Stability of functional inequality in digital metric space

Abstract

In the present article, the Hyers–Ulam stability of the following inequality is analyzed:

$$ \textstyle\begin{cases} d (f(\imath +\jmath ), \ (f(\imath )+ \ f(\jmath )) )\leq d (\rho _{1}((f(\imath +\jmath )+ f(\imath - \jmath ),\ 2f(\imath )) ) \\ \hphantom{ d (f(\imath +\jmath ), \ (f(\imath )+ \ f(\jmath )) )\leq}{}+ d (\rho _{2} (2f (\frac{\imath +\jmath}{2} ), \ (f(\imath )+ f(\jmath )) ) ) \end{cases} $$

(0.1)

in the setting of digital metric space, where $\rho _{1}$ and $\rho _{2}$ are fixed nonzero complex numbers with $1>\sqrt{2}|\rho _{1}|+|\rho _{2}|$ by using fixed point and direct approach.

1 Introduction and preliminaries

In the realm of mathematical analysis, Hyers–Ulam stability has emerged as a fundamental concept with far-reaching implications across various disciplines. The notion of stability, originally introduced by Donald H. Hyers [12] in 1941 and independently by Ulam [36] in the 1940s, has since garnered immense attention due to its profound implications in functional equations and related areas of mathematics.

Hyers–Ulam stability deals with the continuity of mappings that approximately satisfy functional equations. Specifically, it investigates the behavior of solutions to functional equations under small perturbations, elucidating the robustness of these solutions in a broader context. This stability concept has found applications in diverse fields such as physics, engineering, economics, and computer science, underscoring its interdisciplinary relevance.

The historical development of Hyers–Ulam stability reflects a journey of deep mathematical inquiry and discovery. From its inception to modern advancements, researchers have delved into refining stability criteria, exploring stability in various function spaces and extending stability concepts to dynamic systems and beyond.

The stability problem of functional equations originated from a question of Ulam [36] concerning the stability of group homomorphisms. The functional equation

$$ f(x+y)=f(x)+f(y) $$

(1.1)

is called Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [12] gave the first answer to the question of Ulam for Banach spaces. For additive mappings, Aoki [1] generalized the Hyerś theorem and Rassias for linear mappings. Many authors worked on different functional equations.

In 2003, Smajdor [35] examined the stability of the functional equation

$$ 3f \biggl(f \biggl(\frac{x+y+z}{3} \biggr) \biggr)+f(x)+f(y)+f(z)=2 \biggl[f \biggl( \frac{x+y}{2} \biggr)+f \biggl(\frac{y+z}{2} \biggr)+f \biggl( \frac{x+z}{2} \biggr) \biggr]. $$

They also gave some applications to a multivalued version of the above equation.

In 2014, Mortici et al. [22] investigated the Hyers–Ulam stability problems for the functional equation $F(x)-g(x)F(h(x))=0$, and in the same year, Mo Jung [21] proved Hyers–Ulam stability of the linear functional equation $f(\varphi (x)) = g(x).f(x)$ in a single variable on a complete metric group.

In 2015, Park [25] investigated the additive ρ-functional inequalities and proved the Hyers–Ulam stability of additive ρ-functional inequalities in complex Banach spaces.

Recently, in 2023, Nawaz et al. [23] analyzed the Hyers–Ulam stability of the following cubic ρ-functional inequalities and quartic ρ-functional inequalities:

$$ \textstyle\begin{cases} \Vert f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)- \rho (4f (x+ \frac{y}{2} ) \\ \quad {}+4 (f (x-\frac{y}{2} )-f(x+y)-f(x-y) ) \\ \quad {}-6f(x),r ) \Vert \geq \frac{r}{r+\varphi (x, y)} \end{cases} $$

and

$$ \textstyle\begin{cases} \Vert f(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y)- \rho (8f (x+\frac{y}{2} ) \\ \quad {}+8 (f (x-\frac{y}{2} )-2f(x+y)-2f(x-y) ) \\ \quad {}-12f(x)+3f(y),r ) \Vert \geq \frac{r}{r+\varphi (x, y)} \end{cases} $$

in the setting of fuzzy matrix.

The Ulam–Hyers stability idea is very important in realistic problems in numerical analysis, biology, and economics. The logistic equation (both differential and difference), wave solution of a reaction diffusion system, the SIS epidemic model, the Cournot model in economics, and the reaction diffusion equation are all generalized to nonlinear systems.

In the year 2022, Pachaiyappan et al. [24] proposed a new method in image security system: they used m-cubic and m-quartic functional equations for encryption and decryption.

So, we have many interesting results related to the same problem and its applications [2, 7, 10, 11, 13–15, 26, 28–31, 34].

The new emerging structure in functional analysis is a digital metric space. The general topology and functional analysis form the foundation of this space. In an n-dimensional digital space, the characteristics of digital topology are described. Rosenfield [32] used the digital topology for researching digital images. Kong [20] described a digital fundamental group of discrete objects. For studying digital continuous function, Boxer [4] defined digital concepts in terms of digital versions. The digital characteristics of 2D and 3D digital images can be studied by utilizing digital topology. Both computer graphics and image processing can use the digital topology.

The Banach fixed point theorem is the foundation of fixed point theory in metric spaces, which offers useful techniques for locating fixed points and is a crucial tool to solve some issues in the fields of engineering and mathematics. As a result, it has been broadly generalized. Until now, this area has seen a number of developments [3, 5, 6, 8, 16]. Sessa [33] introduced the idea of weak compatibility, and Jungck [17, 18] introduced the ideas of compatibility and weak compatibility.

Definition 1.1

[19] Suppose that n is a nonnegative integer and $\mathbb{Z}$ is the set of all integers. Define the set $\mathbb{Z}^{n}$ by

$\mathbb{Z}^{n}$ is known as the set of all lattice points in the Euclidean space having n-dimension.

