Numerical Algorithm for Coupled Fixed Points in Normed Spaces with Applications to Fractional Differential Equations and Economics

Abstract

This paper introduces interpolative enriched cyclic Reich–Rus–Ćirić operators in normed spaces, expanding existing contraction principles by integrating interpolation and cyclic conditions. This class of operators addresses mappings with discontinuities or non-self mappings, enhancing the applicability of fixed-point theory to more complex problems. This class of operators expands on existing cyclic contractions, including interpolative Kannan mappings, interpolative Reich–Rus–Ćirić contractions, and other known contractions in the literature. We demonstrate the existence and uniqueness of fixed points for these operators and provide an example to illustrate our findings. Moreover, we discuss the applications of our results in solving nonlinear integral equations. Furthermore, we introduce the idea of a coupled interpolative enriched cyclic Reich–Rus–Ćirić operator and establish the existence of a strongly coupled fixed-point theorem for this contraction. Finally, we provide an application to fractional differential equations to show the validity of the main result.

1. Introduction and Preliminaries

In the study of cyclic operators, fixed-point results play an important role by providing the essential requirements for establishing the existence and uniqueness of fixed points. These requirements depend on the specific characteristics of the examined cyclic operators. For example, the Banach fixed-point theorem [1] asserts that a unique fixed point exists for contraction mappings within a complete metric space, ensuring both existence and uniqueness under strict contractive conditions. Alternatively, the Brouwer theorem guarantees the existence of at least one fixed point for continuous transformations defined on a compact and convex set, highlighting its importance in situations where the uniqueness of the fixed point is not always established. Understanding and applying these theorems appropriately allows researchers to analyze complex systems effectively, facilitating developments across various related areas such as differential equations [2], optimization [3], and economic modeling [4].

Let $(\mathcal{K},d)$ be any metric space. An operator $\mathrm{{\rm Y}}:\mathcal{K}\to \mathcal{K}$ is known as a Banach contraction if $\exists k\in [0,1)$ such that for all $w,r\in \mathcal{K}$ the following inequality holds:

$$d(\mathrm{{\rm Y}}w,\mathrm{{\rm Y}}r)\le kd(w,r).$$

A core result in metric fixed-point theory is the Banach fixed-point result, which establishes that all contraction operators defined over a certain metric space $(\mathcal{K},d)$ have a unique fixed point, provided that $(\mathcal{K},d)$ is complete. This theorem guarantees that such mappings, which progressively reduce the distance between points, will tend towards a unique fixed point within the space. Acknowledging that contraction mappings are naturally continuous on $\mathcal{K}$ is essential. This continuity stems from the definition of contraction, which forces the operator to bring points closer together, thereby avoiding any sudden jumps or discontinuities. Nevertheless, exploring contractive conditions where continuity is not required opened up a new direction in this field. In 1968, Kannan [5] launched a fixed-point result for Kannan contractions that do not necessitate continuity. Specifically, an operator $\mathrm{{\rm Y}}:\mathcal{K}\to \mathcal{K}$ is referred to as a Kannan contraction if $\exists k\in [0,{\displaystyle \frac{1}{2}})$ such that $\forall w,r\in \mathcal{K}$ the following condition holds:

$$d(\mathrm{{\rm Y}}w,\mathrm{{\rm Y}}r)\le k\left[d\right(w,\mathrm{{\rm Y}}w)+d(r,\mathrm{{\rm Y}}r\left)\right].$$

Kannan’s work started with the exploration of contractive conditions that do not ensure the continuity of $\mathrm{{\rm Y}}$. For further findings on this topic, we refer readers to the works in [6,7] with the references mentioned therein.

Several results in this direction provide guarantees under less restrictive conditions and can sometimes establish the existence of multiple fixed points. For instance, the Schauder fixed-point theorem declares that continuous mappings on a closed, convex subset of a Banach space will always possess at least one fixed point. While the Kakutani fixed-point theorem assures the presence of a fixed point for continuous mappings defined on a non-empty, convex, and compact subset of a locally convex topological vector space. Although these theorems provide assurance of the existence of fixed points, they do not always offer a practical method for locating them, and observation of the fixed points, in practice, can be challenging or sometimes infeasible. The uniqueness of fixed points is crucial in many contexts, emphasizing the importance of results that specifically ensure this.

Kirk et al. [8] increased the importance and scope of the Banach theorem by incorporating cyclic contractions. This advancement broadened the classical definition of contractive operators by combining the cyclic contractive mappings, showing that these mappings likewise guarantee the existence of a unique fixed point. More information regarding this can be found in [9,10]. This extension expanded the importance of mappings for which fixed points can be assured. It created a fresh framework for analyzing the existence and uniqueness of fixed points in a wide range of mathematical contexts. For any given $q\in \mathbb{N}$ and a self-mapping $\mathrm{{\rm Y}}$ on $\mathcal{K}$, a finite family $\{{\aleph }_l\subseteq \mathcal{K}:l=1,2,\cdots ,q\}$ is referred to as a cyclic representation of $\mathcal{K}$ over $\mathrm{{\rm Y}}$ if

• $\mathcal{K}={\bigcup }_{l=1}^q{\aleph }_l$;
• $\mathrm{{\rm Y}}\left({\aleph }_1\right)\subseteq {\aleph }_2,\mathrm{{\rm Y}}\left({\aleph }_2\right)\subseteq {\aleph }_3,\cdots ,\mathrm{{\rm Y}}\left({\aleph }_{q-1}\right)\subseteq {\aleph }_q$ and $\mathrm{{\rm Y}}\left({\aleph }_q\right)\subseteq {\aleph }_1$.

Kirk et al. in [8] stated their fixed-point theorem as follows:

Theorem 1

([8]). Consider a complete metric space $(\mathcal{K},d)$ with $\{{\aleph }_1,\cdots ,{\aleph }_q\}$ a finite class of non-empty closed subsets of $\mathcal{K}$ and a mapping $\mathrm{{\rm Y}}:{\bigcup }_{l=1}^q{\aleph }_l\to {\bigcup }_{l=1}^q{\aleph }_l$. Further, suppose the following assumptions hold:

• $\{{\aleph }_l:l=1,2,\cdots ,q\}$ constitutes a cyclic representation of ${\bigcup }_{l=1}^q{\aleph }_l$ over Υ;
• ∃$k\in [0,1)$ such that for all $w\in {\aleph }_l$ and $r\in {\aleph }_{l+1}$,

  $$d(\mathrm{{\rm Y}}w,\mathrm{{\rm Y}}r)\le kd(w,r),$$

  where ${\aleph }_{q+1}={\aleph }_1$.

Then, Y possesses a unique fixed point $w^{*}\in {\bigcap }_{l=1}^q{\aleph }_l$.

Berinde and Păcurar [11], in 2020, initiated a novel and generalized type of contractive mapping named enriched contractions. This type of mapping extends beyond the familiar Banach contractions to encompass a range of nonexpansive mappings studied in the literature. Their study showed that each enriched contraction has a unique fixed point. This fixed point can be approximated by employing an appropriate Krasnoselskii iteration scheme in Banach spaces.

Definition 1

([11]). Let $(\mathcal{K},\parallel ·\parallel )$ be a normed space. Then, a mapping $\mathrm{{\rm Y}}:\mathcal{K}\to \mathcal{K}$ is named an enriched contraction if there exists $b\ge 0$ and $\upsilon \in [0,b+1)$ such that

$$\parallel b(w-r)+\mathrm{{\rm Y}}w-\mathrm{{\rm Y}}r\parallel \le \upsilon \parallel w-r\parallel ,\forall w,r\in \mathcal{K}.$$

(1)

Enriched contraction mappings are of great importance because they encompass both the Banach contraction mappings, as well as nonexpansive mappings. Although non-expansive mappings do not invariably ensure the existence of fixed points, enriched contractions ensure the presence of a unique fixed point.

Suppose $\mathcal{B}$ is a convex set within a normed space $\mathcal{K}$, $\upsilon \in (0,1]$, and $\mathrm{{\rm Y}}:\mathcal{B}\to \mathcal{B}$. A mapping ${\mathrm{{\rm Y}}}_{\upsilon }:\mathcal{B}\to \mathcal{B}$ defined by ${\mathrm{{\rm Y}}}_{\upsilon }\left(w\right)=(1-\upsilon )w+\upsilon \mathrm{{\rm Y}}\left(w\right)$ is referred to as an averaged operator of $\mathrm{{\rm Y}}$. It is important to note that the set of all fixed points of the averaged operator ${\mathrm{{\rm Y}}}_{\upsilon }$ is identical to the set of all fixed points of the operator $\mathrm{{\rm Y}}$. A key question is whether an operator $\mathrm{{\rm Y}}$, defined on the union of q nonempty closed subsets $\{{\aleph }_1,\dots ,{\aleph }_q\}$ in a normed space $(\mathcal{K},\parallel ·\parallel )$, and which forms a cyclic representation of ${\cup }_{l=1}^q{\aleph }_l$ with respect to ${\mathrm{{\rm Y}}}_{\upsilon }$ for some $\upsilon$, ensures the existence of a fixed point. This question was positively answered by Abbas et al., who demonstrated a fixed-point theorem for generalized enriched cyclic contractions [12]. For more details on enriched contraction mappings, see [13,14,15,16,17,18,19] and the references therein. Very recently, Salisu et al. [20] introduced enriched multi-valued mappings in geodesic spaces and established fixed-point results in CAT(0) spaces, along with an approximation scheme. Zhou et al. [21] developed fixed-point theorems for generalized convex orbital Lipschitz operators and proposed an iterative algorithm for approximation.

