Applying Theorems on b-Metric Spaces to Differential and Integral Equations Through Connected-Image Contractions

Abstract

This paper introduces a new concept of a connected-image set for a mapping, which extends the notion of edge-preserving properties with respect to mapping. We also present novel definitions of connected-image contractions, with a focus on fixed-point theorems involving auxiliary functions in b-metric spaces. The relationships between these mathematical concepts are explored, along with their applications to solving differential and integral equations. In particular, we discuss existence results for solving integral equations and second-order ordinary differential equations with inhomogeneous Dirichlet boundary conditions, as well as theorems related to contractions of the integral type.

1. Introduction

Differential and integral equations play a vital role in agricultural economics by modeling key processes such as crop growth, resource allocation, and pest control. These equations allow people to study how changes in inputs, such as water, fertilizers, and labor, affect agricultural outcomes like crop yield and profitability. Differential equations provide valuable insights into the dynamics of economic systems, as illustrated in [1,2], enabling predictions about growth trends and resource use. On the other hand, integral equations are crucial for managing resources, assessing soil moisture, and optimizing irrigation practices, as demonstrated in [3,4,5]. The existence of solutions to these equations is often supported by fixed-point theorems, as shown in [6,7,8], ensuring that these models are reliable for practical agricultural applications. These mathematical tools help us understand and improve farming practices, leading to more efficient and sustainable agriculture.

In a general metric space, the distance function satisfies non-negativity, symmetry, and the strict triangle inequality. Banach’s fixed-point theorem in these spaces guarantees the existence of a unique fixed point for contraction mappings on complete metric spaces, forming a critical tool in various mathematical applications. Expanding on this, b-metric spaces, introduced by Czerwik in 1993 [9], generalize the strict triangle inequality by allowing a relaxation factor $s\ge 1$, as defined below.

Definition 1 

([9]). Let X be a nonempty set. A function $d_s:X\times X\to [0,\infty )$ is called a b-metric on X if, for all $u,v,w\in X$ and for some constant $s\ge 1$, it satisfies the following conditions:

• $d_s(u,v)=0$ if and only if $u=v$;
• $d_s(u,v)=d_s(v,u)$;
• $d_s(u,w)\le s[d_s(u,v)+d_s(v,w)].$

Additionally, the pair $(X,d_s)$ is known as a b-metric space.

It is important to note that every metric space can be considered a b-metric space with a fixed constant value of $s=1$. Later, in 1998, Czerwik [10] introduced the concept of Cauchy sequences in b-metric spaces, enhancing traditional notions of convergence.

Definition 2 

([10]). Let $(X,d_s)$ be a b-metric space and $\left\{u_n\right\}$ be a sequence in X. For any $\epsilon >0$ and $n,m\in \mathbb{N}$, the following are true:

• $\left\{u_n\right\}$ is Cauchy if and only if there exists $N\in \mathbb{N}$ such that $d_s(u_n,u_m)<\epsilon$ for all $n,m\ge N$.
• $\left\{u_n\right\}$ converges to $u\in X$ if and only if there exists $N\in \mathbb{N}$ such that $d_s(u_n,u)<\epsilon$ for all $n\ge N$.
• $(X,d_s)$ is complete if and only if every Cauchy sequence converges in X.

To clarify this concept, we present some examples of b-metric spaces below.

Example 1 

([6]). Let $X=C[0,L]$, where $L>0$, be the set of all real continuous functions on $[0,L]$. The fuction $d_s:C[0,L]\times C[0,L]\to [0,\infty )$ is defined by

$${\displaystyle d_s(u\left(t\right),v\left(t\right))=\underset{0\le t\le L}{sup}{|u\left(t\right)-v\left(t\right)|}^2,u\left(t\right),v\left(t\right)\in C[0,L].}$$

Thus, $(C[0,L],d_s)$ is a complete b-metric space with $s=2$.

In 2008, Jachymski [11] significantly advanced the field by introducing fixed-point theorems for metric spaces endowed with a directed graph $\mathcal{G}$. He defined the concept of $\mathcal{G}$-continuity, which facilitates the extension of the well-known Banach contraction principle to spaces with graphs. This generalization allows for the application of contraction mappings in more complex settings, enriching the study of mathematical analysis.

Definition 3 

([11]). Let X be endowed with a directed graph $\mathcal{G}=\left(V\left(\mathcal{G}\right),E\left(\mathcal{G}\right)\right)$ such that the set $V\left(\mathcal{G}\right)$ of its vertices corresponds to X, and the set $E\left(\mathcal{G}\right)$ of its edges contains all loops but no parallel edges. For any $x\in X,$ a function $\gamma :X\to X$ is said to be $\mathcal{G}$-continuous at x if $\gamma x_n\to \gamma x$ for each sequence $\left\{x_n\right\}$ in X with $x_n\to x$ and $(x_n,x_{n+1})\in E\left(\mathcal{G}\right)$ for all $n\in \mathbb{N}.$ In addition, we say that γ is $\mathcal{G}$-continuous if it is $\mathcal{G}$-continuous at every point in X.

Subsequently, in 2023, Suebcharoen et al. [7] studied fixed-point theorems that utilize auxiliary functions in spaces with directed graphs, emphasizing how these functions aid in establishing fixed-point results within complex structures. Their research was inspired by earlier studies, particularly the work of Karapınar et al. [12], which demonstrated the effectiveness of auxiliary functions in fixed-point theorems for fractional differential equations. Together, these contributions highlight the significant role of auxiliary functions in the advancement of fixed-point theory, motivating us to modify and apply these concepts to b-metric spaces.

This paper is organized as follows: Section 2 introduces the concepts of connected-image set contractions, and examines the existence of fixed points for auxiliary functions. Section 3 demonstrates how these concepts can be applied to establish the existence of solutions for differential and integral equations. Finally, this work aims to provide a clear understanding of these topics and their practical use.

2. Main Results

We start this section by defining key concepts related to connected-image contractions.

Definition 4. 

Let $(X,d_s)$ be a b-metric space endowed with a directed graph $\mathcal{G}$, and let $\gamma :X\to X$ be a function. The connected-image set is defined as

$$D\left(\gamma \right):=\{u\in X:(u,\gamma u)\in E(\mathcal{G}\left)\right\},$$

with the following properties:

• If $u\in D\left(\gamma \right)$, then $\gamma u\in D\left(\gamma \right)$ for all $u\in X$,
• If $(u,v)\in E\left(\mathcal{G}\right)$ and $v\in D\left(\gamma \right)$, then $(u,\gamma v)\in E\left(\mathcal{G}\right)$ for all $u,v\in X.$

Additionally, we define the fixed-point set of γ as

$$F\left(\gamma \right):=\{u\in X:\gamma u=u\},$$

which contains all fixed points of γ.

Remark 1. 

Let $X\ne ⌀$ be endowed with a directed graph $\mathcal{G}=\left(V\right(\mathcal{G}),E(\mathcal{G}\left)\right)$, and let γ be a self-mapping on X. If the edge set $E\left(\mathcal{G}\right)$ is transitive edge-preserving with respect to γ, then $D\left(\gamma \right)$ is a connected-image set.

The next example will demonstrate that $D\left(\gamma \right)$ is a connected-image set, while $E\left(\mathcal{G}\right)$ is not transitive edge-preserving with respect to $\gamma$.

Example 2. 

Let $X=\{0,1,2,3\}$ be endowed with a directed graph $\mathcal{G}=V\left(\mathcal{G}\right)$, where γ is a self-mapping defined as follows:

$$\gamma 0=0,\gamma 1=2,\gamma 2=1,\gamma 3=3.$$

The edge set $E\left(\mathcal{G}\right)$ is given by:

$$E\left(\mathcal{G}\right)=\left\{\right(0,1),(0,2),(1,1),(2,2),(1,2),(2,1),(1,3),(2,3\left)\right\}.$$

Since $(1,\gamma 1)=(1,2)$ and $(2,\gamma 2)=(2,1)$ are both in $E\left(\mathcal{G}\right)$, it follows that $1,2\in D\left(\gamma \right)$. Consequently, we also have $(\gamma 1,{\gamma }^{2}1)=(2,1)$ and $(\gamma 2,{\gamma }^{2}2)=(1,2)$ in $E\left(\mathcal{G}\right)$, which implies that $\gamma 1,\gamma 2\in D\left(\gamma \right)$. Additionally, since $(0,1),(0,2),(1,1),(1,2),(2,1),(2,2)\in E\left(\mathcal{G}\right)$ and $1,2\in D\left(\gamma \right)$, we can conclude that $(0,\gamma 1),(0,\gamma 2),(1,\gamma 1),(1,\gamma 2),(2,\gamma 1),(2,\gamma 2)\in E\left(\mathcal{G}\right).$ Therefore, $D\left(\gamma \right)$ is a connected-image set. However, while $(0,1),(1,3)\in E\left(\mathcal{G}\right)$, the pair $(0,3)\notin E\left(\mathcal{G}\right)$. This indicates that $E\left(\mathcal{G}\right))$ is not transitive edge-preserving with respect to γ.

Let $(X,d_s,\gamma ,\mathcal{G})$ represent a structure with the following properties throughout this work:

• $X\ne ⌀$, and $(X,d_s)$ is a b-metric space;
• X is endowed with a directed graph $\mathcal{G}=\left(V\right(\mathcal{G}),E(\mathcal{G}\left)\right)$;
• $\gamma$ is a self-mapping.

Definition 5. 

On $(X,d_s,\gamma ,\mathcal{G})$, the sequence $\left\{x_n\right\}$ is called a ${\mathcal{G}}_{\gamma }$-sequence if it satisfies the following conditions:

(S1)

$x_n=\gamma x_{n-1}={\gamma }^n{x}_0$;

(S2)

$(x_m,x_n)\in E\left(\mathcal{G}\right)$ for all $n\ge m\ge 0$.

