Analysis of a Class of Stochastic Animal Behavior Models under Specific Choice Preferences

Abstract

The behavior of animals can be studied in two ways: experimentally, in labs or in the field, or theoretically, via modeling. Extensive research on animal behavior in probabilistic learning circumstances has produced findings that are consistent with so-called occurrence studies. However, such cases can be classified into four different events, depending on the location of the food and the side chosen by the animals. This article sought to overcome these limitations by offering a generic stochastic model under these conditions that can be utilized to analyze the broad range of models that have been reported in the literature. We explored the existence, uniqueness, and stability results of the proposed model using well-known fixed-point methods. Additionally, we present some examples to highlight the significance of our results.

1. Introduction and Preliminaries

Animal behavior is a lively and intriguing field of study within science. Many of the fundamental concepts of animal behavior are inspired by ecology, physiology, and evolution, despite their roots in comparative psychology and ethology (see [1]). Numerous fields within animal behavior study can be applied to real-world challenges, including monitoring well-being indicators for animals in wildlife parks, research labs, and farms (see [2]), behavioral difficulties in pet animals (see [3,4]), and the cognitive implications of environmental control (see [5]).

Light prediction research projects and cooperative games are two types of experimental settings that are often utilized in decision-making research and have served as the background for several studies. The findings of studies that were conducted within these two contexts are significantly different. Investigations on the former have shown that decision-making attitudes are remarkably resilient, which results in the development of a profusion of incredibly exact behavioral models (see [6]). Regrettably, research on the latter seems to have produced more uncertainty than clarity (see [7,8]).

Mathematical psychology is a branch of psychology that studies how to describe observations, reasoning, and consciousness using mathematics. When a substantial volume of quantitative data has been amassed, mathematical models for scientific processes can assist in the evolution of science. This aggregation may steer the construction of models and assess the sufficiency of their intermediate phases. In turn, these models are typically advantageous for organizing and analyzing experimental data and proposing novel routes for investigation. Few fields of psychology have the same amount and diversity of data required for model construction as for learning. Numerous efforts to develop quantitative models for learning processes have provided evidence for this assertion (for details, see [9,10]).

Recent research within mathematical psychology has shown that a simple learning experiment behaves stochastically. Consequently, it is not a novel phenomenon (for more information, see [11,12,13,14,15]).

To illustrate the learning behavior of animals in more than one event, Epstein proposed the following model [16] in 1967:

$$\mathfrak{H}\left(x\right)=\left(e^x{(1+e^x)}^{-1}\right)\mathfrak{H}\left(k_1\left(x\right)\right)+{\left(1+e^x\right)}^{-1}\mathfrak{H}\left(k_2\left(x\right)\right)$$

(1)

for all $x\in \mathfrak{D}$ and where $k_1,k_2:\mathfrak{D}\to \mathbb{R}$ are given mappings and $\mathfrak{H}:\mathfrak{D}\to \mathbb{R}$ is an unknown function. The bilateral Laplace transformation was then used to determine the solution of the aforementioned problem.

In [11,12], the authors examined the motion of a tropical fish in specific circumstances while observing both habit formation and reinforcement extinction behavior. They asserted that four separate outputs are associated with such actions: right non-reward, right reward, left reward, and left non-reward.

In [17], Turab and Sintunavarat utilized the above idea to explain the behavior of a paradise fish in a two-choice situation using the following stochastic equation:

$$\mathfrak{H}\left(x\right)=x\mathfrak{H}({\wp }_{1}x+1-{\wp }_1)+(1-x)\mathfrak{H}\left({\wp }_{2}x\right)$$

(2)

for all $x\in \mathfrak{D}$ and where $0<{\wp }_1\le {\wp }_2<1$ and $\mathfrak{H}:\mathfrak{D}\to \mathbb{R}$ is an unknown function.

Later, the authors proposed the following functional equation to study the traumatic avoidance learning behavior of mongrel dogs (for details, see [18]):

$$\mathfrak{H}\left(x\right)=x\mathfrak{H}({\wp }_{1}x+(1-{\wp }_1){\ell }_1)+(1-x)\mathfrak{H}({\wp }_{2}x+(1-{\wp }_2){\ell }_2)$$

(3)

for all $x\in \mathfrak{D}$ and where $0<{\wp }_1\le {\wp }_2<1$, ${\ell }_1,{\ell }_2\in [0,1],$ and $\mathfrak{H}:\mathfrak{D}\to \mathbb{R}$ is an unknown function. On the other hand, Berinde and Khan [19] and Turab and Sintunavarat [20] used different approaches and extended the work presented in [21] by proposing the following general functional equation:

$$\mathfrak{H}\left(x\right)=x\mathfrak{H}\left({\varrho }_1\left(x\right)\right)+(1-x)\mathfrak{H}\left({\varrho }_2\left(x\right)\right)$$

(4)

for all $x\in \mathfrak{D}$ and where $\mathfrak{H}:\mathfrak{D}\to \mathbb{R}$ is an unknown function and ${\varrho }_1,{\varrho }_2:\mathfrak{D}\to \mathbb{R}$ are given mappings.

Numerous further investigations on human (see [22,23,24,25,26,27]) and animal (see [28,29,30,31]) behavior within probability learning settings have generated diverse findings.

Most research on the study of animal responses in two-option scenarios has only looked at the animals’ progression toward a single alternative (for examples, see [16,17,18,19,20]). On the other hand, in [11,12], the authors categorized such reactions into four groups by concentrating on the food position and the selected side. To fill that void, we proposed the following stochastic model:

$$\mathfrak{H}\left(x\right)={\mathfrak{Y}}_1\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_4\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right),$$

(5)

for all $x\in \mathfrak{D}$ and where $\mathfrak{H}:\mathfrak{D}\to \mathbb{R}$ is an unknown function and ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },{\mathfrak{G}}_4^{\gamma }:\mathfrak{D}\to \mathfrak{D}$ are the four given responses. Additionally:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\mathfrak{Y}}_1\left(x\right)& =\zeta \mathfrak{V}\left(x\right),\hfill \\ \hfill {\mathfrak{Y}}_2\left(x\right)& =(1-\zeta )\mathfrak{V}\left(x\right),\hfill \\ \hfill {\mathfrak{Y}}_3\left(x\right)& =\zeta (1-\mathfrak{V}(x\left)\right),\hfill \\ \hfill {\mathfrak{Y}}_4\left(x\right)& =(1-\zeta )(1-\mathfrak{V}(x\left)\right),\hfill \end{array}\right.\end{array}$$

(6)

where $\mathfrak{V}:\mathfrak{D}\to \mathfrak{D}$ represents the corresponding probability of the event with the fixed proportion of the trials within $0\le \zeta \le 1$.

