Existence and Uniqueness of Solutions to Fractional Differential Equations in Complex-Valued Suprametric Spaces

Abstract

This study focuses on establishing the existence and uniqueness of solutions for nonlinear fractional differential equations through the application of fixed-point methods in complex-valued suprametric spaces. In order to accomplish this, novel cyclic and interpolative contractive conditions are formulated within the complex-valued suprametric setting, leading to the derivation of several common fixed-point theorems. The obtained results extend and encompass a variety of known fixed-point theorems in complex-valued metric spaces as particular instances. In addition, meaningful and non-trivial examples are presented to highlight the effectiveness and practical relevance of the developed theoretical framework.

1. Introduction

Fixed-point (FP) theory is a fundamental and widely studied area of nonlinear analysis, with significant applications in differential equations, optimization, and applied sciences. It is commonly classified into three principal branches: discrete, topological, and metric FP theory. Among these, metric FP theory plays a central role, as it investigates the existence and uniqueness of the FPs of mappings defined on metric spaces (MSs). The concept of an MS was formalized by Fréchet in 1906 [1], providing a rigorous framework to quantify the distance between elements of a set. An MS is characterized by a distance function satisfying non-negativity, symmetry, and the triangle inequality. Since its inception, this notion has been extensively generalized to overcome the limitations of classical settings and to address increasingly complex mathematical models.

One of the earliest and most significant generalizations is the partial MS, introduced by Matthews [2], in which the self-distance of a point may be nonzero. This structure has proven particularly useful in theoretical computer science, especially in domain theory and the semantics of programming languages. Another important extension is the b-MS, introduced by Bakhtin [3], where the triangle inequality is relaxed by incorporating a constant coefficient. This relaxation allows for the analysis of problems where the classical triangle inequality is too restrictive.

Further developments include rectangular MSs, introduced by Branciari [4], where the triangle inequality is replaced by a rectangular inequality involving four points. Berzig [5] later proposed suprametric spaces (SMSs), which further relax the triangle inequality and provide a more flexible framework for FP analysis. These spaces have been successfully applied to nonlinear integral equations and matrix equations, demonstrating their effectiveness in handling complex structures. Subsequently, Berzig [6,7,8] extended this concept to generalized SMSs and b-SMSs by introducing additional relaxations.

In a parallel direction, Azam et al. [9] introduced complex-valued metric spaces (C-VMSs), in which the distance function takes values in the complex plane rather than the real numbers. This extension enables the modeling of phenomena involving oscillatory behavior, phase shifts, and complex dynamical systems, where real-valued metrics may not be sufficient. Later, Rouzkard et al. [10] incorporated rational-type contractive conditions into C-VMSs, enhancing their applicability. The study of FP theory in C-VMS was significantly advanced by Hussain et al. [11], who established common FP (CFP) results under suitable contractive conditions and demonstrated their applicability to integral equations. Panda et al. [12] combined the ideas of SMSs and C-VMSs to introduce complex-valued suprametric spaces (C-VSMSs), providing a unified and more general framework. Within this setting, they established several CFP results for rational-type contractions and demonstrated their applicability to complex nonlinear integral equations. Further contributions in this direction can be found in [13,14,15].

On the other hand, the generalization of contractive conditions plays a fundamental role in FP theory. Fisher [16] introduced a contractive condition involving rational expressions, thereby generalizing the classical notion of contraction proposed by Banach [17]. In recent decades, considerable attention has been devoted to extending and refining classical contractive conditions by formulating more general and flexible contraction principles. Kirk et al. [18] introduced the concept of cyclic contractions to study mappings that alternate between distinct subsets of a MS. Subsequently, Dwivedi [19] established FP theorems for cyclic contractions in the setting of b-MSs. Abbas et al. [20] obtained FP results for cocyclic mappings in C-VMSs.

More recently, interpolative contractions, introduced by Karapınar [21], have provided a powerful framework for bridging different types of contractive conditions and capturing more subtle nonlinear behaviors. This idea was further generalized to interpolative Reich–Rus–Ćirić type contractions in partial MSs [22], and later extended to include rational-type interpolative contractions by Fulga [23] and to complex-valued settings by Alharbi et al. [24].

Although significant progress has been made in this area, a notable gap still exists in the current literature. Most existing results consider cyclic contractions and interpolative contractions in isolation, and their unified treatment within the setting of C-VSMSs remains largely underdeveloped. In particular, many of the established contractive conditions lack the flexibility required to simultaneously encompass the following aspects:

• The cyclic structure of mappings between subsets;
• The interpolative nature of nonlinear contractions;
• The generalized geometry induced by C-VSMSs.

As a result, the applicability of current FP results is limited when dealing with nonlinear problems that exhibit hybrid or mixed behaviors.

Motivated by these limitations, this paper aims to develop new cyclic and interpolative contractive conditions within the framework of C-VSMSs. The principal contributions of this study are summarized as follows:

• We establish three classes of contractive conditions in C-VSMSs, namely cyclic contractions involving rational expressions, interpolative contractions, and interpolative-rational hybrid contractions.
• The proposed contractive conditions generalize and extend some well-known contractions, including classical, rational, and interpolative types.
• The obtained CFP results are supported by nontrivial examples, demonstrating their validity and illustrating the generality of the proposed approach.
• The developed theory applies to a broader class of nonlinear mappings, particularly those exhibiting hybrid structural behavior.

Furthermore, to illustrate the applicability and significance of the obtained results, we employ the established FP theorems to study the existence of solutions for nonlinear fractional differential equations. In this setting, the framework of C-VSMSs offers a natural and effective environment for dealing with complex-valued solutions and nonlocal phenomena, which are often challenging to handle within classical metric approaches. This demonstrates that the proposed methodology is not only a theoretical extension, but also a useful analytical tool for addressing meaningful problems arising in applied mathematics.

2. Preliminaries

In this section, we present the fundamental concepts and definitions that form the foundation of our study. We begin with generalized metric structures, including SMSs, and then extend these ideas to complex-valued settings. These frameworks provide the necessary mathematical tools to formulate and analyze cyclic, interpolative, and rational contractive conditions in complete C-VSMS. Relevant examples are also provided to illustrate these structures and to demonstrate their applicability in supporting the main FP results presented in this research work.

Berzig [5] defined the SMS using the following approach.

Definition 1 

([5]). Let $\mathsf{\Omega }\ne \varnothing$ and $\eta \ge 0$. Let $d:\mathsf{\Omega }\times \mathsf{\Omega }\to {\mathbb{R}}^{+}$ denote a function that meets the below-mentioned criteria

(i) 

$0\le d(\varkappa ,\varsigma )$ and $d(\varkappa ,\varsigma )=0$ ⟺ $\varkappa =\varsigma$;

(ii) 

$d(\varkappa ,\varsigma )=d(\varsigma ,\varkappa );$

(iii) 

$d(\varkappa ,\varsigma )\le d(\varkappa ,\nu )+d(\nu ,\varsigma )+\eta d(\varkappa ,\nu )d(\nu ,\varsigma ).$

For all $\varkappa ,\varsigma ,\nu \in \mathsf{\Omega },$ $(\mathsf{\Omega },d)$ is called an SMS.

The introduction of the nonlinear term $\eta d(\varkappa ,\nu )d(\nu ,\varsigma )$ in SMS allows the relaxation of the classical triangle inequality. This extension is particularly useful when dealing with iterative processes where accumulated errors are multiplicative rather than additive. Such behavior naturally appears in nonlinear operator equations and iterative schemes.

Example 1. 

Let $\mathsf{\Omega }=\left\{a,b,c\right\}$ and define a function $d:\mathsf{\Omega }\times \mathsf{\Omega }\to {\mathbb{R}}^{+}$ by

$$d(a,b)=d(b,a)=0.5$$

$$d(a,c)=d(c,a)=1$$

$$d(b,c)=d(c,b)=2,$$

$$d(a,a)=d(b,b)=d(c,c)=0.$$

Let us take $\eta =1.$ Then, $\left(\mathsf{\Omega },d\right)$ forms an SMS but not an MS, since the triangle inequality of MSs is violated. For instance,

$$d(b,c)=2>1.5=d(b,a)+d(a,c).$$

Let us define a partial ordering $\precsim$ on $\mathbb{C}$ in this way.

$$z_1\precsim z_2\iff Re\left(z_1\right)⩽Re\left(z_2\right),Im\left(z_1\right)⩽Im\left(z_2\right).$$

$\forall z_1,z_2\in \mathbb{C}.$ Consequently,

$$z_1\precsim z_2$$

provided that at least one of the following conditions is satisfied:

$$\begin{array}{ccc}\hfill \left(a\right)Re\left(z_1\right)& =& Re\left(z_2\right),Im\left(z_1\right)<Im\left(z_2\right),\hfill \\ \hfill \left(b\right)Re\left(z_1\right)& <& Re\left(z_2\right),Im\left(z_1\right)=Im\left(z_2\right),\hfill \\ \hfill \left(c\right)Re\left(z_1\right)& <& Re\left(z_2\right),Im\left(z_1\right)<Im\left(z_2\right),\hfill \\ \hfill \left(d\right)Re\left(z_1\right)& =& Re\left(z_2\right),Im\left(z_1\right)=Im\left(z_2\right).\hfill \end{array}$$

This partial order allows comparison of complex distances and plays a crucial role in defining convergence and contractive conditions in C-VMSs.

Azam et al. [9] defined the idea of C-VMS in this fashion.