Lemma 1. 

On $(X,d_s,\gamma ,\mathcal{G})$, if $D\left(\gamma \right)$ is a nonempty connected-image set, then there exists a ${\mathcal{G}}_{\gamma }$-sequence.

Proof. 

Assume that $D\left(\gamma \right)\ne ⌀$. There exists an element $x_0\in X$ such that $(x_0,\gamma x_0)\in E\left(\mathcal{G}\right)$. We define a sequence $\left\{x_n\right\}$ in X as

$$x_n=\gamma x_{n-1}={\gamma }^n{x}_0$$

for all $n\ge 1$. It follows from the initial condition that $(x_0,x_1)=(x_0,\gamma x_0)\in E\left(\mathcal{G}\right)$. According to the definition of $D\left(\gamma \right)$, we can conclude that $(x_n,x_{n+1})=({\gamma }^n{x}_0,{\gamma }^{n+1}{x}_0)\in E\left(\mathcal{G}\right)$ for all $n\ge 0$. Moreover, since $(x_{n+1},\gamma x_{n+1})\in E\left(\mathcal{G}\right)$, we can assert that $(x_n,x_{n+2})\in E\left(\mathcal{G}\right)$. By repeating this reasoning, we deduce that

$$(x_m,x_n)\in E\left(\mathcal{G}\right)$$

for all $n\ge m\ge 0.$ □

Next, we introduce the class $\mathcal{A}\left(X\right)$ as the collection of all auxiliary functions $h:X\times X\to [0,{\displaystyle \frac{1}{s}}]$ that satisfy the condition

$$\mathrm{if}\underset{n\to \infty }{lim}h(x_n,y_n)={\displaystyle \frac{1}{s}},\mathrm{then}\underset{n\to \infty }{lim}{d}_s(x_n,y_n)=0$$

for every pair of sequences $\left\{x_n\right\}$ and $\left\{y_n\right\}$ in X such that the sequence $\left\{d_s(x_n,y_n)\right\}$ is decreasing, where $(X,d_s)$ is a b-metric space with $s\ge 1$.

Example 3. 

Consider the function $h:\mathbb{R}\times \mathbb{R}\to [0,{\displaystyle \frac{1}{s}}]$ with $s\ge 1$, defined for all $x,y\in \mathbb{R}$ as

$$h(x,y)={\displaystyle \frac{e^{-d_s(x,y)}}{s}}.$$

It follows that $h\in \mathcal{A}\left(X\right)$.

Let $\psi :[0,\infty )\to [0,\infty )$ be a function that satisfies the following properties:

• $\psi$ is an increasing and continuous function;
• $\psi \left(t\right)=0$ if and only if $t=0$;
• For any constant $C>0$, $\psi \left(Ct\right)=C\psi \left(t\right)$ for all $t\ge 0$.

We will refer to the collection of all functions $\psi$ that meet these criteria as $\mathsf{\Psi }$. Now, we are ready to present the connected-image contractions of types R and M.

Definition 6. 

On $(X,d_s,\gamma ,\mathcal{G})$, a mapping γ is said to be a connected-image contraction of type R if for each $x,y\in X$, the following hold:

• $D\left(\gamma \right)$ is a connected-image set;
• There exists $h\in \mathcal{A}\left(X\right)$ and $\psi \in \mathsf{\Psi }$ such that $(x,y)\in E\left(\mathcal{G}\right)$ and

  $$\psi \left(d_s(\gamma x,\gamma y)\right)\le h(x,y)\psi \left(R(x,y)\right),$$

  where $R:X\times X\to [0,\infty )$ is defined by

  $$\begin{array}{cc}\hfill R(x,y)=max& \{{\displaystyle \frac{d_s(x,\gamma x)d_s(\gamma y,y)}{d_s(x,y)}}+|d_s(x,y)-d_s(x,\gamma x)|,\hfill \\ \hfill & d_s(x,y)+|d_s(x,\gamma x)-d_s(y,\gamma y)|,\hfill \\ \hfill & d_s(x,\gamma x)+|d_s(x,y)-d_s(y,\gamma y)|,\hfill \\ \hfill & d_s(y,\gamma y)+|d_s(x,y)-d_s(x,\gamma x)|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x,\gamma y)+d_s(y,\gamma x)+s|d_s(x,\gamma x)-d_s(y,\gamma y)|}{2s}}\}.\hfill \end{array}$$

Definition 7. 

On $(X,d_s,\gamma ,\mathcal{G})$, a mapping γ is said to be a connected-image contraction of type M if for each $x,y\in X$, the following hold:

• $D\left(\gamma \right)$ is a connected-image set;
• There exists $h\in \mathcal{A}\left(X\right)$ and $\psi \in \mathsf{\Psi }$ such that $(x,y)\in E\left(\mathcal{G}\right)$ and

  $$\psi \left(d_s(\gamma x,\gamma y)\right)\le h(x,y)\psi \left(M(x,y)\right),$$

  where $M:X\times X\to [0,\infty )$ is defined by

  $$\begin{array}{cc}\hfill M(x,y)=max\{& {\displaystyle \frac{d_s(x,\gamma x)[1+d_s(y,\gamma y)]}{1+d_s(x,y)}}+|d_s(x,y)-d_s(x,\gamma x)|,\hfill \\ \hfill & {\displaystyle \frac{d_s(y,\gamma y)[1+d_s(x,\gamma x)]}{1+d_s(x,y)}}+|d_s(x,y)-d_s(x,\gamma x)|,\hfill \\ \hfill & d_s(x,y)+|d_s(x,\gamma x)-d_s(y,\gamma y)|\}.\hfill \end{array}$$

Lemma 2. 

On $(X,d_s,\gamma ,\mathcal{G})$, let γ be an a connected-image contraction of type R with $D\left(\gamma \right)\ne \varnothing$. Then, we obtain a ${\mathcal{G}}_{\gamma }$-sequence $\left\{x_n\right\}$ such that, if $d_s(x_n,x_{n+1})\ne 0$ for all $n\ge 0$, then $\underset{n\to \infty }{lim}{d}_s(x_n,x_{n+1})=0.$

Proof. 

According to Lemma 1, we have a ${\mathcal{G}}_{\gamma }$-sequence $\left\{x_n\right\}$, and since $\gamma$ is a connected-image contraction of type R, for each $n\ge 0$ and $s\ge 1$,

$$\begin{array}{ccc}\hfill \psi \left(d_s(x_{n+1},x_{n+2})\right)& =& \psi \left(d_s(\gamma x_n,\gamma x_{n+1})\right)\hfill \\ & \le & h(x_n,x_{n+1})\psi \left(R(x_n,x_{n+1})\right)\hfill \\ & \le & {\displaystyle \frac{1}{s}}\psi \left(R(x_n,x_{n+1})\right)\hfill \\ & \le & \psi \left(R(x_n,x_{n+1})\right).\hfill \end{array}$$

(1)

Furthermore, a straightforward calculation shows that

$$\begin{array}{cc}\hfill R(x_n,x_{n+1})=max& \{{\displaystyle \frac{d_s(x_n,\gamma x_n)d_s(\gamma x_{n+1},x_{n+1})}{d_s(x_n,x_{n+1})}}+|d_s(x_n,x_{n+1})-d_s(x_n,\gamma x_n)|,\hfill \\ \hfill & d_s(x_n,x_{n+1})+|d_s(x_n,\gamma x_n)-d_s(x_{n+1},\gamma x_{n+1})|,\hfill \\ \hfill & d_s(x_n,\gamma x_n)+|d_s(x_n,x_{n+1})-d_s(x_{n+1},\gamma x_{n+1})|,\hfill \\ \hfill & d_s(x_{n+1},\gamma x_{n+1})+|d_s(x_n,x_{n+1})-d_s(x_n,\gamma x_n)|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x_n,\gamma x_{n+1})+d_s(x_{n+1},\gamma x_n)+s|d_s(x_n,\gamma x_n)-d_s(x_{n+1},\gamma x_{n+1})|}{2s}}\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =max& \{{\displaystyle \frac{d_s(x_n,x_{n+1})d_s(x_{n+2},x_{n+1})}{d_s(x_n,x_{n+1})}}+|d_s(x_n,x_{n+1})-d_s(x_n,x_{n+1})|,\hfill \\ \hfill & d_s(x_n,x_{n+1})+|d_s(x_n,x_{n+1})-d_s(x_{n+1},x_{n+2})|,\hfill \\ \hfill & d_s(x_n,x_{n+1})+|d_s(x_n,x_{n+1})-d_s(x_{n+1},x_{n+2})|,\hfill \\ \hfill & d_s(x_{n+1},x_{n+2})+|d_s(x_n,x_{n+1})-d_s(x_n,x_{n+1})|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x_n,x_{n+2})+d_s(x_{n+1},x_{n+1})+s|d_s(x_n,x_{n+1})-d_s(x_{n+1},x_{n+2})|}{2s}}\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =max& \{d_s(x_{n+2},x_{n+1}),\hfill \\ \hfill & d_s(x_n,x_{n+1})+|d_s(x_n,x_{n+1})-d_s(x_{n+1},x_{n+2})|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x_n,x_{n+2})+s|d_s(x_n,x_{n+1})-d_s(x_{n+1},x_{n+2})|}{2s}}\}.\hfill \end{array}$$

If we denote

$${\delta }_n=d_s(x_n,x_{n+1}),$$

since $d_s(x_n,x_{n+2})\le s[d_s(x_n,x_{n+1})+d_s(x_{n+1},x_{n+2})],s\ge 1$, we obtain

$$\begin{array}{c}\hfill R(x_n,x_{n+1})\le max\left\{{\delta }_{n+1},{\delta }_n+|{\delta }_n-{\delta }_{n+1}|,{\displaystyle \frac{{\delta }_n+{\delta }_{n+1}+|{\delta }_n-{\delta }_{n+1}|}{2}}\right\}.\end{array}$$