The primary contributions of this manuscript are as follows:

• We propose a general functional equation with more than two options, focusing on food location and the chosen side;
• Our proposed model generalizes many models in the existing literature and opens a new avenue of research within mathematical psychology and learning theory;
• We used the Banach fixed-point theorem to establish the existence of a unique solution to the proposed model (for further information on fixed-point theory, see [32,33,34,35,36]);
• We also discuss Hyers–Ulam and Hyers–Ulam–Rassias stability for the proposed system;
• We present some examples to show the importance of our results within this field of study.

In order to progress, the following result is necessary.

Theorem 1

([37]). Let $(\mathfrak{D},d)$ be a complete metric space and $\mathcal{K}:\mathfrak{D}\to \mathfrak{D}$ be a mapping that satisfies:

$$d(\mathcal{K}\mu ,\mathcal{K}\upsilon )\le \Lambda d(\mu ,\upsilon ),\forall \mu ,\upsilon \in \mathfrak{D},$$

(7)

for $\Lambda <1$. Then, $\mathcal{K}$ has a unique fixed point. Moreover, for each ${\mu }_0\in \mathfrak{D},$ the iteration process ${\left\{{\mu }_n\right\}}_{n\in \mathbb{N}}$ in $\mathfrak{D}$ that is defined by ${\mu }_n=\mathcal{K}{\mu }_{n-1}$ converges to the unique fixed point of $\mathcal{K}$.

2. Main Results

Throughout this paper, we let $\mathfrak{D}=[0,1].$ Before proving the main results, we highlight the following assumptions.

$\left({\tilde{\mathcal{O}}}_1\right):$

Let ${\mathfrak{T}}_{\mathbb{R}}^C$ be a class containing all continuous real-value functions $\mathfrak{H}:\mathfrak{D}\to \mathbb{R}$ satisfying $\mathfrak{H}\left(0\right)=0$ and $∥\mathfrak{H}∥={sup}_{\tilde{\tau }}\left[\left|\mho \mathfrak{H}\left(\tilde{\tau }\right)\right|\backslash \left|\mho \left(\tilde{\tau }\right)\right|\right]<\infty ,$ where $\mho \mathfrak{H}\left(\tilde{\tau }\right)=\mathfrak{H}\left(\mu \right)-\mathfrak{H}\left(\upsilon \right)$ and $\mho \left(\tilde{\tau }\right)=\mu -\upsilon$ with $\mu \ne \upsilon$. There is also a subset ${\mathfrak{C}}_B^S\ne \varnothing$ of ${\mathfrak{P}}^S:=\left\{\mathfrak{H}\in {\mathfrak{T}}_{\mathbb{R}}^C\mid \mathfrak{H}\left(1\right)\le 1\right\}$, such that $({\mathfrak{C}}_B^S,\parallel ·\parallel )$ is a Banach space (for details, see [17]);

$\left({\tilde{\mathcal{O}}}_2\right):$

Let ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },{\mathfrak{G}}_4^{\gamma }:\mathfrak{D}\to \mathfrak{D}$ be contraction mappings with contractive coefficients ${\kappa }_1-{\kappa }_4$, respectively, with ${\mathfrak{G}}_3^{\gamma }\left(0\right)=0={\mathfrak{G}}_4^{\gamma }\left(0\right)$;

$\left({\tilde{\mathcal{O}}}_3\right):$

Let $\mathfrak{V}:\mathfrak{D}\to \mathfrak{D}$ be a non-expansive mapping with $\left|\mathfrak{V}\left(x\right)\right|\le {\kappa }_5$ for ${\kappa }_5>0$ and $\mathfrak{V}\left(0\right)=0$;

$\left({\tilde{\mathcal{O}}}_4\right):$

The mappings ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },{\mathfrak{G}}_4^{\gamma }:\mathfrak{D}\to \mathfrak{D}$ satisfy the following conditions:

$$\begin{array}{c}\hfill \left|{\mathfrak{G}}_1^{\gamma }\left(\mu \right)-{\mathfrak{G}}_3^{\gamma }\left(\upsilon \right)\right|\le {\kappa }_6\left|\mu -\upsilon \right|and\left|{\mathfrak{G}}_2^{\gamma }\left(\mu \right)-{\mathfrak{G}}_4^{\gamma }\left(\upsilon \right)\right|\le {\kappa }_7\left|\mu -\upsilon \right|\end{array}$$

for all $\mu ,\upsilon \in \mathfrak{D}$ with $\mu \ne \upsilon$ and where ${\kappa }_6,{\kappa }_7\in [0,1)$;

$\left({\tilde{\mathcal{O}}}_5\right):$

Let $\alpha :{\mathfrak{C}}_B^S\to [0,\infty )$ be a fixed function. Then, for every $\mathfrak{H}\in {\mathfrak{C}}_B^S$ that satisfies $d(\mathfrak{MH},\mathfrak{H})\le \alpha \left(\mathfrak{H}\right)$, there is a unique ${\mathfrak{H}}^{\star }\in {\mathfrak{C}}_B^S$, such that $\mathfrak{M}{\mathfrak{H}}^{\star }={\mathfrak{H}}^{\star }$ and $d(\mathfrak{H},{\mathfrak{H}}^{\star })\le \theta \alpha \left(\mathfrak{H}\right)$ for $\theta >0$;

$\left({\tilde{\mathcal{O}}}_6\right):$

Let $\chi >0$. Then, for every $\mathfrak{H}\in {\mathfrak{C}}_B^S$ that satisfies $d(\mathfrak{MH},\mathfrak{H})\le \chi$, there is a unique ${\mathfrak{H}}^{\star }\in {\mathfrak{C}}_B^S$, such that $\mathfrak{M}{\mathfrak{H}}^{\star }={\mathfrak{H}}^{\star }$ and $d(\mathfrak{H},{\mathfrak{H}}^{\star })\le \rho \chi$ for $\rho >0$.

We begin with the result stated below.

Theorem 2.