Definition 2 

([9]). Let $\mathsf{\Omega }\ne \varnothing$ and $d:\mathsf{\Omega }\times \mathsf{\Omega }\to \mathbb{C}$ satisfy

(i) 

$0\precsim d(\varkappa ,\varsigma )$ and $d(\varkappa ,\varsigma )=0$ ⟺ $\varkappa =\varsigma$;

(ii) 

$d(\varkappa ,\varsigma )=d(\varsigma ,\varkappa );$

(iii) 

$d(\varkappa ,\varsigma )\precsim d(\varkappa ,\nu )+d(\nu ,\varsigma ).$

For all $\varkappa ,\varsigma ,\nu \in \mathsf{\Omega },$ $(\mathsf{\Omega },d)$ is called a C-VMS.

The extension to complex-valued metrics enables the incorporation of both magnitude and directional (phase) information. This is essential in applications where the solution space is inherently complex-valued, such as signal processing, quantum mechanics, and complex differential equations.

Example 2 

([9]). Let $\mathsf{\Omega }=[0,1]$ and $\varkappa ,\varsigma \in \mathsf{\Omega }.$ Define $d:\mathsf{\Omega }\times \mathsf{\Omega }\to \mathbb{C}$ by

$$d(\varkappa ,\varsigma )=\left\{\begin{array}{c}0,\mathit{if}\varkappa =\varsigma ,\\ {\displaystyle \frac{i}{2}},\mathit{if}\varkappa \ne \varsigma .\end{array}\right.$$

Then $(\mathsf{\Omega },d)$ is a C-VMS.

Panda et al. [12] proposed the definition of a C-VSMS in this form.

Definition 3 

([12]). Let $\mathsf{\Omega }\ne \varnothing ,$ $\eta \ge 0$ and $d:\mathsf{\Omega }\times \mathsf{\Omega }\to \mathbb{C}$ satisfy

(i) 

$0\precsim d(\varkappa ,\varsigma )$ and $d(\varkappa ,\varsigma )=0$ ⟺ $\varkappa =\varsigma$;

(ii) 

$d(\varkappa ,\varsigma )=d(\varsigma ,\varkappa );$

(iii) 

$d(\varkappa ,\varsigma )\precsim d(\varkappa ,\nu )+d(\nu ,\varsigma )+\eta d(\varkappa ,\nu )d(\nu ,\varsigma ).$

For all $\varkappa ,\varsigma ,\nu \in \mathsf{\Omega }$. It follows that $(\mathsf{\Omega },d)$ constitutes a C-VSMS.

The concept of C-VSMS combines the advantages of suprametric structures and complex-valued ordering, providing a more flexible framework for studying nonlinear contractions. This structure is particularly suitable for developing generalized contractive conditions such as interpolative and rational contractions, which may fail under classical metric assumptions.

Example 3. 

Consider $\mathsf{\Omega }=C\left(\left[0,1\right],\mathbb{C}\right)$ and define

$$d(f,g)=\underset{t\in [0,1]}{sup}\left|f\left(t\right)-g\left(t\right)\right|\left(1+i\right).$$

Then $\left(\mathsf{\Omega },d\right)$ forms a C-VSMS. This type of structure is useful in studying complex-valued integral equations.

Remark 1. 

When η is set to zero in Definition 3, the concept of a C-VSMS becomes that of a C-VMS.

These notions of convergence and Cauchy sequences are consistent with the modulus of complex numbers and ensure the completeness of the space, which is essential for the application of FP principles.

Lemma 1 

([12]). A sequence $\left\{{\varkappa }_j\right\}$ in C-VSMS $(\mathsf{\Omega },d)$ is said to converges to ϰ iff

$$\left|d({\varkappa }_j,\varkappa )\right|\to 0$$

as $j\to \infty .$

Lemma 2 

([12]). A sequence $\left\{{\varkappa }_j\right\}$ in C-VSMS $(\mathsf{\Omega },d)$ is a Cauchy sequence iff

$$\left|d({\varkappa }_j,{\varkappa }_{j+m})\right|\to 0$$

as $j\to \infty ,$ for any fixed $m\in \mathbb{N}.$

The concepts introduced in this section will be used to establish new CFP results under cyclic and interpolative–rational contractive conditions. The flexibility of C-VSMS enables the treatment of these contractions in a unified manner, which has not been possible in previously studied settings.

3. Principal Results

In this section, we present the main FP results of the paper in the framework of complete C-VSMSs. Throughout this section, $(\mathsf{\Omega },d)$ denotes a complete C-VSMS. The results are organized into three subsections dealing with cyclic rational contractions, interpolative contractions, and interpolative–rational hybrid contractions.

3.1. Cyclic Rational Contractions in Complex-Valued Suprametric Spaces

In this subsection, we investigate cyclic rational contractive conditions for pairs of self-mappings defined by the union of two nonempty closed subsets of a C-VSMS. This setting naturally models situations where mappings act alternately between distinct subsets, reflecting interdependent or stepwise processes that cannot be captured by classical contraction principles, which typically operate on a single set without structural partitioning. Moreover, classical Banach-type contractions are inadequate for capturing cyclic interactions and become restrictive when the contractive behavior depends on nonlinear relations between distances. To overcome this, we employ a rational-type contraction. Although C-VSMS can be viewed as a special case of cone SMS, the presence of division in the complex setting allows for the natural incorporation of rational expressions. This provides additional flexibility and highlights an advantage of C-VSMS in modeling such nonlinear behaviors. The obtained result not only unifies several known cyclic contraction theorems in MS, SMS, and C-VMS but also significantly extends their scope to the richer structure of C-VSMSs.

Theorem 1. 

Let ($\mathsf{\Omega },d)$ be a complete C-VSMS and let A and B be nonempty closed subsets of Ω. Consider the mappings $\mathcal{M},\mathcal{H}:A\cup B\to A\cup B$ satisfying

$$\mathcal{M}\left(A\right)\subseteq B,\mathcal{M}\left(B\right)\subseteq A\mathrm{and}\mathcal{H}\left(A\right)\subseteq B,\mathcal{H}\left(B\right)\subseteq A.$$

Suppose that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

(1)

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{M}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{M}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

(2)

holds for all $\varkappa \in A,\varsigma \in B$. Then there exists a unique point ${\varkappa }^{*}\in A\cap B$ such that $\mathcal{M}{\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}={\varkappa }^{*}.$

Proof. 

Let ${\varkappa }_0\in A$ be given. Define the sequence {${\varkappa }_j$} by

$${\varkappa }_{2j+1}=\mathcal{M}{\varkappa }_{2j}\in B\mathrm{and}{\varkappa }_{2j+2}=\mathcal{H}{\varkappa }_{2j+1}\in A,$$

∀$j\in \mathbb{N}\cup \left\{0\right\}$. By (1) and (2) with $\varkappa ={\varkappa }_{2j}\in A$ and $\varsigma ={\varkappa }_{2j+1}\in B$, we obtain

$$d({\varkappa }_{2j+1},{\varkappa }_{2j+2})=d(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }_{2j+1})⪯\varpi \mathsf{\Xi }({\varkappa }_{2j},{\varkappa }_{2j+1}),$$

(3)

and

$$\mathsf{\Xi }({\varkappa }_{2j},{\varkappa }_{2j+1})\in \left\{\begin{array}{c}d({\varkappa }_{2j},{\varkappa }_{2j+1}),d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j}),d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1}),\\ {\displaystyle \frac{d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}\end{array}\right\}.$$

In the case where $\mathsf{\Xi }({\varkappa }_{2j},{\varkappa }_{2j+1})=d({\varkappa }_{2j},{\varkappa }_{2j+1}),$ (3) yields

$$d({\varkappa }_{2j+1},{\varkappa }_{2j+2})⪯\varpi d({\varkappa }_{2j},{\varkappa }_{2j+1})$$

∀$j\in \mathbb{N}\cup \left\{0\right\}.$ This leads to the conclusion that

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le \varpi \left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|,$$

∀$j\in \mathbb{N}\cup \left\{0\right\}.$ If

$$\mathsf{\Xi }({\varkappa }_{2j},{\varkappa }_{2j+1})=d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j}),$$

then it follows from (3) that

$$d({\varkappa }_{2j+1},{\varkappa }_{2j+2})⪯\varpi d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})=\varpi d({\varkappa }_{2j},{\varkappa }_{2j+1})$$

from which it follows that

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le \varpi \left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|$$

∀$j\in \mathbb{N}\cup \left\{0\right\}.$ If

$$\mathsf{\Xi }({\varkappa }_{2j},{\varkappa }_{2j+1})=d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1}),$$

then by (3), we obtain

$$d({\varkappa }_{2j+1},{\varkappa }_{2j+2})⪯\varpi d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})=\varpi d({\varkappa }_{2j+1},{\varkappa }_{2j+2})$$

which leads to

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le \varpi \left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|<\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|.$$

Since $\varpi <1,$ this results in a contradiction. Thus, $\mathsf{\Xi }({\varkappa }_{2j},{\varkappa }_{2j+1})\ne d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1}).$ Consider the case where

$$\mathsf{\Xi }({\varkappa }_{2j},{\varkappa }_{2j+1})={\displaystyle \frac{d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}.$$

It follows from (3) that

$$\begin{array}{ccc}\hfill d({\varkappa }_{2j+1},{\varkappa }_{2j+2})& ⪯& \varpi {\displaystyle \frac{d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}\hfill \\ & =& \varpi {\displaystyle \frac{d({\varkappa }_{2j},{\varkappa }_{2j+1})d({\varkappa }_{2j+1},{\varkappa }_{2j+2})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}\hfill \end{array}$$

which yields that

$$\begin{array}{ccc}\hfill \left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|& \le & \varpi {\displaystyle \frac{\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}{\left|1+d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}}\hfill \\ & \le & \varpi \left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\hfill \end{array}$$

because

$$\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|<\left|1+d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|$$

and

$${\displaystyle \frac{\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}{\left|1+d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}}<1.$$

Since $\varpi <1,$ we obtain

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le \varpi \left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|<\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|,$$

which is a contradiction. Consequently, we must have

$$\mathsf{\Xi }({\varkappa }_{2j},{\varkappa }_{2j+1})\ne {\displaystyle \frac{d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}.$$