If ${\delta }_n$ is not a decreasing sequence, then there exists an integer $C\in \mathbb{N}$ such that ${\delta }_C\le {\delta }_{C+1}$. Consequently, we can state that

$$R(x_C,x_{C+1})\le {\delta }_{C+1}.$$

Using the inequality (1) along with the properties of $\psi$, we can obtain

$${\displaystyle \frac{1}{s}}\psi \left({\delta }_{C+1}\right)\le \psi \left({\delta }_{C+1}\right)\le h(x_C,x_{C+1})\psi \left(R(x_C,x_{C+1})\right)\le h(x_C,x_{C+1})\psi \left({\delta }_{C+1}\right)\le {\displaystyle \frac{1}{s}}\psi \left({\delta }_{C+1}\right).$$

Since $d_s(x_n,x_{n+1})\ne 0$ for every $n\ge 0$, we have ${\delta }_{C+1}=d_s(x_{C+1},x_{C+2})>0$. From the inequality above, it follows that

$$h(x_C,x_{C+1})={\displaystyle \frac{1}{s}}.$$

However, since $h\in \mathcal{A}\left(X\right)$, this leads to the conclusion that $d_s(x_C,x_{C+1})=0$, which is a contradiction. Therefore, ${\delta }_n$ must be decreasing, meaning ${\delta }_n>{\delta }_{n+1}$ for all $n\ge 0$. Thus, we can obtain the following inequality:

$$\begin{array}{c}\hfill R(x_n,x_{n+1})\le max\{{\delta }_{n+1},2{\delta }_n-{\delta }_{n+1},{\delta }_n\}=R^{*}\left(n\right).\end{array}$$

Since ${\delta }_n$ is bounded below, the sequence converges, and we have

$$\underset{n\to \infty }{lim}{\delta }_n=L\ge 0.$$

Now, suppose that $L>0$. Due to the properties of $\psi$, we have

$$\underset{n\to \infty }{lim}\psi \left(R^{*}\left(n\right)\right)=\psi \left(L\right)>0.$$

From the inequality (1), it follows that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{s}}\psi \left({\delta }_{n+1}\right)& ={\displaystyle \frac{1}{s}}\psi \left(d_s(x_{n+1},x_{n+2})\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{s}}\psi \left(d_s(\gamma x_n,\gamma x_{n+1})\right)\hfill \\ \hfill & \le \psi \left(d_s(\gamma x_n,\gamma x_{n+1})\right)\hfill \\ \hfill & \le h(x_n,x_{n+1})\psi \left(R(x_n,x_{n+1})\right)\hfill \\ \hfill & \le h(x_n,x_{n+1})\psi \left(R^{*}\left(n\right)\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\psi \left(R^{*}\left(n\right)\right).\hfill \end{array}$$

In the inequality above, taking the limit as $n\to \infty$ gives us

$${\displaystyle \frac{1}{s}}=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{s}}{\displaystyle \frac{\psi \left({\delta }_{n+1}\right)}{\psi \left(R^{*}\left(n\right)\right)}}\le \underset{n\to \infty }{lim}h(x_n,x_{n+1})\le {\displaystyle \frac{1}{s}}.$$

Therefore, we conclude that

$$\underset{n\to \infty }{lim}h(x_n,x_{n+1})={\displaystyle \frac{1}{s}}.$$

According to the definition of auxiliary functions, it follows that

$$\underset{n\to \infty }{lim}{d}_s(x_n,x_{n+1})=\underset{n\to \infty }{lim}{\delta }_n=0,$$

which contradicts the assumption that $L>0$. Hence, $\underset{n\to \infty }{lim}d(x_n,x_{n+1})=L=0$. □

Lemma 3. 

On $(X,d_s,\gamma ,\mathcal{G})$, let γ be a connected-image contraction of type M with $D\left(\gamma \right)\ne \varnothing$. Then, we obtain a ${\mathcal{G}}_{\gamma }$-sequence $\left\{x_n\right\}$ such that, if $d_s(x_n,x_{n+1})\ne 0$ for all $n\ge 0$, then $\underset{n\to \infty }{lim}{d}_s(x_n,x_{n+1})=0.$

Proof. 

According to Lemma 1, we have a ${\mathcal{G}}_{\gamma }$-sequence $\left\{x_n\right\}$, and since $\gamma$ is a connected-image contraction of type M, for each $n\ge 0$ and $s\ge 1$,

$$\begin{array}{ccc}\hfill \psi \left(d_s(x_{n+1},x_{n+2})\right)& =& \psi \left(d_s(\gamma x_n,\gamma x_{n+1})\right)\hfill \\ & \le & h(x_n,x_{n+1})\psi \left(M(x_n,x_{n+1})\right)\hfill \\ & \le & {\displaystyle \frac{1}{s}}\psi \left(M(x_n,x_{n+1})\right)\hfill \\ & \le & \psi \left(M(x_n,x_{n+1})\right).\hfill \end{array}$$

(2)

Furthermore, a straightforward calculation shows that

$$\begin{array}{cc}\hfill M(x_n,x_{n+1})=max\{& {\displaystyle \frac{d_s(x_n,\gamma x_n)[1+d_s(x_{n+1},\gamma x_{n+1})]}{1+d_s(x_n,x_{n+1})}}+|d_s(x_n,x_{n+1})-d_s(x_n,\gamma x_n)|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x_{n+1},\gamma x_{n+1})[1+d_s(x_n,\gamma x_n)]}{1+d_s(x_n,x_{n+1})}}+|d_s(x_n,x_{n+1})-d_s(x_n,\gamma x_n)|,\hfill \\ \hfill & d_s(x_n,x_{n+1})+|d_s(x_n,\gamma x_n)-d_s(x_{n+1},\gamma x_{n+1})|\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =max\{& {\displaystyle \frac{d_s(x_n,x_{n+1})[1+d_s(x_{n+1},x_{n+2})]}{1+d_s(x_n,x_{n+1})}}+|d_s(x_n,x_{n+1})-d_s(x_n,x_{n+1})|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x_{n+1},x_{n+2})[1+d_s(x_n,x_{n+1})]}{1+d_s(x_n,x_{n+1})}}+|d_s(x_n,x_{n+1})-d_s(x_n,x_{n+1})|,\hfill \\ \hfill & d_s(x_n,x_{n+1})+|d_s(x_n,x_{n+1})-d_s(x_{n+1},x_{n+2})|\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =max\{& {\displaystyle \frac{d_s(x_n,x_{n+1})[1+d_s(x_{n+1},x_{n+2})]}{1+d_s(x_n,x_{n+1})}},\hfill \\ \hfill & d_s(x_{n+1},x_{n+2}),d_s(x_n,x_{n+1})+|d_s(x_n,x_{n+1})-d_s(x_{n+1},x_{n+2})|\}.\hfill \end{array}$$

If we denote

$${\delta }_n=d_s(x_n,x_{n+1}),$$

then we obtain

$$\begin{array}{c}\hfill M(x_n,x_{n+1})=max\left\{{\displaystyle \frac{{\delta }_n(1+{\delta }_{n+1})}{1+{\delta }_n}},{\delta }_{n+1},{\delta }_n+|{\delta }_n-{\delta }_{n+1}|\right\}.\end{array}$$

If ${\delta }_n$ is not a decreasing sequence, then there exists an integer $C\in \mathbb{N}$ such that ${\delta }_C\le {\delta }_{C+1}$. Consequently, we can state that

$$\begin{array}{c}\hfill M(x_C,x_{C+1})={\delta }_{C+1}\end{array}$$

Using the inequality (2) along with the properties of $\psi$, we can obtain

$${\displaystyle \frac{1}{s}}\psi \left({\delta }_{C+1}\right)\le \psi \left({\delta }_{C+1}\right)\le h(x_C,x_{C+1})\psi \left(M(x_C,x_{C+1})\right)=h(x_C,x_{C+1})\psi \left({\delta }_{C+1}\right)\le {\displaystyle \frac{1}{s}}\psi \left({\delta }_{C+1}\right).$$

Following the same reasoning as in the proof of Lemma 2, ${\delta }_n$ must be decreasing, meaning ${\delta }_n>{\delta }_{n+1}$ for all $n\ge 0$. Thus, we can obtain the following inequality:

$$\begin{array}{c}\hfill M(x_n,x_{n+1})=max\{{\displaystyle \frac{{\delta }_n(1+{\delta }_{n+1})}{1+{\delta }_n}},{\delta }_{n+1},2{\delta }_n-{\delta }_{n+1}\}.\end{array}$$

Since ${\delta }_n$ is bounded below, the sequence converges, and we have

$$\underset{n\to \infty }{lim}{\delta }_n=K\ge 0,$$

that is,

$$\underset{n\to \infty }{lim}M(x_n,x_{n+1})=K.$$

Now, suppose that $K>0$. Due to the properties of $\psi$, we have

$$\underset{n\to \infty }{lim}\psi \left({\delta }_n\right)=\psi \left(K\right)>0.$$

From the inequality (2), it follows that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{s}}\psi \left({\delta }_{n+1}\right)& ={\displaystyle \frac{1}{s}}\psi \left(d_s(\gamma x_n,\gamma x_{n+1})\right)\hfill \\ \hfill & \le \psi \left(d_s(\gamma x_n,\gamma x_{n+1})\right)\hfill \\ \hfill & \le h(x_n,x_{n+1})\psi \left(M(x_n,x_{n+1})\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\psi \left(M(x_n,x_{n+1})\right).\hfill \end{array}$$

In the inequality above, taking the limit as $n\to \infty$ gives us

$${\displaystyle \frac{1}{s}}=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{s}}{\displaystyle \frac{\psi \left({\delta }_{n+1}\right)}{\psi \left(M(x_n,x_{n+1})\right)}}\le \underset{n\to \infty }{lim}h(x_n,x_{n+1})\le {\displaystyle \frac{1}{s}}.$$

Therefore, we conclude that

$$\underset{n\to \infty }{lim}h(x_n,x_{n+1})={\displaystyle \frac{1}{s}}.$$

According to the definition of auxiliary functions, it follows that

$$\underset{n\to \infty }{lim}{d}_s(x_n,x_{n+1})=\underset{n\to \infty }{lim}{\delta }_n=0,$$

which contradicts the assumption that $K>0$. Hence, $\underset{n\to \infty }{lim}{d}_s(x_n,x_{n+1})=K=0$. □

Lemma 4. 