Consider the model (5)–(6) with the assumptions $\left({\tilde{\mathcal{O}}}_1\right)-\left({\tilde{\mathcal{O}}}_4\right)$. Suppose that ${\Lambda }_1<1,$ where:

$$\begin{array}{c}\hfill {\Lambda }_1:=\left|\zeta \left((1+{\kappa }_5){\kappa }_1+{\kappa }_3+{\kappa }_6\right)+(1-\zeta )\left((1+{\kappa }_5){\kappa }_2+{\kappa }_4+{\kappa }_7\right)\right|.\end{array}$$

(8)

The mapping $\mathfrak{M}:{\mathfrak{C}}_B^S\to {\mathfrak{C}}_B^S$ then has a solution where:

$$\left(\mathfrak{MH}\right)\left(x\right)={\mathfrak{Y}}_1\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_4\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right)$$

(9)

for all $x\in \mathfrak{D}$. Furthermore, the Picard sequence ${\left({\mathfrak{H}}_n\right)}_{n\in \mathbb{N}}$ in ${\mathfrak{C}}_B^S$$\left({\mathfrak{H}}_0\in {\mathfrak{C}}_B^S\right)$ converges to the unique solution of (9), which is defined by:

$$\begin{array}{c}{\mathfrak{H}}_n\left(x\right)={\mathfrak{Y}}_1\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)\\ +{\mathfrak{Y}}_4\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right).\end{array}$$

(10)

Proof. 

We defined a metric $d:{\mathfrak{C}}_B^S\times {\mathfrak{C}}_B^S\to \mathbb{R}$ on ${\mathfrak{C}}_B^S$, which was induced by $∥·∥$. It could be seen that $({\mathfrak{C}}_B^S,d)$ was a complete metric space. Then, we considered the mapping $\mathfrak{M}$ from ${\mathfrak{C}}_B^S$ which was given in (9).

For each $\mathfrak{H},$ it could be seen that $\left(\mathfrak{MH}\right)\left(0\right)=\zeta \mathfrak{H}\left({\mathfrak{G}}_3^{\gamma }\left(0\right)\right)+(1-\zeta )\mathfrak{H}\left({\mathfrak{G}}_4^{\gamma }\left(0\right)\right)=0$. Additionally, $\mathfrak{M}$ was continuous and $∥\mathfrak{MH}∥<\infty$ for all $\mathfrak{H}\in {\mathfrak{C}}_B^S.$ Thus, $\mathfrak{M}$ was a self operator on ${\mathfrak{C}}_B^S.$ Furthermore, it was clear that the fixed point of $\mathfrak{M}$ was the solution of (9). As $\mathfrak{M}$ was a linear mapping (hence ${\mathfrak{H}}_1,{\mathfrak{K}}_2\in {\mathfrak{C}}_B^S$), we could write $∥{\mathfrak{MH}}_1-{\mathfrak{MH}}_2∥=∥\mathfrak{M}({\mathfrak{H}}_1-{\mathfrak{H}}_2)∥$. Thus, for each ${\gamma }_1,{\gamma }_2\in \mathfrak{D}$ with ${\gamma }_1\ne {\gamma }_2,$ we had ${\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}=\mathfrak{M}\left(\mho \tilde{\mathfrak{H}}\left({\gamma }_1\right)-\mho \tilde{\mathfrak{H}}\left({\gamma }_2\right)\right)\backslash \mho \tilde{\gamma },$ where $\mho \tilde{\mathfrak{H}}={\mathfrak{H}}_1-{\mathfrak{H}}_2$ and $\mho \tilde{\gamma }={\gamma }_1-{\gamma }_2$. By using this argument in (9), we could write:

$$\begin{array}{ccc}\hfill {\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}& =& \left[{\mathfrak{Y}}_1\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_1\right)\right)+{\mathfrak{Y}}_2\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_1\right)\right)+{\mathfrak{Y}}_3\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_1\right)\right)\right.\hfill \\ & & +{\mathfrak{Y}}_4\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_1\right)\right)-{\mathfrak{Y}}_1\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_2\right)\right)-{\mathfrak{Y}}_2\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_2\right)\right)\hfill \\ & & \left.-{\mathfrak{Y}}_3\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_2\right)\right)-{\mathfrak{Y}}_4\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_2\right)\right)\right]\mho {\left(\tilde{\gamma }\right)}^{-1}\hfill \\ & =& \left[{\mathfrak{Y}}_1\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_1\right)\right)-{\mathfrak{Y}}_1\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_2\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_1\right)\right)\right.\hfill \\ & & -{\mathfrak{Y}}_2\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_3\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_1\right)\right)-{\mathfrak{Y}}_3\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_2\right)\right)\hfill \\ & & +{\mathfrak{Y}}_4\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_1\right)\right)-{\mathfrak{Y}}_4\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_1\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_2\right)\right)\hfill \\ & & -{\mathfrak{Y}}_1\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_2\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_2\right)\right)-{\mathfrak{Y}}_2\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_2\right)\right)\hfill \\ & & +{\mathfrak{Y}}_3\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_2\right)\right)-{\mathfrak{Y}}_3\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_4\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_2\right)\right)\hfill \\ & & \left.-{\mathfrak{Y}}_4\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_2\right)\right)\right]\mho {\left(\tilde{\gamma }\right)}^{-1}\hfill \end{array}$$

From the above analysis, we could write:

$$\begin{array}{ccc}\hfill \left|{\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}\right|& =& |\left[{\mathfrak{Y}}_1\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_1\right)\right)-{\mathfrak{Y}}_1\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_2\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_1\right)\right)\right.\hfill \\ & & -{\mathfrak{Y}}_2\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_3\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_1\right)\right)-{\mathfrak{Y}}_3\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_2\right)\right)\hfill \\ & & +{\mathfrak{Y}}_4\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_1\right)\right)-{\mathfrak{Y}}_4\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_1\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_2\right)\right)\hfill \\ & & -{\mathfrak{Y}}_1\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_2\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_2\right)\right)-{\mathfrak{Y}}_2\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_2\right)\right)\hfill \\ & & +{\mathfrak{Y}}_3\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_2\right)\right)-{\mathfrak{Y}}_3\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_2\right)\right)+{\mathfrak{Y}}_4\left({\gamma }_1\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_2\right)\right)\hfill \\ & & \left.\left.-{\mathfrak{Y}}_4\left({\gamma }_2\right)\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_2\right)\right)\right]\mho {\left(\tilde{\gamma }\right)}^{-1}\right|\hfill \end{array}$$

By using the assumptions $\left({\tilde{\mathcal{O}}}_1\right)-\left({\tilde{\mathcal{O}}}_4\right)$, we obtained:

$$\begin{array}{ccc}\hfill \left|{\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}\right|& \le & {\kappa }_1{\mathfrak{Y}}_1\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_2{\mathfrak{Y}}_2\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_3{\mathfrak{Y}}_3\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_4{\mathfrak{Y}}_4\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥\hfill \\ & & +\left|\zeta \mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_2\right)\right)-\zeta \mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_1\right)\right)\right|+\left|(1-\zeta )\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_2\right)\right)\right.\hfill \\ & & \left.-(1-\zeta )\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_1\right)\right)\right|+\left|\zeta \mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_1\right)\right)-\zeta \mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_2\right)\right)\right|\hfill \\ & & +\left|(1-\zeta )\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_1\right)\right)-(1-\zeta )\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_2\right)\right)\right|\hfill \\ & \le & {\Lambda }_1∥\mho \tilde{\mathfrak{H}}∥,\hfill \end{array}$$

where ${\Lambda }_1$ is as defined in (8). This produced:

$$\begin{array}{c}\hfill d({\mathfrak{MH}}_1,{\mathfrak{MH}}_2)=∥\mathfrak{M}\left(\mho \tilde{\mathfrak{H}}\right)∥\le {\Lambda }_1∥\mho \tilde{\mathfrak{H}}∥={\Lambda }_{1}d({\mathfrak{H}}_1,{\mathfrak{H}}_2).\end{array}$$

