Two Types of Proximal Connected-Image Contractions in b-Metric Spaces and Applications to Fractional Differential Models

Abstract

In this paper, we introduce a new class of proximal connected-image contractions in the framework of b-metric spaces endowed with a directed graph. By employing auxiliary functions, we establish several existence and uniqueness results for best proximity points and fixed points under appropriate contractive conditions. To demonstrate the applicability of our theoretical findings, we apply the main results to a class of fractional differential equations, showing the effectiveness of the proposed approach.

1. Introduction

Fractional differential equations have emerged as a powerful tool for modeling economic and environmental processes, particularly in agricultural and resource-based systems, since their ability to capture memory and hereditary effects makes them especially suitable for describing long-term dependencies, delayed responses, and cumulative dynamics in economic activities such as investment behavior, production dynamics, market evolution, and risk-controlled financial systems [1,2], as well as in environmental and agricultural applications including crop growth, pest population control, soil moisture evolution, irrigation dynamics, and sustainable resource management, where the nonlocal structure of fractional derivatives accurately reflects historical environmental influences on long-term ecological balance and productivity [3,4]. From a theoretical perspective, the applicability and reliability of fractional models depend critically on the existence and uniqueness of their solutions. In this context, fixed point theory provides one of the most effective analytical frameworks for establishing well-posedness results for fractional differential equations. A wide range of existence and uniqueness results for fractional models has been obtained using fixed point techniques [5,6]. Consequently, the qualitative analysis of fractional differential equations via fixed point methods has become a central topic in modern applied mathematics, forming a rigorous foundation for their use in economic and environmental applications.

Over the past decades, fixed point theory has experienced substantial growth through the introduction of new geometric and algebraic structures that extend the classical metric framework. In a standard metric space, the distance function satisfies non-negativity, symmetry, and the usual triangle inequality, and Banach’s contraction principle guarantees the existence and uniqueness of a fixed point for contraction mappings on complete metric spaces, making it one of the most fundamental tools in nonlinear analysis. To broaden this classical setting, Czerwik [7] introduced in 1993 the notion of a b-metric space by relaxing the strict triangle inequality via a multiplicative constant $s\ge 1$, a generalization that has proved particularly effective in problems involving nonstandard contractions and irregular geometric structures, as defined in Definition 1.

Definition 1

([7]). 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 $a,b,c\in X$ and for some constant $s\ge 1$, it satisfies the following conditions:

•

$d_s(a,b)=0$ if and only if $a=b$,

•

$d_s(a,b)=d_s(b,a)$,

•

$d_s(a,c)\le s[d_s(a,b)+d_s(b,c)].$

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

As a result, a wide range of fixed point results has been established in b-metric spaces, offering enhanced flexibility for existence and uniqueness theory. It is worth observing that every classical metric space is a particular case of a b-metric space corresponding to the choice $s=1$. Subsequently, in 1998, Czerwik [8] extended the analytical structure of b-metric spaces by introducing appropriate notions of Cauchy sequences, convergence, and completeness.

Definition 2

([8]). Let $(X,d_s)$ be a b-metric space and let $\left\{a_n\right\}$ be a sequence in X. For any $\epsilon >0$ and $n,m\in \mathbb{N}$.

•

The sequence $\left\{a_n\right\}$ is said to be Cauchy if there exists $N\in \mathbb{N}$ such that $d_s(a_n,a_m)<\epsilon$ for all $n,m\ge N$.

•

The sequence $\left\{a_n\right\}$ is said to converge to $a\in X$ if there exists $N\in \mathbb{N}$ such that $d_s(a_n,a)<\epsilon$ for all $n\ge N$.

•

The space $(X,d_s)$ is called complete if every Cauchy sequence in X converges to a point of X.

While classical fixed point theorems guarantee the existence of points that remain invariant under a given mapping, many practical problems, particularly in optimization and approximation theory, arise in settings where such mappings do not possess fixed points. In these situations, the theory of best proximity points becomes indispensable, as it ensures the existence of points that minimize the distance between the domain and codomain, especially in the case of cyclic mappings or disjoint sets. Consequently, best proximity theory has emerged as a natural and powerful extension of classical fixed point methods. Alongside this development, fixed point theory has continued to expand through the introduction of new geometric and algebraic structures. In particular, the incorporation of graph-theoretic frameworks has provided a flexible and effective setting for dealing with discontinuous operators, order-type relations, and complex interaction systems. In this direction, Jachymski [9] made a significant contribution in 2008 by establishing fixed point results in metric spaces endowed with a directed graph $\mathcal{G}$ and by introducing the notion of $\mathcal{G}$-continuity, which allows the classical Banach contraction principle to be extended to mappings defined along graph edges. As a result, numerous authors have developed graph-based fixed point theorems that refine and generalize the classical results of Banach and others, thereby broadening the applicability of fixed point techniques to differential and fractional equations. More recently, the use of auxiliary functions, including altering distance functions, simulation functions, and admissible control functions, has further enriched both fixed point and best proximity point theory by enabling the formulation of more general and unifying contraction conditions.

Motivated by the growing interest in fixed point theory on metric-type spaces equipped with graph structures, several recent works have explored the interplay between auxiliary functions, generalized metrics, and directed graphs. In particular, Karapınar et al. [10] employed auxiliary function techniques to establish fixed point results for fractional differential equations, highlighting their effectiveness in nonlocal and nonlinear settings. Building on this idea, Suebcharoen et al. (2023) [6] investigated fixed point theorems in spaces endowed with directed graphs, demonstrating that graph-based structures provide a flexible framework for handling complex dependencies in generalized metric spaces, including b-metric spaces. On the applied side, fractional differential models have received increasing attention due to their ability to capture memory and hereditary effects; for instance, recent works in [11,12] illustrate the relevance of fractional dynamics in realistic biological systems. Collectively, these developments underscore the importance of graph-based fixed point techniques and auxiliary functions in both theoretical and applied contexts. Motivated by these advances, we introduce a new class of proximal connected-image contractions in b-metric spaces endowed with directed graphs and investigate the existence of best proximity points. Our results yield explicit fixed point criteria and are further applied to fractional differential equations, demonstrating the effectiveness of the proposed framework.

2. Main Results

In this section, we develop the theoretical framework needed to establish the existence of best proximity points and fixed points for proximal connected-image contractions in b-metric spaces endowed with directed graphs. Our main goal is to combine the geometric structure of b-metric spaces, the order structure induced by the graph, and the flexibility provided by auxiliary functions into a unified analytical setting. Throughout this work, let $(X,d_s,\mu ,\mathcal{G})$ denote a structure satisfying the following assumptions:

•

$X\ne \varnothing$, and $(X,d_s)$ is a b-metric space with $s\ge 1$,

•

X is endowed with a directed graph $\mathcal{G}=\left(V\right(\mathcal{G}),E(\mathcal{G}\left)\right)$, where $V\left(\mathcal{G}\right)$ denotes the vertex set and $E\left(\mathcal{G}\right)$ denotes the edge set, which contains all loops but no parallel edges,

•

$\mu$ is a mapping from A to B, where A and B are nonempty closed subsets of X.

The distance between A and B is measured by the standard infimum distance

$$d_s(A,B)=\inf \{d_s(a,b):a\in A,b\in B\}.$$

In many applications, this minimal distance is not zero, which prevents the existence of fixed points. This motivates the introduction of best proximity points, which serve as natural substitutes for fixed points in such settings. To facilitate their study, we also define the sets

$$A_0=\{a\in A:d_s(a,b)=d_s(A,B)\mathrm{for}\mathrm{some}b\in B\}$$

and

$$B_0=\{b\in B:d_s(a,b)=d_s(A,B)\mathrm{for}\mathrm{some}a\in A\}.$$

These sets consist precisely of the points that attain the minimal distance between the two sets. To incorporate both the b-metric structure and the graph framework into a single object, we now introduce the notion of a proximal connected-image set, which plays a central role in our approach.

Definition 3.

Given $(X,d_s,\mu ,\mathcal{G})$, for each $x\in X$, we call the set

$$\mathfrak{C}(x,\mu )=\{a\in A:d_s(a,\mu x)=d_s(A,B)\}$$

the proximal connected-image set of x if the mapping μ has the property $CI$, that is,

$$ifa\in \mathfrak{C}(x,\mu )and(x,\mu x)\in E\left(\mathcal{G}\right),then(a,\mu a)\in E\left(\mathcal{G}\right).$$

Property $CI$ guarantees that whenever a point x is connected to its image $\mu x$, every proximal point a of $\mu x$ remains connected to its own image $\mu a$. This condition plays a key role in preserving graph admissibility along proximal sequences. Motivated by the notion of $\mathcal{G}$-continuity in [9] and adapted to our framework, we introduce the following modified concept of continuity for the mapping $\mu$.

Definition 4.

Given $(X,d_s,\mu ,\mathcal{G})$, the mapping μ is said to be ${\mathcal{G}}_P$-continuous if, for every $a\in A$, there exists a sequence $\left\{a_n\right\}\subset A$ such that $(a_n,\mu a_n)\in E\left(\mathcal{G}\right)$ for all $n\in \mathbb{N}$, and

$$a_n\to a\mathit{implies}\mu a_n\to \mu a.$$

This definition ensures that $\mu$ preserves the convergence of sequences that respect the edge structure of the graph, which is essential for passing to the limit in iterative constructions. In particular, if the sequence $\left\{a_n\right\}$ satisfies $\mu a_n=a_{n+1}$ for all $n\in \mathbb{N}$, then ${\mathcal{G}}_P$-continuity coincides with the classical notion of $\mathcal{G}$-continuity. In this case, the sequence $\left\{(a_n,\mu a_n)\right\}=\left\{(a_n,a_{n+1})\right\}$ consists of edges of $\mathcal{G}$, and the preservation of convergence reduces exactly to $\mathcal{G}$-continuity in the sense of [9]. Therefore, ${\mathcal{G}}_P$-continuity extends $\mathcal{G}$-continuity to the proximal and non-self framework.

We now recall the notion of a best proximity point, which generalizes the concept of a fixed point when the domain and codomain do not intersect. A point $x\in A$ is called a best proximity point of $\mu$ if it satisfies

$$d_s(x,\mu x)=d_s(A,B).$$

The set of all best proximity points of $\mu$ is denoted by

$$BP\left(\mu \right)=\{x\in X:x\mathrm{is}\mathrm{a}\mathrm{best}\mathrm{proximity}\mathrm{point}\mathrm{of}\mu \}.$$

Remark 1.

Given $(X,d_s,\mu ,\mathcal{G})$, if $x\in \mathfrak{C}(x,\mu )$, then $x\in BP\left(\mu \right)$.