Recently, Karapinar [22] proposed his famous and novel Kannan-type contractive mapping, termed interpolative Kannan-type contractive mapping, which generalized Kannan’s fixed-point theorem by incorporating interpolation concepts as well as interpolative Reich–Rus–Ćirić contraction in [23]. These techniques have been effectively used in other related studies to generalize various contraction types [24,25,26,27]. The interpolative method has proven to be a powerful approach in fixed-point theory, facilitating the creation of new types of contractive mappings and the development of innovative fixed-point theorems. A mapping $\mathrm{{\rm Y}}:\mathcal{K}\to \mathcal{K}$ is defined as an interpolative Reich–Rus–Ćirić contraction if there exist constants $k\in [0,1)$, $\alpha >0$, and $\beta \ge 0$ with $\alpha +\beta <1$ for which the following condition is satisfied:

$$d(\mathrm{{\rm Y}}w,\mathrm{{\rm Y}}r)\le kd{(w,r)}^{\beta }d{(w,\mathrm{{\rm Y}}w)}^{\alpha }d{(r,\mathrm{{\rm Y}}r)}^{1-\alpha -\beta },$$

for all $w,r\in \mathcal{K}$ with $w,r\notin Fix\left(\mathrm{{\rm Y}}\right)$.

It has been proven that every interpolative Reich–Rus–Ćirić contraction mapping in a complete metric space has a unique fixed point [23]. For further results in this direction, see [18,28,29,30].

The notion of coupled fixed points was first initiated by Mohamed et al. [31]. The later development of a coupled contraction mapping theorem by Choudhury and Maity [32] significantly advanced the field. This sparked a wave of research into coupled fixed-point results, with notable contributions from [10,33].

Definition 2

([31]). Let $\mathcal{K}\ne \varphi$ and $\mathrm{{\rm Y}}:\mathcal{K}\times \mathcal{K}\to \mathcal{K}$ be a mapping. An element $(w,r)\in \mathcal{K}\times \mathcal{K}$ is considered a coupled fixed point of the mapping Y if it satisfies the following two conditions:

$$\mathrm{{\rm Y}}(w,r)=w\mathrm{and}\mathrm{{\rm Y}}(r,w)=r.$$

(2)

In 2014, Choudhury and Maity [32] established several cyclic coupled fixed-point results employing Kannan-type operators and initiated the notion of strong coupled fixed points. The idea of a strong coupled fixed point, the definition of a cyclic coupled Kannan-type operator, and their associated fixed-point results are outlined as follows:

Definition 3

([32]). In the context of (2), a coupled fixed point is termed a strong coupled fixed point when $w=r$. This condition implies that $\mathrm{{\rm Y}}(w,w)=w$.

Definition 4

([32]). Let ${\aleph }_1$ and ${\aleph }_2$ denote two nonempty subsets of a metric space $(\mathcal{K},d)$. We say that a mapping $\mathrm{{\rm Y}}:\mathcal{K}\times \mathcal{K}\to \mathcal{K}$ is a cyclic coupled Kannan-type contraction over ${\aleph }_1$ and ${\aleph }_2$ if it exhibits cyclic behavior relative to ${\aleph }_1$ and ${\aleph }_2$ satisfies the following inequality for some $k\in \left(0,{\displaystyle \frac{1}{2}}\right)$:

$$d(\mathrm{{\rm Y}}(w,r),\mathrm{{\rm Y}}(x,y))\le k\left[d(w,\mathrm{{\rm Y}}(w,r\left)\right)+d(x,\mathrm{{\rm Y}}(x,y\left)\right)\right],$$

where $w,x\in {\aleph }_1$ and $r,y\in {\aleph }_2$.

Theorem 2

([32]). Consider two nonempty, closed subsets ${\aleph }_1$ and ${\aleph }_2$ within a complete metric space $(\mathcal{K},d)$. If $\mathrm{{\rm Y}}:\mathcal{K}\times \mathcal{K}\to \mathcal{K}$ is a cyclic coupled Kannan-type contraction relative to ${\aleph }_1$ and ${\aleph }_2$, and if ${\aleph }_1\cap {\aleph }_2\ne \varnothing$, then Y possesses a strong coupled fixed point within ${\aleph }_1\cap {\aleph }_2$.

Inspired by the research of Karapınar et al. [22,23], Berinde and Păcurar [11], and Kirk et al. [8], we present a new class of mappings known as interpolative enriched cyclic Reich–Rus–Ćirić contraction mappings. We subsequently establish a fixed-point theorem within the setting of Banach spaces. An example is provided to illustrate the result discussed in this study. As a practical application, we demonstrate both the existence and uniqueness of solutions for a particular class of nonlinear integral equations by utilizing interpolative enriched cyclic Reich–Rus–Ćirić contraction mappings. In addition, we investigate the strong coupled fixed-point result for these mappings, drawing from the studies by Bekri et al. [10] and Choudhury and Maity [32].

The main prospects of interpolative enriched cyclic Reich–Rus–Cirić operators lie in their ability to unify and generalize various contractions and their associated principles, offering a broader framework for fixed-point theory in normed spaces. This is crucial for bridging the gaps between existing results, extending classical theories to broader settings, and handling varying contractive behaviors. By incorporating interpolation techniques with enriched cyclic conditions, they are specifically designed to manage discontinuities and non-self mappings, extending the applicability of fixed-point theory to more complex settings. This study provides insights into solving nonlinear integral equations and coupled systems under generalized contractive conditions, thereby enhancing their utility in both theoretical and practical contexts.

2. Main Results

In this section, we present and examine interpolative enriched cyclic Reich–Rus–Ćirić contraction mappings, focusing on their existence and approximation results. We study a finite collection of nonempty closed subsets $\{{\aleph }_1,\dots ,{\aleph }_q\}$ in a normed space $(\mathcal{K},\parallel ·\parallel )$, where $q\in \mathbb{N}$.

Definition 5.

A mapping $\mathrm{{\rm Y}}:{\bigcup }_{l=1}^q{\aleph }_l\to \mathcal{K}$ is referred to as an interpolative enriched cyclic Reich–Rus–Ćirić contraction mapping if

• $\{{\aleph }_l:l=1,2,\cdots ,q\}$ is a cyclic representation of ${\bigcup }_{l=1}^q{\aleph }_l$ over ${\mathrm{{\rm Y}}}_{\upsilon }$, where $\upsilon ={\displaystyle \frac{1}{b+1}}$;
• $\exists b\in [0,\infty ),k\in [0,1)$ and $\alpha >0,\beta \ge 0$ with $\alpha +\beta <1$ such that

  $$\parallel b(w-r)+\mathrm{{\rm Y}}w-\mathrm{{\rm Y}}r\parallel \le k{(b+1)}^{\beta }{\parallel w-r\parallel }^{\beta }{\parallel w-\mathrm{{\rm Y}}w\parallel }^{\alpha }{\parallel r-\mathrm{{\rm Y}}r\parallel }^{1-\alpha -\beta },$$

  (3)

for all $w\in {\aleph }_l,r\in {\aleph }_{l+1}$ for $1\le l\le q$ with $w,r\notin Fix\left(\mathrm{{\rm Y}}\right)$.

To emphasize the constant in (3), we refer to the interpolative enriched cyclic Reich–Rus–Ćirić contraction mapping $\mathrm{{\rm Y}}$ as a $(b,k,\alpha ,\beta )$-interpolative enriched cyclic Reich–Rus–Ćirić contraction.

Theorem 3.

If $\mathrm{{\rm Y}}:{\bigcup }_{l=1}^q{\aleph }_l\to \mathcal{K}$ is a $(b,k,\alpha ,\beta )$-interpolative enriched cyclic Reich–Rus–Ćirić contraction, then

• $Fix\left(\mathrm{{\rm Y}}\right)=\left\{w^{*}\right\}$, for any $w^{*}\in {\bigcap }_{l=1}^q{\aleph }_l$;
• $\exists \upsilon \in (0,1)$ such that the sequence $\left(w_h\right)$ given by

  $$w_{h+1}=(1-\upsilon )w_h+\upsilon \mathrm{{\rm Y}}{w}_h,\forall h\ge 0,$$

  (4)
  tends towards $w^{*}$, starting from any initial element $w_0\in {\bigcup }_{l=1}^q{\aleph }_l$.

Proof. 

Applying the $(b,k,\alpha ,\beta )$-interpolative enriched cyclic Reich–Rus–Ćirić contraction condition (3), we derive

$$∥\left({\displaystyle \frac{1}{\upsilon }}-1\right)(w-r)+\mathrm{{\rm Y}}w-\mathrm{{\rm Y}}r∥\le k{(b+1)}^{\beta }{\parallel w-r\parallel }^{\beta }{\parallel w-\mathrm{{\rm Y}}w\parallel }^{\alpha }{\parallel r-\mathrm{{\rm Y}}r\parallel }^{1-\alpha -\beta }.$$

Equivalently, we obtain

$$∥{\mathrm{{\rm Y}}}_{\upsilon }w-{\mathrm{{\rm Y}}}_{\upsilon }r∥\le k{∥w-r∥}^{\beta }{∥w-{\mathrm{{\rm Y}}}_{\upsilon }w∥}^{\alpha }{∥r-{\mathrm{{\rm Y}}}_{\upsilon }r∥}^{1-\alpha -\beta }.$$

(5)

