Applications of Fixed Point Results to Fractional Differential and Nonlinear Mixed Volterra–Fredholm Integral Equations

Abstract

This work aims to introduce the concept of graphic rational contractions in the framework of extended $\mathcal{F}$-metric spaces and to establish fixed point theorems related to these mappings. In addition, we define and examine the class of interpolative Ćirić–Reich–Rus-type cyclic contractions in the same setting, deriving several new fixed point results that broaden existing theories. To illustrate the validity and originality of the obtained results, appropriate examples are presented. Furthermore, the developed theoretical results are applied to study the existence of solutions for fractional differential equations and nonlinear mixed Volterra–Fredholm integral equations, highlighting their effectiveness and practical importance.

1. Introduction

The field of fixed point (fp) theory remains one of the most active and captivating branches of modern mathematical analysis, largely due to its deep connections with the structure of metric spaces (mss), which provide a rigorous framework for measuring distances and analyzing convergence, thereby offering an ideal setting for investigating the existence and uniqueness of fps. The notion of a ms, introduced by Fréchet [1], represents one of the central foundations of modern mathematics. Owing to its broad applicability in analysis and related disciplines, this concept has been extended and refined in numerous directions. Over time, various meaningful advancements of mss have been proposed. In this context, Bakhtin [2] introduced an important extension which was subsequently developed by Czerwik [3] under the name of b-ms, where the classical triangle inequality is transformed by incorporating a real number $s\ge 1$. This adjustment significantly broadened the scope of analytical techniques and applications.

Subsequently, Fagin et al. [4] introduced the thought of an s-relaxed_p metric and demonstrated that each of these spaces satisfies the conditions of a b-ms, although the reverse implication does not generally hold. Khamsi et al. [5] later reexamined this framework under the terminology of mss, establishing several fp results in this setting. Branciari [6] further generalized the structure by defining rectangular mss, replacing the conventional triangle inequality by a four-point rectangular inequality. Building on this progress, Jleli et al. [7] put forward the concept of an $\mathcal{F}$-ms, which unifies and extends some earlier thoughts.

In current research, Panda et al. [8] enhanced the conception of $\mathcal{F}$-mss by defining a new structure called an extended $\mathcal{F}$-ms ($\mathcal{EF}$-ms). This new framework generalizes the classical $\mathcal{F}$-metric structure by allowing greater flexibility in the definition of the distance function, thereby accommodating a wider class of mappings and operators. Albargi et al. [9] defined the notion of Kannan-type interpolative cyclic contractions and proved some new fp theorems for such contractions.

Building on these successive generalizations of mss, one of the primary motivations has been to extend cornerstone results in fp theory to more flexible frameworks. Among these, the Banach contraction principle [10] stands as the earliest and most fundamental theorem, guaranteeing the existence and uniqueness of a fp for continuous self-mappings satisfying a contraction condition. Thereafter, Kannan [11] loosened the continuity assumption and developed a fp result subject to a less restrictive contractive condition. Reich [12] unified the contractive conditions of Banach and Kannan by establishing a more general fp theorem, thereby extending the scope of classical contraction principles. This theory has undergone significant generalizations beyond classical ms to accommodate more complex structures and relationships among elements. One such direction was initiated by Jachymski [13], who in 2008 introduced the notion of graphic contractions in mss endowed with a directed graph. This approach arose from the observation that many mappings arising in applied sciences, economics, and computer science naturally preserve certain relational structures, which cannot be captured by the ordinary metric framework alone. By embedding a graph structure on the underlying set, Jachymski [13] unified and extended several existing fp theorems, particularly the BCP to cases where the contractive condition is required only along edges of the graph rather than globally. Subsequently, Bojor [14] advanced the incorporation of graph structures into fp theory and established a result that generalizes Kannan’s fp theorem. Additional insights can be found in [15,16].

In recent decades, many researchers have focused on extending and refining these classical results by introducing more general and flexible contractive conditions, which led to the development of interpolative and hybrid contraction mappings. Kirk et al. [17] presented the notion of cyclic contractions to analyze mappings alternating linking distinct subsets of a ms. Inspired by these advancements, Karapınar [18] proposed the idea of interpolative Kannan-type contractions, providing deeper insights into the relationship among classical and contemporary contraction principles and fostering further progress in generalized metric structures. Subsequently, Karapınar et al. [19] introduced interpolative Reich–Rus–Ćirić type contractions within the framework of partial mss, thereby extending the earlier concept of interpolative Kannan-type contractions. Thangaraj et al. [20] introduced a controlled Kannan iterated function system and developed the concepts of controlled Kannan attractors and multivalued fractals by extending the Kannan fp theorem to new generalized mss. Hussain et al. [21] contributed to the development of $\mathcal{F}$-mss by establishing several generalized fp theorems within this setting. Subsequently, Al-Mazrooei et al. [22] investigated fp problems for rational-type contractions with non-negative parameters in $\mathcal{F}$-mss. In due course, Alnaser et al. [23] additionally expanded this framework by deriving new fp theorems and applying them to the study of some differential equations. In addition, Hussain et al. [24,25,26] extended the theory by solving different fractional differential equations. Additional contributions and related advancements in this area are discussed in [27,28].

The present study is devoted to develop novel fp theorems in the framework of $\mathcal{EF}$-mss, focusing on two main directions:

1. Graphic rational contractions in $\mathcal{EF}$-mss: We introduce the concept of graphic rational contractions, which combine the flexibility of $\mathcal{EF}$-metrics with the structure of directed graphs, allowing the study of self-mappings that preserve relational constraints among elements. For these mappings, we establish several new existence and uniqueness results for fps.

2. Interpolative Ćirić–Reich–Rus-type cyclic contractions: We extend classical cyclic and interpolative contraction principles to $\mathcal{EF}$-mss, unifying and generalizing earlier results such as Kannan, Reich, and cyclic contraction theorems. This framework provides a robust tool for analyzing mappings alternating among subsets of a ms.

To demonstrate the practical relevance of the proposed theory, we provide illustrative examples and apply the results to investigate the solution of fractional differential equations and nonlinear Volterra integral equations, highlighting both theoretical depth and applicability.

The remainder of the research article is organized as follows:

Section 2 presents essential preliminaries, including fundamental notions of ms, $\mathcal{F}$-ms, and an $\mathcal{EF}$-ms, together with classical and generalized contraction mappings. These concepts lay the groundwork for the main theoretical developments.

Section 3 contains the principal results, where we introduce graphic rational contractions and interpolative Ćirić–Reich–Rus-type cyclic contractions in $\mathcal{EF}$-mss, establishing corresponding fp theorems that generalize and amalgamate several prior results.

Section 4 illustrates the applicability of the obtained theorems by solving fractional differential equations and nonlinear Volterra integral equations, highlighting the practical significance of the theoretical findings.

2. Preliminaries

We begin this section by outlining the necessary definitions, supporting lemmas, and foundational concepts that underpin the main results presented later.

The concept of a ms was introduced by Fréchet [1] as follows:

Definition 1 ([1]).

Let $X\ne \varnothing$. A function $d:X\times X\to$$[0,\infty )$ is a metric if, for all $\upsilon ,\varkappa ,z\in X$:

$\left(D_1\right) d(\upsilon ,\varkappa )\ge 0$ and $d(\upsilon ,\varkappa )=0$⟺$\upsilon =\varkappa ;$

$\left(D_2\right) d(\upsilon ,\varkappa )=d(\varkappa ,\upsilon ),$

$\left(D_3\right)$$d(\upsilon ,z)\le d(\upsilon ,\varkappa )+d(\varkappa ,z).$

The couple $(X,d)$ is called a ms.

In [3], Czerwik formulated the definition of a b-ms in such fashion.

Definition 2 ([3]).

Let $X\ne \varnothing$ and $s\ge 1$. A real-valued function $d:X\times X\to$$[0,\infty )$ is said to be a b-metric if it satisfies ($D_1$)-($D_2$), and, for all $\upsilon ,\varkappa ,z\in X$:

$$d(\upsilon ,z)\le s\left[d\right(\upsilon ,\varkappa )+d(\varkappa ,z\left)\right].$$

The pair $(X,d)$ is called a b-ms.

According to Fagin et al. [4] an s-relaxed _p-ms is described as follows:

Definition 3 ([4]).

Let $X\ne \varnothing$, and let $d:X\times X\to [0,+\infty )$ fulfill conditions ($D_1$)-$\left(D_2\right)$ and

(D_sp3)

there exists $s\ge 1$ such that, for any two points $(\upsilon ,\varkappa )\in X\times X$, for every integer $N\in \mathbb{N}$ (set of natural numbers), $N\ge 2$, and for any finite sequence ${\left({\upsilon }_i\right)}_{i=1}^N\subset X,$ with $({\upsilon }_1,{\upsilon }_N)=(\upsilon ,\varkappa )$, the inequality

$$d(\upsilon ,\varkappa )>0\Rightarrow d(\upsilon ,\varkappa )\le s\left(\sum _{i=1}^{N-1}d({\upsilon }_i,{\upsilon }_{i+1})\right),$$

is satisfied. In this case, the ordered pair $(X,d)$ is called an s-relaxed _p-ms.

Jleli et al. [7] introduced a significant extension of the classical notion of a ms through the formulation of the thought of an $\mathcal{F}$-ms.

Let $\mathcal{F}$ represents the family of mappings $\hslash :(0,+\infty )\to \mathbb{R}$ satisfying the following conditions:

(${\mathcal{F}}_1$)

ℏ is non-decreasing, that is, for all $0<t_1<t_2,$ we have $\hslash \left(t_1\right)\le \hslash \left(t_2\right),$

(${\mathcal{F}}_2$)

for any sequence $\left\{t_n\right\}\subseteq (0,+\infty )$,

$$\underset{n\to \infty }{lim}{t}_n=0⟺\underset{n\to \infty }{lim}\hslash \left(t_n\right)=-\infty .$$

Definition 4 ([7]).

Let $X\ne \varnothing$, and let $d:X\times X\to [0,+\infty )$. Suppose there exists a pair $(\hslash ,⋏)\in \mathcal{F}\times [0,+\infty )$ such that d fulfills conditions $\left(D_1\right)$ and $\left(D_2\right)$ and in addition fulfills the following inequality:

(${\mathrm{D}}_{\mathcal{F}3}$)

for any couple $(\upsilon ,\varkappa )\in X\times X,$ for each integer $N\ge 2$, and for any sequence ${\left({\upsilon }_i\right)}_{i=1}^N\subset X,$ with $({\upsilon }_1,{\upsilon }_N)=(\upsilon ,\varkappa ),$ the implication

$$d(\upsilon ,\varkappa )>0\Rightarrow \hslash \left(d(\upsilon ,\varkappa )\right)\le \hslash \left(\sum _{i=1}^{N-1}d({\upsilon }_i,{\upsilon }_{i+1})\right)+⋏,$$

holds. On the basis of these assumptions, the function d is called an $\mathcal{F}$-metric on X, and the couple $(X,d)$ is said to be an $\mathcal{F}$-ms.

Panda et al. [8] gave the idea of an extended $\mathcal{F}$-ms ($\mathcal{EF}$-ms) in this manner.

Definition 5 ([8]).

Let $X\ne \varnothing$, and let $d:X\times X\to [0,+\infty )$. Suppose that there exists a pair $(\hslash ,⋏)\in \mathcal{F}\times [0,+\infty )$ such that d satisfy fulfills conditions $\left(D_1\right)$ and $\left(D_2\right)$ and additionally satisfies the inequality given below:

($D_{\mathcal{E}3}$)

there exists $\rho :X\times X\to [1,+\infty )$ such that, for every couple $(\upsilon ,\varkappa )\in X\times X$ and each integer $N\ge 2$, if ${\left({\upsilon }_i\right)}_{i=1}^N\subset X,$ is a sequence with $({\upsilon }_1,{\upsilon }_N)=(\upsilon ,\varkappa )$, then

$$d(\upsilon ,\varkappa )>0\Rightarrow \hslash \left(d(\upsilon ,\varkappa )\right)\le \hslash \left(\sum _{i=1}^{N-1}d({\upsilon }_i,{\upsilon }_{i+1})\prod _{j=1}^i\rho ({\upsilon }_j,\varkappa )\right)+⋏.$$

Under these assumptions, $(X,d)$ is called an $\mathcal{EF}$-ms.

Note that each $\mathcal{F}$-ms is an $\mathcal{EF}$-ms when the control function $\rho (\upsilon ,\varkappa )=1$ for all $\upsilon ,\varkappa \in X.$ The additional function $\rho (·,·)$ in the extended setting serves as a variable control that adjusts the influence of intermediate distances in the generalized triangle inequality. This extra flexibility allows the $\mathcal{EF}$-metric framework to encompass a wider range of generalized distance structures and to handle situations where uniform control (as in the $\mathcal{F}$-metric case) is too restrictive. Consequently, the concept of $\mathcal{EF}$-ms unifies and extends the metric, b-metric, s-relaxed_p-metric, and $\mathcal{F}$-mss, providing a richer environment for developing new fp results and their applications.