Definition 1.2

[19] Choose two different points $\wp = (\wp _{1}, \wp _{2},\dots , \wp _{n})$ and $z = (z_{1}, z_{2}, \dots , z_{n}) \in \mathbb{Z}^{n}$. Suppose that r is a nonnegative integer such that $1 \leq r \leq n$. One can make two points. ℘ and z are $k_{r}$-adjacent in $\mathbb{Z}^{n}$ if there are at most r indices τ such that $|\wp _{\tau}- z_{\tau}| = 1$, and for all other indices o such that $|p_{\tau} - z_{\tau}| \neq 1$, one has $p_{o} = z_{o}$.

Definition 1.3

[8] For some positive integer n, choose a digital image to be graph $(X, \kappa )$, here κ is the adjacency relation on X. A triple $(X, d, \kappa )$ is a digital metric space; here, d is the metric for X and $(X, \kappa )$ is the digital image.

Example 1.4

[9] $\mathrm{MSS}^{\prime }_{18} = \{c_{0} = (1, 1, 0), c_{1} = (0, 2, 0), c_{2} = (-1, 1, 0), c_{3} = (0, 0, 0), c_{4} = (0, 1, -1), c_{5} = (0, 1, 1)\} \subset Z^{3}$ is a minimal simple closed 18-surface (see Fig. 1).

Figure 1

$\mathrm{MSS}^{\prime }_{18}$

Let $(X, \kappa )$ be a digital image and its subset be $(A, \kappa )$. $(X, A)$ is called digital image pair with κ-adjacency, and when A is a singleton set $\{x_{0}\}$, $(X, x_{0})$ is called pointed digital image.

Definition 1.5

[5] Any sequence $\{x_{r}\}$ of points of the digital metric space $(X, d, \kappa )$ is a Cauchy sequence if, for all $\epsilon > 0$, there is $\zeta \in \mathbb{N}$ such that, for all $r, v > \zeta$,

$$ d(x_{r}, x_{v})< \epsilon . $$

Definition 1.6

[3] Any sequence $\{x_{r}\}$ of points of $(X, d, \kappa )$ (digital metric space) tends to limit $\rho \in D$ if for all $\epsilon > 0$ there exists $\zeta \in \mathbb{N}$ such that, for all $r > \zeta$,

$$ d(x_{r}, \rho )< \epsilon . $$

Definition 1.7

[4] $(X, d, \kappa )$ (digital metric space) is complete if any Cauchy sequence $\{x_{r}\}$ of $(X, d, \kappa )$ converges to the point of $(X, d, \kappa )$.

Definition 1.8

[5] Suppose that $(X, \kappa )$ is any digital image. $f:(X, \kappa ) \longrightarrow (X, \kappa )$ is known as right continuous if

Here, $\rho \in X$.

Definition 1.9

[3] Assume that $(X, d, \kappa )$ is a digital metric space and $f:(X, d, \kappa )\longrightarrow (X, d, \kappa )$ is a digital map. If there is $\zeta \in (0, 1)$ such that

$$ d \bigl(f(\imath ), f(\jmath ) \bigr) \leq \zeta d(\imath , \jmath )\quad \text{for all }\imath , \jmath \in X. $$

(1.2)

Then f is a digital contraction map.

Theorem 1.10

[4] Suppose that $(X, d, \kappa )$ is a complete digital metric space with Euclidean metric in $\mathbb{Z}^{n}$. Assume that $f:(X, d, \kappa )\longrightarrow (X, d, \kappa )$ is a digital contraction map. Then f has a unique fixed point.

Theorem 1.11

Suppose that $(X, d, \kappa )$ is a complete digital metric space and strictly contractive mapping $U: (X, d, \kappa ) \longrightarrow (X, d, \kappa )$ with the Lipschitz constant $L<1$. Then, for each given element $\imath \in X$, there exists a natural number $n_{0}$ such that

• (1) $d(U^{n} \imath , U^{n+1}\imath ) < +\infty$ for all $n \ge n_{0}$;

• (2) The sequence $\{U^{n} \imath \}$ is convergent to a fixed point $\jmath ^{*}$ of U;

• (3) $\jmath ^{*}$ is the unique fixed point of $U \in \textit{G}= \{\jmath \in X | d(U^{n_{0}} \imath , \jmath ) < +\infty \}$;

• (4) $d(\jmath , \jmath ^{*}) \le \frac{1}{1-L} g(\jmath , U\jmath )$ for all $\jmath \in \textit{G}$.

Hyers–Ulam stability investigates the following question: Suppose that one has a function $y(t)$ that is close to solving an equation; is there an exact solution $x(t)$ of the equation that is close to $y(t)$? The following system can be studied mathematically [12]:

$$ \frac{d\imath}{dt}=f(\imath ). $$

(1.3)

It will be Ulam–Hyers stable if (1.3) possesses an exact solution, and if all $\epsilon > 0$, there is $\delta >0$ such that if $\imath _{x}(t)$ is an assumption for the solution of (1.3), then there is $\imath (t)$ that is the exact solution of (1.3), which is close to $\beta _{x}$, i.e.,

$$ \biggl\Vert \frac{d\imath _{0}}{dt}-f\imath _{x}(t) \biggr\Vert < \delta \quad \Rightarrow\quad \bigl\Vert \imath (t)-\imath _{0}(t) \bigr\Vert < \epsilon $$

(1.4)

for all $t >0$.

This definition is relevant as if we are analyzing a Hyers–Ulam stable system, this does not mean we have to reach an exact solution (which is either time-consuming or difficult). All is to get a function that satisfies (1.4). Hyers–Ulam stability guarantees the existence of a close, exact solution.

This paper is organized as follows: By using the fixed point approach, we solve (0.1) and prove the Hyers–Ulam stability of (0.1) in a digital metric space in Sect. 2 and in Sect. 3 by direct approach. In Sect. 4, we provide an example, and Sect. 5 contains applications related to generalized Hyers–Ulam stability. This article ends with Sect. 6, which is conclusion.