In order to ensure that the iterative procedure leading to best proximity points can be properly initiated, we introduce the following initial connected-image set.

Definition 5.

Given $(X,d_s,\mu ,\mathcal{G})$, the initial connected-image set is defined as

$$I_{\mu }=\{x\in A:(x,\mu x)\in E\left(\mathcal{G}\right)\mathit{and}\mu \left(A\right)\subseteq B_0\}.$$

The following lemma guarantees that such an iterative process is always well-defined and converges to a best proximity point under suitable continuity assumptions.

Lemma 1.

Given $(X,d_s,\mu ,\mathcal{G})$, if there exists $a_0\in I_{\mu }$ and μ has the property $CI$, then there exists a sequence $\left\{a_n\right\}\subset A$ such that $a_{n+1}$ belongs to the proximal connected-image set $\mathfrak{C}(a_n,\mu )$ for all $n\ge 0$. Moreover, if μ is ${\mathcal{G}}_P$-continuous and $a_n\to a$ in A, then $a\in BP\left(\mu \right).$

Proof. 

By the assumption, there exists $a_0\in A$ such that $(a_0,\mu a_0)\in E\left(\mathcal{G}\right)$ and $\mu \left(A\right)\subseteq B_0$. Since $\mu a_0\in \mu \left(A\right)$, there exists $a_1\in A$ such that

$$d_s(a_1,\mu a_0)=d_s(A,B),$$

which implies that $a_1\in \mathfrak{C}(a_0,\mu )$. Moreover, from $(a_0,\mu a_0)\in E\left(\mathcal{G}\right)$ and $\mu$ has the property $CI$, it follows that $(a_1,\mu a_1)\in E\left(\mathcal{G}\right)$. Then $\mathfrak{C}(a_0,\mu )$ is the proximal connected-image set. By iterating this argument, we obtain a sequence $\left\{a_n\right\}\subset A$ such that, for every $n\ge 0$,

$$d_s(a_{n+1},\mu a_n)=d_s(A,B),$$

(1)

that is, $a_{n+1}\in \mathfrak{C}(a_n,\mu )$. Consequently, we also have

$$(a_{n+1},\mu a_{n+1})\in E\left(\mathcal{G}\right)\mathrm{for}\mathrm{all}n\ge 0.$$

(2)

Then $\mathfrak{C}(a_n,\mu )$ is the proximal connected-image set for all $n\ge 0$. Now, assume that $\mu$ is ${\mathcal{G}}_P$-continuous and that $a_n\to a$ in A. From (2) and the definition of ${\mathcal{G}}_P$-continuity, it follows that $\mu a_n\to \mu a$. Passing to the limit in (1) as $n\to \infty$, we obtain

$$d_s(a,\mu a)=d_s(A,B).$$

Therefore, $a\in BP\left(\mu \right)$. □

We next introduce a class of auxiliary functions that plays a central role in formulating the contractive condition for proximal connected-image mappings and in establishing our main existence results. Let $\mathfrak{A}$ be the class of all auxiliary functions $\mathcal{A}:X^4\to \left[0,{\displaystyle \frac{1}{s}}\right]$ satisfying

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

(3)

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

Example 1.

The following are examples of functions $\mathcal{A}\in \mathfrak{A}$ when $s\ge 1$, which illustrate both simple and nonlinear choices of auxiliary functions.

(1) 

$\mathcal{A}(a,b,x,y)=c$ for $0<c<1.$

(2) 

$\mathcal{A}(a,b,x,y)=\left\{\begin{array}{cc}{\displaystyle 0}\hfill & \mathit{if}x=y,\hfill \\ {\displaystyle \frac{\arctan (d_s(x,y)+|d_s(x,a)-d_s(y,b)\left|\right)}{s(d_s(x,y)+|d_s(x,a)-d_s(y,b)\left|\right)}}\hfill & \mathit{if}x\ne y.\hfill \end{array}\right.$

Classical contraction principles are often inadequate in graph-based and non-self settings, where proximity is realized through best proximity pairs rather than fixed points. To overcome this limitation, we introduce R- and M-type proximal connected-image contractions, which generalize and refine existing contraction conditions by combining auxiliary control functions with structured distance comparisons on proximal connected-image sets. The R-type contraction measures contractiveness through rational and additive combinations of distances involving both domain and image points, while the M-type contraction further extends this framework by employing a maximum-based control function that allows greater flexibility and covers a wider class of nonlinear behaviors. This distinction enables our approach to unify several known contraction principles and adapt them effectively to b-metric spaces endowed with a directed graph.

Definition 6.

Given $(X,d_s,\mu ,\mathcal{G})$, a mapping μ is said to be a proximal connected-image contraction of type $\mathcal{R}$ if, for each $a,b,x,y\in X$ with $(a,b)\in \mathfrak{C}(x,\mu )\times \mathfrak{C}(y,\mu )$, where $\mathfrak{C}(x,\mu )$ and $\mathfrak{C}(y,\mu )$ are proximal connected-image sets. Then there exists $\mathcal{A}\in \mathfrak{A}$ such that

$$d_s(a,b)\le \mathcal{A}(a,b,x,y)\mathcal{R}(a,b,x,y),$$

where $\mathcal{R}:X^4\to [0,\infty )$ is defined by

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

This definition extends classical contraction conditions by incorporating both the auxiliary function and the geometric structure induced by the proximal connected-image sets and the b-metric. The following lemma describes the fundamental behavior of the iterative sequence generated under such contractive mappings and serves as a key step toward proving the main results.

Lemma 2.

Given $(X,d_s,\mu ,\mathcal{G})$, if μ is a proximal connected-image contraction of type $\mathcal{R}$ and there exists $a_0\in I_{\mu }$ and μ has the property $CI$, then there exists a sequence $\left\{a_n\right\}$ such that

(i) 

if $d_s(a_n,a_{n+1})=0$ for some $n\ge 0$, then $a_n\in BP\left(\mu \right)$,

(ii) 

if $d_s(a_n,a_{n+1})\ne 0$ for all $n\ge 0$, then ${\displaystyle \underset{n\to \infty }{\lim }{d}_s(a_n,a_{n+1})=0}$.

Proof. 

$\left(i\right)$ By Lemma 1, the sequence satisfies $a_{n+1}\in \mathfrak{C}(a_n,\mu )$ for all $n\ge 0$. Suppose that $d_s(a_k,a_{k+1})=0$ for some $k\ge 0$. Then $a_k=a_{k+1}$, and hence $a_k=a_{k+1}\in \mathfrak{C}(a_k,\mu )$. By Remark 1, it follows that $a_k\in BP\left(\mu \right)$.

$\left(ii\right)$ By Lemma 1, there exists a sequence $\left\{a_n\right\}\subset A$ such that $\mathfrak{C}(a_n,\mu )$ and $\mathfrak{C}(a_{n+1},\mu )$ are proximal connected-image sets, and

$$(a_{n+1},a_{n+2})\in \mathfrak{C}(a_n,\mu )\times \mathfrak{C}(a_{n+1},\mu )\mathrm{for}\mathrm{all}n\ge 0.$$

Since $\mu$ is a proximal connected-image contraction of type $\mathcal{R}$, for each $n\ge 0$ and $s\ge 1$, we obtain

$$\begin{array}{cc}\hfill d_s(a_{n+1},a_{n+2})& \le \mathcal{A}(a_{n+1},a_{n+2},a_n,a_{n+1})\mathcal{R}(a_{n+1},a_{n+2},a_n,a_{n+1})\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\mathcal{R}(a_{n+1},a_{n+2},a_n,a_{n+1})\hfill \\ \hfill & \le \mathcal{R}(a_{n+1},a_{n+2},a_n,a_{n+1}).\hfill \end{array}$$

(4)

Moreover, a direct computation shows that

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

Let us define

$${\varphi }_n=d_s(a_n,a_{n+1}),n\ge 0.$$

Since $(X,d_s)$ is a b-metric space with constant $s\ge 1$, we have

$$d_s(a_n,a_{n+2})\le s[d_s(a_n,a_{n+1})+d_s(a_{n+1},a_{n+2})].$$

Hence, it follows that

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

Assume that the sequence $\left\{{\varphi }_n\right\}$ is not decreasing. Then there exists some $K\in \mathbb{N}$ such that

$${\varphi }_K\le {\varphi }_{K+1}.$$

Consequently, we obtain

$$\mathcal{R}(a_{K+1},a_{K+2},a_K,a_{K+1})\le {\varphi }_{K+1}.$$

Using inequality (4), we deduce

$${\displaystyle \frac{1}{s}}{\varphi }_{K+1}\le {\varphi }_{K+1}\le \mathcal{A}(a_{K+1},a_{K+2},a_K,a_{K+1})\mathcal{R}(a_{K+1},a_{K+2},a_K,a_{K+1})\le {\displaystyle \frac{1}{s}}{\varphi }_{K+1}.$$

Since $d_s(a_n,a_{n+1})\ne 0$ for all $n\ge 0$, we have ${\varphi }_{K+1}=d_s(a_{K+1},a_{K+2})>0$, which implies

$$\mathcal{A}(a_{K+1},a_{K+2},a_K,a_{K+1})={\displaystyle \frac{1}{s}}.$$

However, since $\mathcal{A}\in \mathfrak{A}$, this yields

$$d_s(a_{K+1},a_{K+2})=0\mathrm{or}{d}_s(a_K,a_{K+1})=0,$$

which is a contradiction. Therefore, the sequence $\left\{{\varphi }_n\right\}$ must be decreasing; that is,

$${\varphi }_n>{\varphi }_{n+1}\mathrm{for}\mathrm{all}n\ge 0.$$

As a result, we obtain

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

Since $\left\{{\varphi }_n\right\}$ is decreasing and bounded below, it converges. Thus,

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

Now, suppose that $\varphi >0$. Then

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

From inequality (4), it follows that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{s}}{\varphi }_{n+1}& ={\displaystyle \frac{1}{s}}{d}_s(a_{n+1},a_{n+2})\le d_s(a_{n+1},a_{n+2})\hfill \\ \hfill & \le \mathcal{A}(a_{n+1},a_{n+2},a_n,a_{n+1}){\mathcal{R}}^{*}\left(n\right)\le {\displaystyle \frac{1}{s}}{\mathcal{R}}^{*}\left(n\right).\hfill \end{array}$$

Taking the limit as $n\to \infty$ in the above inequality, we obtain

$${\displaystyle \frac{1}{s}}=\underset{n\to \infty }{\lim }{\displaystyle \frac{1}{s}}{\displaystyle \frac{{\varphi }_{n+1}}{{\mathcal{R}}^{*}\left(n\right)}}\le \underset{n\to \infty }{\lim }\mathcal{A}(a_{n+1},a_{n+2},a_n,a_{n+1})\le {\displaystyle \frac{1}{s}}.$$

Therefore,

$$\underset{n\to \infty }{\lim }\mathcal{A}(a_{n+1},a_{n+2},a_n,a_{n+1})={\displaystyle \frac{1}{s}}.$$

By the definition of the auxiliary function $\mathcal{A}$, this implies that either

$$\underset{n\to \infty }{\lim }{d}_s(a_{n+1},a_{n+2})=\underset{n\to \infty }{\lim }{\varphi }_{n+1}=0,$$

or

$$\underset{n\to \infty }{\lim }{d}_s(a_n,a_{n+1})=\underset{n\to \infty }{\lim }{\varphi }_n=0.$$

Both alternatives contradict our earlier assumption that $\varphi >0$. Hence, we must have

$$\underset{n\to \infty }{\lim }{d}_s(a_n,a_{n+1})=\varphi =0.$$

□

To prove the existence of best proximity points, we combine proximal connected-image contractions with ${\mathcal{G}}_P$-continuity. The following result guarantees the existence of at least one best proximity point for $\mu$.