The following diagram (Figure 1) summarizes the hierarchy of extended $\mathcal{F}$-metric spaces.

Definition 6 ([8]).

Let $(X,d)$ be an $\mathcal{EF}$-ms.

(1) $\mathcal{E}$-Convergence: A sequence $\left\{{\upsilon }_n\right\}$ in X is said to be $\mathcal{E}$-convergent to $\upsilon \in X$ if $\left\{{\upsilon }_n\right\}$ is converges to υ with respect to an $\mathcal{EF}$-metric d.

(2) $\mathcal{E}$-Cauchy Sequence: A sequence $\left\{{\upsilon }_n\right\}$ in X is called $\mathcal{E}$-Cauchy, if

$$\underset{n,m\to \infty }{lim}d({\upsilon }_n,{\upsilon }_m)=0.$$

(3) $\mathcal{E}$-Completeness: The space $(X,d)$ is $\mathcal{E}$-complete if every $\mathcal{E}$-Cauchy sequence in X is $\mathcal{E}$-convergent to some element of X.

Remark 1.

It follows directly from the definitions that every $\mathcal{F}$-ms is automatically an $\mathcal{EF}$-ms, however, the reverse implication is not valid in general. In fact, by taking

$$\rho (\upsilon ,\varkappa )=1,\forall \upsilon ,\varkappa \in X,$$

the generalized condition $\left(D_{\mathcal{E}3}\right)$ corresponding to an $\mathcal{EF}$-ms reduces precisely to condition $\left(D_{\mathcal{F}3}\right)$ that characterizes an $\mathcal{F}$-ms. Consequently, the concept of an $\mathcal{EF}$-ms proposed by Panda et al. [8] strictly extends the framework of $\mathcal{F}$-ms introduced by Jleli et al. [7].

Theorem 1.

(Classical Fixed Point Results). Let ($X,d$) be a complete MS.

• Banach Contraction Principle: If $\Re :X\to X$ satisfies

  $$d(\Re \upsilon ,\Re \varkappa )\le \varrho d(\upsilon ,\varkappa ),$$

  for some $\varrho \in [0,1),$ then ℜ has a unique fp.
• Kannan’s Fixed Point Theorem: If $\Re :X\to X$ satisfies

  $$d(\Re \upsilon ,\Re \varkappa )\le \varrho \left(d(\upsilon ,\Re \upsilon )+d(\varkappa ,\Re \varkappa )\right),$$

  for some $\varrho \in [0,{\displaystyle \frac{1}{2}}),$ then ℜ has a unique fp.
• Fixed Point Result for Cyclic Contraction: If $\Re :{\Theta }_1\cup {\Theta }_2\to {\Theta }_1\cup {\Theta }_2$ satisfy the following

  $$\Re \left({\Theta }_1\right)\subseteq {\Theta }_{2}and\Re \left({\Theta }_2\right)\subseteq {\Theta }_1.$$

  and

  $$d(\Re \upsilon ,\Re \varkappa )\le \varrho d(\upsilon ,\varkappa ),$$

  for all $\upsilon \in {\Theta }_1,$$\varkappa \in {\Theta }_2$ and some $\varrho \in [0,1),$ then ℜ has a unique fp in ${\Theta }_1\cap {\Theta }_2$, where ${\Theta }_1$ and ${\Theta }_2$ are two nonempty closed subsets of $X.$
• Fixed Point Theorem for Interpolative Contraction: If $\Re :X\to X$ satisfies

  $$d(\Re \upsilon ,\Re \varkappa )\le \varrho {\left[d(\upsilon ,\Re \upsilon )\right]}^{\beta }{\left[d(\varkappa ,\Re \varkappa )\right]}^{1-\beta },$$

  for all $\upsilon ,\varkappa \in X\backslash Fix(\Re ),$ where $Fix(\Re )=\{\upsilon \in X:\Re \upsilon =\upsilon \},$ and for some $\varrho \in [0,1)$ and $\beta \in (0,1)$, then ℜ has a unique fp in $X.$
• Fixed Point Result for Graphic Contraction: Let $(X,d)$ be a complete ms associated with a directed graph $G=\left(X,E(G\right))$ and let $\Re :X\to X.$ Suppose

(a) ℜ is edge-preserving, that is, for all $\upsilon ,\varkappa \in X$ with $(\upsilon ,\varkappa )\in E\left(G\right),$ we have $(\Re \upsilon ,\Re \varkappa )\in E\left(G\right),$

(b) there exists $\varrho \in (0,1)$ such that, ∀$\upsilon ,\varkappa \in X$ with $(\upsilon ,\varkappa )\in E\left(G\right)$, we have

$$d(\Re \upsilon ,\Re \varkappa )\le \varrho d(\upsilon ,\varkappa ).$$

and there exists ${\upsilon }_0\in X$ with $({\upsilon }_0,\Re {\upsilon }_0)\in E\left(G\right),$ then ℜ has a fp in $X.$ Moreover, if G is weakly connetced, then this fp is unique.

3. Main Results

The primary results of this work are established in this section. We begin by introducing the concept of graphic rational contractions within the framework of $\mathcal{EF}$-mss and establish several fp theorems associated with these self-mappings. Subsequently, we define the notion of interpolative Ćirić–Reich–Rus-type cyclic contractions in the same setting and derive corresponding fp theorems that further broaden and consolidate existing theories.

Let $G=\left(V\left(G\right),E(G\right))$ be a directed graph associated with an $\mathcal{EF}$-ms $(X,d):$

• $V\left(G\right)$ is a nonempty set whose elements are called vertices;
• $E\left(G\right)\subseteq V\left(G\right)\times V\left(G\right)$ is a set of directed edges (or arcs) between the vertices.

The graph G is said to be loop-inclusive if every vertex is connected to itself; that is,

$$\Delta =\left\{\left(\upsilon ,\upsilon \right):\upsilon \in V\left(G\right)\right\}\subseteq E\left(G\right).$$

Furthermore, G is assumed to contain no multiple edges, meaning that between any ordered pair of vertices there exists at most one directed edge.

A path from $\upsilon$ to $\varkappa$ in G of length N (a natural number) is a sequence ${\left\{{\upsilon }_i\right\}}_{i=0}^N$ of $N+1$ vertices such that ${\upsilon }_0=\upsilon$, ${\upsilon }_N=\varkappa$ and $({\upsilon }_{i-1},{\upsilon }_i)\in E\left(G\right),$∀$i=1,\cdots ,N.$ Let $\tilde{G}$ denote the symmetric graph obtained from G by ignoring the direction of edges, i.e.,

$$E(\tilde{G})=E\left(G\right)\cup E\left(G^{-1}\right),$$

where $G^{-1}$ is the converse graph of G, defined by

$$E\left(G^{-1}\right)=\left\{\left(\upsilon ,\varkappa \right)\in X\times X:\left(\varkappa ,\upsilon \right)\in E\left(G\right)\right\}.$$

The graph G is said to be weakly connected if the symmetric graph $\tilde{G}$ is connected, i.e., if for any two distinct vertices $\upsilon ,\varkappa \in V\left(G\right)$, there exists a path from $\upsilon$ to $\varkappa$ in $\tilde{G}.$

Definition 7.

Let $(X,d)$ be an $\mathcal{EF}$-ms equipped with a directed graph G. A self-mapping $\Re :X\to X$ is said to be graphic rational contraction if

(i) ℜ is edge-preserving, that is, for all $\upsilon ,\varkappa \in X$

$$(\upsilon ,\varkappa )\in E\left(G\right)implies(\Re \upsilon ,\Re \varkappa )\in E\left(G\right),$$

(1)

(ii) there exists some ${\varrho }_1,{\varrho }_2\in (0,1)$ with ${\varrho }_1+{\varrho }_2<1$ such that for all $\upsilon ,\varkappa \in X,(\upsilon ,\varkappa )\in E\left(G\right)$, we have

$$d(\Re \upsilon ,\Re \varkappa )\le {\varrho }_{1}d(\upsilon ,\varkappa )+{\varrho }_2{\displaystyle \frac{d(\upsilon ,\Re \upsilon )d(\varkappa ,\Re \varkappa )}{1+d(\upsilon ,\varkappa )}}.$$

(2)

Theorem 2.

Let $(X,d)$ be an $\mathcal{E}$-complete $\mathcal{EF}$-ms endowed with a directed graph $G=\left(X,E\left(G\right)\right)$ and let $\Re :X\to X$ be a graphic rational contraction with constant $\varrho \in (0,1)$. Suppose that the following assertions hold:

(i) there exists a ${\upsilon }_0\in X$ such that $({\upsilon }_0,\Re {\upsilon }_0)\in E\left(G\right),$

(ii) for any initial point ${\upsilon }_0\in X,$ the sequence {${\upsilon }_n$} defined by ${\upsilon }_{n+1}=\Re {\upsilon }_n,$ $n\in \mathbb{N}$ satisfy

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)<{\displaystyle \frac{1}{\varrho }},$$

where $\varrho ={\displaystyle \frac{{\varrho }_1}{1-{\varrho }_2}},$

(iii) either ℜ is G-continuous or, if $\left\{{\upsilon }_n\right\}$$\subseteq X$ is a sequence such that ${\upsilon }_n\to \upsilon$ as $n\to \infty$ and $({\upsilon }_n,{\upsilon }_{n+1})$$\in E\left(G\right),$ then $({\upsilon }_n,\upsilon )$$\in E\left(G\right)$, ∀$n\in \mathbb{N}.$ Then ℜ has a fp.

Moreover, ℜ has a unique fp if in addition

(iv) for any two fps ${\upsilon }^{*}$, ${\upsilon }^{/}\in X,$ there exists a finite directed path in G joining ${\upsilon }^{*}$ and ${\upsilon }^{/}$ (i.e., G is path-connected on the set of fps).

Proof. 

Let ${\upsilon }_0\in X$ with $({\upsilon }_0,\Re {\upsilon }_0)\in E\left(G\right).$ Define the iterative sequence

$${\upsilon }_{n+1}=\Re {\upsilon }_n,$$

(3)

for all $n\in \mathbb{N}\cup \left\{0\right\}.$ By edge-preserving property (1) and the fact $({\upsilon }_n,{\upsilon }_{n+1})\in E\left(G\right)$ for $n=0,$ it follows inductively that

$$({\upsilon }_n,{\upsilon }_{n+1})\in E\left(G\right)$$

for all $n\in \mathbb{N}\cup \left\{0\right\}.$ Apply (2) to successive pairs $({\upsilon }_n,{\upsilon }_{n+1})\in E\left(G\right)$ to obtain

$$d({\upsilon }_{n+1},{\upsilon }_{n+2})=d(\Re {\upsilon }_n,\Re {\upsilon }_{n+1})\le \varrho d({\upsilon }_n,{\upsilon }_{n+1}).$$

By induction,

$$d({\upsilon }_{n+1},{\upsilon }_{n+2})\le {\varrho }^{n}d({\upsilon }_0,{\upsilon }_1)$$

(4)

for all $n\in \mathbb{N}\cup \left\{0\right\}.$ For $m>n,$ consider the chain (${\upsilon }_n,{\upsilon }_{n+1}),\dots ,$(${\upsilon }_{m-1},{\upsilon }_m).$ Then using the $\mathcal{EF}$-metric triangle inequality ($D_{E3}$), we have

$$\hslash \left(d({\upsilon }_n,{\upsilon }_m)\right)\le \hslash \left(\sum _{i=n}^{m-1}d({\upsilon }_i,{\upsilon }_{i+1})\prod _{j=1}^i\rho ({\upsilon }_j,{\upsilon }_m)\right)+⋏.$$

Since

$$\begin{array}{ccc}\hfill \sum _{i=n}^{m-1}d({\upsilon }_i,{\upsilon }_{i+1})\prod _{j=1}^i\rho ({\upsilon }_j,{\upsilon }_m)& =& \rho ({\upsilon }_1,{\upsilon }_m)\rho ({\upsilon }_2,{\upsilon }_m)···\rho ({\upsilon }_n,{\upsilon }_m)d({\upsilon }_n,{\upsilon }_{n+1})\hfill \\ & & +\rho ({\upsilon }_1,{\upsilon }_m)\rho ({\upsilon }_2,{\upsilon }_m)···\rho ({\upsilon }_{n+1},{\upsilon }_m)d({\upsilon }_{n+1},{\upsilon }_{n+2})\hfill \\ & & ·\hfill \\ & & ·\hfill \\ & & ·\hfill \\ & & +\rho ({\upsilon }_1,{\upsilon }_m)\rho ({\upsilon }_2,{\upsilon }_m)···\rho ({\upsilon }_{m-1},{\upsilon }_m)d({\upsilon }_{m-1},{\upsilon }_m).\hfill \end{array}$$