As $0<{\Lambda }_1<1,$ by Theorem 1, we obtained the solution of (9) and the Picard sequence ${\left({\mathfrak{H}}_n\right)}_{n\in \mathbb{N}}$ in ${\mathfrak{C}}_B^S$$\left({\mathfrak{H}}_0\in {\mathfrak{C}}_B^S\right)$ converged to the unique solution of (9). □

For the special case of Theorem 2, we considered the following assumption:

$\left({\tilde{\mathcal{O}}}_2^{\star }\right):$

Let ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },{\mathfrak{G}}_4^{\gamma }:\mathfrak{D}\to \mathfrak{D}$ be contraction mappings with contractive coefficients ${\kappa }_1\le {\kappa }_2\le {\kappa }_3\le {\kappa }_4$, respectively, with ${\mathfrak{G}}_3^{\gamma }\left(0\right)=0={\mathfrak{G}}_4^{\gamma }\left(0\right)$.

Corollary 1.

Consider the model (5) and (6) with the assumptions $\left({\tilde{\mathcal{O}}}_1\right),\left({\tilde{\mathcal{O}}}_2^{\star }\right),\left({\tilde{\mathcal{O}}}_3\right)$, and $\left({\tilde{\mathcal{O}}}_4\right)$. When ${\Lambda }_1^{\star }<1,$ where:

$$\begin{array}{c}\hfill {\Lambda }_1^{\star }:=\left|(2+{\kappa }_5){\kappa }_4+({\kappa }_6+{\kappa }_7)\right|,\end{array}$$

(11)

then the mapping $\mathfrak{M}:{\mathfrak{C}}_B^S\to {\mathfrak{C}}_B^S$ has a solution where:

$$\left(\mathfrak{MH}\right)\left(x\right)={\mathfrak{Y}}_1\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_4\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right)$$

(12)

for all $x\in \mathfrak{D}$. Furthermore, the Picard sequence ${\left({\mathfrak{H}}_n\right)}_{n\in \mathbb{N}}$ in ${\mathfrak{C}}_B^S$$\left({\mathfrak{H}}_0\in {\mathfrak{C}}_B^S\right)$ converges to a unique solution of (12), which is defined by:

$$\begin{array}{c}{\mathfrak{H}}_n\left(x\right)={\mathfrak{Y}}_1\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)\\ +{\mathfrak{Y}}_4\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right).\end{array}$$

(13)

The conditions $\left({\tilde{\mathcal{O}}}_1\right)-\left({\tilde{\mathcal{O}}}_4\right)$ were sufficient but not necessary to claim the existence of a unique solution to the proposed model (9). Next, we imposed the following key assumption.

$\left({\tilde{\mathcal{O}}}_4^{\star }\right):$

Points ${\xi }_1^{\star },{\xi }_2^{\star }\in \mathfrak{D}$ exist such that ${\mathfrak{G}}_1^{\gamma }\left({\xi }_1^{\star }\right)={\mathfrak{G}}_3^{\gamma }\left({\xi }_1^{\star }\right)$ and ${\mathfrak{G}}_2^{\gamma }\left({\xi }_2^{\star }\right)={\mathfrak{G}}_4^{\gamma }\left({\xi }_2^{\star }\right)$.

Theorem 3.

Consider the model (5)–(6) with the assumptions $\left({\tilde{\mathcal{O}}}_1\right)-\left({\tilde{\mathcal{O}}}_3\right)$ and $\left({\tilde{\mathcal{O}}}_4^{\star }\right)$. Suppose that ${\Lambda }_2<1,$ where:

$$\begin{array}{c}\hfill {\Lambda }_2:=\left|\zeta \left((1+{\kappa }_5){\kappa }_1+2{\kappa }_3\right)+(1-\zeta )\left((1+{\kappa }_5){\kappa }_2+2{\kappa }_4\right)\right|,\end{array}$$

(14)

then the mapping $\mathfrak{M}:{\mathfrak{C}}_B^S\to {\mathfrak{C}}_B^S$ has a solution where:

$$\left(\mathfrak{MH}\right)\left(x\right)={\mathfrak{Y}}_1\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_4\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right)$$

(15)

for all $x\in \mathfrak{D}$. Furthermore, the Picard sequence ${\left({\mathfrak{H}}_n\right)}_{n\in \mathbb{N}}$ in ${\mathfrak{C}}_B^S$$\left({\mathfrak{H}}_0\in {\mathfrak{C}}_B^S\right)$ converges to a unique solution of (15), which is defined by:

$$\begin{array}{c}{\mathfrak{H}}_n\left(x\right)={\mathfrak{Y}}_1\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)\\ +{\mathfrak{Y}}_4\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right).\end{array}$$

(16)

Proof. 

This theorem followed the same line of reasoning as Theorem 2. The sections that varied from the preceding theorem are highlighted below. For each ${\gamma }_1,{\gamma }_2\in \mathfrak{D}$ with ${\gamma }_1\ne {\gamma }_2,$ we obtained ${\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}=\mathfrak{M}\left(\mho \tilde{\mathfrak{H}}\left({\gamma }_1\right)-\mho \tilde{\mathfrak{H}}\left({\gamma }_2\right)\right)\backslash \mho \tilde{\gamma },$ where $\mho \tilde{\mathfrak{H}}={\mathfrak{H}}_1-{\mathfrak{H}}_2$ and $\mho \tilde{\gamma }={\gamma }_1-{\gamma }_2$. Thus, using the assumptions $\left({\tilde{\mathcal{O}}}_1\right)-\left({\tilde{\mathcal{O}}}_3\right)$, we could write:

$$\begin{array}{ccc}\hfill \left|{\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}\right|& \le & {\kappa }_1{\mathfrak{Y}}_1\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_2{\mathfrak{Y}}_2\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_3{\mathfrak{Y}}_3\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_4{\mathfrak{Y}}_4\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥\hfill \\ \hfill & & \left|\zeta \mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\gamma }_2\right)\right)-\zeta \mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\xi }_1^{\star }\right)\right)\right|+\left|(1-\zeta )\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_2^{\gamma }\left({\gamma }_2\right)\right)\right.\hfill \\ \hfill & & \left.-(1-\zeta )\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_1^{\gamma }\left({\xi }_2^{\star }\right)\right)\right|\left|\zeta \mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\xi }_1^{\star }\right)\right)-\zeta \mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_3^{\gamma }\left({\gamma }_2\right)\right)\right|\hfill \\ \hfill & & +\left|(1-\zeta )\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\xi }_2^{\star }\right)\right)-(1-\zeta )\mho \tilde{\mathfrak{H}}\left({\mathfrak{G}}_4^{\gamma }\left({\gamma }_2\right)\right)\right|.\hfill \end{array}$$

(17)

Then, utilizing the condition $\left({\tilde{\mathcal{O}}}_4^{\star }\right)$, we obtained the following results.