Theorem 1.

Let $(X,d_s,\mu ,\mathcal{G})$ be a given structure, where μ is a proximal connected-image contraction of type $\mathcal{R}$ and is ${\mathcal{G}}_P$-continuous. If there exists $a_0\in I_{\mu }$ and μ has the property $CI$, then $BP\left(\mu \right)\ne \varnothing$.

Proof. 

By Lemma 1, there exists a sequence $\left\{a_n\right\}\subset A$ such that, for all $n\ge 0$,

$$\begin{array}{c}\hfill d_s(a_{n+1},\mu a_n)=d_s(A,B),\end{array}$$

(5)

that is, $a_{n+1}\in \mathfrak{C}(a_n,\mu )$. Consequently,

$$\begin{array}{c}\hfill (a_{n+1},\mu a_{n+1})\in E\left(\mathcal{G}\right)\mathrm{for}\mathrm{all}n\ge 0.\end{array}$$

(6)

Then $\mathfrak{C}(a_n,\mu )$ is the proximal connected-image set for all $n\ge 0$. If $d_s(a_k,a_{k+1})=0$ for some $k\ge 0$, then by Lemma 2 we immediately obtain $a_k\in BP\left(\mu \right)$. Hence, we may assume that

$$d_s(a_n,a_{n+1})\ne 0\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

Again by Lemma 2, we have

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

(7)

We now show that $\left\{a_n\right\}$ is a Cauchy sequence. Suppose, on the contrary, that $\left\{a_n\right\}$ is not Cauchy. Then there exists $ϵ>0$ such that for every $k\in \mathbb{N}$, there are integers $n_k>m_k\ge k$ with $n_k$ minimal satisfying

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

(8)

From (8) and the b-triangular inequality, we have

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

Rewriting this yields

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

Letting $k\to \infty$ and using (7), we obtain

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

Next, we have

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

(9)

Applying the b-triangular inequality gives

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

(10)

Substituting (8) and (10) into (9) gives

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

Thus,

$$\underset{k\to \infty }{\lim \sup } \mathcal{R}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\le ϵ.$$

From (5) and (6),

$$(a_{m_k+1},a_{n_k})\in \mathfrak{C}(a_{m_k},\mu )\times \mathfrak{C}(a_{n_k-1},\mu ),$$

and hence,

$$\begin{array}{cc}\hfill d_s(a_{m_k+1},a_{n_k})& \le \mathcal{A}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\mathcal{R}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\mathcal{R}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1}).\hfill \end{array}$$

Since (7), letting $k\to \infty$ gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{s}}ϵ& \le \underset{k\to \infty }{\lim \sup } d_s(a_{m_k+1},a_{n_k})\hfill \\ \hfill & \le \underset{k\to \infty }{\lim \sup } \mathcal{A}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\mathcal{R}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\hfill \\ \hfill & \le \underset{k\to \infty }{\lim \sup } \mathcal{A}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})ϵ\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}ϵ.\hfill \end{array}$$

Thus,

$$\underset{k\to \infty }{\lim \sup } \mathcal{A}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})={\displaystyle \frac{1}{s}},$$

which yields

$$\underset{k\to \infty }{\lim }{d}_s(a_{m_k+1},a_{n_k})=0\mathrm{or}\underset{k\to \infty }{\lim }{d}_s(a_{m_k},a_{n_k-1})=0.$$

(11)

Finally, using (8) and the b-triangular inequality,

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

and similarly,

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

Letting $k\to \infty$ and using (7) and (11), we obtain

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

contradicting (8). Therefore, $\left\{a_n\right\}$ is Cauchy. By the completeness of X and A is closed, there exists $a\in A$ such that

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

By the ${\mathcal{G}}_P$-continuity of $\mu$ and Lemma 1, we conclude that a is a best proximity point of $\mu$, that is, $a\in BP\left(\mu \right)$. □

To capture a contractive behavior governed by rational and mixed distance expressions, we define another class of proximal connected-image contractions of type $\mathcal{M}$ as follows.

Definition 7.

Given $(X,d_s,\mu ,\mathcal{G})$, a mapping μ is said to be a proximal connected-image contraction of type $\mathcal{M}$ if, for each $a,b,x,y\in X$ with $(a,b)\in \mathfrak{C}(x,\mu )\times \mathfrak{C}(y,\mu )$, where $\mathfrak{C}(x,\mu )$ and $\mathfrak{C}(y,\mu )$ are proximal connected-image sets. Then there exists $\mathcal{A}\in \mathfrak{A}$ such that

$$d_s(a,b)\le \mathcal{A}(a,b,x,y)\mathcal{M}(a,b,x,y),$$

where $\mathcal{M}:X^4\to [0,\infty )$ is defined by

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

The following lemma establishes two fundamental properties of the iterative sequence generated by a proximal connected-image contraction of type $\mathcal{M}$.

Lemma 3.

Given $(X,d_s,\mu ,\mathcal{G})$, if μ is a proximal connected-image contraction of type $\mathcal{M}$ and there exists $a_0\in I_{\mu }$ and μ has the property $CI$, then there exists a sequence $\left\{a_n\right\}$ such that

(i) 

if $d_s(a_n,a_{n+1})=0$ for some $n\ge 0$, then $a_n\in BP\left(\mu \right)$,

(ii) 

if $d_s(a_n,a_{n+1})\ne 0$ for all $n\ge 0$, then $\underset{n\to \infty }{\lim }{d}_s(a_n,a_{n+1})=0$.

Proof. 

$\left(i\right)$ The proof is identical to that of Lemma 2.

$\left(ii\right)$ By Lemma 1, there exists a sequence $\left\{a_n\right\}\subset A$ such that $\mathfrak{C}(a_n,\mu )$ and $\mathfrak{C}(a_{n+1},\mu )$ are proximal connected-image sets, and

$$(a_{n+1},a_{n+2})\in \mathfrak{C}(a_n,\mu )\times \mathfrak{C}(a_{n+1},\mu )\mathrm{for}\mathrm{all}n\ge 0.$$

Since $\mu$ is a proximal connected-image contraction of type $\mathcal{M}$, for each $n\ge 0$ and $s\ge 1$, we obtain

$$\begin{array}{ccc}\hfill d_s(a_{n+1},a_{n+2})& \le & \mathcal{A}(a_{n+1},a_{n+2},a_n,a_{n+1})\mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1})\hfill \\ & \le & {\displaystyle \frac{1}{s}}\mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1})\hfill \\ & \le & \mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1}).\hfill \end{array}$$

(12)

Moreover, a direct computation shows that

$$\begin{array}{c}\mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1})\hfill \\ \begin{array}{cc}\hfill =\max \{& {\displaystyle \frac{d_s(a_n,a_{n+1})[1+d_s(a_{n+1},a_{n+2})]}{1+d_s(a_n,a_{n+1})}},\hfill \\ \hfill & d_s(a_{n+1},a_{n+2}),d_s(a_n,a_{n+1})+|d_s(a_n,a_{n+1})-d_s(a_{n+1},a_{n+2})|{\displaystyle \}}.\hfill \end{array}\hfill \end{array}$$

Let us define

$${\varphi }_n=d_s(a_n,a_{n+1}),n\ge 0.$$

We have

$$\begin{array}{c}\hfill \mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1})=\max \left\{{\displaystyle \frac{{\varphi }_n(1+{\varphi }_{n+1})}{1+{\varphi }_n}},{\varphi }_{n+1},{\varphi }_n+|{\varphi }_n-{\varphi }_{n+1}|\right\}.\end{array}$$

Assume that the sequence $\left\{{\varphi }_n\right\}$ is not decreasing. Then there exists an integer $K\in \mathbb{N}$ such that ${\varphi }_K\le {\varphi }_{K+1}$. Consequently, we obtain

$$\begin{array}{c}\hfill \mathcal{M}(a_{K+1},a_{K+2},a_K,a_{K+1})={\varphi }_{K+1}.\end{array}$$

Using inequality (12), we have

$${\displaystyle \frac{1}{s}}{\varphi }_{K+1}\le {\varphi }_{K+1}\le \mathcal{A}(a_{K+1},a_{K+2},a_K,a_{K+1})\mathcal{M}(a_{K+1},a_{K+2},a_K,a_{K+1})\le {\displaystyle \frac{1}{s}}{\varphi }_{K+1}.$$

By applying the same argument as in the proof of Lemma 2, it follows that $\left\{{\varphi }_n\right\}$ must be a decreasing sequence, that is, ${\varphi }_n>{\varphi }_{n+1}$ for all $n\ge 0$. Hence, we obtain

$$\begin{array}{c}\hfill \mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1})=\max \left\{{\displaystyle \frac{{\varphi }_n(1+{\varphi }_{n+1})}{1+{\varphi }_n}},{\varphi }_{n+1},2{\varphi }_n-{\varphi }_{n+1}\right\}.\end{array}$$

Since $\left\{{\varphi }_n\right\}$ is bounded below, the sequence converges, and therefore

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

which implies

$$\underset{n\to \infty }{\lim }\mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1})=\varphi .$$