On $(X,d_s,\gamma ,\mathcal{G})$, if $F\left(\gamma \right)\ne ⌀$, then $D\left(\gamma \right)\ne ⌀$.

Proof. 

Suppose $F\left(\gamma \right)\ne ⌀$. Then, there exists some $u\in X$ such that $(u,\gamma u)=(u,u)\in E\left(\mathcal{G}\right)$, which implies that $D\left(\gamma \right)\ne ⌀$. □

Here, we outline the conditions necessary for the existence of fixed points for connected-image contractions of types R and M.

Theorem 1. 

On $(X,d_s,\gamma ,\mathcal{G})$, let γ be a connected-image contraction of type R with $\mathcal{G}$-continuity. Then, we have

$$D\left(\gamma \right)\ne ⌀\iff F\left(\gamma \right)\ne ⌀.$$

Proof. 

The sufficient condition is satisfied, as established in Lemma 4. We will now demonstrate the necessary condition. According to Lemma 1, we have a sequence

$$x_n=\gamma x_{n-1}={\gamma }^n{x}_0$$

such that

$$(x_m,x_n)\in E\left(\mathcal{G}\right)$$

(3)

for all $n\ge m\ge 0.$ If $0=d_s(x_k,x_{k+1})=d_s(x_k,\gamma x_k)$ for some $k\in \mathbb{N}$, then $x_k$ is a fixed point. Conversely, let us assume that $d_s(x_n,x_{n+1})\ne 0$ for all $n\in \mathbb{N}$. According to Lemma 2, for each $n\ge 0$, we have

$$\underset{n\to \infty }{lim}{d}_s(x_n,x_{n+1})=0.$$

(4)

The sequence $\left\{x_n\right\}$ must be a Cauchy sequence, as we will demonstrate below. To assume otherwise, suppose that $\left\{x_n\right\}$ is not Cauchy. This means there exists some $ϵ>0$ such that, for every $k\in \mathbb{N}$, we can find indices $n_k$ and $m_k$ in $\mathbb{N}$ with $n_k>m_k\ge k$, where $n_k$ is the smallest integer that satisfies the following conditions:

$$d_s(x_{m_k},x_{n_k})\ge ϵ\mathrm{and}{d}_s(x_{m_k},x_{n_k-1})<ϵ.$$

(5)

From (5) and applying the b-triangular inequality, we obtain

$$ϵ\le d_s(x_{m_k},x_{n_k})\le s(d_s(x_{m_k},x_{m_k+1})+d_s(x_{m_k+1},x_{n_k})).$$

Using the properties of $\psi$, we can express this as

$${\displaystyle \frac{1}{s}}\psi \left(ϵ\right)=\psi \left({\displaystyle \frac{ϵ}{s}}\right)\le \psi \left({\displaystyle \frac{d_s(x_{m_k},x_{n_k})}{s}}\right)\le \psi (d_s(x_{m_k},x_{m_k+1})+d_s(x_{m_k+1},x_{n_k})).$$

Taking the limit as $k\to \infty$ along with (4), we obtain

$${\displaystyle \frac{1}{s}}\psi \left(ϵ\right)\le \underset{k\to \infty }{lim\; sup}\psi \left(d_s(x_{m_k+1},x_{n_k})\right).$$

We now conclude that

$R(x_{m_k},x_{n_k-1})$

$$\begin{array}{cc}\hfill =max& \{{\displaystyle \frac{d_s(x_{m_k},\gamma x_{m_k})d_s(\gamma x_{n_k-1},x_{n_k-1})}{d_s(x_{m_k},x_{n_k-1})}}+|d_s(x_{m_k},x_{n_k-1})-d_s(x_{m_k},\gamma x_{m_k})|,\hfill \\ \hfill & d_s(x_{m_k},x_{n_k-1})+|d_s(x_{m_k},\gamma x_{m_k})-d_s(x_{n_k-1},\gamma x_{n_k-1})|,\hfill \\ \hfill & d_s(x_{m_k},\gamma x_{m_k})+|d_s(x_{m_k},x_{n_k-1})-d_s(x_{n_k-1},\gamma x_{n_k-1})|,\hfill \\ \hfill & d_s(x_{n_k-1},\gamma x_{n_k-1})+|d_s(x_{m_k},x_{n_k-1})-d_s(x_{m_k},\gamma x_{m_k})|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x_{m_k},\gamma x_{n_k-1})+d_s(x_{n_k-1},\gamma x_{m_k})+s|d_s(x_{m_k},\gamma x_{m_k})-d_s(x_{n_k-1},\gamma x_{n_k-1})|}{2s}}\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =max& \{{\displaystyle \frac{d_s(x_{m_k},x_{m_k+1})d_s(x_{n_k},x_{n_k-1})}{d_s(x_{m_k},x_{n_k-1})}}+|d_s(x_{m_k},x_{n_k-1})-d_s(x_{m_k},x_{m_k+1})|,\hfill \\ \hfill & d_s(x_{m_k},x_{n_k-1})+|d_s(x_{m_k},x_{m_k+1})-d_s(x_{n_k-1},x_{n_k})|,\hfill \\ \hfill & d_s(x_{m_k},x_{m_k+1})+|d_s(x_{m_k},x_{n_k-1})-d_s(x_{n_k-1},x_{n_k})|,\hfill \\ \hfill & d_s(x_{n_k-1},x_{n_k})+|d_s(x_{m_k},x_{n_k-1})-d_s(x_{m_k},x_{m_k+1})|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x_{m_k},x_{n_k})+d_s(x_{n_k-1},x_{m_k+1})+s|d_s(x_{m_k},x_{m_k+1})-d_s(x_{n_k-1},x_{n_k})|}{2s}}\}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le max& \{{\displaystyle \frac{d_s(x_{m_k},x_{m_k+1})d_s(x_{n_k},x_{n_k-1})}{d_s(x_{m_k},x_{n_k-1})}}+d_s(x_{m_k},x_{n_k-1})+d_s(x_{m_k},x_{m_k+1}),\hfill \\ \hfill & d_s(x_{m_k},x_{n_k-1})+d_s(x_{m_k},x_{m_k+1})+d_s(x_{n_k-1},x_{n_k}),\hfill \\ \hfill & d_s(x_{m_k},x_{m_k+1})+d_s(x_{m_k},x_{n_k-1})+d_s(x_{n_k-1},x_{n_k}),\hfill \\ \hfill & d_s(x_{n_k-1},x_{n_k})+d_s(x_{m_k},x_{n_k-1})+d_s(x_{m_k},x_{m_k+1}),\hfill \\ \hfill & {\displaystyle \frac{d_s(x_{m_k},x_{n_k})+d_s(x_{n_k-1},x_{m_k+1})+s{d}_s(x_{m_k},x_{m_k+1})+s{d}_s(x_{n_k-1},x_{n_k})}{2s}}\}.\hfill \end{array}$$

(6)

By applying the b-triangular inequality, we obtain

$$\begin{array}{cc}\hfill d_s(x_{m_k},x_{n_k})+d_s(x_{n_k-1},x_{m_k+1})& \le s(d_s(x_{m_k},x_{n_k-1})+d_s(x_{n_k-1},x_{n_k}))\hfill \\ \hfill & +s(d_s(x_{n_k-1},x_{m_k})+d_s(x_{m_k},x_{m_k+1}))\hfill \\ \hfill & \le sϵ+s{d}_s(x_{n_k-1},x_{n_k}))+sϵ+s{d}_s(x_{m_k},x_{m_k+1})\hfill \\ \hfill & =2sϵ+s{d}_s(x_{n_k-1},x_{n_k}))+s{d}_s(x_{m_k},x_{m_k+1}).\hfill \end{array}$$

(7)

By substituting (5) and (7) into the inequality (6), we obtain

$$\begin{array}{cc}\hfill R(x_{m_k},x_{n_k-1})\le max& \{{\displaystyle \frac{d_s(x_{m_k},x_{m_k+1})d_s(x_{n_k},x_{n_k-1})}{d_s(x_{m_k},x_{n_k-1})}}+ϵ+d_s(x_{m_k},x_{m_k+1}),\hfill \\ \hfill & ϵ+d_s(x_{m_k},x_{m_k+1})+d_s(x_{n_k-1},x_{n_k}),\hfill \\ \hfill & d_s(x_{m_k},x_{m_k+1})+ϵ+d_s(x_{n_k-1},x_{n_k}),\hfill \\ \hfill & d_s(x_{n_k-1},x_{n_k})+ϵ+d_s(x_{m_k},x_{m_k+1}),\hfill \\ \hfill & {\displaystyle \frac{2sϵ+2s{d}_s(x_{n_k-1},x_{n_k}))+2s{d}_s(x_{m_k},x_{m_k+1})}{2s}}\}.\hfill \end{array}$$

Using (4) and the properties of $\psi$, we conclude that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; sup}\psi \left(R(x_{m_k},x_{n_k-1})\right)\le \psi \left(ϵ\right).\end{array}$$