2 Solution of functional inequality: a fixed point approach (0.1)

Lemma 2.1

Assume $f: (X, d, \kappa )\longrightarrow (G, d, \kappa )$ to be a mapping satisfying $f(0)=0$ also

$$ \textstyle\begin{cases} d (f(\imath +\jmath ), (f( \imath )+ f(\jmath ) ) ) \\ \quad \leq d (\rho _{1} ( (f( \imath +\jmath )+ f(\imath - \jmath ), 2f(\imath ) ) ) ) \\ \qquad {} +d (\rho _{2} (2f (\frac{\imath +\jmath}{2} ), (f(\imath )+ f(\jmath ) ) ) )+\varphi (\imath , \jmath ) \end{cases} $$

(2.1)

for all $\imath , \jmath \in (X, d, \kappa )$. Then f is additive.

Proof

Assume that f satisfies (2.1).

Substituting $\imath =0=\jmath$ in inequality (2.1), one obtains

$$ g \bigl(f(0) \bigr)\leq 0, $$

which gives $f(0)=0$.

Again, by substituting $\imath =\jmath$ in the same inequality, one has

$$ d \bigl(f(2\imath ), 2f(\imath ) \bigr)\leq \vert \rho _{1} \vert .d \bigl(f(2\imath ), 2f( \imath ) \bigr), $$

and thus $f(2\imath )=2f(\imath )$ for all $\imath \in (X, d, \kappa )$ since $|\rho _{1}|<1$. Therefore,

$$ f \biggl(\frac{\imath}{2} \biggr)=\frac{1}{2}f(\imath ) $$

(2.2)

for all $\imath \in (X, d, \kappa )$.

(2.1) and (2.2) give

$$ \textstyle\begin{cases} d (f(\imath +\jmath ), ( f(\imath )+ f( \jmath ) ) )\leq d (\rho _{1} ( (f(\imath +\jmath )+ f(\imath - \jmath ) ), 2 f(\imath ) ) \\ \hphantom{d (f(\imath +\jmath ), ( f(\imath )+ f( \jmath ) ) )\leq}{} +d (\rho _{2} (2f (\frac{\imath +\jmath}{2} ), (f(\imath )+ f(\jmath ) ) ) ) \\ \hphantom{d (f(\imath +\jmath ), ( f(\imath )+ f( \jmath ) ) )} = \vert \rho _{1} \vert d ( (f(\imath +\jmath )+ f( \imath -\jmath ) ), 2f( \imath ) ), \\ \hphantom{d (f(\imath +\jmath ), ( f(\imath )+ f( \jmath ) ) )\leq}{} + \vert \rho _{2} \vert d (f(\imath +\jmath ), (f( \imath )- f(\jmath ) ) ). \end{cases} $$

Hence

$$ (1-|\rho _{2})d \bigl(f(\imath +\jmath ), \bigl(f( \imath )+ f(\jmath ) \bigr) \bigr)\leq \vert \rho _{1} \vert d ( \bigl(f(\imath +\jmath )+ \bigl(f(\imath - \jmath ) \bigr), 2f(\imath ) \bigr) $$

(2.3)

for all $\imath , \jmath \in (X, d, \kappa )$.

Assuming $\imath +\jmath =\gamma$ and $\imath -\jmath =\omega$ in (2.3), one obtains

$$\begin{aligned}& \bigl(1- \vert \rho _{2} \vert \bigr)d \biggl( \biggl(f(\gamma )- f \biggl(\frac{\gamma +\omega}{2} \biggr) \biggr), f \biggl(\frac{\gamma -\omega}{2} \biggr) \biggr) \\& \quad \leq \vert \rho _{1} \vert d \biggl( \bigl(f(\gamma )+ f( \omega ) \bigr), 2f \biggl( \frac{\gamma +\omega}{2} \biggr) \biggr). \end{aligned}$$

Therefore,

$$\begin{aligned}& \frac{1}{2} \bigl(1- \vert \rho _{2} \vert \bigr)d \bigl( \bigl(f(\gamma +\omega )+ f(\omega - \gamma ) \bigr), 2f(\gamma ) \bigr) \\& \quad \leq \vert \rho _{1} \vert d \bigl(f(\gamma + \omega ), \bigl(f(\omega )+ f(\gamma ) \bigr) \bigr) \big) \end{aligned}$$

(2.4)

for all $\gamma , \omega \in (X, d, \kappa )$.

From (2.3) and (2.4), one has

$$ \bigl(1- \vert \rho _{2} \vert \bigr)^{2}d \bigl(f( \imath +\jmath ), \bigl(f(\imath )+ f( \jmath ) \bigr) \bigr)\leq \bigl( \vert \rho _{1} \vert \bigr)^{2} d \bigl(f(\imath +\jmath ), \bigl(f(\imath )+f(\jmath ) \bigr) \bigr) \big) $$

for all $\imath , \jmath \in (X, d, \kappa )$.