Suppose, on the contrary, that $\varphi >0$. Then

$$\underset{n\to \infty }{\lim }{\varphi }_n=\varphi >0.$$

From inequality (12), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{s}}{\varphi }_{n+1}& ={\displaystyle \frac{1}{s}}{d}_s(a_{n+1},a_{n+2})\hfill \\ \hfill & \le d_s(a_{n+1},a_{n+2}))\hfill \\ \hfill & \le \mathcal{A}(a_{n+1},a_{n+2},a_n,a_{n+1})\mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1})\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1}).\hfill \end{array}$$

Taking the limit as $n\to \infty$ in the above inequality yields

$${\displaystyle \frac{1}{s}}=\underset{n\to \infty }{\lim }{\displaystyle \frac{1}{s}}{\displaystyle \frac{{\varphi }_{n+1}}{\mathcal{M}(a_{n+1},a_{n+2},a_n,a_{n+1})}}\le \underset{n\to \infty }{\lim }\mathcal{A}(a_{n+1},a_{n+2},a_n,a_{n+1})\le {\displaystyle \frac{1}{s}}.$$

Therefore,

$$\underset{n\to \infty }{\lim }\mathcal{A}(a_{n+1},a_{n+2},a_n,a_{n+1})={\displaystyle \frac{1}{s}}.$$

By the definition of auxiliary functions, it follows that

$$\underset{n\to \infty }{\lim }{d}_s(a_{n+1},a_{n+2})=\underset{n\to \infty }{\lim }{\varphi }_{n+1}=0$$

or

$$\underset{n\to \infty }{\lim }{d}_s(a_n,a_{n+1})=\underset{n\to \infty }{\lim }{\varphi }_n=0,$$

which contradicts the assumption that $\varphi >0$. Hence,

$$\underset{n\to \infty }{\lim }{d}_s(a_n,a_{n+1})=\varphi =0.$$

□

Using the convergence properties established in the previous lemma for proximal connected-image contractions of type $\mathcal{M}$, we now prove the existence of best proximity points under the additional assumption of ${\mathcal{G}}_P$-continuity.

Theorem 2.

Let $(X,d_s,\mu ,\mathcal{G})$ be a given structure, where μ is a proximal connected-image contraction of type $\mathcal{M}$ and is ${\mathcal{G}}_P$-continuous. If there exists $a_0\in I_{\mu }$ and μ has the property $CI$, then $BP\left(\mu \right)\ne \varnothing$.

Proof. 

The proof follows the same framework as that of Theorem 1. Hence, there exists a sequence $\left\{a_n\right\}\subset A$ such that, $\mathfrak{C}(a_n,\mu )$ is the proximal connected-image set for all $n\ge 0$. If $d_s(a_k,a_{k+1})=0$ for some $k\ge 0$, then by Lemma 3 we immediately obtain $a_k\in BP\left(\mu \right)$. Hence, we may assume that

$$d_s(a_n,a_{n+1})\ne 0\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

Again by Lemma 3, we have

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

(13)

We now show that $\left\{a_n\right\}$ is a Cauchy sequence. Suppose, on the contrary, that $\left\{a_n\right\}$ is not Cauchy. Then there exists $ϵ>0$ such that for every $k\in \mathbb{N}$, there are integers $n_k>m_k\ge k$ with $n_k$ minimal satisfying

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

(14)

By applying the same line of reasoning as in the proof of Theorem 1, we obtain that

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

Using (13), we deduce that

$$\begin{array}{c}\hfill \underset{k\to \infty }{\lim \sup } \mathcal{M}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\le ϵ.\end{array}$$

We observe that $(a_{m_k+1},a_{n_k})\in \mathfrak{C}(a_{m_k},\mu )\times \mathfrak{C}(a_{n_k-1},\mu )$. Hence, we conclude that

$$\begin{array}{cc}\hfill d_s(a_{m_k+1},a_{n_k})& \le \mathcal{A}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\mathcal{M}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\hfill \\ \hfill & \le {\displaystyle \frac{1}{s}}\mathcal{M}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1}).\hfill \end{array}$$

Since (13), letting $k\to \infty$ in the above inequality yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{s}}ϵ& \le \underset{k\to \infty }{\lim \sup } d_s(a_{m_k+1},a_{n_k})\hfill \\ \hfill & \le \underset{k\to \infty }{\lim \sup } \mathcal{A}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\mathcal{M}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})\hfill \\ \hfill & \le \underset{k\to \infty }{\lim \sup } \mathcal{A}(a_{m_k+1},a_{n_k},a_{m_k},a_{n_k-1})ϵ\le {\displaystyle \frac{1}{s}}ϵ.\hfill \end{array}$$

Following the same arguments as those used in the proof of Theorem 1, it follows that the sequence $\left\{a_n\right\}$ is a Cauchy sequence in $(X,d_s)$. By the completeness of the b-metric space and A is closed, there exists an element $a\in A$ such that

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

(15)

By the ${\mathcal{G}}_P$-continuity of $\mu$ and Lemma 1, we conclude that a is a best proximity point of $\mu$, that is, $a\in BP\left(\mu \right)$. □

The following theorem establishes the uniqueness of the best proximity point under the hypotheses of Theorem 1 or Theorem 2, provided an additional coincidence condition is satisfied.

Theorem 3.

Assume that all the hypotheses of Theorem 1 (or equivalently, Theorem 2) hold. Suppose further that $(x,y)\in \mathfrak{C}(x,\mu )\times \mathfrak{C}(y,\mu )$ with $x\ne y$. Then μ admits a unique best proximity point.

Proof. 

Let $(x,y)\in \mathfrak{C}(x,\mu )\times \mathfrak{C}(y,\mu )$. Suppose, by contradiction, that $x\ne y$; then $d_s(x,y)>0.$ Since $(x,y)\in \mathfrak{C}(x,\mu )\times \mathfrak{C}(y,\mu )$, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{s}}{d}_s(x,y)\le d_s(x,y)\le \mathcal{A}(x,y,x,y)\mathcal{R}(x,y,x,y)\le {\displaystyle \frac{1}{s}}\mathcal{R}(x,y,x,y)={\displaystyle \frac{1}{s}}{d}_s(x,y).\end{array}$$

Hence, it follows that $\mathcal{A}(x,y,x,y)={\displaystyle \frac{1}{s}}$, which implies $d_s(x,y)=0$. This contradicts the assumption that x and y are distinct points. Therefore, we must have $x=y$, and thus $\mu$ has a unique best proximity point. □

To further illustrate and validate our main results, we now present the following example.

Example 2.

Consider the following setting:

(i) 

Let $X=({\mathbb{R}}^2,d_s)$ be a b-metric space endowed with the directed graph $\mathcal{G}$ defined by

$$E\left(\mathcal{G}\right)=\left\{\right((a,b),(x,y))\in X\times X:y\ge b\},$$

where the b-metric $d_s$ is given by

$$d_s\left((a_1,b_1),(a_2,b_2)\right)=|a_1-a_2{|}^2+{|b_1-b_2|}^2.$$

(ii) 

Define the sets

$$A=\left\{\left(0.5,b\right):0\le b\le 0.5\right\}$$

and

$$B=\left\{\left(1.5,b\right):0\le b\le 0.5\right\}\cup \left\{\left(a,0.5\right):1.5\le a\le 2\right\}.$$

(iii) 

Define the mapping $\mu :A\to B$ by

$$\mu \left(0.5,b\right)=\left(1.5,0.5\ln (b+1)\right),\mathit{for}\mathit{all}\left(0.5,b\right)\in A.$$

Since $\left(0.5,0\right)\in A$, we have

$$\left(\left(0.5,0\right),\mu \left(0.5,0\right)\right)=\left(\left(0.5,0\right),\left(1.5,0\right)\right)\in E\left(\mathcal{G}\right).$$

Moreover, since $\mu \left(A\right)\subseteq B_0$, it follows that $(0.5,0)\in I_{\mu }$. Clearly, the mapping $\mu$ is ${\mathcal{G}}_P$-continuous. Next, we show that $\mathfrak{C}(x,\mu )$ and $\mathfrak{C}(y,\mu )$ are proximal connected-image sets for all $x,y\in A$. Let $\left(0.5,x\right),\left(0.5,a\right)\in A$. Since

$$\left(\left(0.5,x\right),\mu \left(0.5,x\right)\right)=\left(\left(0.5,x\right),\left(1.5,0.5\ln (x+1)\right)\right)\in E\left(\mathcal{G}\right),$$

we obtain $x\ge 0.5\ln (x+1)$ for all $x\in [0,0.5]$. If $\left(0.5,u\right)\in \mathfrak{C}(x,\mu )$, then

$$d_s\left(\left(0.5,a\right),\left(1.5,0.5\ln (x+1)\right)\right)=d_s(A,B)=1,$$

which implies $a=0.5\ln (x+1)$. For all $x\in [0,0.5]$, we have $0.5\ln (x+1)\ge 0$ then

$$\begin{array}{cc}\hfill 0.5\ln (x+1)& \ge \ln (0.5\ln (x+1)+1)\hfill \\ \hfill a& \ge \ln (a+1)\ge 0.5\ln (a+1).\hfill \end{array}$$

Therefore,

$$\left(\left(0.5,a\right),\mu \left(0.5,a\right)\right)\in E\left(\mathcal{G}\right),$$

and hence $\mathfrak{C}(x,\mu )$ is a proximal connected-image set. Likewise, $\mathfrak{C}(y,\mu )$ satisfies the same property. Now, define the function $\mathcal{A}\in \mathfrak{A}$ by

$$\mathcal{A}(a,b,x,y)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}x=y,\hfill \\ {\displaystyle \frac{1}{2}}{\displaystyle \frac{\arctan \left(d_s(x,y)+|d_s(x,a)-d_s(y,b)|\right)}{d_s(x,y)+|d_s(x,a)-d_s(y,b)|}}\hfill & \mathrm{if}x\ne y.\hfill \end{array}\right.$$