Next, take $w_0\in {\bigcup }_{l=1}^q{\aleph }_l$. Then, there exists $l\in \{1,\dots ,q\}$ such that $w_0\in {\aleph }_l$. As $\{{\aleph }_l:l=1,2,\dots ,q\}$ is a cyclic representation of ${\bigcup }_{l=1}^q{\aleph }_l$ with respect to ${\mathrm{{\rm Y}}}_{\upsilon }$, so $w_1={\mathrm{{\rm Y}}}_{\upsilon }{w}_0\in {\aleph }_{l+1}$. Using (5), we have

$$\begin{array}{cc}\hfill ∥{\mathrm{{\rm Y}}}_{\upsilon }{w}_0-{\mathrm{{\rm Y}}}_{\upsilon }{w}_1∥& \le k\parallel w_0-w_1{\parallel }^{\beta }{∥w_0-{\mathrm{{\rm Y}}}_{\upsilon }{w}_0∥}^{\alpha }{∥w_1-{\mathrm{{\rm Y}}}_{\upsilon }{w}_1∥}^{1-\alpha -\beta }\hfill \\ \hfill \Rightarrow \parallel w_1-w_2\parallel & \le k\parallel w_0-w_1{\parallel }^{\beta }{∥w_0-w_1∥}^{\alpha }{∥w_1-w_2∥}^{1-\alpha -\beta }\hfill \\ \hfill \parallel w_1-w_2{\parallel }^{\alpha +\beta }& \le k\parallel w_0-w_1{\parallel }^{\alpha +\beta }\hfill \\ \hfill \Rightarrow \parallel w_1-w_2\parallel & \le k^{{\displaystyle \frac{1}{\alpha +\beta }}}\parallel w_0-w_1\parallel .\hfill \end{array}$$

Further, this implies that

$$∥w_1-w_2∥\le k∥w_0-w_1∥,$$

(6)

where $w_2={\mathrm{{\rm Y}}}_{\upsilon }{w}_1$. Inductively, we get

$$∥w_h-w_{h+1}∥\le k^h∥w_0-w_1∥.$$

(7)

Similarly, for any $m,h\in \mathbb{N}$ with $m>0$, we obtain

$$\begin{array}{cc}\hfill ∥w_h-w_{h+m}∥& \le \sum _{k=h}^{h+m-1}∥w_k-w_{k+1}∥\hfill \\ & \le (k^h+k^{h+1}+k^{h+2}+\cdots +k^{h+m-1})∥w_0-w_1∥\hfill \\ & \le (k^h+k^{h+1}+k^{h+2}+\cdots )∥w_0-w_1∥\hfill \\ & =k^h\left({\displaystyle \frac{1-k^m}{1-k}}\right)∥w_0-w_1∥\hfill \\ & \le \left({\displaystyle \frac{k^h}{1-k}}\right)∥w_0-w_1∥.\hfill \end{array}$$

As $k\in [0,1)$, this shows $\left\{w_h\right\}$ is a Cauchy sequence in ${\bigcup }_{l=1}^q{\aleph }_l$. Since ${\bigcup }_{l=1}^q{\aleph }_l$ is a complete subspace of $\mathcal{K}$, therefore $\left\{w_h\right\}$ converges to some $w^{*}\in {\bigcup }_{l=1}^q{\aleph }_l$. Using Assumption 1, which states that $\{{\aleph }_l:l=1,2,\dots ,q\}$ is a cyclic representation of ${\bigcup }_{l=1}^q{\aleph }_l$ over ${\mathrm{{\rm Y}}}_{\upsilon }$, $\left\{w_h\right\}$ contains an infinite number of terms in each ${\aleph }_l$ for every $l\in \{1,\dots ,q\}$. Therefore, $w^{*}\in {\bigcap }_{l=1}^q{\aleph }_l$. We will now demonstrate that $w^{*}$ is a fixed point of ${\mathrm{{\rm Y}}}_{\upsilon }$. From (5), it follows that

$$\begin{array}{cc}\hfill ∥w^{*}-{\mathrm{{\rm Y}}}_{\upsilon }{w}^{*}∥& \le ∥w^{*}-w_{h+1}∥+∥{\mathrm{{\rm Y}}}_{\upsilon }{w}^{*}-w_{h+1}∥\hfill \\ & =∥w^{*}-w_{h+1}∥+∥{\mathrm{{\rm Y}}}_{\upsilon }{w}^{*}-{\mathrm{{\rm Y}}}_{\upsilon }{w}_h∥\hfill \\ & \le ∥w^{*}-w_{h+1}∥+\hfill \\ & k\parallel w^{*}-w_h{\parallel }^{\beta }{∥w^{*}-{\mathrm{{\rm Y}}}_{\upsilon }{w}^{*}∥}^{\alpha }{∥w_h-{\mathrm{{\rm Y}}}_{\upsilon }{w}_h∥}^{1-\alpha -\beta }.\hfill \end{array}$$

This gives

$$∥w^{*}-{\mathrm{{\rm Y}}}_{\upsilon }{w}^{*}∥\le ∥w^{*}-w_{h+1}∥+k{\parallel w^{*}-w_h\parallel }^{\beta }{∥w^{*}-{\mathrm{{\rm Y}}}_{\upsilon }{w}^{*}∥}^{\alpha }{∥w_h-w_{h+1}∥}^{1-\alpha -\beta }.$$

Taking the limit as $h\to \infty$, we obtain $\parallel w^{*}-{\mathrm{{\rm Y}}}_{\upsilon }{w}^{*}\parallel =0$, demonstrating that $w^{*}$ is indeed a fixed point of ${\mathrm{{\rm Y}}}_{\upsilon }$. To demonstrate the uniqueness of the fixed point, assume to the contrary that $\exists r^{*}\in {\bigcap }_{l=1}^q{\aleph }_l$ with $w^{*}\ne r^{*}$ for which ${\mathrm{{\rm Y}}}_{\upsilon }{r}^{*}=r^{*}$. According to (5), we have

$$\begin{array}{cc}\hfill ∥w^{*}-r^{*}∥& =∥{\mathrm{{\rm Y}}}_{\upsilon }{w}^{*}-{\mathrm{{\rm Y}}}_{\upsilon }{r}^{*}∥\hfill \\ & \le k\parallel w^{*}-r^{*}{\parallel }^{\beta }{∥w^{*}-{\mathrm{{\rm Y}}}_{\upsilon }{w}^{*}∥}^{\alpha }{∥r^{*}-{\mathrm{{\rm Y}}}_{\upsilon }{r}^{*}∥}^{1-\alpha -\beta }\hfill \\ & =k\parallel w^{*}-r^{*}{\parallel }^{\beta }{∥w^{*}-w^{*}∥}^{\alpha }{∥r^{*}-r^{*}∥}^{1-\alpha -\beta }\hfill \\ & =0.\hfill \end{array}$$

This shows that $∥w^{*}-r^{*}∥=0$, that is $w^{*}=r^{*}$. Hence, the fixed point of ${\mathrm{{\rm Y}}}_{\upsilon }$ is unique, which implies that the fixed point of $\mathrm{{\rm Y}}$ is also unique. □

We derive Theorem 2.2 from Abbas et al. [26] as a corollary of our result.

Corollary 1.

Let $(\mathcal{K},\parallel ·\parallel )$ be a Banach space and $\mathrm{{\rm Y}}:{\bigcup }_{l=1}^q{\aleph }_l\to \mathcal{K}$ be a $(b,k,\alpha ,0)$-interpolative enriched cyclic Reich–Rus–Ćirić contraction. Then, Y possesses a unique fixed point.

Proof. 

This outcome directly follows as an immediate impact of Theorem 3, which demonstrates that a $(b,k,\alpha ,0)$-interpolative enriched cyclic Reich–Rus–Ćirić contraction $\mathrm{{\rm Y}}$ on ${\bigcup }_{l=1}^q{\aleph }_l$ within a Banach space $(\mathcal{K},\parallel ·\parallel )$ ensures the presence of a single fixed point. □

By setting $b=0$ in Theorem 3, we derive the following result from Edraou et al. [28] as a consequence of our findings.

Corollary 2.

Suppose that $(\mathcal{K},\parallel ·\parallel )$ is a Banach space and $\mathrm{{\rm Y}}:{\bigcup }_{l=1}^q{\aleph }_l\to \mathcal{K}$ is a $(0,k,\alpha ,\beta )$-interpolative enriched cyclic Reich–Rus–Ćirić contraction. Then, Y possesses a unique fixed point.

Proof. 

The conclusion is derived from Theorem 3, which guarantees the uniqueness of the fixed point for such mappings. □

By setting ${\bigcup }_{l=1}^q{\aleph }_l=\mathcal{K}$ in Corollary 2, we recover Theorem 2.2 from Karapinar et al. [23] within the framework of Banach spaces.

Corollary 3.

Consider $(\mathcal{K},\parallel ·\parallel )$ as a Banach space and $\mathrm{{\rm Y}}:\mathcal{K}\to \mathcal{K}$ a $(0,k,\alpha ,\beta )$-interpolative Reich–Rus–Ćirić contraction. Then, Y possesses a unique fixed point.

We obtain Theorem 2.0.4 from [12] as a corollary of our result.

Corollary 4.

Given a Banach space $(\mathcal{K},\parallel ·\parallel )$ and $\mathrm{{\rm Y}}:{\bigcup }_{l=1}^q{\aleph }_l\to \mathcal{K}$ is a $(b,k,0,0)$-interpolative enriched cyclic Reich–Rus–Ćirić contraction, then Y possesses a unique fixed point.

Setting $b=0$ in Corollary 1 yields the following result:

Corollary 5.

Suppose that $(\mathcal{K},\parallel ·\parallel )$ is a Banach space and $\mathrm{{\rm Y}}:{\bigcup }_{l=1}^q{\aleph }_l\to \mathcal{K}$ is a $(0,k,\alpha ,0)$-interpolative enriched cyclic Reich–Rus–Ćirić contraction, then Y possesses a unique fixed point.

If we assume ${\bigcup }_{l=1}^q{\aleph }_l=\mathcal{K}$ in Corollary 2, we obtain Theorem 2.2 of Karapinar [22] in the context of Banach spaces.

Corollary 6.

Suppose that $(\mathcal{K},\parallel ·\parallel )$ is a Banach space and $\mathrm{{\rm Y}}:\mathcal{K}\to \mathcal{K}$ is a $(0,k,\alpha ,0)$-interpolative Reich–Rus–Ćirić contraction, then Y possesses a unique fixed point.