Hence, $\sqrt{2}|\rho _{1}|+|\rho _{2}|<1$, $f(\imath +\jmath )=f(\imath )+f(\jmath )$ for all $\imath , \jmath \in (X, d, \kappa )$. Therefore, f is additive. □

Theorem 2.2

Assume $\varphi :X^{2}\rightarrow [0, \infty )$ such that there is $\zeta < 1$ with

$$ \varphi \biggl(\frac{\imath}{2}, \frac{\jmath}{2} \biggr)\leq \frac{\zeta}{2}\varphi (\imath , \jmath ) $$

(2.5)

for all $\imath , \jmath \in (X, d, \kappa )$. Suppose that $f:(X, d, \kappa )\longrightarrow (G, d, \kappa )$ is a mapping satisfying $f(0)=0$ and

$$ \textstyle\begin{cases} d (f(\imath +\jmath ), (f( \imath )+ f(\jmath ) ) )\leq d (\rho _{1} ( (f( \imath +\jmath )+ f(\imath - \jmath ), 2f(\imath ) ) ) ) \\ \hphantom{d (f(\imath +\jmath ), (f( \imath )+ f(\jmath ) ) )\leq}{} +d (\rho _{2} (2f (\frac{\imath +\jmath}{2} ), (f(\imath )+ f(\jmath ) ) ) )+\varphi (\imath , \jmath ) \end{cases} $$

(2.6)

for all $\imath , \jmath \in (X, d, \kappa )$. Then there exists an additive mapping $\mathbb{A} : (X, d, \kappa )\longrightarrow (G, g, \kappa )$ such that

$$ d \bigl(f(\imath ), \mathbb{A}(\imath ) \bigr)\leq \frac{\zeta}{2(1-\zeta )(1- \vert \rho _{1} \vert )}\varphi ( \imath , \imath ) $$

for all $\imath \in (X, d, \kappa )$.

Proof

Substituting $\imath = \jmath$ in (2.6), one obtains

$$ \bigl(1- \vert \rho _{1} \vert \bigr) d \bigl(f(2 \imath ), 2f(\imath ) \bigr)\leq \varphi (\imath , \imath ) $$

(2.7)

for all $\imath \in (X, d, \kappa )$.

From the above equation, one has

$$ d \biggl(f(\imath ), 2f \biggl(\frac{\imath}{2} \biggr) \biggr)\leq \frac{1}{(1- \vert \rho _{1} \vert )} \varphi \biggl(\frac{\imath}{2}, \frac{\imath}{2} \biggr) \leq \frac{\zeta}{2(1- \vert \rho _{1} \vert )}\varphi ( \imath , \imath ). $$

Therefore $d(f, Uf)\leq \frac{\zeta}{2(1-|\rho _{1}|)}$.

From Theorem 1.11, one has $\mathbb{A} : (X, d, \kappa )\longrightarrow (G, d, \kappa )$, which satisfies the following:

(1) $\mathbb{A}$ is a fixed point of U, i.e.,

$$ \frac{1}{2}\mathbb{A}(\imath )= \mathbb{A} \biggl( \frac{\imath}{2} \biggr) $$

(2.8)

for all $\imath \in (X, d, \kappa )$. Mapping $\mathbb{A}$ is a unique fixed point of U in

$$ W= \bigl\{ j \in \mathbb{A} : d(f, j) < \infty \bigr\} , $$

which gives $\mathbb{A}$ is unique, which satisfies (2.8) such that there is $\nu \in (0, +\infty )$ that satisfies

$$ d \bigl(f(\imath ), \mathbb{A}(\imath ) \bigr)\leq \nu \varphi ( \imath , \imath ) $$

for all $\imath \in (X, d, \kappa )$;

(2) $d(U^{s} f, \mathbb{A}) \longrightarrow 0$ as $s \rightarrow +\infty$. That gives the equality

$$ \lim_{s\to +\infty} 2^{s} f \biggl(\frac{\imath}{2^{s}} \biggr) = \mathbb{A}(\imath ) $$

for all $\imath \in (X, d, \kappa )$;

(3) $d(f, \mathbb{A}) \leq \frac{1}{1-\zeta} d(f, Uf)$, which gives the inequality

$$ d(f, \mathbb{A}) \leq \frac{\zeta}{(2-2\zeta )(1- \vert \rho _{1} \vert )} \varphi (\imath , \imath ) $$

for all $\imath \in (X, d, \kappa )$.

From (2.5) and (2.6), one has

$$ \textstyle\begin{cases} d (\mathbb{A}(\imath +\jmath ), \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) )= \lim_{s\to \infty} 2^{s} d (f (\frac{\imath +\jmath}{2^{s}} ), (f ( \frac{\imath}{2^{s}} )+ f (\frac{\jmath}{2^{s}} ) ) ) \\ \hphantom{d (\mathbb{A}(\imath +\jmath ), \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) )} \leq \lim_{s\to +\infty} 2^{s} \vert \rho _{1} \vert d ( (f ( \frac{\imath +\jmath}{2^{s}} ) +f ( \frac{\imath -\jmath}{2^{s}} ) ), 2f ( \frac{\imath}{2^{s}} ) ) \\ \hphantom{d (\mathbb{A}(\imath +\jmath ), \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) )=}{} + \lim_{s\to +\infty} 2^{s} \vert \rho _{2} \vert d (2f ( \frac{\imath +\jmath}{2^{s+1}} ), (f ( \frac{\imath}{2^{s}} )+ f (\frac{\jmath}{2^{s}} ) ) ) \\ \hphantom{d (\mathbb{A}(\imath +\jmath ), \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) )=}{} + \lim_{s \to +\infty} 2^{s}\varphi ( \frac{\imath}{2^{s}}, \frac{\jmath}{2^{s}} ) \\ \hphantom{d (\mathbb{A}(\imath +\jmath ), \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) )} =d ( \vert \rho _{1} \vert (\mathbb{A}(\imath +\jmath )+ \mathbb{A}( \imath -\jmath ), 2\mathbb{A}(\imath ) ) ) \\ \hphantom{d (\mathbb{A}(\imath +\jmath ), \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) )=}{}+d ( \vert \rho _{2} \vert (2\mathbb{A} ( \frac{\imath +\jmath}{2} ), (\mathbb{A} (\imath )+ \mathbb{A} (\jmath ) ) ) ). \end{cases} $$