By using the inequality (4), we have

$$\begin{array}{ccc}\hfill \sum _{i=n}^{m-1}d({\upsilon }_i,{\upsilon }_{i+1})\prod _{j=1}^i\rho ({\upsilon }_j,{\upsilon }_m)& \le & \rho ({\upsilon }_1,{\upsilon }_m)\rho ({\upsilon }_2,{\upsilon }_m)···\rho ({\upsilon }_n,{\upsilon }_m){\varrho }^{n}d({\upsilon }_0,{\upsilon }_1)\hfill \\ & & +\rho ({\upsilon }_1,{\upsilon }_m)\rho ({\upsilon }_2,{\upsilon }_m)···\rho ({\upsilon }_{n+1},{\upsilon }_m){\varrho }^{n+1}d({\upsilon }_0,{\upsilon }_1)\hfill \\ & & ·\hfill \\ & & ·\hfill \\ & & ·\hfill \\ & & +\rho ({\upsilon }_1,{\upsilon }_m)\rho ({\upsilon }_2,{\upsilon }_m)···\rho ({\upsilon }_{m-1},{\upsilon }_m){\varrho }^{m-1}d({\upsilon }_0,{\upsilon }_1)\hfill \\ & =& d({\upsilon }_0,{\upsilon }_1)\sum _{i=n}^{m-1}{\varrho }^i\prod _{j=1}^i\rho ({\upsilon }_j,{\upsilon }_m).\hfill \end{array}$$

(5)

By substituting inequality (5) into (4) and using the property (${\mathcal{F}}_1$), we have

$$\begin{array}{ccc}\hfill \hslash \left(d({\upsilon }_n,{\upsilon }_m)\right)& \le & \hslash \left(\sum _{i=n}^{m-1}d({\upsilon }_i,{\upsilon }_{i+1})\prod _{j=1}^i\rho ({\upsilon }_j,{\upsilon }_m)\right)+⋏\hfill \\ & \le & \hslash \left(d({\upsilon }_0,{\upsilon }_1)\sum _{i=n}^{m-1}{\varrho }^i\prod _{j=1}^i\rho ({\upsilon }_j,{\upsilon }_m)\right)+⋏.\hfill \end{array}$$

(6)

Since ${sup}_{m\ge 1}{lim}_{n\to \infty }\rho \left({\upsilon }_n,{\upsilon }_m\right)\varrho <1,$ the series ${\sum }_{n=1}^{\infty }{\varrho }^n{\displaystyle \prod _{i=1}^n}\rho ({\upsilon }_i,{\upsilon }_m)$ converges (by the ratio test). Let

$$S=\sum _{n=1}^{\infty }{\varrho }^n\prod _{i=1}^n\rho ({\upsilon }_i,{\upsilon }_m)$$

and

$$S_n=\sum _{j=1}^n{\varrho }^j\prod _{i=1}^j\rho ({\upsilon }_i,{\upsilon }_m).$$

Since

$$\sum _{i=n}^{m-1}d({\upsilon }_i,{\upsilon }_{i+1})\prod _{j=1}^i\rho ({\upsilon }_j,{\upsilon }_m)\le d({\upsilon }_0,{\upsilon }_1)\left[\sum _{j=1}^{m-1}{\varrho }^j\prod _{i=1}^j\rho ({\upsilon }_i,{\upsilon }_m)-\sum _{j=1}^{n-1}{\varrho }^j\prod _{i=1}^j\rho ({\upsilon }_i,{\upsilon }_m)\right]$$

(7)

for $m>n.$ Since ${\sum }_{j=1}^{m-1}{\varrho }^j{\displaystyle \prod _{i=1}^j}\rho ({\upsilon }_i,{\upsilon }_m)-{\sum }_{j=1}^{n-1}{\varrho }^j{\displaystyle \prod _{i=1}^j}\rho ({\upsilon }_i,{\upsilon }_m)$ converges, so there exists some $N\in \mathbb{N}$ such that

$$0<\left(\sum _{j=1}^{m-1}{\varrho }^j\prod _{i=1}^j\rho ({\upsilon }_i,{\upsilon }_m)-\sum _{j=1}^{n-1}{\varrho }^j\prod _{i=1}^j\rho ({\upsilon }_i,{\upsilon }_m)\right)d({\upsilon }_0,{\upsilon }_1)<\delta ,$$

(8)

for $n\ge N.$ Since ${lim}_{t\to 0^{+}}\hslash \left(t\right)=-\infty$ by condition (${\mathcal{F}}_2$), for the fixed number $\hslash \left(ϵ\right)-⋏\in \mathbb{R},$ there is $\delta >0$ such that

$$0<t<\delta ⟹\hslash \left(t\right)<\hslash \left(ϵ\right)-⋏.$$

(9)

Now by (7), (8) and (9), we have

$$\begin{array}{ccc}\hfill \hslash \left(\sum _{i=n}^{m-1}d({\upsilon }_i,{\upsilon }_{i+1})\prod _{j=1}^i\rho ({\upsilon }_j,{\upsilon }_m)\right)& \le & \hslash \left(d({\upsilon }_0,{\upsilon }_1)\left[\begin{array}{c}\sum _{j=1}^{m-1}{\varrho }^j{\displaystyle \prod _{i=1}^j}\rho ({\upsilon }_i,{\upsilon }_m)\\ -\sum _{j=1}^{n-1}{\varrho }^j{\displaystyle \prod _{i=1}^j}\rho ({\upsilon }_i,{\upsilon }_m)\end{array}\right]\right)\hfill \\ & <& \hslash \left(ϵ\right)-⋏,\hfill \end{array}$$

(10)

for $m>n\ge N.$ Thus, by (6) and (10), we have

$$\begin{array}{ccc}\hfill d({\upsilon }_n,{\upsilon }_m)& >& 0⟹\hslash \left(d({\upsilon }_n,{\upsilon }_m)\right)\le \hslash \left(\sum _{i=n}^{m-1}d({\upsilon }_i,{\upsilon }_{i+1})\prod _{j=1}^i\rho ({\upsilon }_j,{\upsilon }_m)\right)+⋏\hfill \\ & <& \hslash \left(ϵ\right).\hfill \end{array}$$

This shows, by condition (${\mathcal{F}}_1$) that for all $m>n\ge N,$ we have $d({\upsilon }_n,{\upsilon }_m)<ϵ,$ proving that the sequence {${\upsilon }_n$} is $\mathcal{E}$-Cauchy. As $(X,d)$ is $\mathcal{E}$-complete, a point ${\upsilon }^{*}\in X$ exists such that $\left\{{\upsilon }_n\right\}$ is $\mathcal{E}$-convergent to ${\upsilon }^{*}$ under the $\mathcal{EF}$-metric, i.e.,

$$\underset{n\to \infty }{lim}d({\upsilon }_n,{\upsilon }^{*})=0.$$

(11)

Next, we show that ${\upsilon }^{*}$ is a fp of self-mapping ℜ$:X\to X.$ Suppose ℜ is G-continuous. Since

$${\upsilon }_n\to {\upsilon }^{*},\mathrm{as}n\to \infty ,$$

by G-continuity of $\Re ,$ we have

$$\Re {\upsilon }_n\to \Re {\upsilon }^{*},\mathrm{as}n\to \infty .$$

Taking the limit in (3) as $n\to \infty ,$ we obtain

$$\Re {\upsilon }^{*}={\upsilon }^{*}.$$

Hence, ${\upsilon }^{*}$ is a fp of $\Re .$ Further if $\left\{{\upsilon }_n\right\}$$\subseteq X$ is a sequence such that ${\upsilon }_n\to {\upsilon }^{*}$ as $n\to \infty$ and $({\upsilon }_n,{\upsilon }_{n+1})$$\in E\left(G\right),$ then by the assumption, we get $({\upsilon }_n,{\upsilon }^{*})$ $\in E\left(G\right)$, ∀$n\in \mathbb{N}.$ Now apply the contraction (2) to each pair $({\upsilon }_n,{\upsilon }^{*})\in E\left(G\right).$ That yields by (2)

$$d(\Re {\upsilon }_n,\Re {\upsilon }^{*})\le \varrho d({\upsilon }_n,{\upsilon }^{*}).$$

By ($D_{E3}$), we have

$$\begin{array}{ccc}\hfill \hslash \left(d({\upsilon }^{*},\Re {\upsilon }^{*})\right)& \le & \hslash \left(\begin{array}{c}d({\upsilon }^{*},{\upsilon }_{n+1})\rho ({\upsilon }^{*},\Re {\upsilon }^{*})\\ +d({\upsilon }_{n+1},\Re {\upsilon }^{*})\rho ({\upsilon }^{*},\Re {\upsilon }^{*})\rho ({\upsilon }^{*},{\upsilon }_{n+1})\end{array}\right)+⋏\hfill \\ & =& \hslash \left(\begin{array}{c}d({\upsilon }^{*},{\upsilon }_{n+1})\rho ({\upsilon }^{*},\Re {\upsilon }^{*})\\ +d(\Re {\upsilon }_n,\Re {\upsilon }^{*})\rho ({\upsilon }^{*},\Re {\upsilon }^{*})\rho ({\upsilon }^{*},{\upsilon }_{n+1})\end{array}\right)+⋏\hfill \\ & \le & \hslash \left(\begin{array}{c}d({\upsilon }^{*},{\upsilon }_{n+1})\rho ({\upsilon }^{*},\Re {\upsilon }^{*})\\ +\varrho d({\upsilon }_n,{\upsilon }^{*})\rho ({\upsilon }^{*},\Re {\upsilon }^{*})\rho ({\upsilon }^{*},{\upsilon }_{n+1})\end{array}\right)+⋏.\hfill \end{array}$$

(12)

Since

$$\underset{n\to \infty }{lim}\hslash \left(\begin{array}{c}d({\upsilon }^{*},{\upsilon }_{n+1})\rho ({\upsilon }^{*},\Re {\upsilon }^{*})\\ +\varrho d({\upsilon }_n,{\upsilon }^{*})\rho ({\upsilon }^{*},\Re {\upsilon }^{*})\rho ({\upsilon }^{*},{\upsilon }_{n+1})\end{array}\right)+⋏=-\infty .$$

So, by (12) and (${\mathcal{F}}_2$), we have $d({\upsilon }^{*},\Re {\upsilon }^{*})=0$, i.e. ${\upsilon }^{*}=\Re {\upsilon }^{*}$. Thus, ${\upsilon }^{*}\in X$ is a fp of $\Re .$ Now we prove the uniqueness of fp. We start by supposing that, for the sake of contradiction, that ℜ has two fps, that are, ${\upsilon }^{*},{\upsilon }^{/}\in X$ such that ${\upsilon }^{*}=\Re {\upsilon }^{*}$ and ${\upsilon }^{/}=\Re {\upsilon }^{/}$ but ${\upsilon }^{*}\ne {\upsilon }^{/}.$ By assumption (iv), there exists a finite directed path {$u_j{\}}_{j=0}^M$ such that $u_0={\upsilon }^{*},$$u_M={\upsilon }^{/}$ and $\left(u_{j-1},u_j\right)\in E\left(G\right)$ for all $j=1,2,\dots ,M$. We prove by induction that

$$d({\upsilon }^{*},u_j)=0,$$

for all $j=1,2,\dots ,M.$

For $j=1$, since $\left({\upsilon }^{*},u_1\right)=\left(u_0,u_1\right)\in E\left(G\right),$ applying the contraction condition (2) gives

$$d({\upsilon }^{*},u_1)=d(\Re {\upsilon }^{*},\Re u_1)\le {\varrho }_{1}d({\upsilon }^{*},u_1)+{\varrho }_2{\displaystyle \frac{d({\upsilon }^{*},\Re {\upsilon }^{*})d(u_1,\Re u_1)}{1+d({\upsilon }^{*},u_1)}}.$$

Since ${\upsilon }^{*}$ is a fp of $\Re ,$ then $d({\upsilon }^{*},\Re {\upsilon }^{*})=0.$ Therefore,

$$d({\upsilon }^{*},u_1)\le {\varrho }_{1}d({\upsilon }^{*},u_1).$$

Since $0<{\varrho }_1<1,$ this inequality implies $d({\upsilon }^{*},u_1)=0.$ Hence, ${\upsilon }^{*}=u_1.$ Assume now that

$${\upsilon }^{*}=u_{k-1},$$

for some $k\in \{2,3,\dots ,M\}.$ We need to show ${\upsilon }^{*}=u_k$. Since $\left(u_{k-1},u_k\right)\in E\left(G\right)$ and $u_{k-1}={\upsilon }^{*}$ (by the induction hypothesis), we have

$$\left({\upsilon }^{*},u_k\right)\in E\left(G\right).$$

Applying (2) and using the fact that $d({\upsilon }^{*},\Re {\upsilon }^{*})=0,$ we have

$$\begin{array}{ccc}\hfill d\left({\upsilon }^{*},u_k\right)& =& d(\Re {\upsilon }^{*},\Re u_k)\le {\varrho }_{1}d({\upsilon }^{*},u_k)+{\varrho }_2{\displaystyle \frac{d({\upsilon }^{*},\Re {\upsilon }^{*})d(u_k,\Re u_k)}{1+d({\upsilon }^{*},u_k)}}\hfill \\ & =& {\varrho }_{1}d({\upsilon }^{*},u_k)+{\varrho }_2\left(0\right)\hfill \\ & =& {\varrho }_{1}d({\upsilon }^{*},u_k).\hfill \end{array}$$

Because $0<{\varrho }_1<1,$ we must have $d\left({\upsilon }^{*},u_k\right)=0,$ which implies ${\upsilon }^{*}=u_k.$ By induction, we conclude that ${\upsilon }^{*}=u_M.$ Since $u_M={\upsilon }^{/},$ by the definition of the directed path, we have ${\upsilon }^{*}={\upsilon }^{/}.$ This contradicts our initial assumption that ${\upsilon }^{*}\ne {\upsilon }^{/}.$ Hence, fp is unique. □

As a special case of our main Theorem 2, we recover the well-known fp result established by Panda et al. [8] for contraction mappings in $\mathcal{EF}$-mss. This demonstrates that our theorem not only generalizes existing results but also unifies earlier contraction principles within a broader graph-theoretic framework.