Thus, considering all possible cases, it follows that

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le \varpi \left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|,$$

(4)

∀$j\in \mathbb{N}\cup \left\{0\right\}.$ Similarly, using (1) and (2) with $\varkappa ={\varkappa }_{2j+2}\in A$ and $\varsigma ={\varkappa }_{2j+1}\in B$, we obtain

$$\begin{array}{ccc}\hfill d({\varkappa }_{2j+2},{\varkappa }_{2j+3})& =& d(\mathcal{H}{\varkappa }_{2j+1},\mathcal{M}{\varkappa }_{2j+2})\hfill \\ & =& d(\mathcal{M}{\varkappa }_{2j+2},\mathcal{H}{\varkappa }_{2j+1})⪯\varpi \mathsf{\Xi }({\varkappa }_{2j+2},{\varkappa }_{2j+1})\hfill \end{array}$$

(5)

and

$$\mathsf{\Xi }({\varkappa }_{2j+2},{\varkappa }_{2j+1})\in \left\{\begin{array}{c}d({\varkappa }_{2j+2},{\varkappa }_{2j+1}),d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2}),d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1}),\\ {\displaystyle \frac{d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2})d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j+2},{\varkappa }_{2j+1})}}\end{array}\right\}.$$

Now, we analyze each possibility for $\mathsf{\Xi }({\varkappa }_{2j+2},{\varkappa }_{2j+1}).$ If

$$\mathsf{\Xi }({\varkappa }_{2j+2},{\varkappa }_{2j+1})=d({\varkappa }_{2j+2},{\varkappa }_{2j+1}),$$

then from (5), we obtain

$$d({\varkappa }_{2j+2},{\varkappa }_{2j+3})⪯\varpi d({\varkappa }_{2j+2},{\varkappa }_{2j+1})=\varpi d({\varkappa }_{2j+1},{\varkappa }_{2j+2}),$$

yielding

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le \varpi \left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|.$$

∀$j\in \mathbb{N}\cup \left\{0\right\}.$ If

$$\mathsf{\Xi }({\varkappa }_{2j+2},{\varkappa }_{2j+1})=d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2}),$$

then from (5), we have

$$d({\varkappa }_{2j+2},{\varkappa }_{2j+3})⪯\varpi d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2})=\varpi d({\varkappa }_{2j+2},{\varkappa }_{2j+3})$$

∀$j\in \mathbb{N}\cup \left\{0\right\}.$ Moreover, it shows that

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le \varpi \left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|<\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|$$

which is a contradiction. Hence, this case cannot occur. If

$$\mathsf{\Xi }({\varkappa }_{2j+2},{\varkappa }_{2j+1})=d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1}),$$

then by (5), we obtain

$$d({\varkappa }_{2j+2},{\varkappa }_{2j+3})⪯\varpi d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})=\varpi d({\varkappa }_{2j+1},{\varkappa }_{2j+2})$$

giving

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le \varpi \left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|,$$

for all $\mathbb{N}\cup \left\{0\right\}.$ If

$$\mathsf{\Xi }({\varkappa }_{2j+2},{\varkappa }_{2j+1})={\displaystyle \frac{d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2})d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j+2},{\varkappa }_{2j+1})}},$$

then by (5), we obtain

$$\begin{array}{ccc}\hfill d({\varkappa }_{2j+2},{\varkappa }_{2j+3})& ⪯& \varpi {\displaystyle \frac{d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2})d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j+2},{\varkappa }_{2j+1})}}\hfill \\ & =& \varpi {\displaystyle \frac{d({\varkappa }_{2j+2},{\varkappa }_{2j+3})d({\varkappa }_{2j+1},{\varkappa }_{2j+2})}{1+d({\varkappa }_{2j+2},{\varkappa }_{2j+1})}}\hfill \end{array}$$

from which it follows that

$$\begin{array}{ccc}\hfill \left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|& \le & \varpi {\displaystyle \frac{\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}{\left|1+d({\varkappa }_{2j+2},{\varkappa }_{2j+1})\right|}}\hfill \\ & \le & \varpi \left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|.\hfill \end{array}$$

Since

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|<\left|1+d({\varkappa }_{2j+2},{\varkappa }_{2j+1})\right|,$$

it follows that

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le \varpi \left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|<\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|,$$

which, because $\varpi <1,$ leads to a contradiction. Therefore, the only remaining possibility is

$$\mathsf{\Xi }({\varkappa }_{2j+2},{\varkappa }_{2j+1})\ne {\displaystyle \frac{d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2})d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j+2},{\varkappa }_{2j+1})}}.$$

Hence, considering all cases, we conclude

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le \varpi \left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|$$

(6)

for all $\mathbb{N}\cup \left\{0\right\}.$ Finally, combining (4) and (6), we obtain

$$\left|d({\varkappa }_j,{\varkappa }_{j+1})\right|\le \varpi \left|d({\varkappa }_{j-1},{\varkappa }_j)\right|$$

for all $\mathbb{N}\cup \left\{0\right\}.$ Hence

$$\left|d({\varkappa }_j,{\varkappa }_{j+1})\right|\le \varpi \left|d({\varkappa }_{j-1},{\varkappa }_j)\right|\le {\varpi }^2\left|d({\varkappa }_{j-2},{\varkappa }_{j-1})\right|\le \cdots \le {\varpi }^j\left|d({\varkappa }_0,{\varkappa }_1)\right|$$

(7)

∀$\mathbb{N}\cup \left\{0\right\}.$ Let $m>j$. Then we have

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_j,{\varkappa }_m\right)\right|& \le & \left|d\left({\varkappa }_j,{\varkappa }_{j+1}\right)\right|+\left|d\left({\varkappa }_{j+1},{\varkappa }_m\right)\right|+\eta \left|d\left({\varkappa }_j,{\varkappa }_{j+1}\right)\right|\left|d\left({\varkappa }_{j+1},{\varkappa }_m\right)\right|\hfill \\ & =& \left|d\left({\varkappa }_j,{\varkappa }_{j+1}\right)\right|+\left(1+\eta \left|d\left({\varkappa }_j,{\varkappa }_{j+1}\right)\right|\right)\left|d\left({\varkappa }_{j+1},{\varkappa }_m\right)\right|.\hfill \end{array}$$

(8)

Applying the same inequality recursively to $\left|d\left({\varkappa }_{j+1},{\varkappa }_m\right)\right|,$ we obtain

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{j+1},{\varkappa }_m\right)\right|& \le & \left|d\left({\varkappa }_{j+1},{\varkappa }_{j+2}\right)\right|+\left|d\left({\varkappa }_{j+2},{\varkappa }_m\right)\right|+\eta \left|d\left({\varkappa }_{j+1},{\varkappa }_{j+2}\right)\right|\left|d\left({\varkappa }_{j+2},{\varkappa }_m\right)\right|\hfill \\ & =& \left|d\left({\varkappa }_{j+1},{\varkappa }_{j+2}\right)\right|+\left(1+\eta \left|d\left({\varkappa }_{j+1},{\varkappa }_{j+2}\right)\right|\right)\left|d\left({\varkappa }_{j+2},{\varkappa }_m\right)\right|,\hfill \end{array}$$

and similarly for $\left|d\left({\varkappa }_{j+2},{\varkappa }_m\right)\right|,$ and so on, until

$$\begin{array}{ccc}\hfill \left|d\left({\varkappa }_{m-2},{\varkappa }_m\right)\right|& \le & \left|d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)\right|+\left|d\left({\varkappa }_{m-1},{\varkappa }_m\right)\right|+\eta \left|d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)\right|\left|d\left({\varkappa }_{m-1},{\varkappa }_m\right)\right|\hfill \\ & =& \left|d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)\right|+\left(1+\eta \left|d\left({\varkappa }_{m-2},{\varkappa }_{m-1}\right)\right|\right)\left|d\left({\varkappa }_{m-1},{\varkappa }_m\right)\right|.\hfill \end{array}$$

By recursively substituting each inequality into the preceding one in (8) and simplifying, we obtain

$$|d\left({\varkappa }_j,{\varkappa }_m\right)|\le {\displaystyle \sum _{k=j}^{m-1}}|d\left({\varkappa }_j,{\varkappa }_{j+1}\right)|{\displaystyle \prod _{i=j}^{k-1}}\left(1+\eta \left|d\left({\varkappa }_i,{\varkappa }_{i+1}\right)\right|\right).$$

Using the inequality in (7), it follows that

$$|d\left({\varkappa }_j,{\varkappa }_m\right)|\le |d\left({\varkappa }_0,{\varkappa }_1\right)|{\displaystyle \sum _{k=j}^{m-1}}{\varpi }^j{\displaystyle \prod _{i=j}^{k-1}}\left(1+\eta {\varpi }^i\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\right).$$

(9)

Observe that $\left(1+\eta {\varpi }^i\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\right)\ge 1$, ∀$i.$ Therefore,

$${\displaystyle \prod _{i=j}^{k-1}}\left(1+\eta {\varpi }^i\left|d\left({\varkappa }_0,{\varkappa }_1\right)\right|\right)\ge 1,$$

which allows the series to be expressed in the following form:

$$|d\left({\varkappa }_0,{\varkappa }_1\right)|{\displaystyle \sum _{k=j}^{m-1}}{\varpi }^j{\displaystyle \prod _{i=j}^{k-1}}\left(1+\eta {\varpi }^i|d\left({\varkappa }_0,{\varkappa }_1\right)|\right)\ge |d\left({\varkappa }_0,{\varkappa }_1\right)|{\displaystyle \sum _{k=j}^{m-1}}{\varpi }^j.$$

(10)