Therefore,

$$ \textstyle\begin{cases} d (\mathbb{A}(\imath +\jmath ), ( \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) ) )\leq d (\rho _{1} ( \mathbb{A}(\imath +\jmath )+ \mathbb{A}(\imath -\jmath ), 2 \mathbb{A}(\imath ) ) ) \\ \hphantom{d (\mathbb{A}(\imath +\jmath ), ( \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) ) )\leq}{} +d (\rho _{2} (2\mathbb{A} (\frac{\imath +\jmath}{2} ), (\mathbb{A}(\imath )+ \mathbb{A}(\jmath ) ) ) ) \end{cases} $$

for all $\imath , \jmath \in (X, d, \kappa )$. From (2.1), $\mathbb{A} : (X, d, \kappa )\longrightarrow (G, d, \kappa )$ is additive. □

Corollary 2.3

Suppose that $z > 1$, ϑ is a positive real number, and $f: (X, d, \kappa )\longrightarrow (G, d, \kappa )$, which satisfies $f(0)=0$ and

$$ \textstyle\begin{cases} d (f(\imath +\jmath ), (f( \imath )+f(\jmath ) ) ) \\ \quad \leq d (\rho _{1} ( (f( \imath +\jmath )+ f(\imath - \jmath ) ), 2f(\imath ) ) ) \\ \qquad {} + (\rho _{2} (2f (\frac{\imath +\jmath}{2} ), (f(\imath )+ f(\jmath ) ) ) )+\vartheta ( \Vert \imath \Vert ^{z}, \Vert \jmath \Vert ^{z} ) \end{cases} $$

(2.9)

for all $\imath , \jmath \in (X, d, \kappa )$. Then there is an additive mapping $\mathbb{A} : (X, d, \kappa )\longrightarrow (G, d, \kappa )$ such that

$$ d \bigl(f(\imath ), \mathbb{A}(\imath ) \bigr)\leq \frac{2\vartheta}{(2^{z}-2)(1- \vert \rho _{1} \vert )} \Vert \imath \Vert ^{z} $$

for all $\imath \in (X, d, \kappa )$.

Proof

For proof, follow the steps of (2.2) and choose $\phi (\imath , \jmath )=\vartheta (\|\imath \|^{z}, \|\jmath \|^{z})$ for all $\imath , \jmath \in (X, d, \kappa )$. Then one can assume $\zeta =2^{1-z}$ to obtain the required result. □

Theorem 2.4

Suppose $\varphi :X^{2}\rightarrow [0, \infty )$ such that there is $\zeta < 1$ having

$$ \varphi (\imath , \jmath )\leq 2\zeta \varphi \biggl( \frac{\imath}{2}, \frac{y}{2} \biggr) $$

for all $\imath , \jmath \in (X, d, \kappa )$. Let $f: (X, d, \kappa )\longrightarrow (G, d, \kappa )$, which satisfies $f(0)=0$ and (2.6). Then $\mathbb{A}: (X, d, \kappa )\longrightarrow (G, d, \kappa )$ exists there, which is a unique additive mapping such that

$$ d \bigl(f(\imath ), \mathbb{A}(\imath ) \bigr)\leq \frac{1}{(2-2\zeta )(1- \vert \rho _{1} \vert )}\varphi ( \imath , \imath ) $$

for all $\imath \in (X, d, \kappa )$.

Proof

Following from the proof of Theorem 2.2, suppose that $(X, d, \kappa )$ is a digital metric space and $U: (X, d, \kappa ) \longrightarrow (X, d, \kappa )$ is linear such that

$$ Uj(\imath )=\frac{1}{2}j(2\imath ) $$

for all $\imath \in (X, d, \kappa )$.

From (2.7), one gets

$$ d \biggl(f(\imath ), \frac{1}{2}f(2\imath ) \biggr)\leq \frac{1}{2(1- \vert \rho _{1} \vert )}\varphi (\imath , \imath ) $$

for all $\imath \in (X, d, \kappa )$.

For the rest of the proof, follow the steps in (2.2). □

Corollary 2.5

Suppose that $z < 1$, ϑ is a positive real number, and $f: (X, d, \kappa )\longrightarrow (G, d, \kappa )$, which satisfies $f(0)=0$, also (2.9). Then there exists an additive mapping $\mathbb{A}:(X, d, \kappa )\longrightarrow (G, d, \kappa )$ that satisfies

$$ d \bigl(f(\imath ), \mathbb{A}(\imath ) \bigr) \leq \frac{2\vartheta}{(2-2^{z})(1- \vert \rho _{1} \vert )} \Vert \imath \Vert ^{z} $$

for all $\imath \in (X, d, \kappa )$.

Proof

For proof, follow the steps of (2.4) and choose $\phi (\imath , \jmath )=\vartheta (\|\imath \|^{z}, \|\jmath \|^{z})$ for all $\imath , \jmath \in (X, d, \kappa )$. Then, by assuming $\zeta =2^{z-1}$, one can obtain the required result. □

3 Solution of functional inequality by direct method (0.1)

Theorem 3.1

Assume $\varphi :X^{2}\rightarrow [0, \infty )$ such that

$$ \Omega (\imath , \jmath )=\sum^{\infty}_{s=1}2^{s} \varphi \biggl( \frac{\imath}{2^{s}, \frac{\jmath}{2^{s}}} \biggr)< +\infty $$

(3.1)

for all $\imath , \jmath \in (X, d, \kappa )$. Suppose that $f:(X, d, \kappa )\longrightarrow (G, d, \kappa )$ satisfies $f(0)=0$ and also (2.6). Then there exists a unique additive mapping $\mathbb{A}\mathbbm{ }: (X, d, \kappa )\longrightarrow (G, d, \kappa )$ such that

$$ d \bigl(f(\imath ), A(\imath ) \bigr)\leq \frac{1}{2(1- \vert \rho _{1} \vert )} \Omega (\imath , \imath ) $$

(3.2)

for all $\imath \in (X, d, \kappa )$.