Corollary 1 ([8]).

Let $(X,d)$ be an $\mathcal{E}$-complete $\mathcal{EF}$-ms. Suppose $\Re :X\to X$ satisfies

$$d(\Re \upsilon ,\Re \varkappa )\le \varrho d(\upsilon ,\varkappa ),forall\upsilon ,\varkappa \in X$$

for some constant $\varrho \in (0,1).$ Then ℜ has a unique fp in $X.$

Proof. 

Apply Theorem 2 with the directed graph $G=\left(X,X\times X\right).$ □

Remark 2.

(i) By defining the control function $\rho :X\times X\to [1,+\infty )$ as $\rho (\upsilon ,\varkappa )=1$ in Definition 5, the $\mathcal{EF}$-ms reduces to a standard $\mathcal{F}$-ms. Consequently, Theorem 2 coincides with the main result of Jleli et al. [7].

(ii) If we choose $\hslash \left(t\right)=lnt,$ for $t>0$ and $⋏=lns,$ for $s>1,$ and define $\rho (\upsilon ,\varkappa )=1$ in Definition 5, then the $\mathcal{EF}$-ms reduces to a b-ms. Considering the graph $G=\left(X,X\times X\right),$ Theorem 2 specializes to the leading fp theorem of Czerwik [3].

(iii) Defining $\rho (\upsilon ,\varkappa )=1$ and choosing $\hslash \left(t\right)=lnt,$ for $t>0$ with $⋏=0$ in Definition 5, the $\mathcal{EF}$-ms reduces to a classical ms. In this case, Theorem 2 recovers the well-known fp theorem of Jachymski [13].

(iv) Finally, by taking $G=\left(X,X\times X\right)$ along with the setting in (iii), Theorem 2 reduces to the classical Banach contraction principle [10].

Example 1.

Let $X=\left\{1,2,3\right\}$. Define the $\mathcal{EF}$-metric $d:X\times X\to {\mathbb{R}}^{+}$ (non-negative real numbers) by

$$d(\upsilon ,\varkappa )=\left|\upsilon -\varkappa \right|,$$

$\upsilon ,\varkappa \in X$ and the function $\rho :X\times X\to [1,+\infty )$ by $\rho (\upsilon ,\varkappa )=1+{\displaystyle \frac{1}{6}}\upsilon +{\displaystyle \frac{1}{12}}\varkappa .$ Then $(X,d)$ is an $\mathcal{EF}$-ms with $\hslash \left(t\right)=lnt$ for $t>0$ and $⋏=0$. Moreover, the set X is finite, so an $\mathcal{EF}$-ms is $\mathcal{E}$-complete. Define the directed graph $G=\left(X,E\left(G\right)\right)$ by specifying the edge set

$$E\left(G\right)=\left\{(2,1),(3,1)\right\}\cup \left\{(1,1),(2,2),(3,3)\right\},$$

(i.e., edges $2\to 1$ and $3\to 1,$ plus self-loops). The structure of the directed graph $G=\left(\Psi ,E\left(G\right)\right)$ is illustrated in Figure 2.

Define the self-mapping $\Re :X\to X$ by

$$\Re \left(1\right)=1,\Re \left(2\right)=1,\Re \left(3\right)=2.$$

Now if $(\upsilon ,\varkappa )\in E\left(G\right),$ then $\left(\Re \upsilon ,\Re \varkappa \right)$ equals either $(1,1)$ or $(1,1)$ or $(2,1),$ each of which is in $E\left(G\right).$ So (1) holds. Now if

$$(\upsilon ,\varkappa )=(2,1)\in E\left(G\right),$$

then

$$d\left(\Re 2,\Re 1\right)=d\left(1,1\right)=0<0.5=0.5·d(2,1).$$

If

$$(\upsilon ,\varkappa )=(3,1)\in E\left(G\right),$$

then

$$d\left(\Re 3,\Re 1\right)=d\left(2,1\right)=1=0.5·d(3,1).$$

Thus, the inequality (2) holds with $\varrho ={\varrho }_1=0.5\in (0,1)$ and ${\varrho }_2=0$. Hence, the self-mapping $\Re :X\to X$ is a graphic rational contraction. Take ${\upsilon }_0=2.$ Then

$$\left({\upsilon }_0,\Re {\upsilon }_0\right)=(2,1)\in E\left(G\right).$$

So condition (i) of Theorem 2 is satisfied. For any initial ${\upsilon }_0\in X,$ the Picard sequence ${\upsilon }_{n+1}=\Re {\upsilon }_n,$ for all $n\in \mathbb{N},$ from any start becomes eventually $\left\{1,1,\dots \right\}$ or $\left\{2,1,1,\dots \right\}.$ So ${\upsilon }_n\to 1,$ and ${\upsilon }_m\to 1$ as $n,m\to \infty .$ Thus, we have

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)=\rho \left(1,1\right)=1+{\displaystyle \frac{1}{6}}\left(1\right)+{\displaystyle \frac{1}{12}}1={\displaystyle \frac{7}{6}}+{\displaystyle \frac{1}{12}}.$$

Therefore,

$$\underset{m.n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)={\displaystyle \frac{7}{6}}+{\displaystyle \frac{1}{12}}\left(1\right)={\displaystyle \frac{15}{12}}<2={\displaystyle \frac{1}{\varrho }}.$$

Thus, the condition (ii) holds. Since X is finite, so every function $X\to X$ is continuous with respect to $d.$ Hence, ℜ is continuous and therefore G-continuous. So condition (iii) holds. All hypotheses of the Theorem 2 are satisfied, so ℜ has a fp. Indeed $\Re \left(1\right)=1$. However, the main result of Panda et al. [8] (the Banach-type contraction theorem in $\mathcal{EF}$-mss) is not applicable here, since the contractive condition fails for the non-edge pair $(\upsilon ,\varkappa )=(3,2).$ Indeed,

$$d\left(\Re 3,\Re 2\right)=d\left(2,1\right)=1>0.5=0.5·d(3,2),$$

showing that no global contraction constant $\varrho <1$ can satisfy the required inequality for all pairs in $X\times X.$ Hence, the result of [8] cannot be applied, while our graph-based theorem remains valid.

Example 2.

Let $X=\left\{1,2,3,4,5,6,7,8\right\}$. Define the $\mathcal{EF}$-metric $d:X\times X\to {\mathbb{R}}^{+}$ by

$$d(\upsilon ,\varkappa )=\left|\upsilon -\varkappa \right|,$$

$\upsilon ,\varkappa \in X$ and the function $\rho :X\times X\to [1,+\infty )$ by $\rho (\upsilon ,\varkappa )=1+2\upsilon +3\varkappa .$ Then $(X,d)$ is an $\mathcal{EF}$-ms with $\hslash \left(t\right)=lnt$ for $t>0$ and $⋏=0$. Moreover, the set X is finite, so an $\mathcal{EF}$-ms is $\mathcal{E}$-complete. Define the directed graph $G=\left(X,E\left(G\right)\right)$ by specifying the edge set

$$E\left(G\right)=\left\{(1,1)\right\}\cup \left\{(2,1),(3,1),(4,1),(5,1),(6,1),(7,1),(8,1)\right\}.$$

So every vertex $2,3,\dots ,8$ has a directed edge to 1, and 1 has a self-loop.

Define the self-mapping $\Re :X\to X$ by

$$\Re \left(\upsilon \right)=1$$

for every $\upsilon \in X.$ Now for $(\upsilon ,\varkappa )\in E\left(G\right),$ we have

$$\left(\Re \upsilon ,\Re \varkappa \right)=\left(1,1\right)\in E\left(G\right).$$

Thus, ℜ is edge-preserving. Now if $(\upsilon ,\varkappa )\in E\left(G\right),$ then either $\upsilon =\varkappa =1$ (so $d\left(\upsilon ,\varkappa \right)=0$) or $\varkappa =1$ and $\upsilon \in \left\{2,3,\dots ,8\right\}$ (so $d\left(\upsilon ,\varkappa \right)=\left|\upsilon -\varkappa \right|\ge 1$). In every case

$$d\left(\Re \upsilon ,\Re \varkappa \right)=d\left(1,1\right)=0\le \varrho d(\upsilon ,\varkappa ),$$

for $\varrho ={\varrho }_1={\displaystyle \frac{1}{28}}\in (0,1)$ and ${\varrho }_2=0.$ Thus, the inequality (2) holds. Hence, the self-mapping $\Re :X\to X$ is a graphic contraction. Let ${\upsilon }_0=2.$ Then $\Re {\upsilon }_0=1$ and $(2,1)\in E\left(G\right)$ by definition. So condition (i) of Theorem 2 is satisfied. For any initial ${\upsilon }_0\in X,$ define the Picard sequence ${\upsilon }_{n+1}=\Re {\upsilon }_n,$ for all $n\in \mathbb{N}.$ Because ℜ is constant equal to 1, we have

$${\upsilon }_1=1,{\upsilon }_2=\Re 1=1,{\upsilon }_3=1,\dots$$

So ${\upsilon }_n\to 1$ and ${\upsilon }_m\to 1,$ as $m,n\to \infty .$ Thus, we have

$$\underset{m.n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)=\rho \left(1,1\right)=1+2\left(1\right)+3\left(1\right)=3+3.$$

Therefore,

$$\underset{m.n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)=6<28={\displaystyle \frac{1}{\varrho }}.$$

Thus, the condition (ii) holds. The mapping ℜ is constant, hence continuous and therefore G-continuous. So condition (iii) holds. All hypotheses of the Theorem 2 are satisfied; hence,

$$\Re \left(1\right)=1$$

so 1 is a fp. Figure 3 displays the directed graph $G=\left(\Psi ,E\left(G\right)\right)$, with its vertices and edge relations.

Definition 8.

Let $(X,d)$ be an $\mathcal{EF}$-ms and let ${\Theta }_1,{\Theta }_2\subset X$ be non-empty subsets. A self mapping ℜ$:{\Theta }_1\cup {\Theta }_2\to {\Theta }_1\cup {\Theta }_2$ is said to be an interpolative Ćirić–Reich–Rus-type cyclic contraction if there exist $\varrho \in (0,1)$ and positive reals $\beta ,\gamma$ such that $\beta +\gamma <1$ and

$$d(\Re \upsilon ,\Re \varkappa )\le \varrho {\left[d(\upsilon ,\varkappa )\right]}^{\gamma }{\left[d(\upsilon ,\Re \upsilon )\right]}^{\beta }{\left[d(\varkappa ,\Re \varkappa )\right]}^{1-\beta -\gamma },$$

(13)

for all $(\upsilon ,\varkappa )\in {\Theta }_1\times {\Theta }_2$ with $\upsilon ,\varkappa \notin Fix(\Re )$, where $Fix(\Re )$ represents the collection of all fps of ℜ.

Example 3.