From (3), we can infer that $(x_{m_k},x_{n_k-1})\in E\left(\mathcal{G}\right)$ for each $k\in \mathbb{N}$. Consequently, we have

$$\begin{array}{cc}\hfill \psi \left(d_s(x_{m_k+1},x_{n_k})\right)& =\psi \left(d_s(\gamma x_{m_k},\gamma x_{n_k-1})\right)\hfill \\ \hfill & \le h(x_{m_k},x_{n_k-1})\psi \left(R(x_{m_k},x_{n_k-1})\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\psi \left(R(x_{m_k},x_{n_k-1})\right).\hfill \end{array}$$

Since $\underset{n\to \infty }{lim}{d}_s(x_n,x_{n+1})=0$, applying the previous equality as $k\to \infty$ yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{s}}\psi \left(ϵ\right)& \le \underset{k\to \infty }{lim\; sup}\psi \left(d_s(x_{m_k+1},x_{n_k})\right)\hfill \\ \hfill & \le \underset{k\to \infty }{lim\; sup}h(x_{m_k},x_{n_k-1})\psi \left(R(x_{m_k},x_{n_k-1})\right)\hfill \\ \hfill & \le \underset{k\to \infty }{lim\; sup}h(x_{m_k},x_{n_k-1})\psi \left(ϵ\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\psi \left(ϵ\right).\hfill \end{array}$$

As a result, we find that

$${\displaystyle \underset{k\to \infty }{lim\; sup}h(x_{m_k},x_{n_k-1})=\frac{1}{s}.}$$

This implies that

$$\underset{k\to \infty }{lim}{d}_s(x_{m_k},x_{n_k-1})=0$$

(8)

By employing (5) and the b-triangular inequality, we obtain

$$0<ϵ\le d_s(x_{m_k},x_{n_k})\le s(d_s(x_{m_k},x_{n_k-1})+d_s(x_{n_k-1},x_{n_k})).$$

Taking the limit as $k\to \infty$ in the above inequality, we utilize (3) and (8) to conclude that

$$\underset{k\to \infty }{lim}{d}_s(x_{m_k},x_{n_k})=0,$$

which contradicts (5). Therefore, the sequence $\left\{x_n\right\}$ must be Cauchy in $(X,d_s)$. According to the completeness of the b-metric space, there exists an element $x\in X$ such that

$$\underset{n\to \infty }{lim}{d}_s(x_n,x)=0.$$

Given the $\mathcal{G}$-continuity of $\gamma$, we have

$${\displaystyle x=\underset{n\to \infty }{lim}{x}_{n+1}=\underset{n\to \infty }{lim}\gamma x_n=\gamma \left(\underset{n\to \infty }{lim}{x}_n\right)=\gamma x.}$$

Thus, we conclude that $x\in F\left(\gamma \right)$. □

Theorem 2. 

On $(X,d_s,\gamma ,\mathcal{G})$, if γ is a connected-image contraction of type M and satisfies at least one of the following statements, then we conclude that

$$D\left(\gamma \right)\ne ⌀\iff F\left(\gamma \right)\ne ⌀.$$

(a)

γ is $\mathcal{G}$-continuous;

(b)

If a sequence $\left\{u_n\right\}$ converges to $u\in X$ and satisfies the condition $(u_n,u_{n+1})\in E\left(\mathcal{G}\right)$ for every n, then there exists a subsequence $\left\{u_{n_k}\right\}$ from $\left\{u_n\right\}$ such that $(u_{n_k},u)\in E\left(\mathcal{G}\right)$ for every k, and

$$\mathrm{if}\underset{n\to \infty }{lim}h(x_n,y_n)={\displaystyle \frac{1}{s}},\mathrm{then}\underset{n\to \infty }{lim}{d}_s(\gamma x_n,\gamma y_n)=0.$$

Proof. 

The sufficient condition is satisfied, as established in Lemma 4. We will now demonstrate the necessary condition. According to Lemma 1, we have the sequence

$$x_n=\gamma x_{n-1}={\gamma }^n{x}_0$$

such that

$$(x_m,x_n)\in E\left(\mathcal{G}\right)$$

(9)

for all $n\ge m\ge 0.$ If $0=d_s(x_k,x_{k+1})=d_s(x_k,\gamma x_k)$ for some $k\in \mathbb{N}$, then $x_k$ is a fixed point. Conversely, let us assume that $d_s(x_n,x_{n+1})\ne 0$ for all $n\in \mathbb{N}$. According to Lemma 3, for each $n\ge 0$, we have

$$\underset{n\to \infty }{lim}{d}_s(x_n,x_{n+1})=0.$$

(10)

The sequence $\left\{x_n\right\}$ must be a Cauchy sequence, as we will demonstrate below. To assume otherwise, suppose that $\left\{x_n\right\}$ is not Cauchy. This means there exists some $ϵ>0$ such that, for every $k\in \mathbb{N}$, we can find indices $n_k$ and $m_k$ in $\mathbb{N}$ with $n_k>m_k\ge k$, where $n_k$ is the smallest integer that satisfies the following conditions:

$$d_s(x_{m_k},x_{n_k})\ge ϵ\mathrm{and}{d}_s(x_{m_k},x_{n_k-1})<ϵ.$$

By applying the reasoning from the proof of Theorem 1, we can derive that

$M(x_{m_k},x_{n_k-1})$

$$\begin{array}{cc}\hfill =max& \{{\displaystyle \frac{d_s(x_{m_k},\gamma x_{m_k})[1+d_s(x_{n_k-1},\gamma x_{n_k-1})]}{1+d_s(x_{m_k},x_{n_k-1})}}+|d_s(x_{m_k},x_{n_k-1})-d_s(x_{m_k},\gamma x_{m_k})|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x_{n_k-1},\gamma x_{n_k-1})[1+d_s(x_{m_k},\gamma x_{m_k})]}{1+d_s(x_{m_k},x_{n_k-1})}}+|d(x_{m_k},x_{n_k-1})-d_s(x_{m_k},\gamma x_{m_k})|,\hfill \\ \hfill & d_s(x_{m_k},x_{n_k-1})+|d_s(x_{m_k},\gamma x_{m_k})-d_s(x_{n_k-1},\gamma x_{n_k-1})|\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le max& \{{\displaystyle \frac{d_s(x_{m_k},x_{m_k+1})[1+d_s(x_{n_k-1},x_{n_k})]}{1+d_s(x_{m_k},x_{n_k-1})}}+ϵ+d_s(x_{m_k},x_{m_k+1}),\hfill \\ \hfill & {\displaystyle \frac{d_s(x_{n_k-1},x_{n_k})[1+d_s(x_{m_k},x_{m_k+1})]}{1+d_s(x_{m_k},x_{n_k-1})}}+ϵ+d_s(x_{m_k},x_{m_k+1}),\hfill \\ \hfill & ϵ+d_s(x_{m_k},x_{m_k+1})+d_s(x_{n_k-1},x_{n_k})\}.\hfill \end{array}$$

Using (10) along with the properties of $\psi$, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; sup}\psi \left(M(x_{m_k},x_{n_k-1})\right)\le \psi \left(ϵ\right).\end{array}$$

From (9), we can infer that $(x_{m_k},x_{n_k-1})\in E\left(\mathcal{G}\right)$ for each $k\in \mathbb{N}$. Consequently, we have

$$\begin{array}{cc}\hfill \psi \left(d_s(x_{m_k+1},x_{n_k})\right)& =\psi \left(d_s(\gamma x_{m_k},\gamma x_{n_k-1})\right)\hfill \\ \hfill & \le h(x_{m_k},x_{n_k-1})\psi \left(M(x_{m_k},x_{n_k-1})\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\psi \left(M(x_{m_k},x_{n_k-1})\right).\hfill \end{array}$$

Since $\underset{n\to \infty }{lim}{d}_s(x_n,x_{n+1})=0$, applying the previous equality as $k\to \infty$ yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{s}}\psi \left(ϵ\right)& \le \underset{k\to \infty }{lim\; sup}\psi \left(d_s(x_{m_k+1},x_{n_k})\right)\hfill \\ \hfill & \le \underset{k\to \infty }{lim\; sup}h(x_{m_k},x_{n_k-1})\psi \left(M(x_{m_k},x_{n_k-1})\right)\hfill \\ \hfill & \le \underset{k\to \infty }{lim\; sup}h(x_{m_k},x_{n_k-1})\psi \left(ϵ\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\psi \left(ϵ\right).\hfill \end{array}$$

By utilizing the arguments presented in the proof of Theorem 1, we can conclude that the sequence $\left\{x_n\right\}$ must be Cauchy in $(X,d_s)$. Based on the completeness of the b-metric space, there exists an element $x\in X$ such that

$$\underset{n\to \infty }{lim}{d}_s(x_n,x)=0.$$

(11)

Assuming that the statement $\left(a\right)$ holds, according to the $\mathcal{G}$-continuity of $\gamma$, we have

$${\displaystyle x=\underset{n\to \infty }{lim}{x}_{n+1}=\underset{n\to \infty }{lim}\gamma x_n=\gamma \left(\underset{n\to \infty }{lim}{x}_n\right)=\gamma x,\mathrm{that}\mathrm{is},x\in F\left(\gamma \right).}$$