Proof

Substituting $\imath =\jmath$ in (2.6), one obtains

$$ \bigl(1- \vert \rho _{1} \vert \bigr)d \bigl(f(2 \imath ), 2f(\imath ) \bigr)\leq \varphi (\imath , \imath ) $$

(3.3)

for all $\imath \in (X, d, \kappa )$. Therefore,

$$ d \biggl(f(\imath ), 2f \biggl(\frac{\imath}{2} \biggr) \biggr)\leq \frac{1}{(1- \vert \rho _{1} \vert )}\varphi \biggl(\frac{\imath}{2}, \frac{\imath}{2} \biggr) $$

for all $\imath \in (X, d, \kappa )$. Thus,

$$ \textstyle\begin{cases} d (2^{i}f ( \frac{\imath}{2^{i}} ), 2^{ \upsilon}f (\frac{\imath}{2^{\upsilon}} ) )\leq \sum^{\upsilon -1}_{s=i}2^{s}d (2^{s}f ( \frac{\imath}{2^{s}} )-2^{s+1}f ( \frac{\imath}{ 2^{\upsilon +1}} ) ) \\ \hphantom{d (2^{i}f ( \frac{\imath}{2^{i}} ), 2^{ \upsilon}f (\frac{\imath}{2^{\upsilon}} ) )} \leq \sum^{\upsilon -1}_{s=i}\frac{2^{s}}{2((1- \vert \rho _{1} \vert )} \varphi (\frac{\imath}{2^{s+1}}, \frac{\imath}{2^{s+1}} ) \end{cases} $$

(3.4)

for all nonnegative integers υ and i having $\upsilon > i$ and all $\beta \in (X, d, \kappa )$. From (3.4) one gets that the sequence $2^{s}f (\frac{\imath}{2^{s}} )$ is Cauchy for all $\beta \in (X, d, \kappa )$. Since $(G, d, \kappa )$ is complete, the sequence $2^{s}f (\frac{\imath}{2^{s}} )$ converges. Therefore, one can define $\mathbb{A} : (X, d, \kappa ) \longrightarrow (G, d, \kappa )$ by

$$ \mathbb{A}(\imath )=\lim_{s\to +\infty} 2^{s}f \biggl( \frac{\imath}{2^{s}} \biggr) $$

for all $\imath \in (X, d, \kappa )$. Moreover, substituting $i=0$ and letting $\lim_{\upsilon \to +\infty}$ in (3.4), one obtains (3.2).

From (2.6) and (3.5), one has

$$ \textstyle\begin{cases} d (\mathbb{A}(\imath +\jmath ), (A(\imath )+ A( \jmath ) ) ) \\ \quad \leq \lim_{s\to +\infty} 2^{s} d (f ( \frac{\imath +\jmath}{2^{s}} ), (f (\frac{\imath}{2^{s}} )+ f (\frac{\jmath}{2^{s}} ) ) ) \\ \quad \leq \lim_{s\to +\infty} 2^{s} \vert \rho _{1} \vert ( (f ( \frac{\imath +\jmath}{2^{s}} )+ f ( \frac{\imath -\jmath}{2^{s}} ) ) , 2f ( \frac{\imath}{2^{s}} ) ) \\ \qquad {} + \lim_{s\to +\infty} 2^{s} \vert \rho _{2} \vert (2f ( \frac{\imath +\jmath}{2^{s+1}} ), (f ( \frac{\imath}{2^{s}} )+ f (\frac{\jmath}{2^{s}} ) ) ) \\ \qquad {} + \lim_{s\to +\infty} 2^{s}\varphi ( \frac{\imath}{2^{s}}, \frac{\jmath}{2^{s}} ) \\ \quad =d (\rho _{1} ( (\mathbb{A}(\imath +\jmath )+ \mathbb{A}( \imath -\jmath ) ), 2\mathbb{A}(\imath ) ) ) \\ \qquad {}+d (\rho _{2} (2\mathbb{A} (\frac{\imath +\jmath}{2} ), (\mathbb{A}(\imath )+ \mathbb{A}(\jmath ) ) ) ) \end{cases} $$

for all $\imath , \jmath \in (X, d, \kappa )$. Thus,

$$ \textstyle\begin{cases} d (\mathbb{A}(\imath +\jmath ), ( \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) ) )\leq d (\rho _{1} ( \mathbb{A}(\imath +\jmath )+ \mathbb{A}(\imath -\jmath ), 2 \mathbb{A}(\imath ) ) ), \\ \hphantom{d (\mathbb{A}(\imath +\jmath ), ( \mathbb{A}( \imath )+ \mathbb{A}(\jmath ) ) )\leq}{}+d (\rho _{2} (2\mathbb{A} (\frac{\imath +\jmath}{2} ), ( \mathbb{A}(\imath )+ \mathbb{A}(\jmath ) ) ) ) \end{cases} $$

for all $\imath , \alpha \jmath \in (X, d, \kappa )$. From Lemma 2.1, $\mathbb{A}$ is additive.

Now, let another additive mapping be $\mathbb{G} : (X, d, \kappa ) \longrightarrow (X, d, \kappa )$ satisfying (3.2). Then one gets

$$ \textstyle\begin{cases} d (\mathbb{A}(\imath ), \mathbb{G}(\imath ) )= d ( 2^{s}\mathbb{A} (\frac{\imath}{2^{s}} ), 2^{s}\mathbb{G} (\frac{\imath}{2^{s}} ) ) \\ \hphantom{d (\mathbb{A}(\imath ), \mathbb{G}(\imath ) )} \leq d ( 2^{s}\mathbb{A}\mathbbm{ } (\frac{\imath}{2^{s}} ), 2^{s}f (\frac{\imath}{2^{s}} ) )+d ( 2^{s} \mathbb{G} (\frac{\imath}{2^{s}} ), 2^{s}f ( \frac{\imath}{2^{s}} ) ) \\ \hphantom{d (\mathbb{A}(\imath ), \mathbb{G}(\imath ) )} \leq \frac{2^{s}}{1- \vert \rho _{1} \vert }\Omega (\frac{\imath}{2^{s}}, \frac{\imath}{2^{s}} ) \end{cases} $$

that approaches zero when $g\to \infty$ for all $\imath \in (X, d, \kappa )$. Therefore, one gets $\mathbb{A}(\beta )=\mathbb{G}(\beta )$ for all $\imath \in (X, d, \kappa )$, which proves the uniqueness of $\mathbb{A}$. □