Since ${\displaystyle \sum _{k=j}^{m-1}}{\varpi }^j$ is a finite geometric series with the first term ${\varpi }^j$ and the common ratio $\varpi ,$ its sum is given by

$${\displaystyle \sum _{k=j}^{m-1}}{\varpi }^j={\varpi }^j{\displaystyle \frac{\left(1-{\varpi }^{m-j}\right)}{1-\varpi }}.$$

As $m\to \infty$, ${\varpi }^{m-j}\to 0,$ so the series converges to

$${\displaystyle \frac{{\varpi }^j}{1-\varpi }},$$

Since $\varpi <1,$ it follows that ${\displaystyle \frac{{\varpi }^j}{1-\varpi }}$→ 0 as $j\to \infty .$ Letting $j,m\to \infty$ in (8) and applying the established bounds, we obtain

$$\underset{j,m\to \infty }{lim}|d\left({\varkappa }_j,{\varkappa }_m\right)|=0,$$

which shows that $\left\{{\varkappa }_j\right\}$ is a Cauchy sequence. As $\mathsf{\Omega }$ is complete, so $\exists {\varkappa }^{*}\in \mathsf{\Omega }$ such that ${\varkappa }_j\to {\varkappa }^{*}$ as $j\to \infty .$ Therefore,

$$\underset{j\to \infty }{lim}{\varkappa }_j={\varkappa }^{*}.$$

(11)

Since {${\varkappa }_{2j}$} is a subsequence in A and A is closed, its limit ${\varkappa }^{*}$ belong to $\overline{A}=A.$ Similarly, {${\varkappa }_{2j+1}$} is in B and B is closed, so ${\varkappa }^{*}\in \overline{B}=B.$ Consequently, ${\varkappa }^{*}\in A\cap B.$ From (1) and (2) with $\varkappa ={\varkappa }^{*}\in A$ and $\varsigma ={\varkappa }_{2j+1}\in B$, we have

$$\begin{array}{ccc}\hfill d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+d\left({\varkappa }_{2j+2},\mathcal{M}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)d\left({\varkappa }_{2j+2},\mathcal{M}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+d\left(\mathcal{H}{\varkappa }_{2j+1},\mathcal{M}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)d\left(\mathcal{H}{\varkappa }_{2j+1},\mathcal{M}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+d\left(\mathcal{M}{\varkappa }^{*},\mathcal{H}{\varkappa }_{2j+1}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)d\left(\mathcal{M}{\varkappa }^{*},\mathcal{H}{\varkappa }_{2j+1}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi \mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)+\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right),\hfill \end{array}$$

(12)

which entails that

$$\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|\le \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|+\varpi \left|\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|+\eta \varpi \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|\left|\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|,$$

(13)

and

$$\begin{array}{ccc}\hfill \mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)& \in & \left\{\begin{array}{c}d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right),d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*}),d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1}),\\ {\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})}{1+d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)}}\end{array}\right\}\hfill \\ & =& \left\{\begin{array}{c}d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right),d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*}),d({\varkappa }_{2j+1},{\varkappa }_{2j+2}),\\ {\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})d({\varkappa }_{2j+1},{\varkappa }_{2j+2})}{1+d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)}}\end{array}\right\}.\hfill \end{array}$$

By taking the limit as $j\to +\infty$ in the preceding inequality, we deduce the following result

$$\underset{j\to \infty }{lim}\left|d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|=0,$$

and

$$\underset{j\to \infty }{lim}\eta \varpi \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|\left|\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|=0,$$

because ${lim}_{j\to \infty }\left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|=0.$ Therefore, it remains to find

$$\underset{j\to \infty }{lim}\varpi \left|\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|.$$

We consider the following four possibilities. If $\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)=d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right).$ From (13), we obtain

$$\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|\le \underset{j\to \infty }{lim}\varpi \left|d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|=0$$

which entails that ${\varkappa }^{*}=\mathcal{M}{\varkappa }^{*}.$ If $\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)=d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*}),$ then (13) gives

$$\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|\le \varpi \left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|$$

which is impossible since $\varpi <1.$ Hence, this case is ruled out. If $\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)=d({\varkappa }_{2j+1},{\varkappa }_{2j+2}),$ then

$$\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|\le \underset{j\to \infty }{lim}\varpi \left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|=0,$$

so ${\varkappa }^{*}=\mathcal{M}{\varkappa }^{*}.$ Now, if

$$\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)={\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})d({\varkappa }_{2j+1},{\varkappa }_{2j+2})}{1+d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)}}.$$

By (13), we obtain

$$\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|\le \varpi \underset{j\to \infty }{lim}{\displaystyle \frac{\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}{\left|1+d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|}}=0,$$

which again implies ${\varkappa }^{*}=\mathcal{M}{\varkappa }^{*}.$ Thus, ${\varkappa }^{*}$ is a FP of $\mathcal{H}$. Likewise, from (1) and (2) with $\varkappa ={\varkappa }_{2j}\in A$ and $\varsigma ={\varkappa }^{*}\in B$, we obtain

$$\begin{array}{ccc}\hfill d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)+d\left({\varkappa }_{2j+1},\mathcal{H}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)d\left({\varkappa }_{2j+1},\mathcal{H}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)+d\left(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)d\left(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }^{*}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi \mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right)+\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right),\hfill \end{array}$$

which further yields

$$\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|\le \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|+\varpi \left|\mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right)\right|+\eta \varpi \left|d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|\left|\mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right)\right|,$$

(14)

where

$$\begin{array}{ccc}\hfill \mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right)& \in & \left\{d\left({\varkappa }_{2j},{\varkappa }^{*}\right),d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j}),d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}),{\displaystyle \frac{d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})}{1+d\left({\varkappa }_{2j},{\varkappa }^{*}\right)}}\right\}\hfill \\ & =& \left\{d\left({\varkappa }_{2j},{\varkappa }^{*}\right),d({\varkappa }_{2j},{\varkappa }_{2j+1}),d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}),{\displaystyle \frac{d({\varkappa }_{2j},{\varkappa }_{2j+1})d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})}{1+d\left({\varkappa }_{2j},{\varkappa }^{*}\right)}}\right\}.\hfill \end{array}$$

Taking the limit as $j\to +\infty$, we obtain

$$\underset{j\to \infty }{lim}\left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|=0$$

and

$$\underset{j\to \infty }{lim}\eta \varpi \left|d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|\left|\mathsf{\Xi }\left(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }^{*}\right)\right|=0,$$

because ${lim}_{j\to \infty }\left|d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|=0.$ Consequently, it remains to determine

$$\underset{j\to \infty }{lim}\varpi \left|\mathsf{\Xi }\left(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }^{*}\right)\right|.$$

We consider four possible scenarios for $\mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right)$. If $\mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right)=d\left({\varkappa }_{2j},{\varkappa }^{*}\right).$ From (14), we obtain

$$\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|\le \varpi \underset{j\to \infty }{lim}\left|d\left({\varkappa }_{2j},{\varkappa }^{*}\right)\right|=0,$$

which indicates that ${\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}.$ If $\mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right)=d({\varkappa }_{2j},{\varkappa }_{2j+1}).$ Then (14) gives

$$\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|\le \varpi \underset{j\to \infty }{lim}\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|=0,$$

and hence, ${\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}.$ If $\mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right)=d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*}).$ From (14), we obtain

$$\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|\le \varpi \left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|,$$

which is impossible since $\varpi <1.$ Therefore, this case is ruled out. If

$$\mathsf{\Xi }\left({\varkappa }_{2j},{\varkappa }^{*}\right)={\displaystyle \frac{d({\varkappa }_{2j},{\varkappa }_{2j+1})d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})}{1+d\left({\varkappa }_{2j},{\varkappa }^{*}\right)}}.$$

By (14), we obtain

$$\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|\le \varpi \underset{j\to \infty }{lim}{\displaystyle \frac{\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|}{\left|1+d\left({\varkappa }_{2j},{\varkappa }^{*}\right)\right|}}=0,$$

which indicates that ${\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}.$ Hence, ${\varkappa }^{*}$ is a FP of $\mathcal{H}.$ Consequently, ${\varkappa }^{*}$ is a CFP of $\mathcal{H}$ and $\mathcal{M}$ in $A\cap B$. To establish the uniqueness of ${\varkappa }^{*},$ assume to the contrary that there is one more CFP ${\varkappa }^{\prime }$ of $\mathcal{H}$ and $\mathcal{M}$ in $A\cap B$, i.e., ${\varkappa }^{\prime }=\mathcal{H}{\varkappa }^{\prime }=\mathcal{M}{\varkappa }^{\prime }$ with ${\varkappa }^{*}\ne {\varkappa }^{\prime }.$ Then, from (1) and (2) with $\varkappa ={\varkappa }^{*}\in A$ and $\varkappa ={\varkappa }^{\prime }\in B$, we have

$$d({\varkappa }^{*},{\varkappa }^{\prime })=d(\mathcal{M}{\varkappa }^{*},\mathcal{H}{\varkappa }^{\prime })⪯\varpi \mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }^{\prime }\right)$$

(15)

where

$$\begin{array}{ccc}\hfill \mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }^{\prime }\right)& \in & \left\{d({\varkappa }^{*},{\varkappa }^{\prime }),d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*}),d({\varkappa }^{\prime },\mathcal{H}{\varkappa }^{\prime }),{\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})d({\varkappa }^{\prime },\mathcal{H}{\varkappa }^{\prime })}{1+d({\varkappa }^{*},{\varkappa }^{\prime })}}\right\}\hfill \\ & =& \left\{d({\varkappa }^{*},{\varkappa }^{\prime }),d({\varkappa }^{*},{\varkappa }^{*}),d({\varkappa }^{\prime },{\varkappa }^{\prime }),{\displaystyle \frac{d({\varkappa }^{*},{\varkappa }^{*})d({\varkappa }^{\prime },{\varkappa }^{\prime })}{1+d({\varkappa }^{*},{\varkappa }^{\prime })}}\right\}.\hfill \end{array}$$

Then we have only $\mathsf{\Xi }\left({\varkappa }^{*},{\varkappa }^{\prime }\right)=d({\varkappa }^{*},{\varkappa }^{\prime }),$ which implies by (15) that

$$\left|d({\varkappa }^{*},{\varkappa }^{\prime })\right|\le \varpi \left|d({\varkappa }^{*},{\varkappa }^{\prime })\right|,$$

which is a contradiction because $\varpi <1$. Hence ${\varkappa }^{*}={\varkappa }^{\prime }.$ Hence CFP is unique in $A\cap B$. □

Example 4. 