We will now provide an example to illustrate Theorem 3. This example demonstrates how the theorem is applied in a specific scenario, highlighting the implementation of the theoretical results.

Example 1.

Consider $\mathcal{K}={\mathbb{R}}^2$ with the usual norm. Take $\mathrm{{\rm Y}}:{\aleph }_1\cup {\aleph }_2\to {\mathbb{R}}^2$ defined by

$$\mathrm{{\rm Y}}\left(w\right)=\left\{\begin{array}{cc}\left(-2w_1,w_2\right)\hfill & \mathrm{if}w=\left(w_1,w_2\right)\in {\aleph }_1,\hfill \\ \left(w_1,-2w_2\right)\hfill & \mathrm{if}w=\left(w_1,w_2\right)\in {\aleph }_2,\hfill \end{array}\right.$$

where

$${\aleph }_1=\{(w,0);w\in \mathbb{R}\}\mathrm{and}{\aleph }_2=\{(0,w);w\in \mathbb{R}\}.$$

This example is crucial, as it highlights the significance of our study. It demonstrates that Y is neither an interpolative Reich–Rus–Ćirić contraction nor a cyclic interpolative Reich–Rus–Ćirić contraction. To show that Y is not an interpolative Reich–Rus–Ćirić contraction, assume the contrary that Y is an interpolative Reich–Rus–Ćirić contraction. Thus, there exist constants $k\in [0,1)$, $\alpha >0$, and $\beta \ge 0$ with $\alpha +\beta <1$ for which the following condition is satisfied:

$$d(\mathrm{{\rm Y}}w,\mathrm{{\rm Y}}r)\le kd{(w,r)}^{\beta }d{(w,\mathrm{{\rm Y}}w)}^{\alpha }d{(r,\mathrm{{\rm Y}}r)}^{1-\alpha -\beta },$$

for all $w,r\in \mathcal{K}$ with $w,r\notin Fix\left(\mathrm{{\rm Y}}\right)$. But for $w=(1,0),r=(0,1)\in \mathcal{K}$, we obtain

$$\begin{array}{ccc}\hfill d\left(\mathrm{{\rm Y}}\right(1,0),\mathrm{{\rm Y}}(0,1\left)\right)& \le & kd{((1,0),(0,1))}^{\beta }d{((1,0),\mathrm{{\rm Y}}(1,0))}^{\alpha }d{((0,1),\mathrm{{\rm Y}}(0,1))}^{1-\alpha -\beta }\hfill \\ \hfill \Rightarrow d\left(\right(-2,0),(0,-2\left)\right)& \le & kd{((1,0),(0,1))}^{\beta }d{((1,0),(-2,0))}^{\alpha }d{((0,1),(0,-2))}^{1-\alpha -\beta }\hfill \\ \hfill \Rightarrow 2\sqrt{2}& \le & k{\left[\sqrt{2}\right]}^{\beta }{3}^{\alpha }{3}^{1-\alpha -\beta }\hfill \\ \hfill \Rightarrow 2^{1-\beta }& \le & k{3}^{1-\beta }\hfill \\ \hfill \Rightarrow 1& \le & k{\left[{\displaystyle \frac{3}{2}}\right]}^{1-\beta }.\hfill \end{array}$$

However, when β is very close to 1, we observe that ${\left[{\displaystyle \frac{3}{2}}\right]}^{1-\beta }\to 1$, leading to $1\le k$, which is a contradiction, since $k\in [0,1)$. This contradiction confirms that Y is not an interpolative Reich–Rus–Ćirić contraction. Next, it follows directly from the definition of Y that for $(1,0)\in {\aleph }_1$, we have $\mathrm{{\rm Y}}(1,0)=(-2,0)\notin {\aleph }_2$. Similarly, for $(0,1)\in {\aleph }_2$, we find $\mathrm{{\rm Y}}(0,1)=(0,-2)\notin {\aleph }_1$. This establishes that Y is also not a cyclic interpolative Reich–Rus–Ćirić contraction. This example highlights the distinctiveness and relevance of our results. It shows that existing frameworks, such as those by Karapinar et al. [23] and Edraou et al. [28], are not applicable to this scenario. This underscores the necessity of our study in addressing gaps and extending the theory to new classes of contractions. Next, we will demonstrate that Y qualifies as a cyclic enriched interpolative Reich–Rus–Ćirić contraction.

If $b=2$, then $\upsilon ={\displaystyle \frac{1}{3}}$ and we obtain

$${\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}w=\left\{\begin{array}{cc}\left(0,w_2\right)\hfill & \mathrm{if}w=\left(w_1,w_2\right)\in {\aleph }_1,\hfill \\ \left(w_1,0\right)\hfill & \mathrm{if}w=\left(w_1,w_2\right)\in {\aleph }_2.\hfill \end{array}\right.$$

It is straightforward to verify that $\{{\aleph }_1,{\aleph }_2\}$ represents a cyclic representation of ${\aleph }_1\cup {\aleph }_2$ over ${\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}$, as demonstrated by

$$\begin{array}{cc}\hfill {\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}\left({\aleph }_1\right)& =\{{\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}\left(w\right):w\in {\aleph }_1\}\hfill \\ & =\{(0,w_2):w=(w_1,w_2)\in {\aleph }_1\}\hfill \\ & =\{(0,w_2):w_2\in \mathbb{R}\}\subseteq {\aleph }_2.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill {\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}\left({\aleph }_2\right)& =\{{\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}\left(w\right):w\in {\aleph }_2\}\hfill \\ & =\{(w_1,0):w=(w_1,w_2)\in {\aleph }_2\}\hfill \\ & =\{(w_1,0):w_1\in \mathbb{R}\}\subseteq {\aleph }_1.\hfill \end{array}$$

In addition, it is clear that Y is a $(2,{\displaystyle \frac{1}{5}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{3}})$-interpolative enriched cyclic Reich–Rus–Ćirić contraction. Observe that inequality (3) is equivalent to inequality (5), which becomes

$$\parallel {\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}w-{\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}r\parallel \le {\displaystyle \frac{1}{5}}{\parallel w-r\parallel }^{{\displaystyle \frac{1}{3}}}\parallel w-{\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}{w\parallel }^{{\displaystyle \frac{1}{2}}}{\parallel r-{\mathrm{{\rm Y}}}_{{\displaystyle \frac{1}{3}}}r\parallel }^{{\displaystyle \frac{1}{6}}}.$$

So, for all $w\in {\aleph }_1$ and $r\in {\aleph }_2$, we obtain

$$\begin{array}{cc}\hfill \parallel (0,0)-(0,0)\parallel & \le {\displaystyle \frac{1}{5}}\parallel (w_1,-w_2){\parallel }^{{\displaystyle \frac{1}{3}}}\parallel (w_1,0){\parallel }^{{\displaystyle \frac{1}{2}}}{\parallel (0,w_2)\parallel }^{{\displaystyle \frac{1}{6}}}\hfill \\ \hfill 0& \le {\displaystyle \frac{1}{5}}{(w_1^2+w_2^2)}^{{\displaystyle \frac{1}{6}}}|w_1{|}^{{\displaystyle \frac{1}{2}}}{|w_2|}^{{\displaystyle \frac{1}{6}}}.\hfill \end{array}$$

Hence, all the requirements of Theorem 3 are satisfied. As a result, Y possesses a unique fixed point $w^{*}$ in ${\bigcap }_{l=1}^q{\aleph }_l$.

3. Application to the Existence of Solutions for Nonlinear Integral Equations

In the following section, we apply Theorem 3 to investigate both the existence and uniqueness of solutions for nonlinear integral equations. Specifically, we analyze the following nonlinear integral equation:

$$x\left(\mathbb{a}\right)={\int }_0^{\mathbb{k}}G(\mathbb{a},\mathbb{u})\mathfrak{b}(\mathbb{u},x\left(\mathbb{u}\right))d\mathbb{u};\mathbb{a}\in [0,\mathbb{k}],$$

(8)

where $\mathbb{k}>0$, and let $\mathfrak{b}:[0,\mathbb{k}]\times \mathbb{R}\to \mathbb{R}$ and $G:[0,\mathbb{k}]\times [0,\mathbb{k}]\to [0,\infty )$ be continuous functions. Define $\mathcal{K}=\mathcal{C}\left(\right[0,\mathbb{k}\left]\right)$ as the set of real continuous functions on $[0,\mathbb{k}]$. Consider the function $d:\mathcal{X}\times \mathcal{K}\to \mathbb{R}$ given by

$$d_{\infty }(\mathfrak{b},\mathfrak{g})=\underset{\mathbb{a}\in [0,\mathbb{k}]}{max}|\mathfrak{b}\left(\mathbb{a}\right)-\mathfrak{g}\left(\mathbb{a}\right)|,\mathfrak{b},\mathfrak{g}\in \mathcal{K}.$$

(9)

It is known that $\left(\mathcal{K},d_{\infty }\right)$ is a complete metric space. Let $\mathfrak{g},\mathfrak{e}\in \mathcal{K}$ and ${\alpha }_0,{\beta }_0\in \mathbb{R}$ for which

$${\alpha }_0\le \mathfrak{g}\left(\mathbb{a}\right)\le \mathfrak{e}\left(\mathbb{a}\right)\le {\beta }_0,\mathbb{a}\in [0,\mathbb{k}].$$

(10)

Consider for all $\mathbb{a}\in [0,\mathbb{k}],x\in \mathcal{C}\left(\right[0,\mathbb{k}\left]\right)$ and $\upsilon \in (0,1]$, we have

$$\mathfrak{g}\left(\mathbb{a}\right)\le \upsilon {\int }_0^{\mathbb{k}}G(\mathbb{a},\mathbb{u})\mathfrak{b}(\mathbb{u},\mathfrak{e}\left(\mathbb{u}\right))d\mathbb{u}+(1-\upsilon )x\left(\mathbb{u}\right),$$

