfill \\ \hfill & \le \frac{k}{2}\rho (MG\left(t\right)-Mf\left(t\right))+\frac{k}{2}\rho (M{G}_1\left(t\right)-Mf\left(t\right))\hfill \\ \hfill & \le \frac{k^2\theta \left(\psi \left(t\right)\right)}{2-k\gamma }.\hfill \end{array}$$

Then, (19) holds for $n=0$. Next, if (19) holds for $n\in {\mathbb{N}}_0$, then

$$\begin{array}{cc}\hfill \rho \left(M{T}^{n+1}G\left(t\right)-M{T}^{n+1}{G}_1\left(t\right)\right)& \le \frac{1}{\left|M\right|}\sum _{i\notin I}{\alpha }_i\rho \left(\left(M{T}^{n}G\circ {\phi }_i-M{T}^n{G}_1\circ {\phi }_i\right)\left(\psi \left(t\right)\right)\right)\hfill \\ \hfill & \le \frac{1}{\left|M\right|}\sum _{i\notin I}{\alpha }_i{\gamma }^n\frac{k^2\theta (\psi \circ {\phi }_i\circ \psi \left(t\right))}{2-k\gamma }\hfill \\ \hfill & \le {\gamma }^{n+1}\frac{k^2\theta \left(\psi \left(t\right)\right)}{2-k\gamma }.\hfill \end{array}$$

Thus, by induction, we have shown (19). Therefore, for every $n\in {\mathbb{N}}_0$ and every $t\in X$, we have

$$\begin{array}{cc}\hfill \rho (MG\left(t\right)-M{G}_1\left(t\right))& =\rho (M{T}^{n}G\left(t\right)-M{T}^n{G}_1\left(t\right))\hfill \\ \hfill & \le {\gamma }^n\frac{k^2\theta \left(\psi \left(t\right)\right)}{2-k\gamma }.\hfill \end{array}$$

Letting n tend to $+\infty$, we obtain $G\left(t\right)=G_1\left(t\right)$ for every $t\in X$. This finishes the proof. □

Using Theorem 2, we can show the stability of various linear functional equations. For instance, we can prove the stability of the following Cauchy inhomogeneous functional equation:

$$f(x_1+x_2)-f\left(x_1\right)-f\left(x_2\right)=D(x_1,x_2).$$

(20)

Corollary 1.

Assume that $\parallel \parallel$ is a norm on X, $L\in {\mathbb{R}}_{+}$, and $p,q\in [-\infty ,s]$ with $s=$$min\left(1,2-\frac{ln\left(k\right)}{ln\left(2\right)}\right)$. Assume also that $D:X^2\to Y_{\rho }$ is a given symmetric and biadditive mapping, and $f:X\to Y_{\rho }$ satisfies

$$\rho \left(f(x_1+x_2)-f\left(x_1\right)-f\left(x_2\right)-D(x_1,x_2)\right)\le L\left(\parallel x_1{\parallel }^p+{\parallel x_2\parallel }^q\right),$$

for every $x_1,x_2\in X$. Then, there is a unique solution $G:X\to Y_{\rho }$ to the Cauchy inhomogeneous Equation (20) such that

$$\rho \left(2f\left(x\right)-2G\left(x\right)\right)\le \frac{kL(\parallel x{\parallel }^p+\parallel x{\parallel }^q)}{2-2^{r-1}k},x\in X,$$

(21)

with $r=max(p,q)$.

Proof. 

Here, we have $m=3$, $n=2$. Define $\psi :X\to X^2$ by $\psi \left(x\right)=(x,x)$, $x\in X$, and for every $i=1,2,3$, the linear mappings ${\phi }_i:X^2\to X$ and $A_i:Y_{\rho }\to Y_{\rho }$ by ${\phi }_1(x_1,x_2):=x_1+x_2$, ${\phi }_2(x_1,x_2):=x_1$, ${\phi }_3(x_1,x_2):=x_2$ and $A_1\left(y\right)=-A_2\left(y\right)=-A_3\left(y\right):=y$. Then, (see the next remark) condition (7) holds. Putting $I=\{2,3\}$, ${\omega }_1=2^r$ with $r=max(p,q)$, ${\alpha }_1=1$, $M=-2$, and $\theta (x_1,x_2)=L\left(\parallel x_1{\parallel }^p+{\parallel x_2\parallel }^q\right)$ for $x_1,x_2\in X$; from Theorem 2, we obtain that there is a unique solution $G:X\to Y_{\rho }$ to Equation (20), which satisfies (21), as desired. □

In a simplified situation when $X=\mathbb{R}$ and the modular is a norm, Corollary 1 has the following form.

Corollary 2.