Let $\mathsf{\Omega }=\left[0,1\right]\subset \mathbb{R}.$ Define $d:\mathsf{\Omega }\times \mathsf{\Omega }\to \mathbb{C}$ by

$$d(\varkappa ,\varsigma )=\left|\varkappa -\varsigma \right|+i\left|\varkappa -\varsigma \right|,$$

for all $\varkappa ,\varsigma \in \mathsf{\Omega }$ and $i^2=-1.$ Then $\left(\mathsf{\Omega },d\right)$ is a complete C-VSMS with $\eta \ge 0.$ Let

$$A=\left[0,{\displaystyle \frac{1}{2}}\right]\mathrm{and}B=\left[{\displaystyle \frac{1}{2}},1\right].$$

Clearly, A and B are nonempty, closed, and $A\cap B=\left\{{\displaystyle \frac{1}{2}}\right\}.$ Define $\mathcal{M},\mathcal{H}:A\cup B\to A\cup B$ by

$$\mathcal{M}\left(\varkappa \right)={\displaystyle \frac{1}{2}}$$

and

$$\mathcal{H}\left(\varkappa \right)={\displaystyle \frac{1}{2}}$$

for all $\varkappa \in \mathsf{\Omega }.$ Then

$$\mathcal{M}\left(A\right)\subseteq B,\mathcal{M}\left(B\right)\subseteq A,$$

and

$$\mathcal{H}\left(A\right)\subseteq B,\mathcal{H}\left(B\right)\subseteq A.$$

So the cyclic condition is satisfied. Now for any $\varkappa \in A,\varsigma \in B,$

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )=d\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)=0+0i.$$

Hence,

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{M}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{M}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

holds for any $\varpi \in [0,1).$ Thus, condition (1) and (2) is satisfied. Thus ${\displaystyle \frac{1}{2}}$ is the unique CFP of mappings $\mathcal{M}$ and $\mathcal{H}.$

Corollary 1. 

Let A and B be nonempty closed subsets of Ω. Consider mapping $\mathcal{H}:A\cup B\to A\cup B$ satisfying

$$\mathcal{H}\left(A\right)\subseteq B,\mathcal{H}\left(B\right)\subseteq A.$$

Suppose that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

and

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{H}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{H}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

holds for all $\varkappa \in A,\varsigma \in B$. Then, $\mathcal{H}$ admits a unique FP in $A\cap B.$

Consequently, the main result of Abdou [13] follows as a particular case of Theorem 1 when $A=B=\mathsf{\Omega }$.

Corollary 2 

([13]). Let $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

and

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{M}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{M}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a unique CFP in $\mathsf{\Omega }.$

Proof. 

Take $A=B=\mathsf{\Omega }$ in Theorem 1. □

Corollary 3. 

Let $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma )$$

and

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{H}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{H}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

for all $\varkappa ,\varsigma \in \mathsf{\Omega }$. Then, $\mathcal{H}$ admits a unique FP in $\mathsf{\Omega }.$

Proof. 

Take $\mathcal{M}=\mathcal{H}$ in above corollary. □

3.2. Fixed-Point Results for Interpolative Contractions

This subsection is devoted to interpolative CFP theorems in C-VSMSs, where the contraction condition is formulated through a nonlinear interpolation of distances. Such conditions model situations in which the contractive behavior is governed by a balance between multiple distance terms, rather than a single linear bound as in classical contractions. Consequently, Banach-type contractions may fail to apply, as they cannot capture these intermediate or mixed behaviors. To overcome this limitation, the interpolative framework provides greater flexibility by allowingf for weighted nonlinear combinations of distances. Notably, this type of contraction has not previously been explored in the setting of C-VSMSs. Under mild assumptions, we establish the existence of a CFP for a pair of self-mappings, thereby extending the applicability of FP theory in this generalized space.

Theorem 2. 

Let $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $p,q\in (0,1)$ with $p+q<1$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[d(\varkappa ,\varsigma )\right]}^{1-p-q},$$

(16)

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M}$,$\mathcal{H}\}$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a CFP in $\mathsf{\Omega }.$

Proof. 

Let ${\varkappa }_0\in \mathsf{\Omega }$ be given. Define the sequence {${\varkappa }_j$} by

$${\varkappa }_{2j+1}=\mathcal{M}{\varkappa }_{2j}\mathrm{and}{\varkappa }_{2j+2}=\mathcal{H}{\varkappa }_{2j+1},$$

∀$j\in \mathbb{N}\cup \left\{0\right\}$. If for some j we have ${\varkappa }_{2j}\in Fix\left\{\mathcal{M},\mathcal{H}\right\},$ then

$${\varkappa }_{2j}=\mathcal{M}{\varkappa }_{2j}=\mathcal{H}{\varkappa }_{2j},$$

hence, ${\varkappa }_{2j}$ is a CFP. Thus, without loss of generality, we assume

$${\varkappa }_{2j}\notin Fix\left\{\mathcal{M},\mathcal{H}\right\}$$

$\forall j\in \mathbb{N}\cup \left\{0\right\}.$ By (16) with $\varkappa ={\varkappa }_{2j}$ and $\varsigma ={\varkappa }_{2j+1}$, we have

$$d(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }_{2j+1})⪯\varpi {\left[d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})\right]}^p·{\left[d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})\right]}^q·{\left[d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^{1-p-q},$$

that is,

$$\begin{array}{ccc}\hfill d({\varkappa }_{2j+1},{\varkappa }_{2j+2})& ⪯& \varpi {\left[d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^p·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^q·{\left[d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^{1-p-q}\hfill \\ & =& \varpi {\left[d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^{1-q}·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^q,\hfill \end{array}$$

from which it follows that

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le \varpi {\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^{1-q}·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q,$$

which further implies that

$${\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^{1-q}\le \varpi {\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^{1-q}.$$

Taking both sides to power ${\displaystyle \frac{1}{1-q}}$ (since $1-q>0$), we obtain

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le {\varpi }^{{\displaystyle \frac{1}{1-q}}}\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|,$$

(17)

∀$j\in \mathbb{N}\cup \left\{0\right\}$. Similarly, from (16) with $\varkappa ={\varkappa }_{2j+2}$ and $\varsigma ={\varkappa }_{2j+1}$, we have

$$\begin{array}{ccc}\hfill d(\mathcal{H}{\varkappa }_{2j+1},\mathcal{M}{\varkappa }_{2j+2})& =& d(\mathcal{M}{\varkappa }_{2j+2},\mathcal{H}{\varkappa }_{2j+1})⪯\varpi {\left[d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2})\right]}^p\hfill \\ & & ·{\left[d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})\right]}^q·{\left[d({\varkappa }_{2j+2},{\varkappa }_{2j+1})\right]}^{1-p-q}\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill d({\varkappa }_{2j+2},{\varkappa }_{2j+3})& ⪯& \varpi {\left[d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right]}^p·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^q·{\left[d({\varkappa }_{2j+2},{\varkappa }_{2j+1})\right]}^{1-p-q}\hfill \\ & =& \varpi {\left[d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right]}^p·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^{1-p},\hfill \end{array}$$

from which it follows that

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le \varpi {\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^{1-p},$$

which further implies

$${\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|}^{1-p}\le \varpi {\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^{1-p}.$$

Taking both sides to power ${\displaystyle \frac{1}{1-p}}$ (since $1-p>0$), we obtain

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le {\varpi }^{{\displaystyle \frac{1}{1-p}}}\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|.$$

(18)

Let

$${\mu }_1={\varpi }^{{\displaystyle \frac{1}{1-q}}}\mathrm{and}{\mu }_2={\varpi }^{{\displaystyle \frac{1}{1-p}}}.$$

Since $\varpi <1$ and $1-p, 1-q\in (0,1),$ we have ${\mu }_1,{\mu }_2<1.$ Then the inequalities (17) and (18) becomes

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le {\mu }_1\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|,$$

(19)

and

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le {\mu }_2\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|.$$

(20)

Let $\mu =max\left\{{\mu }_1,{\mu }_2\right\}<1.$ Then from (19) and (20), we conclude

$$\left|d({\varkappa }_j,{\varkappa }_{j+1})\right|\le \mu \left|d({\varkappa }_{j-1},{\varkappa }_j)\right|$$

for all $j\in \mathbb{N}.$ Hence,

$$\left|d({\varkappa }_j,{\varkappa }_{j+1})\right|\le \mu \left|d({\varkappa }_{j-1},{\varkappa }_j)\right|\le {\mu }^2\left|d({\varkappa }_{j-2},{\varkappa }_{j-1})\right|\le \cdots \le {\mu }^j\left|d({\varkappa }_0,{\varkappa }_1)\right|,$$