(11)

and

$$\mathfrak{e}\left(\mathbb{a}\right)\ge \upsilon {\int }_0^{\mathbb{k}}G(\mathbb{a},\mathbb{u})\mathfrak{b}(\mathbb{u},\mathfrak{g}\left(\mathbb{u}\right))d\mathbb{u}+(1-\upsilon )x\left(\mathbb{u}\right).$$

(12)

Moreover, for each $\mathbb{u}\in [0,\mathbb{k}]$, the mapping $\mathfrak{b}(\mathbb{u},·)$ is a non-increasing function, i.e., for all $w,r\in \mathbb{R}$, with $w\ge r$ and for each $\mathbb{u}\in [0,\mathbb{k}]$, we have

$$\mathfrak{b}(\mathbb{u},w)\le \mathfrak{b}(\mathbb{u},r).$$

(13)

Further,

$$\underset{\mathbb{a}\in [0,\mathbb{k}]}{sup}{\int }_0^{\mathbb{k}}G(\mathbb{a},\mathbb{u})d\mathbb{u}\le 1.$$

(14)

Finally, suppose that, for $\upsilon \in (0,1],k\in [0,1),\alpha >0,\beta \ge 0$ with $\alpha +\beta <1$, and for all $\mathbb{u}\in [0,\mathbb{k}],w,r\in \mathbb{R}$ with ( $w\le {\beta }_0$ and $r\ge {\alpha }_0$) or ( $w\ge {\alpha }_0$ and $\left.r\le {\beta }_0\right)$, we have

$$|\mathfrak{b}(\mathbb{u},w)-\mathfrak{b}(\mathbb{u},r)|\le {\displaystyle \frac{k}{{\upsilon }^{\beta }}}\left({\parallel w-r\parallel }^{\beta }{|w-\mathrm{{\rm Y}}w|}^{\alpha }{|r-\mathrm{{\rm Y}}r|}^{1-\alpha -\beta }-(1-\upsilon )(w-r)\right).$$

(15)

Let us consider the following set

$$X=:\{x\in \mathcal{C}([0,\mathbb{k}]):\mathfrak{g}(\mathbb{a})\le x(\mathbb{a})\le \mathfrak{e}(\mathbb{a});\mathbb{a}\in [0,\mathbb{k}\left]\right\}.$$

(16)

We have the following result:

Theorem 4.

Given the assumptions outlined in conditions (11)–(16) the nonlinear integral Equation (8) has a unique solution $x^{*}$ within the space X.

Proof. 

Define ${\mathcal{A}}_{\mathfrak{e}}$ and ${\mathcal{A}}_{\mathfrak{g}}$ by

$$\begin{array}{cc}{\mathcal{A}}_{\mathfrak{e}}=\{x\in \mathcal{K}:x\left(\mathbb{u}\right)\le \mathfrak{e}\left(\mathbb{u}\right),\hfill & \forall \mathbb{u}\in [0,\mathbb{k}]\},\hfill \\ {\mathcal{A}}_{\mathfrak{g}}=\{x\in \mathcal{K}:x\left(\mathbb{u}\right)\ge \mathfrak{g}\left(\mathbb{u}\right),\hfill & \forall \mathbb{u}\in [0,\mathbb{k}]\}\hfill \end{array}$$

which are the closed subsets of $\mathcal{K}$. Define the mapping $\mathrm{{\rm Y}}:{\mathcal{A}}_{\mathfrak{e}}\cup {\mathcal{A}}_{\mathfrak{g}}\to \mathcal{K}$ by

$$\mathrm{{\rm Y}}x\left(\mathbb{a}\right)={\int }_0^{\mathbb{k}}G(\mathbb{a},\mathbb{u})\mathfrak{b}(\mathbb{u},x\left(\mathbb{u}\right))d\mathbb{u};\mathbb{a}\in [0,\mathbb{k}].$$

For $\upsilon \in (0,1]$, we have

$${\mathrm{{\rm Y}}}_{\upsilon }x\left(\mathbb{a}\right)=\upsilon {\int }_0^{\mathbb{k}}G(\mathbb{a},\mathbb{u})\mathfrak{b}(\mathbb{u},x\left(\mathbb{u}\right))d\mathbb{u}+(1-\upsilon )u\left(\mathbb{a}\right);\mathbb{a}\in [0,\mathbb{k}].$$

Let $x\in {\mathcal{A}}_{\mathfrak{e}}$. Then, by using (13), we have

$$G(\mathbb{a},\mathbb{u})\mathfrak{b}(\mathbb{a},x(\mathbb{u}\left)\right)\ge G(\mathbb{a},\mathbb{u})\mathfrak{b}(\mathbb{a},\mathfrak{e}(\mathbb{u}\left)\right),\forall \mathbb{a},\mathbb{u}\in [0,\mathbb{k}].$$

It follows from (11) that

$$\begin{array}{cc}\hfill \upsilon {\int }_0^{\mathbb{k}}G(\mathbb{a},\mathbb{u})\mathfrak{b}(\mathbb{a},x\left(\mathbb{u}\right))d\mathbb{u}+(1-\upsilon )x\left(\mathbb{a}\right)& \ge \upsilon {\int }_0^{\mathbb{k}}G(\mathbb{a},\mathbb{u})\mathfrak{b}(\mathbb{a},\mathfrak{e}\left(\mathbb{u}\right))d\mathbb{u}+(1-\upsilon )x\left(\mathbb{a}\right)\hfill \\ & \ge \mathfrak{g}\left(\mathbb{a}\right).\hfill \end{array}$$

Thus, ${\mathrm{{\rm Y}}}_{\upsilon }\left(x\right)\ge \mathfrak{g}$, and this implies

$${\mathrm{{\rm Y}}}_{\upsilon }\left({\mathcal{A}}_{\mathfrak{e}}\right)\subseteq {\mathcal{A}}_{\mathfrak{g}}.$$

In a similar fashion, we obtain the following:

$${\mathrm{{\rm Y}}}_{\upsilon }\left({\mathcal{A}}_{\mathfrak{g}}\right)\subseteq {\mathcal{A}}_{\mathfrak{e}}.$$

Given the conditions (14) and (15), it follows that

$$\begin{array}{cc}\hfill \left|{\mathrm{{\rm Y}}}_{\upsilon }x-{\mathrm{{\rm Y}}}_{\upsilon }y\right|\le & \upsilon {\int }_0^{\mathbb{k}}G(\mathbb{a},\mathbb{u})\left(\right|\mathfrak{b}(\mathbb{u},x\left(\mathbb{u}\right))-\mathfrak{b}(\mathbb{a},y\left(\mathbb{u}\right))|\hfill \\ & +(1-\upsilon )\left(x\right(\mathbb{u})-y(\mathbb{u}\left)\right))d\mathbb{u}\hfill \\ \hfill \le & {\displaystyle \frac{k}{{\upsilon }^{\beta -1}}}{\int }_0^{\mathbb{k}}{G(\mathbb{a},\mathbb{u})|x-y|}^{\beta }{|x-\mathrm{{\rm Y}}x|}^{\alpha }{|y-\mathrm{{\rm Y}}y|}^{1-\alpha -\beta }d\mathbb{u}\hfill \\ \hfill \Rightarrow \underset{\mathbb{a}\in [0,\mathbb{k}]}{max}\left|{\mathrm{{\rm Y}}}_{\upsilon }x\left(\mathbb{a}\right)-{\mathrm{{\rm Y}}}_{\upsilon }y\left(\mathbb{a}\right)\right|\le & {\displaystyle \frac{k}{{\upsilon }^{\beta -1}}}{\left(\underset{\mathbb{a}\in [0,\mathbb{k}]}{max}|x\left(\mathbb{a}\right)-y\left(\mathbb{a}\right)|\right)}^{\beta }\times \hfill \\ & {\left(\underset{\mathbb{a}\in [0,\mathbb{k}]}{max}|x\left(\mathbb{a}\right)-\mathrm{{\rm Y}}x\left(\mathbb{a}\right)|\right)}^{\alpha }\times \hfill \\ & {\left(\underset{\mathbb{a}\in [0,\mathbb{k}]}{max}|y\left(\mathbb{a}\right)-\mathrm{{\rm Y}}y\left(\mathbb{a}\right)|\right)}^{1-\alpha -\beta }{\int }_0^{\mathbb{k}}\mathfrak{G}(\mathbb{a},\mathbb{u})d\mathbb{u}.\hfill \end{array}$$

Therefore, we conclude that

$$d_{\infty }\left({\mathrm{{\rm Y}}}_{\upsilon }x,{\mathrm{{\rm Y}}}_{\upsilon }y\right)\le k{\left[d_{\infty }(x,y)\right]}^{\beta }{\left[d_{\infty }\left(x,{\mathrm{{\rm Y}}}_{\upsilon }x\right)\right]}^{\alpha }{\left[d_{\infty }\left(y,{\mathrm{{\rm Y}}}_{\upsilon }y\right)\right]}^{1-\alpha -\beta }.$$

Since $\mathrm{{\rm Y}}$ meets all the requirements of Theorem 3, so the integral Equation (8) has a unique solution $x^{*}$ in X. □

4. Coupled Fixed-Point Results

This section introduces coupled cyclic mapping, focusing on the coupled interpolative enriched cyclic Reich–Rus–Ćirić contraction, and shows that such mappings have strong coupled fixed points.

Definition 6.