Now, let us consider the case where the statement $\left(b\right)$ holds. If we assume that x is not a fixed point of $\gamma$, it follows that $\gamma x\ne x$, and consequently, $d_s(\gamma x,x)>0$. According to Lemma 1 and Equation (11), we have $(x_{n_k},x)\in E\left(\mathcal{G}\right)$ for all $k\in \mathbb{N}$. Thus, we can derive

$$d_s(x,\gamma x)\le s[d_s(x,\gamma x_{n_k})+d_s(\gamma x_{n_k},\gamma x)].$$

This leads to the conclusion that

$$d_s(x,\gamma x)-s{d}_s(x,\gamma x_{n_k})\le s{d}_s(\gamma x_{n_k},\gamma x).$$

The definition of $\psi$ effectively demonstrates that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{s}}\psi (d_s(x,\gamma x)-s{d}_s(x,\gamma x_{n_k}))& \le & {\displaystyle \frac{1}{s}}\psi \left(s{d}_s(\gamma x_{n_k},\gamma x)\right)\hfill \\ & =& \psi \left(d_s(\gamma x_{n_k},\gamma x)\right)\hfill \\ & \le & h(x_{n_k},x)\psi \left(M(x_{n_k},x)\right)\hfill \\ & \le & {\displaystyle \frac{1}{s}}\psi \left(M(x_{n_k},x)\right),\hfill \end{array}$$

(12)

where

$$\begin{array}{cc}\hfill M(x_{n_k},x)=max\{& {\displaystyle \frac{d_s(x_{n_k},\gamma x_{n_k})[1+d_s(x,\gamma x)]}{1+d_s(x_{n_k},x)}}+|d_s(x_{n_k},x)-d_s(x_{n_k},\gamma x_{n_k})|,\hfill \\ \hfill & {\displaystyle \frac{d_s(x,\gamma x)[1+d_s(x_{n_k},\gamma x_{n_k})]}{1+d_s(x_{n_k},x)}}+|d_s(x_{n_k},x)-d_s(x_{n_k},\gamma x_{n_k})|,\hfill \\ \hfill & d_s(x_{n_k},x)+|d_s(x_{n_k},\gamma x_{n_k})-d_s(x,\gamma x)|\}.\hfill \end{array}$$

In the equation above, as we take $k\to \infty$ and apply Lemma 3 along with (11), we obtain

$$\underset{k\to \infty }{lim}M(x_{n_k},x)=d_s(x,\gamma x)>0.$$

Utilizing the properties of $\psi$, we find

$$\underset{k\to \infty }{lim}\psi \left(M(x_{n_k},x)\right)=\psi \left(d_s(x,\gamma x)\right)>0.$$

Next, taking $k\to \infty$ in (12) yields

$$\underset{k\to \infty }{lim}h(x_{n_k},x)={\displaystyle \frac{1}{s}}.$$

According to assumption $\left(b\right)$, we have

$$d_s(x,\gamma x)=\underset{k\to \infty }{lim}{d}_s(\gamma x_{n_k},\gamma x)=0,$$

which leads to a contradiction. Consequently, we conclude that $\gamma x=x$, establishing that $\gamma$ has x as one of its fixed points. □

Theorem 3. 

Let us apply all the conditions from Theorem 1 (Theorem 2) under the assumption that $(x,y)\in E\left(\mathcal{G}\right)$ for all $x,y\in F\left(\gamma \right)$ with $x\ne y$. $D\left(\gamma \right)$ is nonempty if and only if there exists a unique fixed point of γ.

Proof. 

The sufficient condition is satisfied, as established in Lemma 4. We will now demonstrate the necessary condition. Let $D\left(\gamma \right)$ be nonempty and let $x,y\in F\left(\gamma \right)$. Suppose that $x\ne y$; then, we have $d_s(x,y)>0.$ Since $(x,y)\in E\left(\mathcal{G}\right)$, we can conclude that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{s}}\psi \left(d_s(\gamma x,\gamma y)\right)\le \psi \left(d_s(\gamma x,\gamma y)\right)\le h(x,y)\psi \left(R(x,y)\right)\le {\displaystyle \frac{1}{s}}\psi \left(R(x,y)\right)={\displaystyle \frac{1}{s}}\psi \left(d_s(\gamma x,\gamma y)\right).\end{array}$$

Based on the properties of $\psi$, we have $h(x,y)={\displaystyle \frac{1}{s}}$, which implies that $d_s(x,y)=0$. This contradicts the assumption that x and y are distinct points; therefore, x must equal y. □

To further strengthen our key findings, we offer the following illustrative example.

Example 4. 

Let $X=\mathbb{R}$ and define a distance function $d_s$ as follows:

$$d_s(u,v)={(u-v)}^{2}forallu,v\in X.$$

This establishes $(X,d_s)$ as a complete b-metric space with a constant $s=2$. Next, we set

$$h(u,v)={\displaystyle \frac{4}{9}}and\psi \left(t\right)=2tforallt\ge 0.$$

We now define a mapping $\gamma :X\to X$ and a set $E\left(\mathcal{G}\right)$ as follows:

$$\begin{array}{c}\hfill \gamma u=\left\{\begin{array}{cc}{\displaystyle \frac{3}{2}}\hfill & ifu<0,\hfill \\ u+{\displaystyle \frac{3}{2}}\hfill & if0\le u<1,\hfill \\ {\displaystyle \frac{-u+6}{2}}\hfill & if1\le u\le 3,\hfill \\ -2{(u-4)}^2+{\displaystyle \frac{7}{2}}\hfill & ifu>3,\hfill \end{array}\right.\end{array}$$

and

$$E\left(\mathcal{G}\right)=\left\{\right(u,v):u,v\in [-3,3\left]\right\}.$$

We can observe that the mapping γ is $\mathcal{G}$-continuous. We will demonstrate that γ is a connected-image contraction of type R with $D\left(\gamma \right)\ne \varnothing$. Since $(0,\gamma 0)=(0,{\displaystyle \frac{3}{2}})\in E\left(\mathcal{G}\right)$, it follows that $D\left(\gamma \right)\ne \varnothing$. If $u\in D\left(\gamma \right)$, then $u,\gamma u\in [-3,3]$. According to the definition of γ, we have $\gamma u,{\gamma }^{2}u\in [{\displaystyle \frac{3}{2}},{\displaystyle \frac{5}{2}}]$, which implies that $\gamma u\in D\left(\gamma \right)$. For each $u,v\in X$ such that $(u,v)\in E\left(\mathcal{G}\right)$ and $v\in D\left(\gamma \right)$, we know that $u,\gamma v\in [-3,3]$. Consequently, $(u,\gamma v)\in E\left(\mathcal{G}\right)$. To analyze the case when $(u,v)\in E\left(\mathcal{G}\right)$, we observe that $u,v\in [-3,3]$. This necessitates consideration of the following cases:

Case 1: Suppose $u,v\ge 0$ and $u>v$. We can identify the following three subcases:

Subcase (1.1) If $u,v\in [0,1)$, then

$$\begin{array}{cc}\hfill \psi \left(d_s(\gamma u,\gamma v)\right)& =2d_s(\gamma u,\gamma v)=2{(u-v)}^2\le 2({\displaystyle \frac{4}{9}})({\displaystyle \frac{9}{4}})\le 2({\displaystyle \frac{4}{9}})\left({\displaystyle \frac{9}{4}}+|{(u-v)}^2-{\displaystyle \frac{9}{4}}|\right)\hfill \\ \hfill & =2({\displaystyle \frac{4}{9}})\left(d_s(u,\gamma u)+|d_s(u,v)-d_s(v,\gamma v)|\right)\le {\displaystyle \frac{4}{9}}\left(2R(u,v)\right)\hfill \\ \hfill & =h(u,v)\psi \left(R\right(u,v\left)\right).\hfill \end{array}$$

Subcase (1.2) If $u,v\in [1,3]$, then

$$\begin{array}{cc}\hfill \psi \left(d_s(\gamma u,\gamma v)\right)& =2d_s(\gamma u,\gamma v)=2{\left({\displaystyle \frac{u-v}{2}}\right)}^2\le 2({\displaystyle \frac{4}{9}}){(u-v)}^2\hfill \\ \hfill & \le 2({\displaystyle \frac{4}{9}})\left({(u-v)}^2+|{({\displaystyle \frac{3u-6}{2}})}^2-{({\displaystyle \frac{3v-6}{2}})}^2|\right)\hfill \\ \hfill & =2({\displaystyle \frac{4}{9}})\left(d_s(u,v)+|d_s(u,\gamma u)-d_s(v,\gamma v)|\right)\hfill \\ \hfill & \le {\displaystyle \frac{4}{9}}\left(2R(u,v)\right)=h(u,v)\psi \left(R(u,v)\right).\hfill \end{array}$$

Subcase (1.3) If $u\in [1,3]$ and $v\in [0,1)$, then

$$\begin{array}{cc}\hfill \psi \left(d_s(\gamma u,\gamma v)\right)& =2d_s(\gamma u,\gamma v)=2{\left({\displaystyle \frac{u+2v-3}{2}}\right)}^2\le 2({\displaystyle \frac{4}{9}})({\displaystyle \frac{9}{4}})\hfill \\ \hfill & \le 2({\displaystyle \frac{4}{9}})\left({\displaystyle \frac{9}{4}}+|{(u-v)}^2-{\left({\displaystyle \frac{3u-6}{2}}\right)}^2|\right)\hfill \\ \hfill & =2({\displaystyle \frac{4}{9}})\left(d_s(v,\gamma v)+|d_s(u,v)-d_s(u,\gamma u)|\right)\hfill \\ \hfill & \le {\displaystyle \frac{4}{9}}\left(2R(u,v)\right)=h(u,v)\psi \left(R(u,v)\right).\hfill \end{array}$$

Case 2: Suppose $u,v<0$. We have $\gamma u=\gamma v={\displaystyle \frac{3}{2}}.$ Therefore, we obtain

$$0=\psi \left(d_s(\gamma u,\gamma v)\right)=\psi \left(0\right)\le h(u,v)\psi \left(R(u,v)\right).$$

Case 3: Suppose $u\ge 0$ and $v<0$. We will consider the following two subcases:

Subcase (3.1) If $u\in [0,1)$ and $v<0$, then

$$\begin{array}{cc}\hfill \psi \left(d_s(\gamma u,\gamma v)\right)& =2d_s(\gamma u,\gamma v)=2u^2\le 2\left({\displaystyle \frac{4}{9}}\right)\left({\displaystyle \frac{9}{4}}\right)\le 2\left({\displaystyle \frac{4}{9}}\right)\left({\displaystyle \frac{9}{4}}+|{(u-v)}^2-{\displaystyle \frac{9}{4}}|\right)\hfill \\ \hfill & =2\left({\displaystyle \frac{4}{9}}\right)\left(d_s(u,\gamma u)+|d_s(u,v)-d_s(v,\gamma v)|\right)\le {\displaystyle \frac{4}{9}}\left(2R(u,v)\right)\hfill \\ \hfill & =h(u,v)\psi \left(R\right(u,v\left)\right).\hfill \end{array}$$

Subcase (3.2) If $u\in [1,3]$ and $v<0$, then

$$\begin{array}{cc}\hfill \psi \left(d_s(\gamma u,\gamma v)\right)& =2d_s(\gamma u,\gamma v)=2{\left({\displaystyle \frac{3-u}{2}}\right)}^2\le 2{\displaystyle \frac{4}{9}}{\left(v-{\displaystyle \frac{3}{2}}\right)}^2\hfill \\ \hfill & \le 2{\displaystyle \frac{4}{9}}\left({(v-{\displaystyle \frac{3}{2}})}^2+|{(u-v)}^2-{({\displaystyle \frac{3u-6}{2}})}^2|\right)\hfill \\ \hfill & =2{\displaystyle \frac{4}{9}}\left(d_s(v,\gamma v)+|d_s(u,v)-d_s(u,\gamma u)|\right)\hfill \\ \hfill & \le {\displaystyle \frac{4}{9}}\left(2R(u,v)\right)=h(u,v)\psi \left(R(u,v)\right).\hfill \end{array}$$

As all conditions of Theorem 1 are fulfilled, it follows that γ has a fixed point, namely 2.

3. Applications to Integral Equations and Ordinary Differential Equations

In this section, we examined how the fixed-point theorems developed in the previous section can be applied to prove the existence of solutions for nonlinear integral equations. Building on this foundation, we also explore their application to ordinary differential equations (ODEs). By applying fixed-point theorems to ODEs, we establish the existence of solutions in this setting as well, showcasing the broad applicability of these theorems. This approach highlights the role of fixed-point theory as a valuable tool for proving the existence of solutions across different types of mathematical equations, from differential to integral.

To introduce the framework for applying the fixed-point theorems to ordinary differential equations, we first define a structure that will help establish the necessary conditions for existence theorems.

Let $X\ne \varnothing$ be a non-empty set, and let $\gamma :X\to X$ be a given mapping. Define $\Gamma$ as a family of functions $\eta :{\mathbb{R}}^2\to \mathbb{R}$ that satisfy the following conditions:

• If $\eta (u,\gamma u)\ge 0$, then $\eta (\gamma u,{\gamma }^{2}u)\ge 0$ for all $u\in X$;
• If $\eta (u,v)\ge 0$ and $\eta (v,\gamma v)\ge 0$, then $\eta (u,\gamma v)\ge 0$ for all $u,v\in X$.

Given $\eta \in \Gamma$, we define ${\mathcal{G}}_{\eta }=(X,E\left({\mathcal{G}}_{\eta }\right))$, where $E\left({\mathcal{G}}_{\eta }\right)$ is the edge set of a directed graph, specified as:

$$E\left({\mathcal{G}}_{\eta }\right)=\{(u,v)\in X\times X:\eta (u,v)\ge 0\}.$$

This configuration creates the foundation for the subsequent analysis. It is now possible to state the following lemma, which links these abstract definitions to the existence of solutions for nonlinear differential and integral equations.

Lemma 5. 

On $(C[0,L],d_s,\gamma ,{\mathcal{G}}_{\eta })$, where $\eta \in \Gamma$, the set $D\left(\gamma \right)=\{u\in C[0,L]:(u,\gamma u)\in E\left({\mathcal{G}}_{\eta }\right)\}$ forms a connected-image set.

Let $L>0$, and and define the distance function $d_s:C[0,L]\times C[0,L]\to [0,\infty )$ as

$$d_s(u,v)=\underset{t\in [0,L]}{sup}{|u\left(t\right)-v\left(t\right)|}^2,$$

for $u,v\in C[0,L]$. The pair $(C[0,L],d_s)$ forms a complete b-metric space with $s=2$. We now provide an example of how Theorem 1 can be applied to ordinary differential equations. Consider the second-order differential equation

$$u^{″}\left(t\right)=g(t,u\left(t\right)),\mathrm{for}\mathrm{all}t\in [0,1],$$

(13)

subject to the Dirichlet conditions

$$u\left(0\right)=\xi u\left(\omega \right)\mathrm{and}u\left(1\right)=0,$$

(14)

where $g:[0,1]\times C[0,1]\to \mathbb{R}$ is a continuous function, $\omega \in (0,1)$ is a fixed point, and $\xi$ is a constant such that $(1-\omega )\xi \ne 1$. It is essential to note that a function $u\in C[0,1]$ is a solution to the differential Equation (13) with the boundary conditions (14) if and only if it satisfies the following integral equation:

$$u\left(t\right)={\int }_0^1\mathsf{\Phi }(t,s)g(s,u\left(s\right))ds+{\displaystyle \frac{\xi (1-t)}{1-(1-\omega )\xi }}{\int }_0^1\mathsf{\Phi }(\omega ,s)g(s,u\left(s\right))ds,$$

where $\mathsf{\Phi }(t,s)$ is the Green’s function for the corresponding boundary value problem:

$$u^{″}\left(t\right)=g(t,u\left(t\right)),\mathrm{for}\mathrm{all}t\in [0,1],$$

with the homogeneous Dirichlet conditions

$$u\left(0\right)=0\mathrm{and}u\left(1\right)=0.$$

The Green’s function $G(t,s)$ for the boundary value problem (13)–(14) is given by the expression

$$G(t,s)=\mathsf{\Phi }(t,s)+{\displaystyle \frac{\xi (1-t)}{1-(1-\eta )\xi }}\mathsf{\Phi }(\omega ,s),$$

(15)

where $\mathsf{\Phi }(t,s)$ is defined as

$$\mathsf{\Phi }(t,s)=\left\{\begin{array}{cc}(t-1)s\hfill & \mathrm{if}0\le s\le t\le 1,\hfill \\ t(s-1)\hfill & \mathrm{if}0\le t\le s\le 1.\hfill \end{array}\right.$$

This Green’s function represents the solution to the second-order differential equation with homogeneous Dirichlet conditions and is used to transform the original boundary value problem into an equivalent integral equation. As such, $G(t,s)$ captures both the dynamics of the differential operator and the specified boundary conditions of the inhomogeneous Dirichlet. Furthermore, it is straightforward to verify the following estimate:

$${\int }_0^1\mid \mathsf{\Phi }(t,s)\mid ds\le {\displaystyle \frac{1}{8}}.$$

This bound ensures that the Green’s function is integrable over the interval $[0,1]$, with a controlled upper bound on its integral. Such a result is crucial for the subsequent analysis of the solution to the differential equation, as it guarantees the existence of a well-behaved integral representation for the solution.

Lemma 6. 

The Green’s function $G(t,s)$ defined in (15) satisfies the following inequality for $t\in [0,1]$:

$${\int }_0^1\mid G(t,s)\mid ds\le {\displaystyle \frac{1}{8}}\left(1+{\displaystyle \frac{\mid \xi \mid }{\mid 1-(1-\omega )\xi \mid }}\right).$$

Proof. 

Begin by expressing the integral of $\mid G(t,s)\mid$ as

$${\int }_0^1\mid G(t,s)\mid ds={\int }_0^1\left|\mathsf{\Phi }(t,s)+{\displaystyle \frac{\xi (1-t)}{1-(1-\omega )\xi }}\mathsf{\Phi }(\omega ,s)\right|ds.$$

The integral can be split into two terms:

$${\int }_0^1\mid G(t,s)\mid ds\le {\int }_0^1\mid \mathsf{\Phi }(t,s)\mid ds+\left|{\displaystyle \frac{\xi (1-t)}{1-(1-\omega )\xi }}\right|{\int }_0^1\mid \mathsf{\Phi }(\omega ,s)\mid ds.$$

Next, apply the known bound on the integral of $\mid \mathsf{\Phi }(t,s)\mid$, which is

$${\int }_0^1\mid \mathsf{\Phi }(t,s)\mid ds\le {\displaystyle \frac{1}{8}}.$$

Thus, the inequality becomes

$${\int }_0^1\mid G(t,s)\mid ds\le {\displaystyle \frac{1}{8}}+\left|{\displaystyle \frac{\xi (1-t)}{1-(1-\omega )\xi }}\right|{\displaystyle \frac{1}{8}}.$$

Finally, since $\mid 1-t\mid \le 1$ for all $t\in [0,1]$, the second term can be bounded as follows:

$${\int }_0^1\mid G(t,s)\mid ds\le {\displaystyle \frac{1}{8}}\left(1+{\displaystyle \frac{\mid \xi \mid }{\mid 1-(1-\omega )\xi \mid }}\right).$$

This completes the proof. □

We now introduce an operator $T:C[0,1]\to C[0,1]$ defined by the expression

$$Tu\left(t\right)={\int }_0^{1}G(t,s)g(s,u\left(s\right))ds,$$

(16)

where $G(t,s)$ represents the Green’s function associated with the differential operator, and $g(s,u(s\left)\right)$ is the given function in the differential equation. It follows that a function $u\in C[0,1]$ is a solution to the differential Equation (13) subject to the boundary conditions (14) if and only if u is a fixed point of the operator T. In other words, solving the boundary value problem is equivalent to finding a fixed point of the operator T.