∀$\mathbb{N}.$ By using the same procedure as we have done in Theorem 1, we can prove that $\left\{{\varkappa }_j\right\}$ is a Cauchy sequence in $\mathsf{\Omega }$. From (16) with $\varkappa ={\varkappa }^{*}$ and $\varsigma ={\varkappa }_{2j+1}$, we have

$$\begin{array}{ccc}\hfill d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+d\left({\varkappa }_{2j+2},\mathcal{M}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)d\left({\varkappa }_{2j+2},\mathcal{M}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+d\left(\mathcal{H}{\varkappa }_{2j+1},\mathcal{M}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)d\left(\mathcal{H}{\varkappa }_{2j+1},\mathcal{M}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+d\left(\mathcal{M}{\varkappa }^{*},\mathcal{H}{\varkappa }_{2j+1}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)d\left(\mathcal{M}{\varkappa }^{*},\mathcal{H}{\varkappa }_{2j+1}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi {\left[d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right]}^p·{\left[d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})\right]}^q·{\left[d({\varkappa }^{*},{\varkappa }_{2j+1})\right]}^{1-p-q}\hfill \\ & & +\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right){\left[d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right]}^p·{\left[d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})\right]}^q·{\left[d({\varkappa }^{*},{\varkappa }_{2j+1})\right]}^{1-p-q}\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi {\left[d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right]}^p·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^q·{\left[d({\varkappa }^{*},{\varkappa }_{2j+1})\right]}^{1-p-q}\hfill \\ & & +\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right){\left[d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right]}^p·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^q·{\left[d({\varkappa }^{*},{\varkappa }_{2j+1})\right]}^{1-p-q},\hfill \end{array}$$

from which it follows that

$$\begin{array}{ccc}\hfill \left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|& \le & \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|+\varpi {\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q·{\left|d({\varkappa }^{*},{\varkappa }_{2j+1})\right|}^{1-p-q}\hfill \\ & & +\eta \varpi \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|{\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q·{\left|d({\varkappa }^{*},{\varkappa }_{2j+1})\right|}^{1-p-q}.\hfill \end{array}$$

(21)

Taking the limit as $j\to +\infty$ in inequality (21), we obtain

$$\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|=0,$$

that is ${\varkappa }^{*}=\mathcal{M}{\varkappa }^{*}.$ Similarly, from (16), with $\varkappa ={\varkappa }_{2j}$ and $\varsigma ={\varkappa }^{*}$, we have

$$\begin{array}{ccc}\hfill d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)+d\left({\varkappa }_{2j+1},\mathcal{H}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)d\left({\varkappa }_{2j+1},\mathcal{H}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)+d\left(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)d\left(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }^{*}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi {\left[d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})\right]}^p·{\left[d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right]}^q·{\left[d({\varkappa }_{2j},{\varkappa }^{*})\right]}^{1-p-q}\hfill \\ & & +\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right){\left[d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})\right]}^p·{\left[d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right]}^q·{\left[d({\varkappa }_{2j},{\varkappa }^{*})\right]}^{1-p-q}\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi {\left[d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^p·{\left[d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right]}^q·{\left[d({\varkappa }_{2j},{\varkappa }^{*})\right]}^{1-p-q}\hfill \\ & & +\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right){\left[d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^p·{\left[d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right]}^q·{\left[d({\varkappa }_{2j},{\varkappa }^{*})\right]}^{1-p-q},\hfill \end{array}$$

from which it follows that

$$\begin{array}{ccc}\hfill \left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|& \le & \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|+\varpi {\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^p·{\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|}^q·{\left|d({\varkappa }_{2j},{\varkappa }^{*})\right|}^{1-p-q}\hfill \\ & & +\eta \varpi \left|d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|{\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^p·{\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|}^q·{\left|d({\varkappa }_{2j},{\varkappa }^{*})\right|}^{1-p-q}.\hfill \end{array}$$

(22)

Taking the limit as $j\to \infty$ in above inequality (22), we obtain

$$\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|=0.$$

Therefore, ${\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}.$ Hence, ${\varkappa }^{*}$ is CFP of the mappings $\mathcal{M}$ and $\mathcal{H}.$ □

Example 5. 

Let $\mathsf{\Omega }=\left\{0,0.25,1\right\}\subset \mathbb{R}.$ Define $d:\mathsf{\Omega }\times \mathsf{\Omega }\to \mathbb{C}$ by

$$d(\varkappa ,\varsigma )=\left|\varkappa -\varsigma \right|+i\left|\varkappa -\varsigma \right|,$$

for all $\varkappa ,\varsigma \in \mathsf{\Omega }$ and $i^2=-1.$ Then, $\left(\mathsf{\Omega },d\right)$ is a complete C-VSMS with $\eta \ge 0.$ Define $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$ by

$$\mathcal{M}\left(0\right)=0,\mathcal{M}\left(0.25\right)=0,\mathcal{M}\left(1\right)=0.25$$

and $\mathcal{H}=\mathcal{M}$. Since

$$Fix\{\mathcal{M},\mathcal{H}\}=\left\{0\right\},$$

thus

$$\mathsf{\Omega }\backslash Fix\{\mathcal{M},\mathcal{H}\}=\left\{0.25,1\right\}.$$

Let

$$p=q={\displaystyle \frac{1}{4}},$$

then

$$p+q={\displaystyle \frac{1}{2}}<1,$$

and choose

$$\varpi ={\displaystyle \frac{1}{2}}\in [0,1).$$

We verify the inequality (16) for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M},\mathcal{H}\}=\left\{0.25,1\right\}.$

• Case 1. If $\varkappa =0.25$ and $\varsigma =1,$ then we have

  $$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )=0.25+0.25i,$$

  $$d(\varkappa ,\mathcal{M}\varkappa )=0.25+0.25i$$

  $$d(\varkappa ,\mathcal{H}\varkappa )=0.75+0.75i$$

  $$d(\varkappa ,\varsigma )=0.75+0.75i.$$

  Thus,

  $${\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[d(\varkappa ,\varsigma )\right]}^{1-p-q}⪰{\displaystyle \frac{1}{2}}\left(0.25+0.25i\right),$$

  for $\varpi ={\displaystyle \frac{1}{2}}.$ Hence,

  $$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[d(\varkappa ,\varsigma )\right]}^{1-p-q}.$$
• Case 2. If $\varkappa =1$ and $\varsigma =0.25.$ This case is symmetric, and the same computation yields

  $$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[d(\varkappa ,\varsigma )\right]}^{1-p-q}.$$

  Hence, all hypotheses of Theorem (2) are satisfied, and the mappings $\mathcal{M}$ and $\mathcal{H}$ admit a unique CFP in $\mathsf{\Omega },$ namely, $\varkappa =0.$

Corollary 4. 

Let $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $p,q\in (0,1)$ with $p+q<1$ such that

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{H}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[d(\varkappa ,\varsigma )\right]}^{1-p-q},$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M}$,$\mathcal{H}\}$. Then, $\mathcal{H}$ admits a FP in $\mathsf{\Omega }.$

Proof. 

Take $\mathcal{M}=\mathcal{H}$ in Theorem 2. □

Corollary 5. 

Let $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $\sigma \in (0,{\displaystyle \frac{1}{2}})$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^{\sigma }·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^{\sigma }·{\left[d(\varkappa ,\varsigma )\right]}^{1-2\sigma },$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M}$,$\mathcal{H}\}$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a CFP in $\mathsf{\Omega }.$

Proof. 

Take $p=q=\sigma$ in Theorem 2, with $2\sigma <1.$ □

3.3. Fixed-Point Results for Interpolative–Rational Hybrid Contractions

In this subsection, we develop CFP results for interpolative–rational hybrid contractions in the framework of C-VSMSs. The contractive condition under consideration blends the features of interpolative contractions, characterized by weighted power-type terms, with rational control functions involving nonlinear distance ratios. This hybrid approach provides a more flexible and unifying mechanism for controlling the behavior of mappings than either interpolative or rational contractions alone. Under appropriate parameter restrictions, we establish the existence of a CFP for a pair of self-mappings defined on a complete C-VSMS.

Theorem 3. 

Let $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $p,q,r\in (0,1)$ with $p+q+r=1$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[{\displaystyle \frac{d(\varkappa ,\mathcal{M}\varkappa )}{1+d(\varkappa ,\mathcal{M}\varkappa )}}d(\varkappa ,\varsigma )\right]}^r,$$

(23)

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M}$,$\mathcal{H}\}$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a CFP in $\mathsf{\Omega }.$

Proof. 

Let ${\varkappa }_0\in \mathsf{\Omega }$ be given. Define the sequence {${\varkappa }_j$} by

$${\varkappa }_{2j+1}=\mathcal{M}{\varkappa }_{2j}\mathrm{and}{\varkappa }_{2j+2}=\mathcal{H}{\varkappa }_{2j+1},$$

∀$j\in \mathbb{N}\cup \left\{0\right\}$. If for some j we have ${\varkappa }_{2j}\in Fix\left\{\mathcal{M},\mathcal{H}\right\},$ then

$${\varkappa }_{2j}=\mathcal{M}{\varkappa }_{2j}=\mathcal{H}{\varkappa }_{2j},$$

hence, ${\varkappa }_{2j}$ is a CFP. Thus, without loss of generality, we assume

$${\varkappa }_{2j}\notin Fix\left\{\mathcal{M},\mathcal{H}\right\}$$

$\forall j\in \mathbb{N}\cup \left\{0\right\}.$ By (23) with $\varkappa ={\varkappa }_{2j}$ and $\varsigma ={\varkappa }_{2j+1}$, we have

$$\begin{array}{ccc}\hfill d(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }_{2j+1})& ⪯& \varpi {\left[d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})\right]}^p·{\left[d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})}{1+d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})}}d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^r\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill d({\varkappa }_{2j+1},{\varkappa }_{2j+2})& ⪯& \varpi {\left[d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^p·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }_{2j},{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^r\hfill \end{array}$$

from which it follows that

$$\begin{array}{ccc}\hfill \left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|& \le & \varpi {\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q\hfill \\ & & ·{\left[\left|{\displaystyle \frac{d({\varkappa }_{2j},{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}\right|\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|\right]}^r\hfill \\ & \le & \varpi {\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q·{\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^r\hfill \\ & =& \varpi {\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^{p+r}·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q.\hfill \end{array}$$