Let $X=[0,1]$. Define $d:X\times X\to [0,+\infty )$ by

$$d(\upsilon ,\varkappa )=|\upsilon -\varkappa |$$

$\forall \upsilon ,\varkappa \in X$ and $\rho :X\times X\to [1,+\infty )$ by $\rho (\upsilon ,\varkappa )=1+\upsilon +\varkappa .$ Select $\hslash \in \mathcal{F}$ by $\hslash \left(t\right)=ln\left(t\right)$ for $t>0$ and $⋏=ln\left(1\right)$, then ($X,d$) is an $\mathcal{EF}$-ms. Define two non-empty subsets

$${\Theta }_1=\left[0,0.60\right]and{\Theta }_2=\left[0.4,1\right].$$

Clearly,

$${\Theta }_1\cup {\Theta }_2=Xand{\Theta }_1\cap {\Theta }_2=\left[0.4,0.6\right]\ne \varnothing .$$

Define a mapping $\Re :{\Theta }_1\cup {\Theta }_2\to {\Theta }_1\cup {\Theta }_2$ by

$$\Re \left(\upsilon \right)=0.5,$$

for all $\upsilon \in X.$ Then

$$\Re \left({\Theta }_1\right)\subseteq {\Theta }_{2}and\Re \left({\Theta }_2\right)\subseteq {\Theta }_1,$$

so ℜ is cyclic. Moreover,

$$Fix(\Re )=\left\{0.5\right\}\subset {\Theta }_1\cap {\Theta }_2.$$

Choose positive constants

$$\varrho =0.5,\beta =0.2,\gamma =0.5,$$

so $\beta +\gamma =0.7<1$ and $\varrho \in (0,1).$ For any $(\upsilon ,\varkappa )$$\in {\Theta }_1\times {\Theta }_2$ with $\upsilon ,\varkappa \notin Fix(\Re ),$ we have

$$d(\Re \upsilon ,\Re \varkappa )=\left|0.5-0.5\right|=0.$$

On the other hand,

$$\varrho {\left[d(\upsilon ,\varkappa )\right]}^{\gamma }{\left[d(\upsilon ,\Re \upsilon )\right]}^{\beta }{\left[d(\varkappa ,\Re \varkappa )\right]}^{1-\beta -\gamma }\ge 0.$$

Hence,

$$d(\Re \upsilon ,\Re \varkappa )\le \varrho {\left[d(\upsilon ,\varkappa )\right]}^{\gamma }{\left[d(\upsilon ,\Re \upsilon )\right]}^{\beta }{\left[d(\varkappa ,\Re \varkappa )\right]}^{1-\beta -\gamma },$$

for all $(\upsilon ,\varkappa )$$\in {\Theta }_1\times {\Theta }_2$ with $\upsilon ,\varkappa \notin Fix(\Re ).$ Therefore, ℜ is an interpolative Ćirić–Reich–Rus-type cyclic contraction on ${\Theta }_1\cup {\Theta }_2.$

Theorem 3.

Let $(X,d)$ be an $\mathcal{E}$-complete $\mathcal{EF}$-ms and let ${\Theta }_1,{\Theta }_2\subset X$ be non-empty subsets. If ℜ$:{\Theta }_1\cup {\Theta }_2\to {\Theta }_1\cup {\Theta }_2$ is an interpolative Ćirić–Reich–Rus-type cyclic contraction. Given any starting point ${\upsilon }_0\in X,$ consider the sequence {${\upsilon }_n$} generated recursively by

$${\upsilon }_{n+1}=\Re {\upsilon }_n,n\in \mathbb{N}.$$

If

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)<{\displaystyle \frac{1}{\omega }},$$

where $\omega ={\varrho }^{{\displaystyle \frac{1}{1-\beta }}}\in (0,1),$ then ℜ has a unique fp in ${\Theta }_1\cap {\Theta }_2.$

Proof. 

Consider the sequence $\left\{{\upsilon }_n\right\}\subset X$ defined recursively as

$${\upsilon }_{n+1}=\Re {\upsilon }_n,$$

$\forall n\in \mathbb{N}\cup \left\{0\right\}.$ Set ${\upsilon }_0\in {\Theta }_1,$ then

$${\upsilon }_1=\Re {\upsilon }_0\in {\Theta }_2$$

$${\upsilon }_2=\Re {\upsilon }_1\in {\Theta }_1$$

$${\upsilon }_3=\Re {\upsilon }_2\in {\Theta }_2$$

$${\upsilon }_4=\Re {\upsilon }_3\in {\Theta }_1$$

and so forth. In view of (13), we obtain

$$\begin{array}{ccc}\hfill d({\upsilon }_{n+1},{\upsilon }_n)& =& d(\Re {\upsilon }_n,\Re {\upsilon }_{n-1})\le \varrho {\left[d({\upsilon }_n,{\upsilon }_{n-1})\right]}^{\gamma }{\left[d({\upsilon }_n,\Re {\upsilon }_n)\right]}^{\beta }\hfill \\ & & ·{\left[d({\upsilon }_{n-1},\Re {\upsilon }_{n-1})\right]}^{1-\beta -\gamma }\hfill \\ & =& \varrho {\left[d({\upsilon }_n,{\upsilon }_{n-1})\right]}^{\gamma }{\left[d({\upsilon }_n,{\upsilon }_{n+1})\right]}^{\beta }{\left[d({\upsilon }_{n-1},{\upsilon }_n)\right]}^{1-\beta -\gamma }\hfill \\ & =& \varrho {\left[d({\upsilon }_{n-1},{\upsilon }_n)\right]}^{1-\beta }{\left[d({\upsilon }_n,{\upsilon }_{n+1})\right]}^{\beta },\hfill \end{array}$$

which implies that

$${\left[d({\upsilon }_n,{\upsilon }_{n+1})\right]}^{1-\beta }\le \varrho {\left[d({\upsilon }_{n-1},{\upsilon }_n)\right]}^{1-\beta },$$

that is,

$$d({\upsilon }_n,{\upsilon }_{n+1})\le \omega d({\upsilon }_{n-1},{\upsilon }_n)$$

(14)

∀$n\in \mathbb{N}\cup \left\{0\right\},$ where $\omega ={\varrho }^{{\displaystyle \frac{1}{1-\beta }}}\in (0,1).$ In the same way, through (13), we have

$$\begin{array}{ccc}\hfill d({\upsilon }_n,{\upsilon }_{n-1})& =& d(\Re {\upsilon }_{n-1},\Re {\upsilon }_{n-2})\le \varrho {\left[d({\upsilon }_{n-1},{\upsilon }_{n-2})\right]}^{\gamma }{\left[d({\upsilon }_{n-1},\Re {\upsilon }_{n-1})\right]}^{\beta }\hfill \\ & & ·{\left[d({\upsilon }_{n-2},\Re {\upsilon }_{n-2})\right]}^{1-\beta -\gamma }\hfill \\ & =& \varrho {\left[d({\upsilon }_{n-1},{\upsilon }_{n-2})\right]}^{\gamma }{\left[d({\upsilon }_{n-1},{\upsilon }_n)\right]}^{\beta }{\left[d({\upsilon }_{n-2},{\upsilon }_{n-1})\right]}^{1-\beta -\gamma }\hfill \\ & =& \varrho {\left[d({\upsilon }_{n-2},{\upsilon }_{n-1})\right]}^{1-\beta }{\left[d({\upsilon }_{n-1},{\upsilon }_n)\right]}^{\beta },\hfill \end{array}$$

which implies that

$${\left[d({\upsilon }_{n-1},{\upsilon }_n)\right]}^{1-\beta }\le \omega {\left[d({\upsilon }_{n-2},{\upsilon }_{n-1})\right]}^{1-\beta },$$

that is,

$$d({\upsilon }_{n-1},{\upsilon }_n)\le \omega d({\upsilon }_{n-2},{\upsilon }_{n-1})$$

(15)

∀$n\in \mathbb{N}\cup \left\{0\right\},$ where $\omega ={\varrho }^{{\displaystyle \frac{1}{1-\beta }}}\in (0,1).$ It follows from (14) and (15) that

$$d({\upsilon }_n,{\upsilon }_{n+1})\le {\omega }^{n}d({\upsilon }_0,{\upsilon }_1)$$

(16)

∀$n\in \mathbb{N}\cup \left\{0\right\}.$ By following the same approach as in the proof of Theorem 2, we conclude that the sequence {${\upsilon }_n$} is $\mathcal{E}$-Cauchy. Because $(X,d)$ is $\mathcal{E}$-complete, there exists a point ${\upsilon }^{*}\in X$ to which the sequence $\left\{{\upsilon }_n\right\}$ converges in the $\mathcal{EF}$-metric; that is,

$$\underset{n\to \infty }{lim}d({\upsilon }_n,{\upsilon }^{*})=0.$$

It is evident that the subsequence {${\upsilon }_{2n}$} lies entirely in ${\Theta }_1,$ while {${\upsilon }_{2n+1}$} belongs to ${\Theta }_2.$ Because both subsequences approach the same point ${\upsilon }^{*},$ we conclude that ${\upsilon }^{*}\in {\Theta }_1\cap {\Theta }_2.$ Next, we verify that ${\upsilon }^{*}$ is indeed a fp of ℜ$:{\Theta }_1\cup {\Theta }_2\to {\Theta }_1\cup {\Theta }_2.$ By ($D_{\mathcal{E}3}$), we get

$$\begin{array}{ccc}\hfill \hslash \left(d(\Re {\upsilon }^{*},{\upsilon }^{*})\right)& \le & \hslash \left(\begin{array}{c}d(\Re {\upsilon }^{*},{\upsilon }_{n+1})\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\\ +d({\upsilon }_{n+1},{\upsilon }^{*})\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\rho ({\upsilon }_{n+1},{\upsilon }^{*})\end{array}\right)+⋏\hfill \\ & =& \hslash \left(\begin{array}{c}d(\Re {\upsilon }^{*},\Re {\upsilon }_n)\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\\ +d({\upsilon }_{n+1},{\upsilon }^{*})\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\rho ({\upsilon }_{n+1},{\upsilon }^{*})\end{array}\right)+⋏\hfill \\ & \le & \hslash \left(\begin{array}{c}\left(\begin{array}{c}\varrho {\left[d({\upsilon }^{*},{\upsilon }_n)\right]}^{\gamma }{\left[d({\upsilon }^{*},\Re {\upsilon }^{*})\right]}^{\beta }\\ ·{\left[d({\upsilon }_n,\Re {\upsilon }_n)\right]}^{1-\beta -\gamma }\end{array}\right)\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\\ +d({\upsilon }_{n+1},{\upsilon }^{*})\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\rho ({\upsilon }_{n+1},{\upsilon }^{*}))\end{array}\right)+⋏\hfill \\ & =& \hslash \left(\begin{array}{c}\left(\begin{array}{c}\varrho {\left[d({\upsilon }^{*},{\upsilon }_n)\right]}^{\gamma }{\left[d({\upsilon }^{*},\Re {\upsilon }^{*})\right]}^{\beta }\\ ·{\left[d({\upsilon }_n,{\upsilon }_{n+1})\right]}^{1-\beta -\gamma }\end{array}\right)\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\\ +d({\upsilon }_{n+1},{\upsilon }^{*})\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\rho ({\upsilon }_{n+1},{\upsilon }^{*}))\end{array}\right)+⋏\hfill \end{array}$$

(17)

Since

$$\underset{n\to \infty }{lim}\hslash \left(\begin{array}{c}\left(\varrho {\left[d({\upsilon }^{*},{\upsilon }_n)\right]}^{\gamma }{\left[d({\upsilon }^{*},\Re {\upsilon }^{*})\right]}^{\beta }{\left[d({\upsilon }_n,{\upsilon }_{n+1})\right]}^{1-\beta -\gamma }\right)\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\\ +d({\upsilon }_{n+1},{\upsilon }^{*})\rho (\Re {\upsilon }^{*},{\upsilon }^{*})\rho ({\upsilon }_{n+1},{\upsilon }^{*}))\end{array}\right)+⋏=-\infty .$$

Hence, using (17) and condition (${\mathcal{F}}_2$), we obtain $d(\Re {\upsilon }^{*},{\upsilon }^{*})=0$, which implies $\Re {\upsilon }^{*}={\upsilon }^{*}$. Therefore, ${\upsilon }^{*}\in X$ is a fp of $\Re .$ Assuming the opposite for contradiction, suppose that ℜ has two fps ${\upsilon }^{*}$ and ${\upsilon }^{/}$, so that

$$\Re {\upsilon }^{*}={\upsilon }^{*}\ne {\upsilon }^{/}=\Re {\upsilon }^{/}.$$

Then

$$d({\upsilon }^{*},{\upsilon }^{/})>0.$$

By (13), we have

$$\begin{array}{ccc}\hfill d({\upsilon }^{*},{\upsilon }^{/})& =& d(\Re {\upsilon }^{*},\Re {\upsilon }^{/})\le \varrho {\left[d({\upsilon }^{*},{\upsilon }^{/})\right]}^{\gamma }{\left[d({\upsilon }^{*},\Re {\upsilon }^{*})\right]}^{\beta }{\left[d({\upsilon }^{/},\Re {\upsilon }^{/})\right]}^{1-\beta -\gamma }\hfill \\ & =& \varrho {\left[d({\upsilon }^{*},{\upsilon }^{/})\right]}^{\gamma }{\left[d({\upsilon }^{*},{\upsilon }^{*})\right]}^{\beta }{\left[d({\upsilon }^{/},{\upsilon }^{/})\right]}^{1-\beta -\gamma },\hfill \end{array}$$

whcih is possible only if ${\upsilon }^{*}={\upsilon }^{/}.$ □

This theorem generalizes classical cyclic contraction results to the framework of $\mathcal{EF}$-mss by introducing an interpolative control involving the parameters $\beta$ and $\gamma .$ The inequality (13) balances the distances $d(\upsilon ,\varkappa ),d(\upsilon ,\Re \upsilon ),$ and $d(\varkappa ,\Re \varkappa )$, allowing the contraction strength to vary adaptively between iterates. The restriction $\beta +\gamma <1$ guarantees that the overall contractive influence remains below unity, leading to convergence of the generated sequence toward a unique fp in ${\Theta }_1\cap {\Theta }_2.$ Hence, this result unifies and extends various cyclic and interpolative fp theorems from metric, $\mathcal{F}$-metric, and $\mathcal{EF}$-metric settings.