$Case1:$

When ${\gamma }_2={\xi }_1^{\star }={\xi }_2^{\star }$, then using (17), we obtained:

$$\begin{array}{ccc}\hfill \left|{\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}\right|& \le & {\kappa }_1{\mathfrak{Y}}_1\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_2{\mathfrak{Y}}_2\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_3{\mathfrak{Y}}_3\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_4{\mathfrak{Y}}_4\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥\hfill \\ & \le & {\Lambda }_2∥\mho \tilde{\mathfrak{H}}∥.\hfill \end{array}$$

$Case2:$

When ${\gamma }_2\ne {\xi }_1^{\star }$ and ${\gamma }_2={\xi }_2^{\star }$, then using (17), we obtained:

$$\begin{array}{ccc}\hfill \left|{\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}\right|& \le & {\kappa }_1{\mathfrak{Y}}_1\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_2{\mathfrak{Y}}_2\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_3{\mathfrak{Y}}_3\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_4{\mathfrak{Y}}_4\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥\hfill \\ & & +{\kappa }_1\zeta |{\gamma }_2-{\xi }_1^{\star }|∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_3\zeta |{\xi }_1^{\star }-{\gamma }_2|∥\mho \tilde{\mathfrak{H}}∥\hfill \\ & \le & {\Lambda }_2∥\mho \tilde{\mathfrak{H}}∥.\hfill \end{array}$$

$Case3:$

When ${\gamma }_2={\xi }_1^{\star }$ and ${\gamma }_2\ne {\xi }_2^{\star }$, then using (17), we obtained:

$$\begin{array}{ccc}\hfill \left|{\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}\right|& \le & {\kappa }_1{\mathfrak{Y}}_1\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_2{\mathfrak{Y}}_2\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_3{\mathfrak{Y}}_3\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_4{\mathfrak{Y}}_4\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥\hfill \\ & & +{\kappa }_2(1-\zeta )|{\gamma }_2-{\xi }_2^{\star }|∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_4(1-\zeta )|{\xi }_2^{\star }-{\gamma }_2|∥\mho \tilde{\mathfrak{H}}∥\hfill \\ & \le & {\Lambda }_2∥\mho \tilde{\mathfrak{H}}∥.\hfill \end{array}$$

$Case4:$

When ${\gamma }_2\ne {\xi }_1^{\star }\ne {\xi }_2^{\star }$, then using (17), we obtained:

$$\begin{array}{ccc}\hfill \left|{\Theta }_{{\gamma }_1\ne {\gamma }_2}^{{\gamma }_1,{\gamma }_2}\right|& \le & {\kappa }_1{\mathfrak{Y}}_1\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_2{\mathfrak{Y}}_2\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_3{\mathfrak{Y}}_3\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_4{\mathfrak{Y}}_4\left({\gamma }_1\right)∥\mho \tilde{\mathfrak{H}}∥\hfill \\ & & +{\kappa }_1\zeta |{\gamma }_2-{\xi }_1^{\star }|∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_3\zeta |{\xi }_1^{\star }-{\gamma }_2|∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_2(1-\zeta )|{\gamma }_2\hfill \\ & & -{\xi }_2^{\star }|∥\mho \tilde{\mathfrak{H}}∥+{\kappa }_4(1-\zeta )|{\xi }_2^{\star }-{\gamma }_2|∥\mho \tilde{\mathfrak{H}}∥\hfill \\ & \le & {\Lambda }_2∥\mho \tilde{\mathfrak{H}}∥,\hfill \end{array}$$

where ${\Lambda }_2$ is as defined in (14). Hence:

$$\begin{array}{c}\hfill d({\mathfrak{MH}}_1,{\mathfrak{MH}}_2)=∥\mathfrak{M}\left(\mho \tilde{\mathfrak{H}}\right)∥\le {\Lambda }_2∥\mho \tilde{\mathfrak{H}}∥={\Lambda }_{2}d({\mathfrak{H}}_1,{\mathfrak{H}}_2).\end{array}$$

As $0<{\Lambda }_2<1,$ by Theorem 1, we obtained the solution of (15) and the Picard sequence ${\left({\mathfrak{H}}_n\right)}_{n\in \mathbb{N}}$ in ${\mathfrak{C}}_B^S$$\left({\mathfrak{H}}_0\in {\mathfrak{C}}_B^S\right)$, as defined in (16), converged to the unique solution of (15). □

The following result followed on from the preceding theorem.

Corollary 2.

Consider the model (5)–(6) with the assumptions $\left({\tilde{\mathcal{O}}}_1\right),\left({\tilde{\mathcal{O}}}_2^{\star }\right),\left({\tilde{\mathcal{O}}}_3\right)$, and $\left({\tilde{\mathcal{O}}}_4^{\star }\right)$. When ${\Lambda }_2^{\star }:(3+{\kappa }_5){\kappa }_4<1$, then the mapping $\mathfrak{M}:{\mathfrak{C}}_B^S\to {\mathfrak{C}}_B^S$ has a solution where:

$$\left(\mathfrak{MH}\right)\left(x\right)={\mathfrak{Y}}_1\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_4\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right)$$

(18)

for all $x\in \mathfrak{D}$. Furthermore, the Picard sequence ${\left({\mathfrak{H}}_n\right)}_{n\in \mathbb{N}}$ in ${\mathfrak{C}}_B^S$$\left({\mathfrak{H}}_0\in {\mathfrak{C}}_B^S\right)$ converges to a unique solution of (18), which is defined by:

$$\begin{array}{c}{\mathfrak{H}}_n\left(x\right)={\mathfrak{Y}}_1\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)\\ +{\mathfrak{Y}}_4\left(x\right){\mathfrak{H}}_{n-1}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right).\end{array}$$

(19)

Remark 1.