Let $a,b,x,y\in A$, where

$$\begin{array}{c}\hfill a=\left(0.5,\overline{a}\right),b=\left(0.5,\overline{b}\right),x=\left(0.5,\overline{x}\right),y=\left(0.5,\overline{y}\right),\end{array}$$

such that $(a,b)\in \mathfrak{C}(x,\mu )\times \mathfrak{C}(y,\mu )$, that is,

$$d_s(a,\mu x)=d_s(A,B)=d_s(b,\mu y).$$

It follows that

$$\overline{a}=0.5\ln (1+\overline{x}),\overline{b}=0.5\ln (1+\overline{y}),$$

where $\overline{x},\overline{y}\in [0,0.5]$. If $a=b$ or $x=y$, the contraction inequality is trivial. Assume $a\ne b$ and $x\ne y$. Then $\overline{a},\overline{b},\overline{x},\overline{y}$ are all distinct. Consequently, we have the following calculation

$$\begin{array}{cc}\hfill d_s(a,b)& =d_s\left((0.5,\overline{a}),(0.5,\overline{b})\right)=d_s\left((0.5,0.5\ln (\overline{x}+1)),(0.5,0.5\ln (\overline{y}+1))\right)\hfill \\ \hfill & =|0.5\ln (\overline{x}+1)-0.5\ln (\overline{y}+1){|}^2=0.25{\left|\ln \left({\displaystyle \frac{\overline{x}+1}{\overline{y}+1}}\right)\right|}^2\hfill \\ \hfill & =0.25{\left|\ln \left(1+{\displaystyle \frac{\overline{x}-\overline{y}}{\overline{y}+1}}\right)\right|}^2\le 0.5\ln (1+|\overline{x}-\overline{y}{|}^2)\le 0.5\arctan \left(\right|\overline{x}-\overline{y}{|}^2)\hfill \\ \hfill & \le 0.5\arctan (|\overline{x}-\overline{y}{|}^2+\left||\overline{x}-\overline{a}{|}^2-{|\overline{y}-\overline{b}|}^2\right|)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\displaystyle \frac{\arctan (|\overline{x}-\overline{y}{|}^2+\left||\overline{x}-\overline{a}{|}^2-{|\overline{y}-\overline{b}|}^2\right|)}{|\overline{x}-\overline{y}{|}^2+\left||\overline{x}-\overline{a}{|}^2-{|\overline{y}-\overline{b}|}^2\right|}}(|\overline{x}-\overline{y}{|}^2+\left||\overline{x}-\overline{a}{|}^2-{|\overline{y}-\overline{b}|}^2\right|)\hfill \\ \hfill & =\mathcal{A}(a,b,x,y)\mathcal{R}(a,b,x,y).\hfill \end{array}$$

Hence, $\mu$ is a proximal connected-image contraction of type $\mathcal{R}$. By Theorem 1, we conclude that $BP\left(\mu \right)\ne \varnothing$, and $\left(0.5,0\right)$ is the best proximity point of $\mu$.

Finally, we consider a special case of our main results by assuming that $A=B=X$. Under this assumption, the best proximity point problem reduces to a fixed point problem. Consequently, the results obtained in the previous sections yield corresponding fixed point theorems. Moreover, in the next section, we shall illustrate the applicability of these fixed point results to fractional differential equations. Under this reduced framework, we introduce the following concepts.

Definition 8.

Given $(X,d_s,\mu ,\mathcal{G})$, a mapping μ is said to be a $CI$-contraction of type $\mathcal{R}$ if the property $CI$ holds for all $x,y\in X$ and there exists a function $\mathcal{A}\in \mathfrak{A}$ such that

$$d_s(\mu x,\mu y)\le \mathcal{A}(\mu x,\mu y,x,y)\mathcal{R}(\mu x,\mu y,x,y),$$

where $\mathcal{R}:X^4\to [0,\infty )$ is defined by

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

Definition 9.

Given $(X,d_s,\mu ,\mathcal{G})$, a mapping μ is said to be a $CI$-contraction of type $\mathcal{M}$ if the property $CI$ holds for all $x,y\in X$ and there exists a function $\mathcal{A}\in \mathfrak{A}$ such that

$$d_s(\mu x,\mu y)\le \mathcal{A}(\mu x,\mu y,x,y)\mathcal{M}(\mu x,\mu y,x,y),$$

where $\mathcal{M}:X^4\to [0,\infty )$ is defined by

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

For a self-mapping $\mu :X\to X$, let $F\left(\mu \right)$ denote the set of all fixed points of $\mu$. In this particular case, since $A=B=X$, it is clear that

$$F\left(\mu \right)=BP\left(\mu \right)\mathrm{and}{I}_{\mu }=\{x\in X:(x,\mu x)\in E\left(\mathcal{G}\right)\}.$$

We are now in a position to derive the corresponding fixed point consequences of our main results.

Corollary 1.

Let $(X,d_s,\mu ,\mathcal{G})$ be given. Suppose that μ is a $CI$-contraction of type $\mathcal{R}$ and is ${\mathcal{G}}_P$-continuous. If there exists $a_0\in I_{\mu }$, then $F\left(\mu \right)\ne \varnothing$.

Corollary 2.

Let $(X,d_s,\mu ,\mathcal{G})$ be given. Suppose that μ is a $CI$-contraction of type $\mathcal{M}$ and is ${\mathcal{G}}_P$-continuous. If there exists $a_0\in I_{\mu }$, then $F\left(\mu \right)\ne \varnothing$.

3. Applications to Nonlinear Fractional Differential Equations

In this section, we extend the fixed point results established earlier to the setting of fractional differential equations. Fractional models naturally arise in diverse scientific fields and are well known for their nonlocal behavior, which makes classical analytical techniques difficult to apply directly. By reformulating the fractional differential equation as an equivalent integral equation and invoking the fixed point theorems developed in the previous sections, we obtain conditions guaranteeing the existence of solutions in the fractional framework. This demonstrates that the fixed point approach remains effective not only for standard differential and integral equations but also for their fractional counterparts, further emphasizing the broad applicability of the theory.

To make the fixed point method applicable to the fractional problem, we introduce the structural requirements that the operator must satisfy. These assumptions ensure that the solution map behaves well enough for the existence theory to hold. Let X be a nonempty set, and let $\gamma :X\to X$ be a given mapping. Define $\mathrm{\Gamma }$ as the collection of all functions $\rho :{\mathbb{R}}^2\to \mathbb{R}$ satisfying:

• For every $u\in X$, if $\rho (u,\gamma u)\ge 0$, then $\rho (\gamma u,{\gamma }^{2}u)\ge 0$,
• For every $u,v\in X$, if $\rho (u,v)\ge 0$ and $\rho (v,\gamma v)\ge 0$, then $\rho (u,\gamma v)\ge 0$.

For each $\rho \in \mathrm{\Gamma }$, we define a directed graph

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

This graph structure forms the basis for the subsequent application of our fixed point theorems. In the fractional setting, a fractional differential equation can be rewritten as an equivalent Volterra-type integral equation. We therefore study the problem in the space $(C[0,1],d_s)$ equipped with the b-metric

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

which is complete for $s=2$. This framework allows us to apply the abstract fixed point results established earlier and to prove the existence of solutions to fractional differential equations. The following consequence illustrates this application in detail.

Here, we discuss the following nonlinear fractional differential equations with nonlocal integral boundary condition of order $\alpha$, where $n<\alpha \le n+1$. Specifically, we consider

$$\left\{\begin{array}{c}{D}_{RL}^{\alpha }u\left(t\right)+f(t,u\left(t\right))=0,0<t<1,\hfill \\ {\displaystyle u^k\left(0\right)=0,k=0,1,2\dots ,n-1u\left(1\right)=\lambda {\int }_0^{\eta }u\left(\tau \right)d\tau ,}\hfill \end{array}\right.$$

(16)

where $\lambda >0$, $\lambda \ne \alpha$, $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a given function, and $D_{RL}^{\alpha }$ denotes the Riemann-Liouville fractional derivative of order $\alpha$, that is,

$$D_{RL}^{\alpha }u\left(t\right)=\left\{\begin{array}{cc}{\displaystyle {\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}{\displaystyle \frac{d^n}{d{x}^n}}{\int }_0^t{\displaystyle \frac{u\left(\tau \right)}{{(x-\tau )}^{\alpha +1-n}}}d\tau }\hfill & \mathrm{if}n-1<\alpha <n,\hfill \\ {\displaystyle \frac{d^n}{d{x}^n}}u\left(t\right)\hfill & \mathrm{if}\alpha =n.\hfill \end{array}\right.$$

Next, we derive the Green’s function associated with the nonlinear fractional Equation (16). This representation will be useful for establishing the existence of solutions.

Lemma 4.

Let $n<\alpha \le n+1$, $0<\eta <1$, and let $y\in C[0,1]$. Consider

$$\left\{\begin{array}{c}{}^{C}D_{RL}^{\alpha }u\left(t\right)+y\left(t\right)=0,0<t<1,\hfill \\ {\displaystyle u^k\left(0\right)=0,k=0,1,2\dots ,n-1u\left(1\right)=\lambda {\int }_0^{\eta }u\left(\tau \right)d\tau ,}\hfill \end{array}\right.$$

(17)

with $\kappa :=1-\lambda {\displaystyle \frac{{\eta }^{\alpha }}{\alpha }}\ne 0$. Then the problem has a unique solution

$$u\left(t\right)={\int }_0^{1}G(t,\tau )y\left(\tau \right)d\tau ,$$

where the Green function is

$$G(t,\tau )={\displaystyle \frac{1}{\kappa \mathrm{\Gamma }\left(\alpha \right)}}\left\{\begin{array}{cc}-\kappa {(t-\tau )}^{\alpha -1}+t^{\alpha -1}\left[{(1-\tau )}^{\alpha -1}-{\displaystyle \frac{\lambda }{\alpha }}{(\eta -\tau )}^{\alpha }\right]\hfill & \mathit{if}0\le \tau \le \min \{t,\eta \},\hfill \\ t^{\alpha -1}\left[{(1-\tau )}^{\alpha -1}-{\displaystyle \frac{\lambda }{\alpha }}{(\eta -\tau )}^{\alpha }\right]\hfill & \mathit{if}t\le \tau \le \eta ,\hfill \\ -\kappa {(t-\tau )}^{\alpha -1}+t^{\alpha -1}{(1-\tau )}^{\alpha -1}\hfill & \mathit{if}\eta \le \tau \le t,\hfill \\ t^{\alpha -1}{(1-\tau )}^{\alpha -1}\hfill & \mathit{if}\max \{t,\eta \}\le \tau \le 1.\hfill \end{array}\right.$$

(18)

Proof. 