Let $\parallel \parallel$ be a complete norm on Y, $\widehat{y}\in Y$, $L\in {\mathbb{R}}_{+}$ and $p,q\in [-\infty ,1]$. Moreover, let $f:\mathbb{R}\to Y$ be continuous at some point $x_0\in \mathbb{R}$ and satisfy

$$∥f(x_1+x_2)-f\left(x_1\right)-f\left(x_2\right)-x_1{x}_2\widehat{y}∥\le L\left(|x_1{|}^p+{\left|x_2\right|}^q\right),$$

for every $x_1,x_2\in \mathbb{R}$. Then, there is a unique vector $y_0\in Y$ with

$$∥f\left(x\right)-x{y}_0-\frac{x^2}{2}\widehat{y}∥\le \frac{L\left(\right|x{|}^p+\left|x{|}^q\right)}{2-2^r},x\in \mathbb{R},$$

(22)

where $r=max(p,q)$.

Proof. 

As we have already noticed just after Definition 3, every norm is a convex modular that satisfies the ${\Delta }_2$-condition with $k=2$. So, we can apply Corollary 1 (with $X=\mathbb{R}$ and $D(x_1,x_2)=x_1{x}_2\widehat{y}$ for $x_1,x_2\in \mathbb{R}$). According to it there is a unique solution $G:\mathbb{R}\to Y$ to the Cauchy inhomogeneous Equation (20) such that

$$∥2f\left(x\right)-2G\left(x\right)∥\le \frac{kL\left(\right|x{|}^p+\left|x{|}^q\right)}{2-2^{r-1}k},x\in \mathbb{R}.$$

(23)

Note that in this case G fulfils the equation

$$G(x_1+x_2)-G\left(x_1\right)-G\left(x_2\right)=x_1{x}_2\widehat{y},x_1,x_2\in \mathbb{R},$$

whence

$$G(x_1+x_2)-\frac{{(x_1+x_2)}^2}{2}\widehat{y}-\left(G\left(x_1\right)-\frac{x_1^2}{2}\widehat{y}\right)-\left(G\left(x_2\right)-\frac{x_2^2}{2}\widehat{y}\right)=0,x_1,x_2\in \mathbb{R}.$$

Hence, the function $G_0:\mathbb{R}\to Y$, given by

$$G_0\left(x\right)=G\left(x\right)-\frac{x^2}{2}\widehat{y},x\in \mathbb{R},$$

is additive. Next, (23) implies that G is bounded on a neigbourhood of $x_0$ and so is $G_0$, which means (see, e.g., [37]) that there is $y_0\in Y$ such that

$$G_0\left(x\right)=x{y}_0,x\in \mathbb{R}.$$

Now, it is easily seen that (23) yields (22). The uniqueness of G implies the uniqueness of $y_0$. □

Example 1.

If $Y_{\rho }$ is a commutative algebra, then the function $D:X^2\to Y_{\rho }$ given by $D(x,y)={\phi }_1\left(x\right){\phi }_1\left(y\right)$ for all $x,y\in X$, where ${\phi }_1:X\to Y_{\rho }$ is a linear mapping, is symmetric and biadditive.

The next remark provides some comments on condition (7).

Remark 2.

(1)

Every constant function $D:X^n\to Y$ satisfies condition (7).

(2)

If $D_1,D_2:X^n\to Y$ satisfy (7), then so does the function ${\alpha }_1{D}_1+{\alpha }_2{D}_2$ for any fixed scalars ${\alpha }_1$, ${\alpha }_2$.

(3)

Consider the situation in Corollary 1 (i.e., when Equation (5) has the form (20)). Then, condition (7) has the form

$$D(x_1+x_2,x_1+x_2)-D(x_1,x_1)-D(x_2,x_2)=D(2x_1,2x_2)-2D(x_1,x_2).$$

(24)

It is easy to check that, for every $h:X\to Y$, the function $D:X^2\to Y$, given by

$$D(x_1,x_2)=h(x_1+x_2)-h\left(x_1\right)-h\left(x_2\right),x_1,x_2\in X,$$

(25)

is a solution to Equation (24). In particular, if D is symmetric and biadditive, then (25) holds with $h\left(x\right)=\frac{1}{2}D(x,x)$ for $x\in X$. Thus, Equation (24) holds for every symmetric and biadditive function $D:X^2\to Y$.

4. Conclusions

We continue the investigation of the stability in the sense of Ulam of the non-homogeneous version of the very general linear functional Equation (5), which was introduced in [38] and generalizes numerous linear functional equations. Here, using the direct method, we show that this equation is stable in the context of complete modular spaces, whenever the modular is assumed to be convex and satisfies the ${\Delta }_2$-condition. The outcome of this study covers most of the known results in the same context.