Since $p+r=1-q,$ we obtain

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le \varpi {\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^{1-q}·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q$$

which implies

$${\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^{1-q}\le \varpi {\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^{1-q}.$$

Taking both sides to power ${\displaystyle \frac{1}{1-q}}$ (since $1-q>0$), we obtain

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le {\varpi }^{{\displaystyle \frac{1}{1-q}}}\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|,$$

(24)

∀$j\in \mathbb{N}\cup \left\{0\right\}$. Similarly, from (23) with $\varkappa ={\varkappa }_{2j+2}$ and $\varsigma ={\varkappa }_{2j+1}$, we have

$$\begin{array}{ccc}\hfill d(\mathcal{H}{\varkappa }_{2j+1},\mathcal{M}{\varkappa }_{2j+2})& =& d(\mathcal{M}{\varkappa }_{2j+2},\mathcal{H}{\varkappa }_{2j+1})⪯\varpi {\left[d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2})\right]}^p\hfill \\ & & ·{\left[d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})\right]}^q·{\left[{\displaystyle \frac{d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2})}{1+d({\varkappa }_{2j+2},\mathcal{M}{\varkappa }_{2j+2})}}d({\varkappa }_{2j+2},{\varkappa }_{2j+1})\right]}^r\hfill \\ & =& \varpi {\left[d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right]}^p·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^q·\hfill \\ & & {\left[{\displaystyle \frac{d({\varkappa }_{2j+2},{\varkappa }_{2j+3})}{1+d({\varkappa }_{2j+2},{\varkappa }_{2j+3})}}d({\varkappa }_{2j+2},{\varkappa }_{2j+1})\right]}^r\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|& \le & \varpi {\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q\hfill \\ & & ·{\left[\left|{\displaystyle \frac{d({\varkappa }_{2j+2},{\varkappa }_{2j+3})}{1+d({\varkappa }_{2j+2},{\varkappa }_{2j+3})}}\right|\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+1})\right|\right]}^r\hfill \\ & \le & \varpi {\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q·{\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+1})\right|}^r\hfill \\ & =& \varpi {\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^{q+r}.\hfill \end{array}$$

Since $q+r=1-p,$ we obtain

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le \varpi {\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^{1-p}$$

which further implies that

$${\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|}^{1-p}\le \varpi {\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^{1-p}.$$

Taking both sides to power ${\displaystyle \frac{1}{1-p}}$ (since $1-p>0$), we obtain

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le {\varpi }^{{\displaystyle \frac{1}{1-p}}}\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|.$$

(25)

Let

$${\mu }_1={\varpi }^{{\displaystyle \frac{1}{1-q}}}\mathrm{and}{\mu }_2={\varpi }^{{\displaystyle \frac{1}{1-p}}}.$$

Since $\varpi <1$ and $1-p,1-q\in (0,1),$ we have ${\mu }_1,{\mu }_2<1.$ Then, the inequalities (24) and (25) becomes

$$\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|\le {\mu }_1\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|,$$

(26)

and

$$\left|d({\varkappa }_{2j+2},{\varkappa }_{2j+3})\right|\le {\mu }_2\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|.$$

(27)

Let $\mu =max\left\{{\mu }_1,{\mu }_2\right\}<1.$ Then, from (26) and (27), we conclude

$$\left|d({\varkappa }_j,{\varkappa }_{j+1})\right|\le \mu \left|d({\varkappa }_{j-1},{\varkappa }_j)\right|$$

for all $j\in \mathbb{N}.$ Hence

$$\left|d({\varkappa }_j,{\varkappa }_{j+1})\right|\le \mu \left|d({\varkappa }_{j-1},{\varkappa }_j)\right|\le {\mu }^2\left|d({\varkappa }_{j-2},{\varkappa }_{j-1})\right|\le \cdots \le {\mu }^j\left|d({\varkappa }_0,{\varkappa }_1)\right|,$$

∀$\mathbb{N}.$ By using the same procedure as we have done in Theorem 1, we can prove that $\left\{{\varkappa }_j\right\}$ is a Cauchy sequence in $\mathsf{\Omega }$. From (23) with $\varkappa ={\varkappa }^{*}$ and $\varsigma ={\varkappa }_{2j+1}$, we have

$$\begin{array}{ccc}\hfill d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+d\left({\varkappa }_{2j+2},\mathcal{M}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)d\left({\varkappa }_{2j+2},\mathcal{M}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+d\left(\mathcal{H}{\varkappa }_{2j+1},\mathcal{M}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)d\left(\mathcal{H}{\varkappa }_{2j+1},\mathcal{M}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+d\left(\mathcal{M}{\varkappa }^{*},\mathcal{H}{\varkappa }_{2j+1}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)d\left(\mathcal{M}{\varkappa }^{*},\mathcal{H}{\varkappa }_{2j+1}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi {\left[d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right]}^p·{\left[d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}{1+d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}}d({\varkappa }^{*},{\varkappa }_{2j+1})\right]}^r\hfill \\ & & +\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right){\left[d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right]}^p·{\left[d({\varkappa }_{2j+1},\mathcal{H}{\varkappa }_{2j+1})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}{1+d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}}d({\varkappa }^{*},{\varkappa }_{2j+1})\right]}^r\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi {\left[d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right]}^p·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}{1+d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}}d({\varkappa }^{*},{\varkappa }_{2j+1})\right]}^r\hfill \\ & & +\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right){\left[d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right]}^p·{\left[d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}{1+d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}}d({\varkappa }^{*},{\varkappa }_{2j+1})\right]}^r\hfill \end{array}$$

which implies

$$\begin{array}{ccc}\hfill \left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|& \le & \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|+\varpi {\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q\hfill \\ & & ·{\left|{\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}{1+d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}}\right|}^r{\left|d({\varkappa }^{*},{\varkappa }_{2j+1})\right|}^r\hfill \\ & & +\eta \varpi \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|{\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|}^p·{\left|d({\varkappa }_{2j+1},{\varkappa }_{2j+2})\right|}^q\hfill \\ & & ·{\left|{\displaystyle \frac{d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}{1+d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})}}\right|}^r{\left|d({\varkappa }^{*},{\varkappa }_{2j+1})\right|}^r.\hfill \end{array}$$

(28)

Taking the limit as $j\to +\infty$ in inequality (28), we obtain

$$\left|d({\varkappa }^{*},\mathcal{M}{\varkappa }^{*})\right|=0,$$

that is ${\varkappa }^{*}=\mathcal{M}{\varkappa }^{*}.$ Similarly, from (23) with $\varkappa ={\varkappa }_{2j}$ and $\varsigma ={\varkappa }^{*}$, we have

$$\begin{array}{ccc}\hfill d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})& ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)+d\left({\varkappa }_{2j+1},\mathcal{H}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)d\left({\varkappa }_{2j+1},\mathcal{H}{\varkappa }^{*}\right)\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)+d\left(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }^{*}\right)+\eta d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)d\left(\mathcal{M}{\varkappa }_{2j},\mathcal{H}{\varkappa }^{*}\right)\hfill \\ & ⪯& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi {\left[d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})\right]}^p·{\left[d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})}{1+d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})}}d({\varkappa }_{2j},{\varkappa }^{*})\right]}^r\hfill \\ & & +\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right){\left[d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})\right]}^p·{\left[d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})}{1+d({\varkappa }_{2j},\mathcal{M}{\varkappa }_{2j})}}d({\varkappa }_{2j},{\varkappa }^{*})\right]}^r\hfill \\ & =& d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)+\varpi {\left[d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^p·{\left[d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }_{2j},{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}d({\varkappa }_{2j},{\varkappa }^{*})\right]}^r\hfill \\ & & +\eta \varpi d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right){\left[d({\varkappa }_{2j},{\varkappa }_{2j+1})\right]}^p·{\left[d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right]}^q\hfill \\ & & ·{\left[{\displaystyle \frac{d({\varkappa }_{2j},{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}d({\varkappa }_{2j},{\varkappa }^{*})\right]}^r\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|& \le & \left|d\left({\varkappa }^{*},{\varkappa }_{2j+2}\right)\right|+\varpi {\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^p·{\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|}^q\hfill \\ & & ·{\left|{\displaystyle \frac{d({\varkappa }_{2j},{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}\right|}^r{\left|d({\varkappa }_{2j},{\varkappa }^{*})\right|}^r\hfill \\ & & +\eta \varpi \left|d\left({\varkappa }^{*},{\varkappa }_{2j+1}\right)\right|{\left|d({\varkappa }_{2j},{\varkappa }_{2j+1})\right|}^p·{\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|}^q\hfill \\ & & ·{\left|{\displaystyle \frac{d({\varkappa }_{2j},{\varkappa }_{2j+1})}{1+d({\varkappa }_{2j},{\varkappa }_{2j+1})}}\right|}^r{\left|d({\varkappa }_{2j},{\varkappa }^{*})\right|}^r.\hfill \end{array}$$

(29)

Taking the limit as $j\to \infty$ in above inequality (29), we obtain

$$\left|d({\varkappa }^{*},\mathcal{H}{\varkappa }^{*})\right|=0.$$

Therefore, ${\varkappa }^{*}=\mathcal{H}{\varkappa }^{*}.$ Hence, ${\varkappa }^{*}$ is CFP of the mappings $\mathcal{M}$ and $\mathcal{H}.$ □

Corollary 6. 

Let $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $p,q,r\in (0,1)$ with $p+q+r=1$ such that

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[{\displaystyle \frac{d(\varkappa ,\mathcal{H}\varkappa )}{1+d(\varkappa ,\mathcal{H}\varkappa )}}d(\varkappa ,\varsigma )\right]}^r,$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\left(\mathcal{H}\right)$. Then, $\mathcal{H}$ admit a FP in $\mathsf{\Omega }.$

Proof. 