Since $n<\alpha \le n+1$, it is well known (see [13]) that the general solution of $D_{RL}^{\alpha }u\left(t\right)=-y\left(t\right)$ is

$$u\left(t\right)=-{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}y\left(\tau \right)d\tau +\sum _{k=1}^{n+1}{C}_k{t}^{\alpha -k}.$$

Using the conditions $u\left(0\right)=u^{\prime }\left(0\right)=\dots =u^{(n-1)}\left(0\right)=0,$ we directly obtain $C_k=0$, for $k=2,3,\dots ,n+1$. Thus

$$u\left(t\right)=-{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}y\left(\tau \right)d\tau +C_1{t}^{\alpha -1}.$$

To compute $C_{n+1}$, we note that

$$u\left(1\right)=-{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^1{(1-\tau )}^{\alpha -1}y\left(\tau \right)d\tau +C_1,$$

and

$${\int }_0^{\eta }u\left(\tau \right)d\tau =-{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^{\eta }{\int }_0^{\nu }{(\nu -\tau )}^{\alpha -1}y\left(\tau \right)d\tau d\nu +C_1{\displaystyle \frac{{\eta }^{\alpha }}{\alpha }}.$$

By reversing the order of integration gives

$${\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^{\eta }{\int }_0^{\nu }{(\nu -\tau )}^{\alpha -1}y\left(\tau \right)d\tau d\nu ={\displaystyle \frac{1}{\alpha \mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^{\eta }{(\eta -\tau )}^{\alpha }y\left(\tau \right)d\tau$$

Using the boundary condition ${\displaystyle u\left(1\right)=\lambda {\int }_0^{\eta }u\left(\tau \right)d\tau }$ we have

$$-{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^1{(1-\tau )}^{\alpha -1}y\left(\tau \right)d\tau +C_1=\lambda \left(-{\displaystyle \frac{1}{\alpha \mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^{\eta }{(\eta -\tau )}^{\alpha }y\left(\tau \right)d\tau +C_1{\displaystyle \frac{{\eta }^{\alpha }}{\alpha }}\right),$$

which implies

$$C_1={\displaystyle \frac{1}{\kappa \mathrm{\Gamma }\left(\alpha \right)}}\left[{\int }_0^1{(1-\tau )}^{\alpha -1}y\left(\tau \right)d\tau -{\displaystyle \frac{\lambda }{\alpha }}{\int }_0^{\eta }{(\eta -\tau )}^{\alpha }y\left(\tau \right)d\tau \right],$$

where $\kappa :=1-\lambda {\displaystyle \frac{{\eta }^{\alpha }}{\alpha }}\ne 0$. Splitting according to the four possible orders of $\tau$, t, and $\eta$ produces the piecewise Green function in (18). This completes the proof. □

Remark 2.

It is worth noting that when $\eta =1$, the integral boundary condition reduces to a local one and the nonlocal effect disappears. In this case, for $n=2$ our result reduces to Theorem 2.1 of [14]. Moreover, when $n=3$, our result recovers Theorem 3.1 of [15] under the corresponding parameter identification.

The following lemma provides an essential estimate for the Green’s function associated with problem (16). This bound will be used later to establish the properties required for the existence results.

Lemma 5.

Let $n<\alpha \le n+1$, $0<\eta <1$, and $\lambda \in \mathbb{R}$ be such that $\kappa :=1-\lambda {\displaystyle \frac{{\eta }^{\alpha }}{\alpha }}\ne 0$. The Green function $G:[0,1]\times [0,1]\to \mathbb{R}$ defined by Equation (18) satisfies

$${\int }_0^1{\left|G(t,\tau )\right|}^{2}d\tau \le {\displaystyle \frac{{\sigma }^2}{\mathrm{\Gamma }{\left(\alpha \right)}^2(2\alpha -1)}},$$

where $\sigma :=1+{\displaystyle \frac{1}{\left|\kappa \right|}}+{\displaystyle \frac{\left|\lambda \right|}{\alpha \left|\kappa \right|}}.$

Proof. 

Let $t,\tau \in [0,1]$. We consider in a first case of $\left|G\right(t,\tau \left)\right|$ that $0\le \tau \le \min \{t,\eta \}$. In this case, we have

$$G(t,\tau )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\left(-{(t-\tau )}^{\alpha -1}+{\displaystyle \frac{t^{\alpha -1}}{\kappa }}\left[{(1-\tau )}^{\alpha -1}-{\displaystyle \frac{\lambda }{\alpha }}{(\eta -\tau )}^{\alpha }\right]\right).$$

Using the triangle inequality and the fact that $0\le \tau ,t,\eta \le 1$, we have

$$\begin{array}{cc}\hfill \left|G\right(t,\tau \left)\right|& \le {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\left({(t-s)}^{\alpha -1}+{\displaystyle \frac{|t^{\alpha -1}|}{\left|\kappa \right|}}{(1-\tau )}^{\alpha -1}+{\displaystyle \frac{|t^{\alpha -1}\lambda |}{\alpha \left|\kappa \right|}}{(\eta -\tau )}^{\alpha }\right)\hfill \\ & \le {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\left({(1-\tau )}^{\alpha -1}+{\displaystyle \frac{1}{\left|\kappa \right|}}{(1-\tau )}^{\alpha -1}+{\displaystyle \frac{\left|\lambda \right|}{\alpha \left|\kappa \right|}}{(1-\tau )}^{\alpha -1}\right)\hfill \\ & ={\displaystyle \frac{\sigma }{\mathrm{\Gamma }\left(\alpha \right)}}{(1-\tau )}^{\alpha -1},\hfill \end{array}$$

where $\sigma :=1+{\displaystyle \frac{1}{\left|\kappa \right|}}+{\displaystyle \frac{\left|\lambda \right|}{\alpha \left|\kappa \right|}}.$ The same estimates can be carried out for the other three cases in (18), yielding the identical bound as

$$\left|G(t,\tau )\right|\le {\displaystyle \frac{\sigma }{\mathrm{\Gamma }\left(\alpha \right)}}{(1-\tau )}^{\alpha -1},0\le \tau ,t\le 1.$$

This implies that

$${\int }_0^1{\left|G(t,\tau )\right|}^{2}d\tau \le {\displaystyle \frac{{\sigma }^2}{\mathrm{\Gamma }{\left(\alpha \right)}^2}}{\int }_0^1{(1-\tau )}^{2\alpha -2}d\tau ={\displaystyle \frac{{\sigma }^2}{\mathrm{\Gamma }{\left(\alpha \right)}^2(2\alpha -1)}},$$

which proves the desired estimate. □

Let us consider the mapping $\mathrm{{\rm Y}}:C[0,1]\to C[0,1]$ given by

$$\left(\mathrm{{\rm Y}}u\right)\left(t\right)={\int }_0^{1}G(t,\tau )f(\tau ,u\left(\tau \right))d\tau .$$

(19)

The kernel $G(t,\tau )$ denotes the Green function associated with the fractional boundary-value problem, while the function f characterizes the nonlinear effects. Rather than working directly with the differential Equation (16), it is often more convenient to interpret a solution as a function that is reproduced by this integral transformation. A function $u\in C[0,1]$ satisfies the fractional problem exactly when it satisfies the fixed point relation $\mathrm{{\rm Y}}u=u$. In this way, the boundary-value problem reduces to locating a fixed point of the operator $\mathrm{{\rm Y}}$.

Lemma 6.

Consider the space $\left(C[0,1],d_s,\mathrm{{\rm Y}},{\mathcal{G}}_{\rho }\right)$ and the operator $\mathrm{{\rm Y}}$ is defined in (19). Assume that one of the following conditions holds.

(H1) 

For every $\tau \in [0,1]$ and all $u,v\in C[0,1]$, there exists $\mathcal{A}\in \mathfrak{A}$ such that

$$\left|f(\tau ,u(\tau \left)\right)-f(\tau ,v(\tau \left)\right)\right|\le {\displaystyle \frac{\mathrm{\Gamma }\left(\alpha \right)\sqrt{2\alpha -1}}{\sigma }}\sqrt{\mathcal{A}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)\mathcal{R}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)}.$$

(H2) 

For every $\tau \in [0,1]$ and all $u,v\in C[0,1]$, there exists $\mathcal{A}\in \mathfrak{A}$ such that

$$\left|f(\tau ,u(\tau \left)\right)-f(\tau ,v(\tau \left)\right)\right|\le {\displaystyle \frac{\mathrm{\Gamma }\left(\alpha \right)\sqrt{2\alpha -1}}{\sigma }}\sqrt{\mathcal{A}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)\mathcal{M}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)}.$$

When condition (H1) is imposed, the operator $\mathrm{{\rm Y}}$ becomes a CI-contraction of type $\mathcal{R}$; on the other hand, if (H2) is used, then $\mathrm{{\rm Y}}$ acts as a CI-contraction of type $\mathcal{M}$.

Proof. 

Using the definition of the operator $\mathrm{{\rm Y}}$, we have

$$d_s(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v)=\underset{t\in [0,1]}{\sup }{\left|{\int }_0^{1}G(t,\tau )\left(f(\tau ,u(\tau \left)\right)-f(\tau ,v(\tau \left)\right)\right)d\tau \right|}^2,$$

which implies

$$d_s(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v)\le \underset{t\in [0,1]}{\sup }{\left({\int }_0^1\left|G(t,\tau )\left(g(\tau ,u(\tau \left)\right)-g(\tau ,v(\tau \left)\right)\right)\right|d\tau \right)}^2.$$

By applying assumption (H1) to the nonlinear term $f(\tau ,u(\tau \left)\right)-f(\tau ,v(\tau \left)\right)$, it follows that

$$d_s(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v)\le {\displaystyle \frac{\mathrm{\Gamma }{\left(\alpha \right)}^2(2\alpha -1)}{{\sigma }^2}}\mathcal{A}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)\mathcal{R}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)\underset{t\in [0,1]}{\sup }{\left({\int }_0^1\left|G(t,\tau )\right|d\tau \right)}^2.$$

Using Lemma 5, we get

$$d_s(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v)\le \mathcal{A}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)\mathcal{R}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v).$$

By the definition of a CI-contraction of type $\mathcal{R}$, condition (H1) ensures that the operator $\mathrm{{\rm Y}}$ possesses this property. When (H2) is invoked instead, the same argument shows that $\mathrm{{\rm Y}}$ operates as a CI-contraction of type $\mathcal{M}$. The proof is complete. □

We are now ready to present the main existence result for the nonlinear fractional differential problem (16), obtained as a consequence of the fixed point framework developed above and Corollaries 1 and 2.

Theorem 4.

Consider the space $\left(C[0,1],d_s,\mathrm{{\rm Y}},{\mathcal{G}}_{\rho }\right)$, where $\rho \in \mathrm{\Gamma }$ and the operator $\mathrm{{\rm Y}}$ is given by (19). Assume that one of the conditions (H1) or (H2) is fulfilled. Suppose further that there exists a function $u_0\in C[0,1]$ for which $\rho \left(u_0\left(t\right),\mathrm{{\rm Y}}{u}_0\left(t\right)\right)\ge 0$ for all $t\in [0,1]$. Under these hypotheses, the operator $\mathrm{{\rm Y}}$ possesses a fixed point $u^{*}\in C[0,1]$, and this fixed point provides a solution of the nonlinear fractional differential problem (16).