Consider a normed space $(\mathcal{K},\parallel ·\parallel )$ and nonempty subsets ${\aleph }_1$ and ${\aleph }_2$ of $\mathcal{K}$. Suppose we have a mapping $\mathrm{{\rm Y}}:\mathcal{K}\times \mathcal{K}\to \mathcal{K}$. We define Y as a coupled interpolative enriched cyclic Reich–Rus–Ćirić contraction with respect to ${\aleph }_1$ and ${\aleph }_2$ if ${\mathrm{{\rm Y}}}_{\upsilon }$, where $\upsilon ={\displaystyle \frac{1}{1+b}}$ is cyclic with respect to ${\aleph }_1$ and ${\aleph }_2$, that is there exists $b\in [0,\infty ),k\in \left[0,1\right)$ and $\alpha >0,\beta \ge 0$ with $\alpha +\beta <1$ such that

$$\begin{array}{cc}\hfill & \left|\right|b(w-x)+\mathrm{{\rm Y}}(w,r)-\mathrm{{\rm Y}}(x,y)\left|\right|\\ & \le k{(1+b)}^{\beta }\left[\right||w-{x\left|\right|]}^{\beta }\left[\right||w-{\mathrm{{\rm Y}}(w,r)\left|\right|]}^{\alpha }\left[\right||x-{\mathrm{{\rm Y}}(x,y)\left|\right|]}^{1-\alpha -\beta },\end{array}$$

(17)

for all $w,x\in {\aleph }_1$ and $r,y\in {\aleph }_2$.

Before presenting the main result in this direction, we offer the following remark.

Remark 1.

For some $\upsilon \in (0,1)$, we defined the following notion that if $\mathrm{{\rm Y}}:{\mathcal{K}}^2\to \mathcal{K}$ is a given mapping, then ${\mathrm{{\rm Y}}}_{\upsilon }:{\mathcal{K}}^2\to \mathcal{K}$ will be defined as ${\mathrm{{\rm Y}}}_{\upsilon }(w,r)=(1-\upsilon )w+\upsilon \mathrm{{\rm Y}}(w,r)$ and is called the average operator of $\mathrm{{\rm Y}}(w,r)$. Further, the set of all strong coupled fixed points of operator Y and ${\mathrm{{\rm Y}}}_{\upsilon }$ coincide with each other. As such, if w is a strong coupled fixed point of Υ, then $\mathrm{{\rm Y}}(w,w)=w$. This implies

$$w=(1-\upsilon )w+\upsilon w=(1-\upsilon )w+\upsilon \mathrm{{\rm Y}}(w,w)={\mathrm{{\rm Y}}}_{\upsilon }(w,w).$$

Similarly, if w is a strong coupled fixed point of ${\mathrm{{\rm Y}}}_{\upsilon }$, then ${\mathrm{{\rm Y}}}_{\upsilon }(w,w)=w$. So, we obtain

$$w={\mathrm{{\rm Y}}}_{\upsilon }(w,w)=(1-\upsilon )w+\upsilon \mathrm{{\rm Y}}(w,w)\Rightarrow \mathrm{{\rm Y}}(w,w)=w.$$

Theorem 5.

Suppose that $(\mathcal{K},\parallel ·\parallel )$ is a Banach space, and let ${\aleph }_1$ and ${\aleph }_2$ be nonempty subsets of $\mathcal{K}$ with ${\aleph }_1\cap {\aleph }_2\ne \varnothing$. If $\mathrm{{\rm Y}}:\mathcal{K}\times \mathcal{K}\to \mathcal{K}$ is a coupled interpolative enriched cyclic Reich–Rus–Ćirić contraction with respect to ${\aleph }_1$ and ${\aleph }_2$, then Y has a strong coupled fixed point in ${\aleph }_1\cap {\aleph }_2$.

Proof. 

Using the expression (17), we obtain

$$\parallel ({\displaystyle \frac{1}{\upsilon }}-1)(w-x)+\mathrm{{\rm Y}}(w,r)-\mathrm{{\rm Y}}(x,y)\parallel \le {\displaystyle \frac{k}{{\upsilon }^{\beta }}}{\parallel w-x\parallel }^{\beta }\parallel w-\mathrm{{\rm Y}}(w,r){\parallel }^{\alpha }{\parallel x-\mathrm{{\rm Y}}(x,y)\parallel }^{1-\alpha -\beta },$$

or equivalently,

$$\parallel {\mathrm{{\rm Y}}}_{\upsilon }(w,r)-{\mathrm{{\rm Y}}}_{\upsilon }{(x,y)\parallel \le k\parallel w-x\parallel }^{\beta }\parallel w-{\mathrm{{\rm Y}}}_{\upsilon }{(w,r)\parallel }^{\alpha }{\parallel x-{\mathrm{{\rm Y}}}_{\upsilon }(x,y)\parallel }^{1-\alpha -\beta }.$$

(18)

Let $w_0\in {\aleph }_1$ and $r_0\in {\aleph }_2$ be arbitrary elements. Define the sequences $\left\{w_h\right\}$ and $\left\{r_h\right\}$ as follows:

$$\left\{\begin{array}{c}{w}_1=(1-\upsilon )w_0+\upsilon \mathrm{{\rm Y}}(r_0,w_0)={\mathrm{{\rm Y}}}_{\upsilon }\left(r_0,w_0\right);w_{h+1}={\mathrm{{\rm Y}}}_{\upsilon }\left(r_h,w_h\right);\hfill \\ r_1={\mathrm{{\rm Y}}}_{\upsilon }\left(w_0,r_0\right);r_{h+1}={\mathrm{{\rm Y}}}_{\upsilon }\left(w_h,r_h\right)\mathrm{for}\mathrm{all}h\in \mathbb{N}.\hfill \end{array}\right.$$

(19)

Then, by utilizing (18) and (19), we obtain

$$\begin{array}{cc}\hfill \parallel w_1-r_2\parallel & =\parallel w_1-{\mathrm{{\rm Y}}}_{\upsilon }\left(w_1,r_1\right)\parallel \hfill \\ & =\parallel {\mathrm{{\rm Y}}}_{\upsilon }\left(r_0,w_0\right)-{\mathrm{{\rm Y}}}_{\upsilon }\left(w_1,r_1\right)\parallel |\hfill \\ & \le k\parallel r_0-w_1{\parallel }^{\beta }{\left[\parallel r_0-{\mathrm{{\rm Y}}}_{\upsilon }\left(r_0,w_0\right)\parallel \right]}^{\alpha }{\left[\parallel w_1-{\mathrm{{\rm Y}}}_{\upsilon }\left(w_1,r_1\right)\parallel \right]}^{1-\alpha -\beta }\hfill \\ & =k\parallel r_0-w_1{\parallel }^{\beta }{\left[\parallel r_0-w_1\parallel \right]}^{\alpha }{\left[\parallel w_1-r_2\parallel \right]}^{1-\alpha -\beta }\hfill \\ \hfill \Rightarrow {\left[\parallel w_1-r_2\parallel \right]}^{\alpha +\beta }& \le k\parallel r_0-w_1{\parallel }^{\alpha +\beta }.\hfill \end{array}$$

This further implies that

$$\parallel w_1-r_2\parallel \le \theta \left|\right|r_0-w_1\left|\right|,$$

(20)

where $\theta =k^{{\displaystyle \frac{1}{\alpha +\beta }}}$ and clearly $\theta \in (0,1)$. Similarly, we have

$$\parallel r_2-w_1\parallel \le \theta \parallel w_0-r_1\parallel ,$$

(21)

where $\theta =k^{{\displaystyle \frac{1}{\alpha +\beta }}}$ and so $\theta \in (0,1)$.

Inductively, using this process for all $h\in \mathbb{N}$, we obtain

$$\parallel w_h-r_{h+1}\parallel \le {\theta }^h\parallel r_0-w_1|\mathrm{and}\parallel r_{h+1}-w_h\parallel \le {\theta }^h\parallel w_0-r_1\parallel ,$$

(22)

for all $h\le m$, where h is even. If m is even, then using (20)–(22), we obtain the following:

$$\begin{array}{cc}\hfill \parallel w_{m+1}-r_{m+2}\parallel & =\parallel {\mathrm{{\rm Y}}}_{\upsilon }\left(r_m,w_m\right)-{\mathrm{{\rm Y}}}_{\upsilon }\left(w_{m+1},r_{m+1}\right)\parallel \hfill \\ & \le k\parallel r_m-w_{m+1}{\parallel }^{\beta }{\left[\parallel r_m-{\mathrm{{\rm Y}}}_{\upsilon }\left(r_m,w_m\right)\parallel \right]}^{\alpha }\times \hfill \\ & {\left[\parallel w_{m+1}-{\mathrm{{\rm Y}}}_{\upsilon }\left(w_{m+1},r_{m+1}\right)\parallel \right]}^{1-\alpha -\beta }\hfill \\ & =k\parallel r_m-w_{m+1}{\parallel }^{\beta }\parallel r_m-w_{m+1}{\parallel }^{\alpha }{\parallel w_{m+1}-r_{m+2}\parallel }^{1-\alpha -\beta }\hfill \\ \hfill \Rightarrow \parallel w_{m+1}-r_{m+2}\parallel & \le \theta \parallel r_m-w_{m+1}\parallel \hfill \\ & \le \theta [{\theta }^n\parallel r_0-w_1\parallel ]\hfill \\ & ={\theta }^{h+1}\parallel r_0-w_1\parallel .\hfill \end{array}$$

Then,

$$\parallel w_{m+1}-r_{m+2}\parallel \le {\theta }^{h+1}\parallel r_0-w_1\parallel .$$

(23)

Similarly, we obtain

$$\parallel r_{m+2}-w_{m+1}\parallel \le {\theta }^{h+1}\parallel w_0-r_1\parallel .$$

(24)

From the inequalities (23) and (24) above, we conclude that for all odd integers h, we obtain

$$\left\{\begin{array}{c}\parallel w_h-r_{h+1}\parallel \le {\theta }^n\parallel r_0-w_1\parallel \hfill \\ \parallel r_h-w_{h+1}\parallel \le {\theta }^h\parallel w_0-r_1\parallel .\hfill \end{array}\right.$$