Corollary 3.2

Assume that $z > 1$ and ϑ is a nonnegative real number and $f: (X, d, \kappa ) \longrightarrow (G, d, \kappa )$, which satisfies $f(0)=0$ and (2.9). Then there exists an additive mapping $\mathbb{A} : (X, d, \kappa ) \longrightarrow (G, d, \kappa )$ that satisfies

$$ d \bigl(f(\imath ), \mathbb{A}(\imath ) \bigr)\leq \frac{2\vartheta}{(2^{z}-2)(1- \vert \rho _{1} \vert )} \Vert \imath \Vert ^{z} $$

for all $\imath \in (X, d, \kappa )$.

Theorem 3.3

Suppose that a function is $\varphi :X^{2}\rightarrow [0, \infty )$ and $f: (X, d, \kappa ) \longrightarrow (G, d, \kappa )$, which satisfy $f(0)=0$, (2.6) and

$$ \Omega (\imath , \jmath )=\sum^{\infty}_{s=0} \frac{1}{2^{s}} \varphi \bigl(2^{s}\imath , 2^{s}\jmath \bigr)< +\infty $$

(3.5)

for all $\imath , \jmath \in (X, d, \kappa )$. Then there exists a unique additive mapping $\mathbb{A} : (X, d, \kappa ) \longrightarrow (G, d, \kappa )$ such that

$$ d \bigl(f(\imath ), \mathbb{A}(\imath ) \bigr)\leq \frac{1}{2(1- \vert \rho _{1} \vert )}\Omega (\imath , \imath ) $$

(3.6)

for all $\imath \in (X, d, \kappa )$.

Proof

From (3.3), one gets

$$ d \biggl(f(\imath ), \frac{1}{2}f(2\imath ) \biggr)\leq \frac{1}{2(1- \vert \rho _{1} \vert )}\varphi (\imath , \imath ) $$

for all $\imath \in (X, d, \kappa )$. Therefore,

$$ \textstyle\begin{cases} d (\frac{1}{2^{\iota}}f (2^{\iota}\imath ), \frac{1}{2^{\upsilon}}f (2^{\upsilon} \imath ) )\leq \sum^{ \upsilon -1}_{s=\iota}2^{s} d (\frac{1}{2^{s}}f (2^{s}\imath ), \frac{1}{2^{s+1}}f (2^{s+1}\imath ) ) \\ \hphantom{d (\frac{1}{2^{\iota}}f (2^{\iota}\imath ), \frac{1}{2^{\upsilon}}f (2^{\upsilon} \imath ) )}\leq \sum^{\upsilon -1}_{s=\iota}\frac{1}{2^{s+1}(1- \vert \rho _{1} \vert )} \varphi (2^{s}\imath , 2^{s}\imath ) \end{cases} $$

(3.7)

for all positive integers υ and ι having $\upsilon > \iota$ and for all $\imath \in (X, d, \kappa )$. (3.7) gives that the sequence $\frac{1}{2^{s}}f(2^{s}\beta )$ is Cauchy for all $\imath \in (X, d, \kappa )$. As $(G, d, \kappa )$ is complete, the sequence $\frac{1}{2^{s}}f(2^{s}\imath )$ converges. Therefore, we can define $\mathbb{A} : (X, d, \kappa ) \longrightarrow (G, d, \kappa )$ as follows:

$$ \mathbb{A}(\imath )=\lim_{s\to +\infty} \frac{1}{2^{s}}f \bigl(2^{s} \imath \bigr) $$

for all $\imath \in (X, d, \kappa )$. Further, substituting $\iota =0$ and letting $\lim_{\upsilon \to +\infty}$ in (3.7), one obtains (3.6).

For the rest of the proof, follow the steps of Theorem 3.1. □

Corollary 3.4

Suppose that $z < 1$ and ϑ is a nonnegative real number and $f: (X, d, \kappa ) \longrightarrow (G, d, \kappa )$ satisfying $f(0)=0$ and (2.9). Then, there exists an additive mapping $\mathbb{A} : (X, d, \kappa ) \longrightarrow (G, d, \kappa )$ that satisfies

$$ d \bigl(f(\imath ), \mathbb{A}(\imath ) \bigr)\leq \frac{2\vartheta}{(2-2^{z})(1- \vert \rho _{1} \vert )} \Vert \imath \Vert ^{z} $$

for all $\imath \in (X, d, \kappa )$.

4 Example

Example 4.1

Suppose that $\psi : R^{2}\rightarrow [0, \infty )$ is a function defined by

$$ \psi (\beta )=\textstyle\begin{cases} 0, & \text{if }x=0; \\ \xi x, & \text{if } \vert x \vert < 1; \\ \xi , & \text{otherwise}, \end{cases} $$

where $\xi > 0$ is a constant. Define a function $j_{a}:Z^{n}\rightarrow Z^{n}$ by

$$ j_{a}(x)=\sum^{+\infty}_{n=0}\psi \biggl(\frac{2^{n}x}{2^{n}} \biggr). $$

Then $j_{a}$ satisfies the equation for all $x\in Z^{n}$. Assume

$$ \textstyle\begin{cases} A_{a}(x) = j_{a}(x), \\ j_{a}(x) = \xi x, \\ j_{a} (\frac{x}{2} ) = \xi \frac{x}{2} \end{cases} $$