Although conditions (H1) and (H2) are stated in a general form, their effectiveness relies on concrete and verifiable choices of the auxiliary function $\mathcal{A}$. In the absence of such structure, these assumptions may become overly flexible and difficult to verify in practice. Therefore, in applications, the function $\mathcal{A}$ should be specified explicitly in advance and chosen in accordance with the structure of the operator $\mathrm{{\rm Y}}$. In what follows, we apply our results to study the existence of solutions for the nonlinear fractional differential Equation (16).

Example 3.

Consider the following nonlocal integral boundary-value problem:

$$D_{RL}^{{\displaystyle \frac{5}{2}}}u\left(t\right)+{\displaystyle \frac{1}{3}}\sqrt{\arctan \left(u^2\left(t\right)\right)}+1=0,t\in [0,1],$$

with the boundary conditions:

$$u\left(0\right)=u^{\prime }\left(0\right)=0,u\left(1\right)=4{\int }_0^{1}u\left(\tau \right)d\tau .$$

Proof. 

We observe that $\alpha ={\displaystyle \frac{5}{2}}, \lambda =4$ and $\eta =1$, which satisfy condition $\kappa =-{\displaystyle \frac{3}{5}}\ne 0$ and by Lemma 5 we obtain

$${\int }_0^1{\left|G(t,\tau )\right|}^{2}d\tau \le {\displaystyle \frac{64}{9\mathrm{\Gamma }{(5/2)}^2}}={\displaystyle \frac{1024}{81\pi }}\approx 4.0241\dots .$$

By applying the Cauchy-Schwarz inequality, we have

$$\begin{array}{cc}\hfill {|\mathrm{{\rm Y}}u\left(t\right)-\mathrm{{\rm Y}}v\left(t\right)|}^2& ={\displaystyle \frac{1}{9}}{\left|{\int }_0^{1}G(t,\tau )\left[\sqrt{\arctan \left(u^2\left(\tau \right)\right)}-\sqrt{\arctan \left(v^2\left(\tau \right)\right)}\right]d\tau \right|}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{9}}\left({\int }_0^1{\left|G(t,\tau )\right|}^{2}d\tau \right)\left({\int }_0^1{|\sqrt{\arctan \left(u^2\left(\tau \right)\right)}-\sqrt{\arctan \left(v^2\left(\tau \right)\right)}|}^{2}d\tau \right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\left({\int }_0^1{|\sqrt{\arctan \left(u^2\left(\tau \right)\right)}-\sqrt{\arctan \left(v^2\left(\tau \right)\right)}|}^{2}d\tau \right).\hfill \end{array}$$

We let $\varphi :[0,\infty )\to [0,\infty )$ by $\varphi \left(t\right)=\sqrt{\arctan \left(t^2\right)}$. We note that

$${\varphi }^{\prime }\left(t\right)={\displaystyle \frac{t}{(1+t^4)\sqrt{\arctan \left(t^2\right)}}}\ge 0,$$

and ${\varphi }^{\prime }\left(0\right)=0$ by continuity. Hence $\varphi$ is non-decreasing on $[0,\infty )$. Direct differentiation yields

$${\varphi }^{″}\left(t\right)={\displaystyle \frac{(1-3t^4)\arctan \left(t^2\right)-t^2}{{(1+t^4)}^2\arctan {\left(t^2\right)}^{3/2}}}.$$

We note that the denominator is strictly positive for $t>0$, so the sign of ${\varphi }^{″}\left(t\right)$ depends on the numerator. Let $x=t^2\ge 0$ and define $p\left(x\right)=(1-3x^2)\arctan \left(x\right)-x.$ We also have $p\left(0\right)=0$ and

$$p^{\prime }\left(x\right)=-{\displaystyle \frac{4x^2}{1+x^2}}-6x\arctan \left(x\right)\le 0,\mathrm{for}\mathrm{all}x\ge 0,$$

Hence $p\left(x\right)\le 0$ for all $x\ge 0$, which implies $\varphi$ is concave on $(0,\infty )$, and by continuity at $t=0$, it is concave on $[0,\infty )$. Since $\varphi$ is concave on $[0,\infty )$ and $\varphi \left(0\right)=0$, it follows that $\varphi$ is sub-additive, that is, $\varphi (a+b)\le \varphi \left(a\right)+\varphi \left(b\right)$ for all $a,b\ge 0$. Since $\varphi$ is non-decreasing and sub-additive, using the fact $\left|\left|u\right(\tau \left)\right|-\left|v\right(\tau \left)\right|\right|\le |u\left(\tau \right)-v\left(\tau \right)|$, we obtain

$$\left|\varphi \left(\right|u\left(\tau \right)\left|\right)-\varphi \left(\right|v\left(\tau \right)\left|\right)\right|\le \varphi \left(\left|\left|u\right(\tau \left)\right|-\left|v\right(\tau \left)\right|\right|\right)\le \varphi \left(\right|u\left(\tau \right)-v\left(\tau \right)\left|\right).$$

That is

$$\left|\sqrt{\arctan \left(\right|u\left(\tau \right){|}^2)}-\sqrt{\arctan \left(\right|v\left(\tau \right){|}^2)}\right|\le \sqrt{\arctan \left(\right|u\left(\tau \right)-v\left(\tau \right){|}^2)}.$$

Consequently, this implies that

$$\begin{array}{c}\hfill {|\mathrm{{\rm Y}}u\left(t\right)-\mathrm{{\rm Y}}v\left(t\right)|}^2\le {\displaystyle \frac{1}{2}}\left({\int }_0^1{\arctan \left(\right|u\left(\tau \right)-v\left(\tau \right)|}^2)d\tau \right)\le {\displaystyle \frac{1}{2}}\arctan \left(d_s(u,v)\right).\end{array}$$

By taking the supremum over $t\in [0,1]$, one has

$$d_s(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v)\le {\displaystyle \frac{1}{2}}\arctan \left(d_s(u,v)\right).$$

This suggests us to introduce

$$\mathcal{A}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}\hfill & \mathrm{if}u=v,\hfill \\ {\displaystyle \frac{1}{2}}{\displaystyle \frac{\arctan \left(d_s(u,v)\right)}{d_s(u,v)}}\hfill & \mathrm{if}u\ne v.\hfill \end{array}\right.$$

Therefore, we arrive at

$$d_s(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v)\le \mathcal{A}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)\mathcal{R}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v).$$

due to the fact that $d_s(u,v)\le \mathcal{R}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v).$ Hence, by applying Theorem 4, $\mathrm{{\rm Y}}$ has a fixed point $u^{*}\in C[0,1]$, which is a solution to the problem. □

After establishing existence results for problem (16), we proceed to study positivity of the solutions. The key ingredient in this analysis is the positivity of the Green function associated with the fractional boundary-value problem.

Lemma 7.

Let $n<\alpha \le n+1$, $0<\eta <1$, and $\lambda >0$. Assume that $\kappa \in (0,1)$. Then the Green function $G(t,\tau )$ given by (18) is a non-negative function for all $(t,\tau )\in [0,1]\times [0,1].$

Proof. 

Since $\mathrm{\Gamma }\left(\alpha \right)>0$ and $\kappa >0$, it suffices to show that the numerator in each branch of $G(t,s)$ is nonnegative.

Case 1:

$0\le \tau \le \min \{t,\eta \}$. Consider

$$G(t,s)={\displaystyle \frac{1}{\kappa \mathrm{\Gamma }\left(\alpha \right)}}\left[-\kappa {(t-\tau )}^{\alpha -1}+t^{\alpha -1}\left[{(1-\tau )}^{\alpha -1}-{\displaystyle \frac{\lambda }{\alpha }}{(\eta -\tau )}^{\alpha }\right]\right].$$

Here, for $0\le \tau \le \eta$, we define $Q:[0,\eta ]\to \mathbb{R}$ by

$$Q\left(\tau \right):={(1-\tau )}^{\alpha -1}-{\displaystyle \frac{\lambda }{\alpha }}{(\eta -\tau )}^{\alpha }.$$

We claim that $Q\left(\tau \right)\ge \kappa {(1-\tau )}^{\alpha -1},$ for $\tau \in [0,\eta ].$ To prove this, we consider

$$\begin{array}{cc}\hfill Q\left(\tau \right)-\kappa {(1-\tau )}^{\alpha -1}& =\left[{(1-\tau )}^{\alpha -1}-{\displaystyle \frac{\lambda }{\alpha }}{(\eta -\tau )}^{\alpha }\right]-\kappa {(1-\tau )}^{\alpha -1}\hfill \\ \hfill & ={\displaystyle \frac{\lambda }{\alpha }}\left[{\eta }^{\alpha }{(1-\tau )}^{\alpha -1}-{(\eta -\tau )}^{\alpha }\right].\hfill \end{array}$$

This suggests us to show that ${\eta }^{\alpha }{(1-\tau )}^{\alpha -1}-{(\eta -\tau )}^{\alpha }\ge 0⟺{\displaystyle \frac{{\eta }^{\alpha }{(1-\tau )}^{\alpha -1}}{{(\eta -\tau )}^{\alpha }}}\ge 1$. We let $\mathrm{\Phi }:[0,\eta ]\to {\mathbb{R}}^{+}$ by

$$\mathrm{\Phi }\left(\tau \right)={\displaystyle \frac{{\eta }^{\alpha }{(1-\tau )}^{\alpha -1}}{{(\eta -\tau )}^{\alpha }}}.$$

By taking differential of $\ln \mathrm{\Phi }\left(\tau \right)$, we see that

$${\displaystyle \frac{{\mathrm{\Phi }}^{\prime }\left(\tau \right)}{\mathrm{\Phi }\left(\tau \right)}}=-{\displaystyle \frac{\alpha -1}{1-\tau }}+{\displaystyle \frac{\alpha }{\eta -\tau }}={\displaystyle \frac{(\alpha -1)(1-\eta )+1-\tau }{(1-\tau )(\eta -\tau )}}.$$

Since $\mathrm{\Phi }\left(\tau \right)>0$, we can conclude that $\mathrm{\Phi }$ is strictly increasing. Therefore, we arrive at

$$\mathrm{\Phi }\left(\tau \right)={\displaystyle \frac{{\eta }^{\alpha }{(1-\tau )}^{\alpha -1}}{{(\eta -\tau )}^{\alpha }}}\ge \mathrm{\Phi }\left(0\right)=1,$$

as expected. That is $Q\left(\tau \right)\ge \kappa {(1-\tau )}^{\alpha -1},$ for $\tau \in [0,\eta ].$ We observe that

$$G(t,\tau )={\displaystyle \frac{1}{\kappa \mathrm{\Gamma }\left(\alpha \right)}}\left[-\kappa {(t-\tau )}^{\alpha -1}+t^{\alpha -1}Q\left(\tau \right)\right]\ge {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\left[t^{\alpha -1}{(1-\tau )}^{\alpha -1}-{(t-\tau )}^{\alpha -1}\right].$$

For $0\le \tau \le t\le 1$, we note that $t(1-\tau )\ge t-\tau$, which implies $G(t,\tau )\ge 0$.