(25)

Given a certain integer m, applying (17) and (25) yields

$$\begin{array}{cc}\hfill \parallel w_{m+1}-r_{m+1}\parallel & =\parallel {\mathrm{{\rm Y}}}_{\upsilon }\left(r_m,w_m\right)-{\mathrm{{\rm Y}}}_{\upsilon }\left(w_m,r_m\right)\parallel \hfill \\ & \le k\parallel r_m-w_m{\parallel }^{\beta }{\parallel r_m-{\mathrm{{\rm Y}}}_{\upsilon }\left(r_m,w_m\right)\parallel }^{\alpha }\times \hfill \\ & \parallel w_m-{\mathrm{{\rm Y}}}_{\upsilon }\left(w_m,r_m\right){\parallel }^{1-\alpha -\beta }\hfill \\ & =k\parallel r_m-w_m{\parallel }^{\beta }\parallel r_m-w_{m+1}{\parallel }^{\alpha }{\parallel w_m-r_{m+1}\parallel }^{1-\alpha -\beta }\hfill \\ & \le {\theta }^{\alpha +\beta }\parallel r_m-w_m{\parallel }^{\beta }{\theta }^{m\alpha }{\parallel w_0-r_1\parallel }^{\alpha }{\theta }^{m(1-\alpha -\beta )}\times \hfill \\ & \parallel r_0-w_1{\parallel }^{1-\alpha -\beta }\hfill \\ & \le {\left({\theta }^m\right)}^{1-\beta }\parallel w_0-r_1{\parallel }^{\alpha }\parallel r_0-w_1{\parallel }^{1-\alpha -\beta }{\parallel r_m-w_m\parallel }^{\beta }\hfill \\ & \le {\theta }^m\parallel r_0-w_1{\parallel }^{1-\beta }{\parallel r_m-w_m\parallel }^{\beta }\hfill \\ & \le {\theta }^m\parallel r_0-w_1{\parallel }^{1-\beta }[{\theta }^{m-1}\parallel r_0-w_1{\parallel }^{1-\beta }{]}^{\beta }{\parallel r_{m-1}-w_{m-1}\parallel }^{\beta }\hfill \\ & \le {\theta }^{2m-1}[\parallel r_0-w_1{\parallel }^{1-\beta }{]}^{1+\beta }{\parallel r_{m-1}-w_{m-1}\parallel }^{\beta }\hfill \\ & ⋮\hfill \\ & \le {\theta }^{{\displaystyle \frac{1}{2}}m(m+1)}[\parallel r_0-w_1{\parallel }^{1-\beta }{]}^{1+m\beta }{\parallel r_0-w_0\parallel }^{\beta }.\hfill \end{array}$$

In a similar manner, we obtain

$$\parallel r_{m+1}-w_{m+1}\parallel \le {\theta }^{{\displaystyle \frac{1}{2}}m(m+1)}[\parallel w_0-r_1{\parallel }^{1-\beta }{]}^{1+m\beta }{\parallel w_0-r_0\parallel }^{\beta }.$$

(26)

Now, by (18), (25) and (26), for all $h\ge 1$, we have

$$\begin{array}{cc}& \parallel w_h-w_{h+1}\parallel +\parallel r_h-r_{h+1}\parallel \hfill \\ & \le \parallel w_h-r_{h+1}\parallel +\parallel r_{h+1}-w_{h+1}\parallel +\parallel r_h-w_{h+1}\parallel +\parallel w_{h+1}-r_{h+1}\parallel \hfill \\ & \le {\theta }^h\parallel r_0-w_1\parallel +{\theta }^{{\displaystyle \frac{1}{2}}h(h+1)}[\parallel w_0-r_1{\parallel }^{1-\beta }{]}^{1+h\beta }\parallel w_0-r_0{\parallel }^{\beta }+{\theta }^h\parallel w_0-r_1\parallel \hfill \\ & +{\theta }^{{\displaystyle \frac{1}{2}}h(h+1)}[\parallel r_0-w_1{\parallel }^{1-\beta }{]}^{1+h\beta }{\parallel r_0-w_0\parallel }^{\beta }\hfill \\ & \le {\theta }^h\left({\displaystyle \genfrac{}{}{0pt}{}{\parallel r_0-w_1\parallel +[\parallel w_0-r_1{\parallel }^{1-\beta }{]}^{1+h\beta }+\parallel w_0-r_1\parallel +}{[\parallel r_0-w_1{\parallel }^{1-\beta }{]}^{1+h\beta }{\parallel r_0-w_0\parallel }^{\beta }}}\right).\hfill \end{array}$$

Since $\theta \in (0,1)$, it follows that as $h\to \infty$, the sequences $\left(w_h\right)$ and $\left(r_h\right)$ are Cauchy sequences in $\mathcal{K}$. With the completeness of $\mathcal{K}$, these sequences converge to certain elements of $w^{*}$ and $r^{*}$, respectively.

Since both ${\aleph }_1$ and ${\aleph }_2$ are closed subsets of $\mathcal{K}$, it follows that $w_h\to w^{*}\in {\aleph }_1$ and $r_h\to r^{*}\in {\aleph }_1$.

Additionally, from (26), we have $\parallel w_h-r_h\parallel \to 0$ as $n\to \infty$. Therefore, $\parallel w-r\parallel =0$, which implies that $w=r$. Hence, $w=r\in {\aleph }_1\cap {\aleph }_2$ and ${\aleph }_1\cap {\aleph }_2\ne \varphi$. To demonstrate the existence of a fixed point, we use (18) to derive

$$\begin{array}{cc}\hfill \parallel w-{\mathrm{{\rm Y}}}_{\upsilon }(w,r)\parallel & \le \parallel w-w_{h+1}\parallel +\parallel w_{h+1}-{\mathrm{{\rm Y}}}_{\upsilon }(w,r)\parallel \hfill \\ & =\parallel w-{\mathrm{{\rm Y}}}_{\upsilon }(r_h,w_h)\parallel +\parallel {\mathrm{{\rm Y}}}_{\upsilon }(r_h,w_h)-{\mathrm{{\rm Y}}}_{\upsilon }(w,r)\parallel \hfill \\ & \le \parallel w-w_{h+1}\parallel +\hfill \\ & k\left|\right|r_h-{w\left|\right|}^{\beta }\parallel r_h-{\mathrm{{\rm Y}}}_{\upsilon }(r_h,w_h){\parallel }^{\alpha }{\parallel w-{\mathrm{{\rm Y}}}_{\upsilon }(w,r)\parallel }^{1-\alpha -\beta }\hfill \\ & =\parallel w-w_{h+1}\parallel +\hfill \\ & k\left|\right|r_h-{w\left|\right|}^{\beta }\parallel r_h-w_{h+1}{\parallel }^{\alpha }{\parallel w-{\mathrm{{\rm Y}}}_{\upsilon }(w,r)\parallel }^{1-\alpha -\beta }.\hfill \end{array}$$

Taking the limit as $h\to \infty$ in the inequality above, we obtain $\parallel w-{\mathrm{{\rm Y}}}_{\upsilon }(w,r)\parallel =0$, thereby establishing that w is a fixed point of ${\mathrm{{\rm Y}}}_{\upsilon }$. Thus, we can deduce that $w={\mathrm{{\rm Y}}}_{\upsilon }(w,w)$, which shows that w is a strong coupled fixed point of ${\mathrm{{\rm Y}}}_{\upsilon }$ and $\mathrm{{\rm Y}}$. □

By setting $b=0$ and $\beta =0$ in (5), we obtain Theorem 11 of [10] as a corollary of our result in the setting of normed spaces.

Corollary 7.

Suppose that $(\mathcal{K},\parallel ·\parallel )$ is a Banach space, and let ${\aleph }_1$ and ${\aleph }_2$ be nonempty subsets of $\mathcal{K}$ with ${\aleph }_1\cap {\aleph }_2\ne \varnothing$. If $\mathrm{{\rm Y}}:\mathcal{K}\times \mathcal{K}\to \mathcal{K}$ is a coupled interpolative cyclic Kannan contraction with respect to ${\aleph }_1$ and ${\aleph }_2$, then Y has a strong coupled fixed point in ${\aleph }_1\cap {\aleph }_2$.

5. Application to Fractional Differential Equations

In this section, we use our main results to explore the existence of solutions for boundary value problems concerning fractional differential equations that incorporate the Caputo fractional derivative.

Suppose that $\mathcal{K}=\mathfrak{C}\left(\right[0,1],\mathbb{R})$ denote the Banach space of all continuous functions mapping from $[0,1]$ to $\mathbb{R}$ equipped with the norm

$$\parallel \varkappa \parallel =\underset{k\in [0,1]}{max}\left|\varkappa \left(k\right)\right|.$$

Now, we will review the fundamental concepts that will be required later.

Definition 7

([34]). For a function v defined on the interval $[a,b]$, the Caputo fractional derivative of order $r>0$ is expressed as follows:

$${(}^c{x}_{a^{+}}^r)v\left(k\right)={\displaystyle \frac{1}{\Gamma (m-r)}}{\int }_a^k{(k-s)}^{m-\varkappa -1}{v}^{\left(m\right)}\left(s\right)ds,(m-1\le r<m,m=\left[r\right]+1),$$

(27)

where $\left[r\right]$ indicates the integer part of the positive real number r, and Γ refers to the gamma function. Suppose the boundary value problem for the fractional order differential equation is defined as

$$\begin{array}{ccc}\hfill & & {}^{c}x_{0+}^r\left(\varkappa \left(k\right)\right)=h(k,\varkappa \left(k\right)),(k\in [0,1],2<r\le 3)\hfill \\ & & \varkappa \left(0\right)=c_0,{\varkappa }^{\prime }\left(0\right)=c,{\varkappa }^{\prime \prime }\left(1\right)=c_1,\hfill \end{array}$$

(28)

where ${}^{c}x_{0+}^r$ signifies that the Caputo fractional derivative of order r, $h:[0,1]\to \mathbb{R}$ is a continuous function, and $c_0,c,c_1$ are real constants.