for all $x \in Z^{n}$. We define a set $S=\{j_{a}:Z^{n}\rightarrow Z^{n}, j_{a}(0)=0\}$ and consider a digital metric space with k-adjacency on $\mathbb{C}$ as described above. Also consider $U:(\mathbb{C}, d, \kappa ) \longrightarrow (\mathbb{C}, d, \kappa )$ such that

$$ Uj_{a}(x)=2j_{a} \biggl(\frac{x}{2} \biggr). $$

Now

$$ \textstyle\begin{cases} \lim_{s\to +\infty}2^{s}j_{a} (\frac{x}{2} )= \lim_{s\to +\infty}2^{s} (\frac{\xi x}{2} ) \\ \hphantom{\lim_{s\to +\infty}2^{s}j_{a} (\frac{x}{2} )} =\lim_{s\to +\infty}2^{s} (\frac{\xi x}{2^{(1-s)}} )=O(x), \end{cases} $$

which is obvious

$$ O \biggl(\frac{x}{2} \biggr)=\frac{1}{2}O(x). $$

Moreover, we have

$$ \textstyle\begin{cases} d (A(x), 2A (\frac{x}{2} ) )= d (\xi x, 2\frac{\xi x}{2} ) \\ \hphantom{d (A(x), 2A (\frac{x}{2} ) )} \leq \psi (\frac{x}{2}, 0 )\leq \frac{\zeta}{2} \psi (x, 0). \end{cases} $$

Therefore,

$$ d(A, UA)\leq \frac{\zeta}{2}. $$

One can also show that

$$ d(A, O)\leq \frac{1}{1-\zeta}d(A, UA). $$

The above results give

$$ d(A, O)\leq \frac{\zeta}{2(1-\zeta )(1- \vert \rho _{1} \vert )}\psi (x, x). $$

Hence all the requirements have been satisfied. So, $O:\mathbb{R}\longrightarrow \mathbb{R}$ satisfies (0.1).

5 Application

We have many applications of Hyers–Ulam stability, such as in nonlinear analysis problems, which include differential equations and integral equations.

Deliberate (1.3), (1.4) and assume

$$ \alpha (r)=\beta (r)+\gamma (r). $$

(5.1)

Also, $\gamma (r)$ is small, therefore linearize it. Substituting (1.3) and (1.4), we get

$$ \gamma (r)\leq \delta f(\beta ) \int \frac{g\beta}{f^{2}(\beta )}. $$

(5.2)

Therefore, one has the following.

Proposition 5.1

If there is a constant ϖ such that (1.3) is locally Hyers–Ulam stable, then

$$ \biggl\vert f(\beta ) \int \frac{g\beta}{f^{2}(\beta )} \biggr\vert < \varpi . $$

(5.3)

Deliberate discrete systems as

$$ f \bigl(\beta (r) \bigr)= \beta (r+1),\quad r=0, 1, 2,\dots , \bigl\vert \Upsilon (r+1)-f \Upsilon (r) \bigr\vert < \delta . $$

(5.4)

Again by assumption

$$ \Upsilon (r)=\beta (r)+\theta (r),\quad r=0, 1, 2, \dots . $$

(5.5)

suppose $\theta (r)$ to be small, therefore linearizing it one obtains the following.

Proposition 5.2

A sufficient condition for (5.4) to be locally Hyers–Ulam stable is that there exists a constant ϖ such that

$$ \biggl\vert \frac{gf(\beta )}{g\beta} \biggr\vert < \varpi < 1. $$

(5.6)

Applications of Hyers–Ulam stability are as follows:

• 1: Wave solution of a reaction diffusion system: Its system is provided by

  $$ \frac{\partial v}{\partial t}=\nabla ^{2}(v)+f(v), $$

  (5.7)

  where $f(u)$ is a differentiable function. A wave solution in one dimension is

  $$ v=v(c),\quad c=-zt+\beta . $$

  So, we obtain

  $$ \biggl(\frac{g^{2}(v(c))}{g(c)^{2}} \biggr)+z \biggl(\frac{g(v)}{g(c)} \biggr)+f(v)=0. $$

  (5.8)

  By following the steps in Proposition 5.1, we obtain that (5.8) is locally Hyers–Ulam stable if there exists ϖ such that

  $$ \biggl(\frac{gf}{gv} \biggr)>\varpi >0. $$

  (5.9)

• 2: Logistic difference equation:

  $$ \beta (r+1)=a\beta (r) \bigl[1-\beta (r) \bigr],\quad a>0, \text{constant }r, \beta \in [0, 1]. $$

  (5.10)

  Utilizing (5.6), (5.10) is locally Hyers–Ulam stable if ϖ exists there such that

  $$ 0< a< \varpi < 1. $$

• 3: Model for economic monopoly having constant output: The case where the market is controlled by a single firm ℘ is monopoly [27]. In these types of models, one typically minimizes the loss. Consider a firm that produces $\mu (r)$ of a certain product at time r and has a profit function

  $$ \wp \bigl(\mu (r) \bigr)=\mu (r) \bigl[s-z\mu (r) \bigr]. $$

  By using a standard approach to Cournot economic dynamical systems having bounded rationality,

  $$ \mu (r+1)=\mu (r)+d\mu (r) \biggl(\frac{g \wp (\mu (r))}{g\mu (r)} \biggr) $$

  (where $d\mu (r)$ is the measure of bounded rationality), at the end we obtain coupled dynamical system

  $$ \mu (r+1)=\mu (r) \bigl[1-d \bigl(2z\mu (r)-s \bigr) \bigr]. $$

  (5.11)

  By rescaling, we find that the system is equivalent to (5.10). Therefore, (5.11) will be locally Hyers–Ulam stable.

6 Conclusion

In the present research paper, we have defined Hyers–Ulam stability for functional equations in the context of digital metric space. We analyzed Hyers–Ulam stability using the direct and fixed point approaches. We provided examples to strengthen our results. Finally, we provided applications for Hyers–Ulam stability.