As a special case of Theorem 3, we now present a fp result for interpolative Ćirić–Reich–Rus-type contractions in an $\mathcal{E}$-complete $\mathcal{EF}$-ms, obtained by removing the cyclic structure (i.e., taking ${\Theta }_1={\Theta }_2=X).$

Corollary 2.

Let $(X,d)$ be an $\mathcal{E}$-complete $\mathcal{EF}$-ms and let $\Re :X\to X$. Suppose the existence of $\varrho \in (0,1)$ and positive reals $\beta ,\gamma$ such that $\beta +\gamma <1$ and

$$d(\Re \upsilon ,\Re \varkappa )\le \varrho {\left[d(\upsilon ,\varkappa )\right]}^{\gamma }{\left[d(\upsilon ,\Re \upsilon )\right]}^{\beta }{\left[d(\varkappa ,\Re \varkappa )\right]}^{1-\beta -\gamma },$$

$\forall \upsilon ,\varkappa \in X$ with $\upsilon ,\varkappa \notin Fix(\Re )$. Given any starting point ${\upsilon }_0\in X,$ consider the sequence {${\upsilon }_n$} generated recursively by

$${\upsilon }_{n+1}=\Re {\upsilon }_n,n\in \mathbb{N}.$$

If

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)<{\displaystyle \frac{1}{\omega }},$$

where $\omega ={\varrho }^{{\displaystyle \frac{1}{1-\beta }}}\in (0,1),$ then ℜ has a unique fp in $X.$

Proof. 

The result follows directly from Theorem 3 by setting ${\Theta }_1={\Theta }_2=X,$ thereby eliminating the cyclic condition. □

Remark 3.

Similar to Remark 2, by defining the control function $\rho :X\times X\to [1,+\infty )$ as

$$\rho (\upsilon ,\varkappa )=1$$

in Definition 5, the $\mathcal{EF}$-ms reduces to an $\mathcal{F}$-ms. Consequently, the corresponding result for interpolative Ćirić–Reich–Rus-type cyclic contractions in the framework of an $\mathcal{F}$-mss can be deduced directly from Theorem 3.

Moreover, analogous results in b-mss and classical mss can be obtained by taking $\hslash \left(t\right)=lnt,$ for $t>0$ and $⋏=lns,$ with $s>1,$ and further letting $⋏=0,$ respectively.

Example 4.

Let $X=\left\{0,0.5,1\right\}\subseteq [0,1]$. Define $d:X\times X\to [0,+\infty )$ by

$$d(\upsilon ,\varkappa )=|\upsilon -\varkappa |$$

∀$\upsilon ,\varkappa \in X$ and $\rho :X\times X\to [1,+\infty )$ by $\rho (\upsilon ,\varkappa )=\upsilon +\varkappa +2.$ Select $\hslash \in \mathcal{F}$ by $\hslash \left(t\right)=ln\left(t\right)$ for $t>0$ and $⋏=0$, then ($X,d$) is an $\mathcal{E}$-complete $\mathcal{EF}$-ms. Take

$${\Theta }_1=\{0,0.5\}and{\Theta }_2=\{0.5,1\}.$$

Then ${\Theta }_1\cup {\Theta }_2=X$ and ${\Theta }_1\cap {\Theta }_2=\left\{0.5\right\}.$ Consider a cyclic mapping $\Re :{\Theta }_1\cup {\Theta }_2\to {\Theta }_1\cup {\Theta }_2$ defined by

$$\Re \left(0\right)=0.5,\Re \left(0.5\right)=0.5,\Re \left(1\right)=0.5,$$

then $Fix(\Re )=\left\{0.5\right\}\subset {\Theta }_1\cap {\Theta }_2.$ Choose positive constants

$$\varrho =0.4,\beta =0.3,\gamma =0.4,$$

so $\beta +\gamma =0.7<1$ and $\varrho \in (0,1).$ For all pairs $(\upsilon ,\varkappa )$ in the set $\{0,1\}$, we find

$$d(\Re \upsilon ,\Re \varkappa )\le \varrho {\left[d(\upsilon ,\varkappa )\right]}^{\gamma }{\left[d(\upsilon ,\Re \upsilon )\right]}^{\beta }{\left[d(\varkappa ,\Re \varkappa )\right]}^{1-\beta -\gamma }.$$

In fact, when $(\upsilon ,\varkappa )=(0,1),$ it holds that

$$\begin{array}{ccc}\hfill d(\Re \upsilon ,\Re \varkappa )& =& d(\Re 0,\Re 1)=\left|0.5-0.5\right|=0\hfill \\ & \le & 0.264=0.4{\left(1\right)}^{0.4}{\left(0.5\right)}^{0.3}{\left(0.5\right)}^{0.3}\hfill \\ & =& \varrho {\left[d(0,1)\right]}^{\gamma }{\left[d(0,\Re 0)\right]}^{\beta }{\left[d(1,\Re 1)\right]}^{1-\beta -\gamma }\hfill \\ & =& \varrho {\left[d(\upsilon ,\varkappa )\right]}^{\gamma }{\left[d(\upsilon ,\Re \upsilon )\right]}^{\beta }{\left[d(\varkappa ,\Re \varkappa )\right]}^{1-\beta -\gamma }.\hfill \end{array}$$

Now define the orbit: for any ${\upsilon }_0\in X,$

$${\upsilon }_{n+1}=\Re {\upsilon }_n.$$

Each sequence converges to $0.5.$

$$0⟼0.5⟼0.5⟼···,1⟼0.5⟼0.5⟼···,0.5⟼0.5.$$

Therefore, ${lim}_{n\to \infty }{\upsilon }_n=c=0.5$ and ${lim}_{m\to \infty }{\upsilon }_m=c=0.5,$ so we have

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)=\rho \left(c,c\right)=0.5+0.5+2\le 3,$$

so,

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)\le 3.$$

Now compute

$$\varpi ={\varrho }^{{\displaystyle \frac{1}{1-\beta }}}={\left(0.4\right)}^{{\displaystyle \frac{1}{1-0.3}}}=0.270,{\displaystyle \frac{1}{\varpi }}=3.70.$$

Hence,

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)\le 3<3.70={\displaystyle \frac{1}{\varpi }}.$$

This fulfills the additional sequence requirement of Theorem 3, and hence, ℜ has a unique fp at $0.5$.

4. Applications

The theory of fps has emerged as a powerful analytical tool for studying the existence and uniqueness of solutions to various types of functional, integral, and differential equations. In recent years, the incorporation of fractional derivatives into mathematical models has attracted significant interest because of their capacity to describe memory and hereditary properties of various complex systems in physics, biology, engineering, and finance. Fractional differential equations (FDEs) provide a natural framework for modeling such phenomena, yet finding their exact analytical solutions often poses significant challenges.

This section employs the fp theorems obtained earlier in the preceding sections to investigate the existence and uniqueness of solutions for a certain class of FDEs. By constructing a suitable operator associated with the given fractional problem and defining an appropriate $\mathcal{EF}$-mss, we demonstrate that the operator satisfies the required contractive condition. Consequently, the solution of the FDE is obtained as the fp of this operator.

Through this method, the effectiveness of fp theory is clearly shown in handling nonlinear fractional systems but also provides a unifying framework that can be extended to various boundary value problems and integral formulations involving fractional derivatives. Thus, the following example/application illustrates the practical significance of our theoretical results in solving real-world fractional models.

Let $\alpha \in (1,2).$ Consider the fractional differential equation:

$$\left\{\begin{array}{c}{D}_t^{\alpha }\left(\upsilon \left(t\right)\right)=g(t,\upsilon \left(t\right)),t\in (0,1)\\ \upsilon \left(0\right)=0,\upsilon \left(1\right)=\underset{0}{\overset{1}{\int }}\upsilon \left(s\right)ds,\end{array}\right.$$

(18)

where

• $D_t^{\alpha }\left(\upsilon \left(t\right)\right)$ denotes a fractional derivative,
• $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a given continuous,
• $\upsilon \left(t\right)$ is the unknown function to be determined.
  The fractional derivative $D_t^{\alpha }\left(\upsilon \left(t\right)\right)$ (with $1<\alpha <2$) leads to the standard Volterra representation

  $$\upsilon \left(t\right)=c_0+c_{1}t+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g(s,\upsilon \left(s\right))ds,$$

  where $c_0$ and $c_1$ are constants determined by the two boundary conditions, and $\mathrm{\Gamma }\left(\alpha \right)$ represents the Gamma function. From $\upsilon \left(0\right)=0,$ we get $c_0=0.$ So

  $$\upsilon \left(t\right)=c_{1}t+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g(s,\upsilon \left(s\right))ds.$$

  (19)
  Use the second (nonlocal) boundary condition $\upsilon \left(1\right)=\underset{0}{\overset{1}{\int }}\upsilon \left(s\right)ds$ to determine $c_1.$ Compute

  $$\upsilon \left(1\right)=c_{1}1+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{\left(1-s\right)}^{\alpha -1}g(s,\upsilon \left(s\right))ds.$$

  (20)
  Compute $\underset{0}{\overset{1}{\int }}\upsilon \left(s\right)ds:$

  $$\begin{array}{ccc}\hfill \underset{0}{\overset{1}{\int }}\upsilon \left(s\right)ds& =& \underset{0}{\overset{1}{\int }}\left[c_{1}s+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{s}{\int }}{\left(s-\tau \right)}^{\alpha -1}g(\tau ,\upsilon \left(\tau \right))d\tau \right]ds\hfill \\ & =& c_1\underset{0}{\overset{1}{\int }}sds+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(s-\tau \right)}^{\alpha -1}g(\tau ,\upsilon \left(\tau \right))d\tau ds.\hfill \end{array}$$

  (21)
  Evaluate the double integral by Fubini:

  $$\begin{array}{ccc}\hfill \underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(s-\tau \right)}^{\alpha -1}g(\tau ,\upsilon \left(\tau \right))d\tau ds& =& \underset{0}{\overset{1}{\int }}g(\tau ,\upsilon \left(\tau \right))\left(\underset{\tau }{\overset{1}{\int }}{\left(s-\tau \right)}^{\alpha -1}ds\right)d\tau \hfill \\ & =& {\displaystyle \frac{1}{\alpha }}\underset{0}{\overset{1}{\int }}{\left(1-\tau \right)}^{\alpha }g(\tau ,\upsilon \left(\tau \right))d\tau .\hfill \end{array}$$

  (22)
  Substituting Equation (22) into Equation (21) and using the fact that $\underset{0}{\overset{1}{\int }}sds={\displaystyle \frac{1}{2}},$ we get

  $$\underset{0}{\overset{1}{\int }}\upsilon \left(s\right)ds={\displaystyle \frac{c_1}{2}}+{\displaystyle \frac{1}{\alpha \mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{\left(1-\tau \right)}^{\alpha }g(\tau ,\upsilon \left(\tau \right))d\tau .$$

  (23)
  Since $\upsilon \left(1\right)=\underset{0}{\overset{1}{\int }}\upsilon \left(s\right)ds$, so comparing the Equation (20) with Equation (23), we obtain

  $$c_1+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{\left(1-s\right)}^{\alpha -1}g(s,\upsilon \left(s\right))ds={\displaystyle \frac{c_1}{2}}+{\displaystyle \frac{1}{\alpha \mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{\left(1-s\right)}^{\alpha }g(s,\upsilon \left(s\right))ds.$$
  Multiply by $2,$ and we get

  $$2c_1+{\displaystyle \frac{2}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{\left(1-s\right)}^{\alpha -1}g(s,\upsilon \left(s\right))ds=c_1+{\displaystyle \frac{2}{\alpha \mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{\left(1-s\right)}^{\alpha }g(s,\upsilon \left(s\right))ds.$$
  Thus,

  $$c_1={\displaystyle \frac{2}{\alpha \mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{\left(1-s\right)}^{\alpha }g(s,\upsilon \left(s\right))ds-{\displaystyle \frac{2}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{\left(1-s\right)}^{\alpha -1}g(s,\upsilon \left(s\right))ds.$$
  Hence,

  $$c_1={\displaystyle \frac{2}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}\left({\displaystyle \frac{{\left(1-s\right)}^{\alpha }}{\alpha }}-{\left(1-s\right)}^{\alpha -1}\right)g(s,\upsilon \left(s\right))ds.$$
  Substituting the value of $c_1$ in the Equation (19), we have

  $$\begin{array}{ccc}\hfill \upsilon \left(t\right)& =& {\displaystyle \frac{2t}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{1}{\int }}\left({\displaystyle \frac{{\left(1-s\right)}^{\alpha }}{\alpha }}-{\left(1-s\right)}^{\alpha -1}\right)g(s,\upsilon \left(s\right))ds\hfill \\ & & +{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g(s,\upsilon \left(s\right))ds.\hfill \end{array}$$
  Thus,

  $$\upsilon \left(t\right)=\underset{0}{\overset{1}{\int }}G(t,s)g(s,\upsilon \left(s\right))ds,$$

  where the Green’s function $G(t,s)$ is:

For $0\le s\le t\le 1,$

$$G(t,s)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\left(t-s\right)}^{\alpha -1}+{\displaystyle \frac{2t}{\mathrm{\Gamma }\left(\alpha \right)}}\left({\displaystyle \frac{{\left(1-s\right)}^{\alpha }}{\alpha }}-{\left(1-s\right)}^{\alpha -1}\right).$$

For $0\le t\le s\le 1,$

$$G(t,s)={\displaystyle \frac{2t}{\mathrm{\Gamma }\left(\alpha \right)}}\left({\displaystyle \frac{{\left(1-s\right)}^{\alpha }}{\alpha }}-{\left(1-s\right)}^{\alpha -1}\right).$$

Let $X=C\left(\right[0,1\left]\right)$ with the sup norm $∥·∥$. Define $d:X\times X\to {\mathbb{R}}^{+}$ by

$$d(\upsilon ,\varkappa )={∥\upsilon -\varkappa ∥}_{\infty }=\underset{t\in [0,1]}{sup}\left|\upsilon \left(t\right)-\varkappa \left(t\right)\right|,$$

for all $\upsilon ,\varkappa \in X$ and $\rho :X\times X\to [1,+\infty )$ by $\rho (\upsilon ,\varkappa )=1.$ Then ($X,d$) is an $\mathcal{E}$-complete $\mathcal{EF}$-ms with $\hslash \left(t\right)=lnt$ and $⋏=0.$

Theorem 4.