Lemma 7. 

On $(C[0,1],d_s,T,{\mathcal{G}}_{\eta })$, where $\eta \in \Gamma$ and T is defined as in (16). Assume that the following condition holds:

(H1)

For all $s\in [0,1]$ and $u,v\in C[0,1]$,

$${\displaystyle |g(s,u\left(s\right))-g(s,v\left(s\right))|\le \frac{1}{\alpha }\sqrt{{\displaystyle \frac{R(u,v)ln(1+d_s(u,v))}{2d_s(u,v)}}},}$$

where

$$\alpha ={\displaystyle \frac{1}{8}}\left(1+{\displaystyle \frac{\mid \xi \mid }{\mid 1-(1-\omega )\xi \mid }}\right)$$

and $d_s(u,v)$ denotes the distance between u and v in the space $C[0,1]$. Under these assumptions, it follows that the operator T is a connected-image contraction of type R.

Proof. 

Let $\psi \left(t\right)={\displaystyle \frac{t}{2}}$ for $t\in [0,\infty )$. For $(u,v)\in E\left({\mathcal{G}}_{\eta }\right)$, the following steps are performed:

$$\psi \left(d_s(Tu,Tv)\right)={\displaystyle \frac{1}{2}}{d}_s(Tu,Tv),$$

where the distance between $Tu\left(t\right)$ and $Tv\left(t\right)$ is given by

$$d_s(Tu,Tv)=\underset{t\in [0,1]}{sup}{|Tu\left(t\right)-Tv\left(t\right)|}^2.$$

Substituting for $Tu\left(t\right)$ and $Tv\left(t\right)$ from the definition of the operator T, the expression becomes

$$d_s(Tu,Tv)=\underset{t\in [0,1]}{sup}|{\int }_0^{1}G(t,s)(g(s,u\left(s\right))-g(s,v\left(s\right)))ds{|}^2.$$

This simplifies to

$$d_s(Tu,Tv)\le \underset{t\in [0,1]}{sup}{\left({\int }_0^1\left|G(t,s)\left(g\right(s,u\left(s\right))-g(s,v\left(s\right)\left)\right)\right|ds\right)}^2.$$

Using the condition $\left(H1\right)$ for the difference between $g(s,u(s\left)\right)$ and $g(s,v(s\left)\right)$, the following inequality holds:

$$|g(s,u\left(s\right))-g(s,v\left(s\right))|\le {\displaystyle \frac{1}{\alpha }}\sqrt{{\displaystyle \frac{R(u,v)ln(1+d_s(u,v))}{2d_s(u,v)}}}.$$

Substituting this inequality into the expression for $d_s(Tu,Tv)$, we obtain

$$d_s(Tu,Tv)\le {\displaystyle \frac{1}{{\alpha }^2}}{\displaystyle \frac{R(u,v)ln(1+d_s(u,v))}{2d_s(u,v)}}\underset{t\in [0,1]}{sup}{\left({\int }_0^1\left|G(t,s)\right|ds\right)}^2.$$

Using Lemma 6, the distance between $Tu\left(t\right)$ and $Tv\left(t\right)$ is therefore bounded by

$$d_s(Tu,Tv)\le R(u,v){\displaystyle \frac{ln(1+d_s(u,v))}{2d_s(u,v)}}.$$

Finally, we define $h(u,v)={\displaystyle \frac{ln(1+d_s(u,v))}{2d_s(u,v)}}$, yielding the inequality

$$\psi \left(d_s(Tu,Tv)\right)={\displaystyle \frac{1}{2}}{d}_s(Tu,Tv)\le h(u,v)\psi \left(R(u,v)\right)$$

where $\psi \left(R\right(u,v\left)\right)$ is a function related to the distance $R(u,v)$. From the definition of a connected-image contraction of type R (see Definition 6), it follows that the operator T is a connected-image contraction of type R. □

Theorem 4. 

On $(C[0,1],d_s,T,{\mathcal{G}}_{\eta })$, where $\eta \in \Gamma$ and T is defined as in (16), suppose that condition $\left(H1\right)$ is satisfied, and that there exists a function $u_0\left(t\right)\in C[0,1]$ such that $\eta (u_0\left(t\right),T{u}_0\left(t\right))\ge 0$ for all $t\in [0,1]$. Then, T has a fixed point $u^{*}\in C[0,1]$, which is a solution to the boundary value problem (13) subject to the conditions (14).

Proof. 

Based on the assumptions, T is a connected-image contraction of type R and $D\left(F\right)\ne \varnothing$. Therefore, since all conditions in Theorem 1 are satisfied, it follows that T has a fixed point $u^{*}\in C[0,1]$, which is a solution to the boundary value problem (13) subject to the boundary conditions (14). □

We now turn our attention to the following integral equation:

$$u\left(t\right)-{\int }_0^{L}G(t,s)K(t,s,u\left(s\right))ds=q\left(t\right),$$

(17)

where $L>0$ and $q:[0,L]\to \mathbb{R}$, $G:[0,L]\times [0,L]\to \mathbb{R}$ and $K:[0,L]\times [0,L]\times C[0,L]\to \mathbb{R}$ are continuous functions. Define a mapping $T:C[0,L]\to C[0,L]$ as

$$Tu\left(t\right)=q\left(t\right)+{\int }_0^{L}G(t,s)K(t,s,u\left(s\right))ds,u\in C[0,L],t,s\in [0,L].$$

(18)

It follows that the solution to the integral Equation (17) is given by the fixed-point equation $Tu=u$.

Lemma 8. 

On $(C[0,L],d_s,T,{\mathcal{G}}_{\eta })$, where $\eta \in \Gamma$ and T is defined as in (18), assume the following conditions are satisfied:

(A1)

For all $t,s\in [0,L]$ and $u,v\in C[0,L]$,

$$|K(t,s,u\left(s\right))-K(t,s,v\left(s\right))|\le \sqrt{{\displaystyle \frac{e^{-d_s(u,v)}R(u,v)}{2}}}.$$

(A2)

For all $t,s\in [0,L]$,

$${\int }_0^L\left|G(t,s)\right|ds\le {\displaystyle \frac{1}{L}}.$$

Then, the mapping T is a connected-image contraction of type R.

Proof. 

Let $\psi \left(t\right)=t$ for each $t\in [0,\infty )$. For $u,v\in C[0,L]$ such that $(u,v)\in E\left({\mathcal{G}}_{\eta }\right)$, conditions $\left(A1\right)$ and $\left(A2\right)$ are applied to demonstrate that T is a connected-image contraction of type R. The steps are as follows:

$$\psi \left(d_s(Tu,Tv)\right)=d_s(Tu,Tv),$$

where

$$d_s(Tu,Tv)=\underset{t\in [0,L]}{sup}{|Tu\left(t\right)-Tv\left(t\right)|}^2.$$

If we substitute the expression for $Tu\left(t\right)$ and $Tv\left(t\right)$ from the definition of the mapping T, we obtain

$$d_s(Tu,Tv)=\underset{t\in [0,L]}{sup}{\left|{\int }_0^{L}G(t,s)K(t,s,u\left(s\right))ds-{\int }_0^{L}G(t,s)K(t,s,v\left(s\right))ds\right|}^2.$$

This simplifies to

$$d_s(Tu,Tv)\le \underset{t\in [0,L]}{sup}{\left({\int }_0^L\left|G(t,s)(K(t,s,u\left(s\right))-K(t,s,v\left(s\right)))\right|ds\right)}^2.$$

Now, using conditions $\left(A1\right)$ and $\left(A2\right)$, the estimate becomes

$$d_s(Tu,Tv)\le {\displaystyle \frac{e^{-d_s(u,v)}R(u,v)}{2}}{L}^2\underset{t\in [0,L]}{sup}{\left({\int }_0^L\left|G(t,s)\right|ds\right)}^2\le {\displaystyle \frac{e^{-d_s(u,v)}}{2}}R(u,v).$$

Thus, the following inequality holds:

$$d_s(Tu,Tv)\le h(u,v)R(u,v),$$

where

$$h(u,v)={\displaystyle \frac{e^{-d_s(u,v)}}{2}}.$$

Since, $\psi \left(t\right)=t$, we conclude that

$$\psi \left(d_s(Tu,Tv)\right)\le h(u,v)\psi \left(R(u,v)\right).$$

According to Lemma 5, this inequality shows that T satisfies the conditions of a connected-image contraction of type R, as required. This completes the proof that T is a connected-image contraction under the given assumptions. □

Therefore, by invoking Lemma 8, the existence of a solution to the integral equation is established.

Theorem 5. 

On $(C[0,L],d_s,T,{\mathcal{G}}_{\eta })$, where $\eta \in \Gamma$ and T is defined as in (18), suppose that conditions $\left(A1\right)$ and $\left(A2\right)$ hold, and that there exists a function $u_0\left(t\right)\in C[0,L]$ such that $\eta (u_0\left(t\right),T{u}_0\left(t\right))\ge 0$ for all $t\in [0,L]$. Under these conditions, T possesses a fixed point $u^{*}\in C[0,L]$, which serves as a solution to the integral Equation (17).

Proof. 

Under these assumptions, T is a connected-image contraction of type R, and $D\left(T\right)\ne \varnothing$. Consequently, all the conditions of Theorem 1 are satisfied, ensuring that T has a fixed point $u^{*}\in C[0,L]$, which provides a solution to the integral Equation (17). □

4. Conclusions

This paper introduces new definitions of connected-image contractions and explores their role in b-metric spaces. It focuses on developing a theoretical framework for fixed-point theorems that utilize auxiliary functions, highlighting their effectiveness in solving differential and integral equations. This work advances our understanding of fixed-point theory and provides a solid foundation for future research and applications in mathematical modeling and decision-making.