Definition 8

([33]). A function $\varkappa \in {\mathfrak{C}}^3([0,1],\mathbb{R})$ for which the ϰ-derivative exists on $[0,1]$ is considered a solution of (28) if it satisfies the equation ${}^{c}x_{0+}^r\left(\varkappa \left(k\right)\right)=h(k,\varkappa \left(k\right))$ on $[0,1]$ along with the conditions $\varkappa \left(0\right)=c_0$, ${\varkappa }^{\prime }\left(0\right)=c$, and ${\varkappa }^{\prime \prime }\left(1\right)=c_1$.

The following lemma will be essential for the subsequent discussion.

Lemma 1

([33]). Let $2<r\le 3$ and let $v:[0,1]\to \mathbb{R}$ be a continuous function. A function ϰ is considered a solution of the fractional integral equation

$$\varkappa \left(k\right)={\displaystyle \frac{1}{\Gamma \left(r\right)}}{\int }_0^k{(k-s)}^{\varkappa -1}v\left(s\right)ds-{\displaystyle \frac{k^2}{2\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}v\left(s\right)ds+c_0+c_{0}k+{\displaystyle \frac{c_1}{2}}{k}^2$$

(29)

if ϰ is a solution to the fractional boundary value problem

$$\begin{array}{c}\hfill {}^{c}x_{0+}^{\varkappa }\left(\varkappa \left(k\right)\right)=v\left(k\right)\varkappa \left(0\right)=c_0,{\varkappa }^{\prime }\left(0\right)=c,{\varkappa }^{″}\left(1\right)=c_1,\end{array}$$

(30)

where ${\varkappa }^{″}\left(1\right)=2c_2+{\displaystyle \frac{1}{\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}v\left(s\right)ds=c_1,$ and $c,c_0$ and $c_2$ are constants in $\mathbb{R}$.

Next, we reformulate the boundary value problem for fractional order differential Equation (28) as a fixed-point problem. To achieve this, we define a self-mapping $T:\mathcal{K}\to \mathcal{K}$ by:

$$\begin{array}{ccc}\hfill T\varkappa \left(k\right)& =& {\displaystyle \frac{1}{\Gamma \left(r\right)}}{\int }_0^k{(k-s)}^{\varkappa -1}h(s,\varkappa \left(s\right))ds\hfill \\ & & -{\displaystyle \frac{k^2}{2\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}h(s,\varkappa \left(s\right))ds+c_0+c_0^{*}k+{\displaystyle \frac{c_1}{2}}{k}^2\hfill \end{array}$$

(31)

where $c_1=2c_2+{\displaystyle \frac{1}{\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}h(s,\varkappa \left(s\right))ds,c_i,c_0^{*}\in \mathbb{R},(i=0,1,2)\mathrm{are}\mathrm{constant}.$

Clearly, the fixed points of mapping T as define in (31) are the solutions of boundary value problem for fractional order differential Equation (28).

Now, we state and prove our main result in this section.

Theorem 6.

Suppose that for all $\varkappa ,y\in I$ there exists $\lambda >0$ such that $\forall k\in [0,1]$,

$$\begin{array}{ccc}\hfill & & |h(k,\varkappa \left(k\right))-h(k,\varkappa \left(k\right))|\le {\lambda |\varkappa \left(k\right)-T\varkappa \left(k\right)|}^{\alpha }·{|y\left(k\right)-Ty\left(k\right)|}^{1-\alpha }\hfill \end{array}$$

(32)

where

$$0<b:=\lambda \left({\displaystyle \frac{1}{\Gamma (r+1)}}+{\displaystyle \frac{1}{2\Gamma (r-1)}}\right)<1.$$

(33)

Then, the Equation (28) has a unique solution in I.

Proof. 

Considering the following and using the given assumption in (32) and (33), we obtain

$$\begin{array}{ccc}& & \left|T\varkappa \right(k)-T\alpha (k\left)\right|\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(r\right)}}{\int }_0^k{(k-s)}^{\varkappa -1}|h(s,\varkappa \left(s\right))-h(s,y\left(s\right))|ds\hfill \\ & & +{\displaystyle \frac{k^2}{2\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}|h(s,\varkappa \left(s\right))-h(s,y\left(s\right))|ds\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(r\right)}}{\int }_0^k{(k-s)}^{\varkappa -1}|h(s,\varkappa \left(s\right))-h(s,y\left(s\right))|ds\hfill \\ & & +{\displaystyle \frac{1}{2\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}|h(s,\varkappa \left(s\right))-h(s,y\left(s\right))|ds\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(r\right)}}{\int }_0^k{(k-s)}^{\varkappa -1}{\lambda |\varkappa \left(k\right)-T\varkappa \left(k\right)|}^{\alpha }·{|\alpha \left(k\right)-Ty\left(k\right)|}^{1-\alpha }ds\hfill \\ & & +{\displaystyle \frac{1}{2\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}{\lambda |\varkappa \left(k\right)-T\varkappa \left(k\right)|}^{\alpha }·{|y\left(k\right)-Ty\left(k\right)|}^{1-\alpha }ds\hfill \\ & =& {\displaystyle \frac{\lambda }{\Gamma \left(r\right)}}{\int }_0^k{(k-s)}^{\varkappa -1}{\left[\right|\varkappa \left(k\right)-T\varkappa \left(k\right)|}^{\alpha }{·|y\left(k\right)-Ty\left(k\right)|}^{1-\alpha }]ds\hfill \\ & & +{\displaystyle \frac{\lambda }{2\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}{\left[\right|\varkappa \left(k\right)-T\varkappa \left(k\right)|}^{\alpha }{·|y\left(k\right)-Ty\left(k\right)|}^{1-\alpha }]ds\hfill \\ & \le & {\displaystyle \frac{\lambda }{\Gamma \left(r\right)}}{\int }_0^k{(k-s)}^{\varkappa -1}\left[\right||\varkappa \left(k\right)-{T\varkappa \left(k\right)\left|\right|}^{\alpha }·\left|\right|y\left(k\right)-{Ty\left(k\right)\left|\right|}^{1-\alpha }]ds\hfill \\ & & +{\displaystyle \frac{\lambda }{2\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}\left[\right||\varkappa \left(k\right)-{T\varkappa \left(k\right)\left|\right|}^{\alpha }·\left|\right|y\left(k\right)-{Ty\left(k\right)\left|\right|}^{1-\alpha }]ds\hfill \\ & =& \left[{\displaystyle \frac{\lambda }{\Gamma \left(r\right)}}{\int }_0^k{(k-s)}^{\varkappa -1}ds+{\displaystyle \frac{\lambda }{2\Gamma (r-2)}}{\int }_0^1{(1-s)}^{\varkappa -3}ds\right]\hfill \\ & & \times \left[\left|\right|\varkappa \left(k\right)-{T\varkappa \left(k\right)\left|\right|}^{\alpha }·\left|\right|y\left(k\right)-{Ty\left(k\right)\left|\right|}^{1-\alpha }\right]\hfill \\ & \le & \lambda \left({\displaystyle \frac{1}{\Gamma (r+1)}}+{\displaystyle \frac{1}{2\Gamma (r-1)}}\right)\left[\right||\varkappa \left(k\right)-{T\varkappa \left(k\right)\left|\right|}^{\alpha }·\left|\right|y\left(k\right)-{Ty\left(k\right)\left|\right|}^{1-\alpha }]\hfill \\ & =& b\left|\right|\varkappa \left(k\right)-{T\varkappa \left(k\right)\left|\right|}^{\alpha }·\left|\right|y\left(k\right)-{Ty\left(k\right)\left|\right|}^{1-\alpha }.\hfill \end{array}$$

Hence, this implies that

$$\parallel T\varkappa -Ty\parallel \le {b\left|\right|\varkappa -T\varkappa \left|\right|}^{\alpha }·\left|\right|y-{Ty\left|\right|}^{1-\alpha }.$$

So, it follows that T is an $(0,k,\alpha ,0)$-interpolative Reich–Rus–Ćirić contraction. Hence, using Corollary 6, the problem (28) has a unique solution. □

6. Conclusions and Future Directions

In this study, we presented a new type of operator within normed spaces, called interpolative enriched cyclic Reich–Rus–Ćirić operators. We established the unique existence of fixed points for these new mappings. The provided example illustrated the theoretical results of these findings. Additionally, we investigated how these mappings can be applied to solve nonlinear integral equations, thereby demonstrating their applications in a wider range of mathematical scenarios. We introduced coupled interpolative enriched cyclic Reich–Rus–Ćirić contractions and proved the existence of strong coupled fixed points. Our findings also yielded several corollaries from the existing literature, demonstrating broad applicability and potential for further research. This study advances fixed-point theory and creates opportunities for applying these results in diverse mathematical and practical fields. Finally, we examined the existence and uniqueness of a solution to fractional differential equations by applying the main result.

Future research could investigate extending interpolative enriched cyclic Reich–Rus–Ćirić contractions to more general settings, such as metric or quasi-metric spaces, beyond the realm of linear normed spaces. Research should also concentrate on developing efficient computational algorithms for identifying fixed points and evaluating the stability of the results when subjected to perturbations. Applications in optimization and various interdisciplinary fields, as well as extensions to higher-dimensional spaces, present promising opportunities for further exploration. Additionally, exploring asymptotic-like contractions and higher-order averaged operators within the context of interpolative enriched cyclic Reich–Rus–Ćirić contractions would be beneficial.