Let ($X,d$) be an $\mathcal{E}$-complete $\mathcal{EF}$-ms endowed with the directed graph $G=\left(X,E\left(G\right)\right)$, where $E\left(G\right)=X\times X.$ Let us assume that the following conditions are satisfied:

(i) the function $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous in t and and Lipschitz in $x,$ that is, for all $t\in \left[0,1\right]$ and $\upsilon ,\varkappa \in \mathbb{R}$

$$\left|g\left(t,\upsilon \right)-g\left(t,\varkappa \right)\right|\le L\left|\upsilon -\varkappa \right|$$

for some constant $L\ge 0.$

(ii) Let $G\left(t,s\right)$ be the Green kernel. Set

$$M=\underset{t\in [0,1]}{sup}\underset{0}{\overset{1}{\int }}\left|G\left(t,s\right)\right|ds,and\varrho =LM.$$

Assume $\varrho \in (0,1)$ (i.e., $LM<1$). Then the boundary-value problem (18) has a unique solution. Moreover the solution is the unique fp of the operator

$$\left(\Re \upsilon \right)\left(t\right)=\underset{0}{\overset{1}{\int }}G(t,s)g(s,\upsilon \left(s\right))ds,$$

and the Picard iterates ${\upsilon }^{(n+1)}=\Re {\upsilon }^{\left(n\right)}$ converge exponentially to this solution for any initial ${\upsilon }^{\left(0\right)}\in C[0,1].$

Proof. 

The operator ℜ is edge-preserving relative to this graph because every ordered pair is an edge: for any $\upsilon ,\varkappa \in X,$$\left(\upsilon ,\varkappa \right)\in E\left(G\right)$ and also $\left(\Re \upsilon ,\Re \varkappa \right)\in E\left(G\right)$ trivially true since $E\left(G\right)=X\times X.$ For ${\upsilon }_1,{\upsilon }_2\in C[0,1],$ we have

$$\begin{array}{ccc}\hfill \left|\left(\Re {\upsilon }_1\right)\left(t\right)-\left(\Re {\upsilon }_2\right)\left(t\right)\right|& =& \left|\underset{0}{\overset{1}{\int }}G(t,s)g(s,{\upsilon }_1\left(s\right))ds-\underset{0}{\overset{1}{\int }}G(t,s)g(s,{\upsilon }_2\left(s\right))ds\right|\hfill \\ & \le & \underset{0}{\overset{1}{\int }}\left|G(t,s)\right|\left|g(s,{\upsilon }_1\left(s\right))ds-g(s,{\upsilon }_2\left(s\right))\right|ds.\hfill \end{array}$$

By the assumption (i), we have

$$\left|\left(\Re {\upsilon }_1\right)\left(t\right)-\left(\Re {\upsilon }_2\right)\left(t\right)\right|\le L{∥{\upsilon }_1-{\upsilon }_2∥}_{\infty }\underset{0}{\overset{1}{\int }}\left|G(t,s)\right|ds.$$

Taking the supremum over $t\in [0,1],$ we have

$$\begin{array}{ccc}\hfill {∥\Re {\upsilon }_1-\Re {\upsilon }_2∥}_{\infty }& \le & L\left(\underset{t\in [0,1]}{sup}\underset{0}{\overset{1}{\int }}\left|G\left(t,s\right)\right|ds\right){∥{\upsilon }_1-{\upsilon }_2∥}_{\infty }\hfill \\ & =& LM{∥{\upsilon }_1-{\upsilon }_2∥}_{\infty }.\hfill \end{array}$$

By hypothesis (ii), we have

$$\varrho =LM<1.$$

Thus

$$d(\Re {\upsilon }_1,\Re {\upsilon }_2)\le \varrho d({\upsilon }_1,{\upsilon }_2).$$

Now choose any ${\upsilon }_0\in X$, then $\left({\upsilon }_0,\Re {\upsilon }_0\right)\in E\left(G\right)$ trivially holds because $E\left(G\right)=X\times X.$ Since $\rho (\upsilon ,\varkappa )=1.$ So

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)=1.$$

Since $\varrho \in (0,1),$ we have $1<{\displaystyle \frac{1}{\varrho }},$ so

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)<{\displaystyle \frac{1}{\varrho }}.$$

Thus, all the conditions of Theorem 2 are satisfied, and the operator has unique fp ${\upsilon }^{*}\in C[0,1]$ which is a solution of the boundary-value problem (18). □

Example 5.

Let $\alpha ={\displaystyle \frac{3}{2}}\in \left(1,2\right).$ Consider the fractional boundary-value problem

$$\left\{\begin{array}{c}{D}_t^{\alpha }\left(\upsilon \left(t\right)\right)={\displaystyle \frac{1}{10}}\left(\upsilon \left(t\right)+sint\right),t\in (0,1)\\ \upsilon \left(0\right)=0,\upsilon \left(1\right)=\underset{0}{\overset{1}{\int }}\upsilon \left(s\right)ds,\end{array}\right.$$

(24)

where $D_t^{\alpha }$ denotes the fractional derivative of order $\alpha ,$ and $\upsilon \left(t\right)$ is the unknown function. Define

$$g(t,\varsigma )={\displaystyle \frac{1}{10}}\left(\varsigma +sint\right),\left(t,\varsigma \right)\in [0,1]\times \mathbb{R}.$$

The function $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous since

• $\varsigma \rightarrowtail \varsigma$ is continuous;
• $t\rightarrowtail sint$ is continuous on $[0,1]$;
• sums and scalar multiples of continuous functions are continuous.

Moreover, for all $\varsigma ,\kappa \in \mathbb{R}$ and $t\in \left[0,1\right],$

$$\left|g\left(t,\varsigma \right)-g\left(t,\kappa \right)\right|={\displaystyle \frac{1}{10}}\left|\varsigma -\kappa \right|.$$

Thus, g is Lipschitz continuous in the second variable with Lipschitz constant

$$L={\displaystyle \frac{1}{10}}.$$

Hence, assumption (i) of Theorem 4 is satisfied. As derived in the theorem, the solution of problem satisfies the equivalent integral equation

$$\upsilon \left(t\right)=\underset{0}{\overset{1}{\int }}G(t,s)g(s,\upsilon \left(s\right))ds,$$

where the Green’s function $G(t,s)$ is:

For $0\le s\le t\le 1,$

$$G(t,s)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\left(t-s\right)}^{\alpha -1}+{\displaystyle \frac{2t}{\mathrm{\Gamma }\left(\alpha \right)}}\left({\displaystyle \frac{{\left(1-s\right)}^{\alpha }}{\alpha }}-{\left(1-s\right)}^{\alpha -1}\right).$$

For $0\le t\le s\le 1,$

$$G(t,s)={\displaystyle \frac{2t}{\mathrm{\Gamma }\left(\alpha \right)}}\left({\displaystyle \frac{{\left(1-s\right)}^{\alpha }}{\alpha }}-{\left(1-s\right)}^{\alpha -1}\right).$$

Since $\alpha ={\displaystyle \frac{3}{2}},$ the Gamma value satisfies

$$\mathrm{\Gamma }\left({\displaystyle \frac{3}{2}}\right)={\displaystyle \frac{\sqrt{\pi }}{2}}.$$

The function $G(t,s)$ is continuous on the compact set $\left[0,1\right]\times \left[0,1\right].$ Therefore,

$$M=\underset{t\in [0,1]}{sup}\underset{0}{\overset{1}{\int }}\left|G\left(t,s\right)\right|ds<\infty .$$

A direct estimate yields

$$M\le 2.$$

Now define

$$\varrho =LM={\displaystyle \frac{1}{10}}\times 2={\displaystyle \frac{1}{5}}<1.$$

Therefore, assumption (ii) holds. Let $X=C\left([0,1]\right)$ endowed with the $\mathcal{EF}$-metric

$$d(\upsilon ,\varkappa )={∥\upsilon -\varkappa ∥}_{\infty }=\underset{t\in [0,1]}{sup}\left|\upsilon \left(t\right)-\varkappa \left(t\right)\right|.$$

Define $\rho :X\times X\to [1,+\infty )$ by $\rho (\upsilon ,\varkappa )=1.$ Then ($X,d$) is an $\mathcal{E}$-complete $\mathcal{EF}$-ms with $\hslash \left(t\right)=lnt$ and $⋏=0.$ All hypotheses of Theorem 4 are satisfied. Hence, the operator $\Re :X\to X$ defined by

$$\left(\Re \upsilon \right)\left(t\right)=\underset{0}{\overset{1}{\int }}G(t,s)g(s,\upsilon \left(s\right))ds,$$

admits a unique fp ${\upsilon }^{*}\in C\left([0,1]\right)$, which is the unique solution of the fractional boundary-value problem (24).

Applications to Integral Equations

This section demonstrates the use of the fp theorems obtained earlier to solve specific integral equations. In particular, we apply the developed theorems to a nonlinear mixed Volterra–Fredholm integral equation and demonstrate that the existence and uniqueness of its solution can be guaranteed within the framework of $\mathcal{EF}$-mss. This application emphasizes the robustness and broad utility of the given fp results in dealing with nonlinear integral problems.

Consider the nonlinear mixed Volterra–Fredholm integral equation

$$\upsilon \left(t\right)=g\left(t\right)+{\int }_a^t{K}_1(t,s,\upsilon \left(s\right))ds+{\int }_a^b{K}_2(t,s,\upsilon \left(s\right))ds,t\in [a,b]$$

(25)

where

• $g\left(t\right):[a,b]\to \mathbb{R}$ is a given continuous function;
• $K_1,K_2:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$ are continuous kernels;
• $\upsilon \left(t\right)$ denotes the unknown function to be solved.

Assume the following hold.

(cond₁): the kernals $K_1$ and $K_2$ are Lipschitz continuous in the third argument; that is,

there exist $L_1,L_2\ge 0$ such that, for all $t,s\in [a,b]$ and $\upsilon ,v\in \mathbb{R}$

$$\left|K_1(t,s,\upsilon )-K_1(t,s,\varkappa )\right|\le L_1\left|\upsilon -\varkappa \right|,$$

$$\left|K_2(t,s,\upsilon )-K_2(t,s,\varkappa )\right|\le L_2\left|\upsilon -\varkappa \right|,$$

(cond₂): the kernal functions $K_1,K_2:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$ are nondecreasing; i.e., if $\upsilon \le \varkappa ,$ then

$$K_1(t,s,\upsilon )\le K_1(t,s,\varkappa ),$$

$$K_2(t,s,\upsilon )\le K_2(t,s,\varkappa ),$$

(cond₃): $\left(L_1+L_2\right)\left(b-a\right)<1,$

(cond₄): there exists some ${\upsilon }_0\in C\left([a,b],\mathbb{R}\right)$ such that

$${\upsilon }_0\left(t\right)\le \left(\Re {\upsilon }_0\right)\left(t\right)$$

for all $t\in [a,b],$ where

$$\left(\Re \upsilon \right)\left(t\right)=g\left(t\right)+{\int }_a^t{K}_1(t,s,\upsilon \left(s\right))ds+{\int }_a^b{K}_2(t,s,\upsilon \left(s\right))ds,$$

(cond₅): the Picard sequence {${\upsilon }_n$} is uniformly bounded in $C\left([a,b],\mathbb{R}\right)$ by some $M>0$ and

$$1+2M<{\displaystyle \frac{1}{\left(L_1+L_2\right)\left(b-a\right)}}.$$

Theorem 5.