The suggested model (5), when associated with (6), is an extension of several models that have been seen in the literature. For example:

•

When we considered $\zeta =0,\mathfrak{V}\left(x\right)=x,$ and ${\mathfrak{G}}_2^{\gamma }\left(1\right)=1,$ then our proposed stochastic Equation (5) with (6) decreased to the functional equation discussed in [17];

•

When we considered $\zeta =1,\mathfrak{V}\left(x\right)=x,$ and ${\mathfrak{G}}_1^{\gamma }\left(1\right)\le 1,$ then our proposed model (5)–(6) reduced to the functional equation discussed in [18];

•

When we considered $\zeta =0$ and $\mathfrak{V}\left(x\right)=x$ and the mappings ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma }:\mathfrak{D}\to \mathfrak{D}$ satisfied the condition $\left({\tilde{\mathcal{O}}}_2\right)$ with ${\mathfrak{G}}_2^{\gamma }\left(1\right)=1,$ then our proposed Equation (5), with regard to (6), generalized the models presented in [19,21].

Remark 2.

We could not predict rapid convergence from the sequence of iterations due to the linear convergence rate of the Picard iteration. We could apply a suitable accelerative approach to overcome this problem (for details, see [38,39,40]).

3. Stability Analysis

The consistency of solutions is essential within computational mathematics. Results can be affected by minor changes in the given dataset. Therefore, it was essential to observe the stability (see [41,42,43,44,45,46,47,48] for details) of the proposed model (5) when associated with (6).

Theorem 4.

Under the premise of Theorem 3, the equation $\mathfrak{MH}=\mathfrak{H},$ where $\mathfrak{M}:{\mathfrak{C}}_B^S\to {\mathfrak{C}}_B^S$, is given by:

$$\left(\mathfrak{MH}\right)\left(x\right)={\mathfrak{Y}}_1\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_4\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right)$$

(20)

for all $\mathfrak{H}\in {\mathfrak{C}}_B^S$ and $x\in \mathfrak{D},$ which has Hyers–Ulam–Rassias stability (as defined in $\left({\tilde{\mathcal{O}}}_5\right)$).

Proof. 

We let $\mathfrak{H}\in {\mathfrak{C}}_B^S$, such that $d(\mathfrak{MH},\mathfrak{H})\le \alpha \left(\mathfrak{H}\right)$. Using Theorem 3, we deduced that there was a unique ${\mathfrak{H}}^{\star }\in {\mathfrak{C}}_B^S$, such that $\mathfrak{M}{\mathfrak{H}}^{\star }={\mathfrak{H}}^{\star }$. Thus, we obtained:

$$\begin{array}{ccc}\hfill d(\mathfrak{H},{\mathfrak{H}}^{\star })& \le & d(\mathfrak{H},\mathfrak{MH})+d(\mathfrak{MH},{\mathfrak{H}}^{\star })\hfill \\ & \le & \alpha \left(\mathfrak{H}\right)+d(\mathfrak{MH},\mathfrak{M}{\mathfrak{H}}^{\star })\hfill \\ & \le & \alpha \left(\mathfrak{H}\right)+{\Lambda }_{2}d(\mathfrak{H},{\mathfrak{H}}^{\star }),\hfill \end{array}$$

where ${\Lambda }_2$ is as defined in (14). So:

$$d(\mathfrak{H},{\mathfrak{H}}^{\star })\le \theta \alpha \left(\mathfrak{H}\right),$$

where $\theta :={(1-{\Lambda }_2)}^{-1}$. □

We could obtain the following outcome about the Hyers–Ulam stability from the above result.

Corollary 3.

Under the hypothesis of Theorem 3, the equation $\mathfrak{MH}=\mathfrak{H},$ where $\mathfrak{M}:{\mathfrak{C}}_B^S\to {\mathfrak{C}}_B^S$, is given by:

$$\left(\mathfrak{MH}\right)\left(x\right)={\mathfrak{Y}}_1\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_1^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_2\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_2^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_3\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_3^{\gamma }\left(x\right)\right)+{\mathfrak{Y}}_4\left(x\right)\mathfrak{H}\left({\mathfrak{G}}_4^{\gamma }\left(x\right)\right)$$

(21)

for all $\mathfrak{H}\in {\mathfrak{C}}_B^S$ and $x\in \mathfrak{D},$ which has Hyers–Ulam stability (as defined in $\left({\tilde{\mathcal{O}}}_6\right)$).

4. Some Illustrative Examples

We now offer the following examples to enhance our key findings.

Example 1.

Consider the following model:

$$\begin{array}{c}\mathfrak{H}\left(x\right)=\zeta x\mathfrak{H}\left({\theta }_{1}x+1-{\theta }_1\right)+(1-\zeta )x\mathfrak{H}\left({\theta }_{2}x+1-{\theta }_2\right)+\zeta (1-x)\mathfrak{H}\left({\theta }_{3}x\right)\\ +(1-\zeta )(1-x)\mathfrak{H}\left({\theta }_{4}x\right),\forall x\in \mathfrak{D},\end{array}$$

(22)

where $0<{\theta }_1<{\theta }_2<{\theta }_3<{\theta }_4<1$ and $\mathfrak{H}:\mathfrak{D}\to \mathbb{R}$. Then, when we considered ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },{\mathfrak{G}}_4^{\gamma }:\mathfrak{D}\to \mathfrak{D}$ by ${\mathfrak{G}}_1^{\gamma }\left(x\right)={\theta }_{1}x+1-{\theta }_1,{\mathfrak{G}}_2^{\gamma }\left(x\right)={\theta }_{2}x+1-{\theta }_2,{\mathfrak{G}}_3^{\gamma }\left(x\right)={\theta }_{3}x,{\mathfrak{G}}_4^{\gamma }\left(x\right)={\theta }_{4}x,and\mathfrak{V}\left(x\right)=x$, (22) was decreased to the proposed model (5).

The above functional equation (22) was used to observe a particular behavior of a rat in a T-maze model (see Figure 1). In that experiment, the rat sometimes chose the same side repeatedly and increased the probability of that event (for details, see [49]).

Figure 1. Behavior of a rat in a T-maze model [50].

The operators ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },$ and ${\mathfrak{G}}_4^{\gamma }$ represent the four possible outcomes, which depended on the food placement and the chosen side (for details, see Table 1).

Table 1. The four events with the corresponding probability of occurrence.