Case 2:

$t\le \tau \le \eta$. In this region, we see that

$$G(t,\tau )={\displaystyle \frac{t^{\alpha -1}}{\kappa \mathrm{\Gamma }\left(\alpha \right)}}Q\left(\tau \right),$$

which directly implies that $G(t,\tau )\ge 0$.

Case 3:

$\eta \le \tau \le t$. Here, we consider

$$G(t,\tau )={\displaystyle \frac{1}{\kappa \mathrm{\Gamma }\left(\alpha \right)}}\left[-\kappa {(t-\tau )}^{\alpha -1}+t^{\alpha -1}{(1-\tau )}^{\alpha -1}\right].$$

As before, $t(1-\tau )\ge t-\tau$ for $0\le \tau \le t\le 1$ and $0<\kappa <1$, we obtain

$$t^{\alpha -1}{(1-\tau )}^{\alpha -1}\ge {(t-\tau )}^{\alpha -1}\ge \kappa {(t-\tau )}^{\alpha -1},$$

hence $G(t,\tau )\ge 0$.

Case 4:

$\max \{t,\eta \}\le \tau \le 1$. In the last region, we have

$$G(t,\tau )={\displaystyle \frac{t^{\alpha -1}{(1-\tau )}^{\alpha -1}}{\kappa \mathrm{\Gamma }\left(\alpha \right)}}.$$

Since all factors are nonnegative, then $G(t,\tau )\ge 0$. This completes the proof. □

Let $\mathcal{P}$ be the class of positive function $\mathcal{P}\subset C[0,1]$ as

$$\mathcal{P}=\left\{u\in C[0,1]\left|u\right(t)\ge 0\mathrm{for}\mathrm{all}t\in [0,1]\right\}.$$

Based on Lemma 7, it is clear that $\mathrm{{\rm Y}}:\mathcal{P}\to \mathcal{P}$, when f is positive. Therefore, based on the setting of Theorem 4, we obtain the existence of a positive solution of the nonlinear fractional differential Equation (16).

Theorem 5.

Consider the space $\left(\mathcal{P},d_s,\mathrm{{\rm Y}},{\mathcal{G}}_{\rho }\right)$ and the operator $\mathrm{{\rm Y}}$ is given by (19). Assume that $\kappa \in (0,1)$ and one of the conditions (H1) or (H2) is fulfilled. Suppose further that there exists a function $u_0\in \mathcal{P}$ for which $\rho \left(u_0\left(t\right),T{u}_0\left(t\right)\right)\ge 0$ for all $t\in [0,1]$. Under these hypotheses, the operator $\mathrm{{\rm Y}}$; possesses a fixed point $u^{*}\in \mathcal{P}$, and this fixed point provides a positive solution of the boundary-value problem (16).

Example 4.

Consider the following nonlocal integral boundary-value problem:

$$D_{RL}^{{\displaystyle \frac{7}{2}}}u\left(t\right)+f(t,u\left(t\right))=0,t\in [0,1],$$

where

$$f(t,u\left(t\right))={\displaystyle \frac{1}{1+u\left(t\right)}},u\left(t\right)\ge 0\mathrm{for}t\in [0,1],$$

together with the boundary conditions:

$$u\left(0\right)=u^{\prime }\left(0\right)=u^{″}\left(0\right)=0,u\left(1\right)={\int }_0^{\eta }u\left(\tau \right)d\tau .$$

Proof. 

We observe that here $\alpha ={\displaystyle \frac{7}{2}}$ and $\lambda =1$, which satisfy condition $\kappa =1-{\displaystyle \frac{2}{7}}{\eta }^{7/2}\in ({\displaystyle \frac{5}{7}},1)$ and by Lemma 5 we obtain

$${\int }_0^1{\left|G(t,\tau )\right|}^{2}d\tau \le {\displaystyle \frac{1}{6\mathrm{\Gamma }{(7/2)}^2}}{\left(1+{\displaystyle \frac{9}{7\left|1-\frac{2}{7}{\eta }^{7/2}\right|}}\right)}^2={\displaystyle \frac{32}{675\pi }}{\left(1+{\displaystyle \frac{9}{7\left|1-\frac{2}{7}{\eta }^{7/2}\right|}}\right)}^2.$$

(20)

Based on the fact that $\kappa =1-{\displaystyle \frac{2}{7}}{\eta }^{7/2}\in ({\displaystyle \frac{5}{7}},1)$, we see that

$${\int }_0^1{\left|G(t,\tau )\right|}^{2}d\tau \le 0.118\dots$$

By applying the Cauchy-Schwarz inequality, we have

$$\begin{array}{cc}\hfill {|\mathrm{{\rm Y}}u\left(t\right)-\mathrm{{\rm Y}}v\left(t\right)|}^2& ={\left|{\int }_0^{1}G(t,\tau )\left[f(\tau ,u(\tau \left)\right)-f(\tau ,v(\tau \left)\right)\right]d\tau \right|}^2\hfill \\ \hfill & \le \left({\int }_0^1{\left|G(t,\tau )\right|}^{2}d\tau \right)\left({\int }_0^1{|f(\tau ,u\left(\tau \right))-f(\tau ,v\left(\tau \right))|}^{2}d\tau \right)\hfill \end{array}$$

Let us consider

$$\begin{array}{cc}\hfill {|f(\tau ,u\left(\tau \right))-f(\tau ,v\left(\tau \right))|}^2& ={\left|\left({\displaystyle \frac{1}{1+u\left(\tau \right)}}-{\displaystyle \frac{1}{1+v\left(\tau \right)}}\right)\right|}^2\hfill \\ \hfill & ={\left|{\displaystyle \frac{u\left(\tau \right)-v\left(\tau \right)}{(1+u(\tau )+v(\tau )+u(\tau \left)v\right(\tau \left)\right)}}\right|}^2\hfill \\ \hfill & \le {\displaystyle \frac{{|u\left(\tau \right)-v\left(\tau \right)|}^2}{1+u^2\left(\tau \right)+v^2\left(\tau \right)}}\hfill \\ \hfill & \le {\displaystyle \frac{{|u\left(\tau \right)-v\left(\tau \right)|}^2}{1+\frac{1}{2}{u}^2\left(\tau \right)+\frac{1}{2}{v}^2\left(\tau \right)-u\left(\tau \right)v\left(\tau \right)}}\hfill \end{array}$$

Using the fact that ${\displaystyle \frac{x}{1+x}}\le \ln (1+x)$, for $x\ge 0$, we get

$${|\mathrm{{\rm Y}}u\left(t\right)-\mathrm{{\rm Y}}v\left(t\right)|}^2\le 2\left({\int }_0^1{\left|G(t,\tau )\right|}^{2}d\tau \right){\int }_0^1\ln \left(1+{\displaystyle \frac{1}{2}}{|u\left(\tau \right)-v\left(\tau \right)|}^2\right)d\tau .$$

(21)

By taking the supremum over $t\in [0,1]$, one has

$$d_s(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v)\le \ln \left(1+{\displaystyle \frac{1}{2}}{d}_s(u,v)\right).$$

Here, we define

$$\mathcal{A}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}\hfill & \mathrm{if}u=v,\hfill \\ {\displaystyle \frac{\ln \left(1+\frac{1}{2}{d}_s(u,v)\right)}{d_s(u,v)}}\hfill & \mathrm{if}u\ne v.\hfill \end{array}\right.$$

Therefore, we arrive at

$$d_s(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v)\le \mathcal{A}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v)\mathcal{R}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v).$$

due to the fact that $d_s(u,v)\le \mathcal{R}(\mathrm{{\rm Y}}u,\mathrm{{\rm Y}}v,u,v).$ Hence, by applying Theorem 5, $\mathrm{{\rm Y}}$ has a fixed point $u^{*}\in \mathcal{P}$, which is a positive solution to the problem. □

Remark 3.

To further illustrate the role of the Green function in the existence of positive solutions, Figure 1 presents a graphical comparison of the Green functions corresponding to Examples 3 and 4. For the parameters in Example 3, the Green function $G(t,\tau )$ changes sign on $[0,1]\times [0,1]$, and therefore the associated integral operator does not preserve positivity. In contrast, for the parameters in Example 4, the Green function is nonnegative for all $(t,\tau )\in [0,1]\times [0,1]$, in agreement with Lemma 7. This comparison explains why the additional assumption $\kappa \in (0,1)$ is required to obtain positive solutions in Theorem 5. More generally, it illustrates that the positivity of solutions is governed by the sign of the associated Green’s function.

To illustrate the theoretical results, we compute numerical solutions of the integral equation using a standard Picard fixed point iteration. The integral equation is discretized by a quadrature-based scheme on a uniform grid, and the iteration is performed until convergence in the supremum norm with tolerance ${10}^{-8}$ or until a maximum of 1000 iterations is reached. The numerical solutions reflect the qualitative properties of the underlying Green’s function and the nonlocal boundary condition

$$u\left(1\right)={\int }_0^{\eta }u\left(\tau \right)d\tau ,$$

which couples the endpoint value to the integral of the solution over the interval $[0,\eta ]$, with the parameter $\eta$ controlling the strength of the nonlocal effect. The numerical results are consistent with the theoretical analysis. In Example 3, where $\kappa \notin (0,1)$, the associated Green’s function changes sign and the numerical solution is not positive. In contrast, in Example 4, where $\kappa \in (0,1)$, the Green’s function is nonnegative and the numerical solutions remain positive for $\eta =0.5$, $0.75$, and 1, as shown in Figure 2.

4. Conclusions

This paper introduces a new class of proximal connected-image contractions in b-metric spaces endowed with directed graphs and investigates the existence of best proximity points and fixed points for such mappings. By employing auxiliary functions and proximal structures, explicit and verifiable conditions guaranteeing the existence and uniqueness of best proximity points are established, extending several classical fixed point results within a graph-based framework. The theoretical results are applied to fractional differential equations by reformulating them as equivalent integral equations. Using the developed proximal fixed point principles, sufficient conditions for the existence of solutions to these fractional problems are obtained. These findings show that the proposed proximal approach effectively captures the nonlocal nature of fractional models and provides a robust tool for analyzing both classical and fractional equations.