Under assumptions (cond₁)–(cond₆) the nonlinear mixed Volterra–Fredholm integral Equation (25) has a unique solution $\upsilon \in C\left([a,b],\mathbb{R}\right).$

Proof. 

Let $X=C\left([a,b],\mathbb{R}\right)$ be equipped with the $\mathcal{EF}$-metric $d:X\times X\to [0,+\infty )$ given by

$$d(\upsilon ,\varkappa )=\underset{t\in [a,b]}{sup}\left|\upsilon \left(t\right)-\varkappa \left(t\right)\right|,$$

and $\rho :X\times X\to [1,+\infty )$ by

$$\rho (\upsilon \left(t\right),\varkappa \left(t\right))=1+\underset{t\in [a,b]}{sup}\left|\upsilon \left(t\right)\right|+\underset{t\in [a,b]}{sup}\left|\varkappa \left(t\right)\right|,$$

then $(X,d)$ form an $\mathcal{E}$-complete $\mathcal{EF}$-ms with $\hslash \left(t\right)=lnt$ for $t>0$ and $⋏=0$. Let G be the graph with vertices X and edges

$$E\left(G\right)=\left\{\left(\upsilon ,\varkappa \right)\in X\times X:\upsilon \left(t\right)\le \varkappa \left(t\right),\forall t\in [a,b]\right\}.$$

Then G is the pointwise order graph which is a directed graph.

For any $\upsilon \in X,$ continuity of g and K (and boundedness on compact domain) imply $\Re \upsilon \in X.$ Now for $\upsilon ,\varkappa \in X,$ with ($\upsilon ,\varkappa$)$\in E\left(G\right).$ Then $\upsilon \left(t\right)\le \varkappa \left(t\right),$ for all $t\in [a,b].$ Thus, by the assumption (cond₂), we have

$$K_1(t,s,\upsilon )\le K_1(t,s,\varkappa ),$$

$$K_2(t,s,\upsilon )\le K_2(t,s,\varkappa ),$$

for all $t,s\in [a,b].$ Hence,

$$\begin{array}{ccc}\hfill \left(\Re \upsilon \right)\left(t\right)& =& g\left(t\right)+{\int }_a^t{K}_1(t,s,\upsilon \left(s\right))ds+{\int }_a^b{K}_2(t,s,\upsilon \left(s\right))ds\hfill \\ & \le & g\left(t\right)+{\int }_a^t{K}_1(t,s,\varkappa \left(s\right))ds+{\int }_a^b{K}_2(t,s,\varkappa \left(s\right))ds\hfill \\ & =& \left(\Re \varkappa \right)\left(t\right),\hfill \end{array}$$

so $\left(\Re \upsilon ,\Re \varkappa \right)\in E\left(G\right).$ Thus, ℜ is edge-preserving. Now for any $\upsilon ,\varkappa \in X,$ with ($\upsilon ,\varkappa$)$\in E\left(G\right)$, we get for $t\in [a,b],$

$$\begin{array}{ccc}\hfill \left|\left(\Re \upsilon \right)\left(t\right)-\left(\Re \varkappa \right)\left(t\right)\right|& \le & \left|{\int }_a^t\left[K_1(t,s,\upsilon \left(s\right))-K_1(t,s,\varkappa \left(s\right))\right]ds\right|\hfill \\ & & +\left|{\int }_a^b\left[K_2(t,s,\upsilon \left(s\right))-K_2(t,s,\varkappa \left(s\right))\right]ds\right|\hfill \\ & \le & {\int }_a^t{L}_1\left|\upsilon \left(s\right)-\varkappa \left(s\right)\right|ds\hfill \\ & & +{\int }_a^b{L}_2\left|\upsilon \left(s\right)-\varkappa \left(s\right)\right|ds\hfill \\ & \le & L_1{\int }_a^t{∥\upsilon -\varkappa ∥}_{\infty }ds\hfill \\ & & +L_2{\int }_a^b{∥\upsilon -\varkappa ∥}_{\infty }ds\hfill \\ & =& L_1\left(t-a\right){∥\upsilon -\varkappa ∥}_{\infty }+L_2\left(b-a\right){∥\upsilon -\varkappa ∥}_{\infty }.\hfill \end{array}$$

Since $t\in [a,b],$$t-a\le b-a,$ we have

$$\left|\left(\Re \upsilon \right)\left(t\right)-\left(\Re \varkappa \right)\left(t\right)\right|\le \left[L_1\left(b-a\right)+L_2\left(b-a\right)\right]{∥\upsilon -\varkappa ∥}_{\infty }$$

Evaluating the supremum over the interval $t\in [a,b]$ gives

$$\begin{array}{ccc}\hfill d(\Re \upsilon ,\Re \varkappa )& =& \underset{t\in [a,b]}{sup}\left|\left(\Re \upsilon \right)\left(t\right)-\left(\Re \varkappa \right)\left(t\right)\right|\le \left[L_1\left(b-a\right)+L_2\left(b-a\right)\right]{∥\upsilon -\varkappa ∥}_{\infty }\hfill \\ & =& \left[L_1\left(b-a\right)+L_2\left(b-a\right)\right]d(\upsilon ,\varkappa )\hfill \\ & =& \varrho d(\upsilon ,\varkappa ).\hfill \end{array}$$

holds for $\varrho =\left[L_1\left(b-a\right)+L_2\left(b-a\right)\right]<1.$ By the assumption (cond₅), there exists ${\upsilon }_0\in X$ with $\left({\upsilon }_0,\Re {\upsilon }_0\right)\in E\left(G\right).$ Define

$${\upsilon }_{n+1}=\Re {\upsilon }_n.$$

Since ℜ is edge-preserving, so

$${\upsilon }_0\le {\upsilon }_1\le {\upsilon }_2\le ···.$$

By (cond₅), the sequence {${\upsilon }_n$} is uniformly bounded and there exists $M>0$ such that

$$\rho \left({\upsilon }_n,{\upsilon }_m\right)\le 1+2M,\forall n,m\in \mathbb{N}.$$

Taking the limit and supremum, we have

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)\le 1+2M.$$

By assumption (cond₃), $\varrho =\left[L_1\left(b-a\right)+L_2\left(b-a\right)\right]<1.$ From (cond₅), we have

$$1+2M<{\displaystyle \frac{1}{\varrho }}.$$

Hence,

$$\underset{m,n\to \infty }{lim\; sup}\rho \left({\upsilon }_n,{\upsilon }_m\right)<{\displaystyle \frac{1}{\varrho }}.$$

Therefore, condition (ii) of Theorem 2 is satisfied. Moreover, since ℜ is Lipschitz in sup-norm, it is G-continuous. Also, if ${\upsilon }_n\to \upsilon$ uniformly and (${\upsilon }_n,{\upsilon }_{n+1})\in E\left(G\right),$ then ${\upsilon }_n\le \upsilon$ for all $n,$ so (${\upsilon }_n,\upsilon )\in E\left(G\right).$ Hence, condition (iii) holds. Thus, ℜ admits a unique fp, representing the unique solution of the nonlinear mixed Volterra–Fredholm integral Equation (25). □

These examples demonstrate that the abstract fp results developed in this work are not only of theoretical value but also directly applicable to mathematical models involving memory effects and hereditary phenomena, such as those described by fractional differential equations. The ability to reformulate such equations as operator equations in a suitable $\mathcal{EF}$-ms provides a systematic approach for proving the existence and uniqueness of solutions. This framework can be readily extended to other models in applied mathematics, including reaction–diffusion systems, population dynamics, and viscoelastic materials.

Example 6.

Consider the nonlinear mixed Volterra–Fredholm integral equation

$$\upsilon \left(t\right)=t+{\int }_0^t{\displaystyle \frac{1}{6}}\upsilon \left(s\right)ds+{\int }_0^1{\displaystyle \frac{1}{15}}\upsilon \left(s\right)ds,t\in [0,1].$$

Here,

$$g\left(t\right)=t,K_1(t,s,\upsilon )={\displaystyle \frac{1}{6}}\upsilon ,K_2(t,s,\upsilon )={\displaystyle \frac{1}{15}}\upsilon .$$

For all $\upsilon ,v\in \mathbb{R},$ we have

$$\left|K_1(t,s,\upsilon )-K_1(t,s,\varkappa )\right|={\displaystyle \frac{1}{6}}\left|\upsilon -\varkappa \right|,$$

$$\left|K_2(t,s,\upsilon )-K_2(t,s,\varkappa )\right|={\displaystyle \frac{1}{15}}\left|\upsilon -\varkappa \right|.$$

Thus,

$$L_1={\displaystyle \frac{1}{6}}and{L}_2={\displaystyle \frac{1}{15}}.$$

Thus, condition (cond₁) is satisfied. Since both kernels $K_1$ and $K_2$ are linear and increasing, we have

$$\upsilon \le \varkappa ,$$

implies

$$K_1(t,s,\upsilon )\le K_1(t,s,\varkappa )$$

$$K_2(t,s,\upsilon )\le K_2(t,s,\varkappa ).$$

Thus, condition (cond₂) is satisfied. Since, the function $g\left(t\right)=t$ is continuous on $[0,1]$. Moreover,

$$\left(L_1+L_2\right)\left(b-a\right)=\left({\displaystyle \frac{1}{6}}+{\displaystyle \frac{1}{15}}\right)\left(1-0\right)={\displaystyle \frac{7}{30}}<1.$$

Hence, the condition (cond₃) holds. Choose ${\upsilon }_0\left(t\right)=0.$ Then,

$$\left(\Re {\upsilon }_0\right)\left(t\right)=t\ge 0={\upsilon }_0\left(t\right),\forall t\in [0,1].$$

Thus,

$${\upsilon }_0\left(t\right)\le \left(\Re {\upsilon }_0\right)\left(t\right)$$

and (cond₄) is satisfied. Define the Picard sequence

$${\upsilon }_{n+1}\left(t\right)=t+{\int }_0^t{\displaystyle \frac{1}{6}}{\upsilon }_n\left(s\right)ds+{\int }_0^1{\displaystyle \frac{1}{15}}{\upsilon }_n\left(s\right)ds.$$

Taking the supremum norm, we obtain

$${∥{\upsilon }_{n+1}∥}_{\infty }\le 1+\left({\displaystyle \frac{1}{6}}+{\displaystyle \frac{1}{15}}\right){∥{\upsilon }_n∥}_{\infty }=1+{\displaystyle \frac{7}{30}}{∥{\upsilon }_n∥}_{\infty }.$$

By induction, it follows that

$${∥{\upsilon }_n∥}_{\infty }\le {\displaystyle \frac{3}{2}},\forall n\in \mathbb{N}.$$

Hence, the Picard sequence is uniformly bounded with $M={\displaystyle \frac{3}{2}}.$ Moreover,

$$1+2M=1+3,$$

and

$${\displaystyle \frac{1}{\varrho }}={\displaystyle \frac{1}{\left(L_1+L_2\right)\left(b-a\right)}}={\displaystyle \frac{30}{7}}\approx 4.2857.$$

Since

$$1+2M<{\displaystyle \frac{1}{\left(L_1+L_2\right)\left(b-a\right)}},$$

so condition (cond₅) is satisfied. All assumptions (cond₁)–(cond₅) of the theorem are satisfied. Therefore, the nonlinear mixed Volterra–Fredholm integral Equation (25) admits a unique solution $\upsilon \in C\left(\left[0,1\right],\mathbb{R}\right).$

5. Conclusions

In this study, we introduced the notion of graphic rational contractions and interpolative Ćirić–Reich–Rus-type cyclic contractions in the setting of $\mathcal{EF}$-mss and obtained some new fp theorems for these generalized contractions. Unlike previous studies, which considered the Banach contraction principle in $\mathcal{EF}$-mss, our work explores graphic and interpolative cyclic contractions in $\mathcal{EF}$-mss for the first time. Furthermore, we provide a fp approach for solving mixed Volterra–Fredholm integral equations in this framework, and our graphic rational contraction theorem serves as a generalization of Banach contraction principle in $\mathcal{EF}$-ms, offering broader applicability under more flexible conditions.

Importantly, our theoretical contributions are broad and flexible: under certain natural simplifications, the developed framework recovers classical fp results, including well-established theorems in standard $\mathcal{F}$-mss, b-mss, and classical mss. This highlights both the generality and foundational relevance of our approach.

To demonstrate the significance and applicability of the results, we provided illustrative examples and validated the practical utility of the derived theorems through their successful application in solving fractional differential equations and mixed Volterra–Fredholm integral equations. Overall, this study not only advances the theory of generalized mss but also offers tools with tangible applications in mathematical analysis and related fields.