Placement of Food	Side Chosen by Rat	Probability of Occurrence
A: Left Side	Left Turn (Food Side)	$\zeta x$
A: Left Side	Right Turn (Non-Food Side)	$(1-\zeta )x$
B: Right Side	Right Turn (Food Side)	$\zeta (1-x)$
B: Right Side	Left Turn (Non-Food Side)	$(1-\zeta )(1-x)$

It is easy to see that the transition operators ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },$ and ${\mathfrak{G}}_4^{\gamma }$ were contraction mappings with contractive coefficients ${\kappa }_1={\theta }_1,$${\kappa }_2={\theta }_2,{\kappa }_3={\theta }_3$ and ${\kappa }_4={\theta }_4$, respectively. We also obtained points ${\xi }_1^{\star },{\xi }_2^{\star }\in [0,1]$, such that ${\mathfrak{G}}_1^{\gamma }\left({\xi }_1^{\star }\right)=\frac{1-{\theta }_1}{{\theta }_1-{\theta }_3}={\mathfrak{G}}_3^{\gamma }\left({\xi }_1^{\star }\right)$ and ${\mathfrak{G}}_2^{\gamma }\left({\xi }_2^{\star }\right)=\frac{1-{\theta }_2}{{\theta }_4-{\theta }_2}={\mathfrak{G}}_4^{\gamma }\left({\xi }_2^{\star }\right)$, where ${\theta }_1-{\theta }_3\ne 0$ and ${\theta }_2-{\theta }_4\ne 0$. Moreover, $\mathfrak{V}:\mathfrak{D}\to \mathfrak{D}$ was a non-expansive mapping and ${\kappa }_5=1$. Additionally, for all $\zeta \in [0,1]$ and ${\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4\in [0,1)$, we obtained ${\Lambda }_2:=|\zeta ({\theta }_1+{\theta }_3)+(1-\zeta )({\theta }_2+{\theta }_4)|<\frac{1}{2}$.

It can be seen that all conditions of Theorem 3 were satisfied. Consequently, the proposed model (23) had a solution. Furthermore, when we picked an initial approximation ${\mathfrak{H}}_0=\mathbf{I}\in {\mathfrak{C}}_B^S$ (whereas $\mathbf{I}$ is an identity function), the next iteration led to the unique solution of (22):

$$\begin{array}{ccc}\hfill {\mathfrak{H}}_1\left(x\right)& =& (x^2-x)\left(\zeta ({\theta }_1-{\theta }_2-{\theta }_3+{\theta }_4)+({\theta }_4-{\theta }_2)\right)+x,\hfill \\ \hfill {\mathfrak{H}}_2\left(x\right)& =& \zeta x{\mathfrak{H}}_1\left({\theta }_{1}x+1-{\theta }_1\right)+(1-\zeta )x{\mathfrak{H}}_1\left({\theta }_{2}x+1-{\theta }_2\right)+\zeta (1-x){\mathfrak{H}}_1\left({\theta }_{3}x\right)\hfill \\ & & +(1-\zeta )(1-x){\mathfrak{H}}_1\left({\theta }_{4}x\right),\hfill \\ & ⋮& \\ \hfill {\mathfrak{H}}_n\left(x\right)& =& \zeta x{\mathfrak{H}}_{n-1}\left({\theta }_{1}x+1-{\theta }_1\right)+(1-\zeta )x{\mathfrak{H}}_{n-1}\left({\theta }_{2}x+1-{\theta }_2\right)+\zeta (1-x){\mathfrak{H}}_{n-1}\left({\theta }_{3}x\right)\hfill \\ & & +(1-\zeta )(1-x){\mathfrak{H}}_{n-1}\left({\theta }_{4}x\right),\hfill \end{array}$$

for all $n\in \mathbb{N}$.

Example 2.

Consider the following model:

$$\begin{array}{ccc}\hfill \mathfrak{H}\left(x\right)& =& \zeta x\mathfrak{H}\left((x+1)/13\right)+(1-\zeta )x\mathfrak{H}\left((3x+4)/44\right)+\zeta (1-x)\mathfrak{H}\left(2x/13\right)\hfill \\ & & +(1-\zeta )(1-x)\mathfrak{H}\left(2x/11\right),\forall x\in \mathfrak{D},\hfill \end{array}$$

(23)

where $\mathfrak{H}:\mathfrak{D}\to \mathbb{R}$. Then, when we considered ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },{\mathfrak{G}}_4^{\gamma }:\mathfrak{D}\to \mathfrak{D}$ by ${\mathfrak{G}}_1^{\gamma }\left(x\right)=(x+1)/13,{\mathfrak{G}}_2^{\gamma }\left(x\right)=(3x+4)/44,{\mathfrak{G}}_3^{\gamma }\left(x\right)=2x/13,{\mathfrak{G}}_4^{\gamma }\left(x\right)=2x/11,and\mathfrak{V}\left(x\right)=x$, (23) was decreased to the proposed model (5).

Here, ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },{\mathfrak{G}}_4^{\gamma }$ were contraction mappings with contractive coefficients ${\kappa }_1=1/13,$${\kappa }_2=3/14, {\kappa }_3=3/44$ and ${\kappa }_4=2/11$, respectively. We also obtained points ${\xi }_1^{\star },{\xi }_2^{\star }\in [0,1]$, such that ${\mathfrak{G}}_1^{\gamma }\left({\xi }_1^{\star }\right)={\mathfrak{G}}_3^{\gamma }\left({\xi }_1^{\star }\right)$ (see Figure 2) and ${\mathfrak{G}}_2^{\gamma }\left({\xi }_2^{\star }\right)={\mathfrak{G}}_4^{\gamma }\left({\xi }_2^{\star }\right)$ (see Figure 3).

Figure 2. The graph of ${\mathfrak{G}}_1^{\gamma }\left(x\right)=\frac{x+1}{13}$ (red) and ${\mathfrak{G}}_3^{\gamma }\left(x\right)=\frac{2x}{13}$ (blue).

Figure 3. The graph of ${\mathfrak{G}}_2^{\gamma }\left(x\right)=\frac{3x+4}{44}$ (red) and ${\mathfrak{G}}_4^{\gamma }\left(x\right)=\frac{2x}{11}$ (blue).

It is clear that $\mathfrak{V}:\mathfrak{D}\to \mathfrak{D}$ was a non-expansive mapping and ${\kappa }_5=1$. Additionally, for all $\zeta \in [0,1]$, we obtained ${\Lambda }_2:=\left(-1005\zeta +1586\right){\left(2002\right)}^{-1}<1$. It can be seen that all conditions of Theorem 3 were satisfied. Consequently, the proposed model (23) had a solution. Furthermore, when we picked an initial approximation ${\mathfrak{H}}_0=\mathbf{I}\in {\mathfrak{C}}_B^S$ (whereas $\mathbf{I}$ is an identity function), the next iteration led to the unique solution of (23):

$$\begin{array}{ccc}\hfill {\mathfrak{H}}_1\left(x\right)& =& \frac{1}{572}[21\zeta x^2-65x^2+156x-24\zeta x],\hfill \\ \hfill {\mathfrak{H}}_2\left(x\right)& =& \frac{1}{187149248}\left[\begin{array}{c}73227{\zeta }^2{x}^3-421850\zeta x^3+604175x^3\\ -356664{\zeta }^2{x}^2+3290144\zeta x^2-6766760x^2\\ +313344{\zeta }^{2}x-4176432\zeta x+13744432x\end{array}\right],\hfill \\ & ⋮& \\ \hfill {\mathfrak{H}}_n\left(x\right)& =& \zeta x{\mathfrak{H}}_{n-1}\left((x+1)/13\right)+(1-\zeta )x{\mathfrak{H}}_{n-1}\left((3x+4)/44\right)+\zeta (1-x){\mathfrak{H}}_{n-1}\left(2x/13\right)\hfill \\ & & +(1-\zeta )(1-x){\mathfrak{H}}_{n-1}\left(2x/11\right)\hfill \end{array}$$

for all $n\in \mathbb{N}$.

Example 3.

Consider the following model:

$$\begin{array}{c}\mathfrak{H}\left(x\right)=\zeta x\mathfrak{H}\left((x+3)23\right)+(1-\zeta )x\mathfrak{H}\left((3x+1)/45\right)+\zeta (1-x)\mathfrak{H}\left(x/5\right)\\ +(1-\zeta )(1-x)\mathfrak{H}\left(6x/28\right),\forall x\in \mathfrak{D},\end{array}$$

(24)

where $\mathfrak{H}:\mathfrak{D}\to \mathbb{R}$. Then, when we considered ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },{\mathfrak{G}}_4^{\gamma }:\mathfrak{D}\to \mathfrak{D}$ by ${\mathfrak{G}}_1^{\gamma }\left(x\right)=(x+3)/23,{\mathfrak{G}}_2^{\gamma }\left(x\right)=(3x+1)/45,{\mathfrak{G}}_3^{\gamma }\left(x\right)=x/5,{\mathfrak{G}}_4^{\gamma }\left(x\right)=6x/28,and\mathfrak{V}\left(x\right)=x$, (24) was decreased to the proposed model (5).

Here, ${\mathfrak{G}}_1^{\gamma },{\mathfrak{G}}_2^{\gamma },{\mathfrak{G}}_3^{\gamma },{\mathfrak{G}}_4^{\gamma }:\mathfrak{D}\to \mathfrak{D}$ were contraction mappings with contractive coefficients ${\kappa }_1=1/23,$${\kappa }_2=1/15, {\kappa }_3=1/5$ and ${\kappa }_4=3/14$, respectively. We also obtained points ${\xi }_1^{\star },{\xi }_2^{\star }\in [0,1]$, such that ${\mathfrak{G}}_1^{\gamma }\left({\xi }_1^{\star }\right)={\mathfrak{G}}_3^{\gamma }\left({\xi }_1^{\star }\right)$ (see Figure 4) and ${\mathfrak{G}}_2^{\gamma }\left({\xi }_2^{\star }\right)={\mathfrak{G}}_4^{\gamma }\left({\xi }_2^{\star }\right)$ (see Figure 5).

Figure 4. The graph of ${\mathfrak{G}}_1^{\gamma }\left(x\right)=\frac{x+1}{23}$ (red) and ${\mathfrak{G}}_3^{\gamma }\left(x\right)=\frac{x}{5}$ (blue).

Figure 5. The graph of ${\mathfrak{G}}_2^{\gamma }\left(x\right)=\frac{3x+1}{45}$ (green) and ${\mathfrak{G}}_4^{\gamma }\left(x\right)=\frac{6x}{28}$ (orange).

It is clear that $\mathfrak{V}:\mathfrak{D}\to \mathfrak{D}$ was a non-expansive mapping and ${\kappa }_5=1$. Additionally, for all $\zeta \in [0,1]$, we obtained ${\Lambda }_2^{\star }:=\frac{6}{7}<1$. It can be seen that all conditions of Corollary 2 were satisfied. Consequently, the proposed model (24) had a solution. Furthermore, when we picked an initial approximation ${\mathfrak{H}}_0=\mathbf{I}\in {\mathfrak{C}}_B^S$ (whereas $\mathbf{I}$ is an identity function), the next iteration led to the unique solution of (23):

$$\begin{array}{ccc}\hfill {\mathfrak{H}}_1\left(x\right)& =& \frac{1}{14490}\left(-2139x^2+3427x-129\zeta x^2+1361\zeta x\right),\hfill \\ \hfill {\mathfrak{H}}_2\left(x\right)& =& \zeta x{\mathfrak{H}}_1\left((x+3)23\right)+(1-\zeta )x{\mathfrak{H}}_1\left((3x+1)/45\right)+\zeta (1-x){\mathfrak{H}}_1\left(x/5\right)\hfill \\ & & +(1-\zeta )(1-x){\mathfrak{H}}_1\left(6x/28\right),\hfill \\ & ⋮& \\ \hfill {\mathfrak{H}}_n\left(x\right)& =& \zeta x{\mathfrak{H}}_{n-1}\left((x+3)23\right)+(1-\zeta )x{\mathfrak{H}}_{n-1}\left((3x+1)/45\right)+\zeta (1-x){\mathfrak{H}}_{n-1}\left(x/5\right)\hfill \\ & & +(1-\zeta )(1-x){\mathfrak{H}}_{n-1}\left(6x/28\right)\hfill \end{array}$$

for all $n\in \mathbb{N}$.

5. Conclusions

Several studies on animal behavior within stochastic learning situations have produced results that are compatible with so-called occurrence investigations. Nevertheless, based on the location of the food source and the side chosen by the animal, such scenarios can be divided into four categories (right reward, right non-reward, left reward, and left non-reward). Based on such circumstances, in this study, we proposed a new stochastic model that can be used to examine a diverse range of animal behavior models under certain conditions. The solutions and stability of the proposed model were investigated using the Banach fixed point theorem. Some examples were also provided to supplement the results. For interested readers, we offer the accompanying open problems.

$Problem1:$

Is there another method for establishing the conclusions of Theorems 2 and 3?

$Problem2:$

Is it possible to prove Theorem 2 without using Condition $\left({\tilde{\mathcal{O}}}_4\right)$?

$Problem3:$

What would happen if an animal did not react to any particular trial?

$Problem4:$

Can we extend the proposed approach to discuss the behavior of animals in situations with more than four choices?