Take $\mathcal{M}=\mathcal{H}$ in Theorem 3. □

Corollary 7. 

Let $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $r\in (0,1)$ such that

$$\begin{array}{ccc}\hfill d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )& ⪯& \varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )·d(\varsigma ,\mathcal{H}\varsigma )\right]}^{{\displaystyle \frac{1-r}{2}}}\hfill \\ & & ·{\left[{\displaystyle \frac{d(\varkappa ,\mathcal{M}\varkappa )}{1+d(\varkappa ,\mathcal{M}\varkappa )}}d(\varkappa ,\varsigma )\right]}^r,\hfill \end{array}$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M}$,$\mathcal{H}\}$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a CFP in $\mathsf{\Omega }.$

Proof. 

Take $p=q={\displaystyle \frac{1-r}{2}}$ in Theorem 3. □

4. Consequences of Main Results

In this section, we present the consequences of our main results by first deriving FP theorems in C-VMSs, followed by their extensions to SMSs.

4.1. Fixed-Point Theorems in Complex-Valued Metric Spaces

Setting $\eta =0$ in Definition 3 reduces the concept of a C-VSMS to that of a C-VMS. This simplification enables us to establish the following results in C-VMSs. Throughout this subsection, we assume that $(\mathsf{\Omega },d)$ is a complete C-VMS.

Corollary 8. 

Let A and B be nonempty closed subsets of Ω. Consider the mappings $\mathcal{M},\mathcal{H}:A\cup B\to A\cup B$ satisfying

$$\mathcal{M}\left(A\right)\subseteq B,\mathcal{M}\left(B\right)\subseteq A\mathrm{and}\mathcal{H}\left(A\right)\subseteq B,\mathcal{H}\left(B\right)\subseteq A.$$

Suppose that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{M}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{M}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

holds for all $\varkappa \in A,\varsigma \in B$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a unique CFP in $A\cap B.$

Proof. 

Take $\eta =0$ in Theorem 1. □

Corollary 9. 

Let let A and B be nonempty closed subsets of Ω. Consider the mapping $\mathcal{H}:A\cup B\to A\cup B$ satisfying

$$\mathcal{H}\left(A\right)\subseteq B,\mathcal{H}\left(B\right)\subseteq A.$$

Suppose that there is a constant $\varpi \in [0,1)$ fulfilling

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{H}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{H}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

∀$\varkappa \in A,\varsigma \in B$. Then, $\mathcal{H}$ admits a unique FP in $A\cap B.$

Proof. 

Take $\mathcal{M}$ = $\mathcal{H}$ in above corollary. □

Now, we derive the following result, which recovers as a special case one of the main results of Hussain et al. [11].

Corollary 10 

([11]). Let $\mathcal{M}$, $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Assume that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{M}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{M}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

for all $\varkappa ,\varsigma \in \mathsf{\Omega }$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a unique CFP in $\mathsf{\Omega }.$

Proof. 

Take $A=B=\mathsf{\Omega }$ in Corollary 8. □

Corollary 11. 

Let $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Assume that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{H}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{H}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

for all $\varkappa ,\varsigma \in \mathsf{\Omega }$. Then, $\mathcal{H}$ admits a unique FP in $\mathsf{\Omega }.$

Proof. 

Take $\mathcal{M}=\mathcal{H}$ in above corollary. □

Corollary 12. 

Let $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $p,q\in (0,1)$ with $p+q<1$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[d(\varkappa ,\varsigma )\right]}^{1-p-q},$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M}$,$\mathcal{H}\}$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a CFP in $\mathsf{\Omega }.$

Corollary 13. 

Let $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $p,q\in (0,1)$ with $p+q<1$ such that

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{H}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[d(\varkappa ,\varsigma )\right]}^{1-p-q},$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\left(\mathcal{H}\right)$. Then, $\mathcal{H}$ has a FP in $\mathsf{\Omega }.$

Proof. 

Take $\mathcal{M}=\mathcal{H}$ in Corollary 12. □

Corollary 14. 

Let $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $\sigma \in (0,{\displaystyle \frac{1}{2}})$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )⪯\varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^{\sigma }·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^{\sigma }·{\left[d(\varkappa ,\varsigma )\right]}^{1-2\sigma },$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M}$,$\mathcal{H}\}$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a CFP in $\mathsf{\Omega }.$

Proof. 

Take $p=q=\sigma$ in Corollary 12 with $2\sigma <1.$ □

4.2. Fixed-Point Outcomes in Suprametric Spaces

If we restrict the set of complex numbers $\mathbb{C}$ to the set of real numbers in Definition 3, that is, by setting the imaginary part to 0, the concept of a C-VSMS reduced to that of an SMS. This simplification allows us to deduce the following outcomes in SMSs. Throughout this subsection, we assume that $(\mathsf{\Omega },d)$ is a complete SMS.

Corollary 15. 

Let A and B be nonempty closed subsets of Ω. Consider the mappings $\mathcal{M},\mathcal{H}:A\cup B\to A\cup B$ satisfying

$$\mathcal{M}\left(A\right)\subseteq B,\mathcal{M}\left(B\right)\subseteq A\mathrm{and}\mathcal{H}\left(A\right)\subseteq B,\mathcal{H}\left(B\right)\subseteq A.$$

Suppose that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )\le \varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{M}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{M}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

holds for all $\varkappa \in A,\varsigma \in B$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a unique CFP in $A\cap B.$

Proof. 

Take $\eta =0$ in Theorem 1. □

Corollary 16. 

Let A and B be nonempty closed subsets of Ω. Consider the mapping $\mathcal{H}:A\cup B\to A\cup B$ satisfying

$$\mathcal{H}\left(A\right)\subseteq B,\mathcal{H}\left(B\right)\subseteq A.$$

Suppose that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )\le \varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{H}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{H}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

holds for all $\varkappa \in A,\varsigma \in B$. Then, $\mathcal{H}$ admits a unique FP in $A\cap B.$

Proof. 

Take $\mathcal{M}$ = $\mathcal{H}$ in above corollary. □

Corollary 17. 

Let $\mathcal{M}$, $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Assume that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )\le \varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{M}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{M}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

for all $\varkappa ,\varsigma \in \mathsf{\Omega }$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a unique CFP in $\mathsf{\Omega }.$

Proof. 

Take $A=B=\mathsf{\Omega }$ in Corollary 15. □

Corollary 18. 

Let $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Assume that there is a constant $\varpi \in [0,1)$ such that

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )\le \varpi \mathsf{\Xi }(\varkappa ,\varsigma ),$$

where

$$\mathsf{\Xi }(\varkappa ,\varsigma )\in \left\{d(\varkappa ,\varsigma ),d(\varkappa ,\mathcal{H}\varkappa ),d(\varsigma ,\mathcal{H}\varsigma ),{\displaystyle \frac{d(\varkappa ,\mathcal{H}\varkappa )d(\varsigma ,\mathcal{H}\varsigma )}{1+d(\varkappa ,\varsigma )}}\right\},$$

for all $\varkappa ,\varsigma \in \mathsf{\Omega }$. Then, $\mathcal{H}$ admits a unique FP in $\mathsf{\Omega }.$

Proof. 

Take $\mathcal{M}=\mathcal{H}$ in the above corollary. □

Corollary 19. 

Let $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $p,q\in (0,1)$ with $p+q<1$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )\le \varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[d(\varkappa ,\varsigma )\right]}^{1-p-q},$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M}$,$\mathcal{H}\}$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a CFP in $\mathsf{\Omega }.$

Corollary 20. 

Let $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $p,q\in (0,1)$ with $p+q<1$ such that

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )\le \varpi {\left[d(\varkappa ,\mathcal{H}\varkappa )\right]}^p·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^q·{\left[d(\varkappa ,\varsigma )\right]}^{1-p-q},$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\left(\mathcal{H}\right)$. Then $\mathcal{H}$ has a FP in $\mathsf{\Omega }.$

Proof. 

Take $\mathcal{M}=\mathcal{H}$ in Corollary 19. □

Corollary 21. 

Let $\mathcal{M},\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$. Suppose that there exist constants $\varpi \in [0,1)$ and $\sigma \in (0,{\displaystyle \frac{1}{2}})$ such that

$$d(\mathcal{M}\varkappa ,\mathcal{H}\varsigma )\le \varpi {\left[d(\varkappa ,\mathcal{M}\varkappa )\right]}^{\sigma }·{\left[d(\varsigma ,\mathcal{H}\varsigma )\right]}^{\sigma }·{\left[d(\varkappa ,\varsigma )\right]}^{1-2\sigma },$$

holds for all $\varkappa ,\varsigma \in \mathsf{\Omega }\backslash Fix\{\mathcal{M}$,$\mathcal{H}\}$. Then, $\mathcal{M}$, $\mathcal{H}$ admit a CFP in $\mathsf{\Omega }.$

Proof. 

Take $p=q=\sigma$ in Corollary 19 with $2\alpha <1.$ □

5. Applications

In this section, we illustrate the effectiveness of the established CFP results by applying them to a class of nonlinear fractional differential equations with nonlocal boundary conditions. The proposed applications demonstrate that the newly introduced cyclic contractive conditions in C-VSMs provide a powerful and flexible tool for analyzing nonlinear problems that cannot be handled by classical metric techniques.

Consider the nonlinear fractional boundary value problem

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

(30)

where $1<\alpha <2$, $D_t^{\alpha }\left(\varkappa \left(t\right)\right)$ denotes the Caputo fractional derivative of order $\alpha ,$ and $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a given continuous function. Using properties of the Caputo derivative, the solution can be written as

$$\varkappa \left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-s\right)}^{\alpha -1}g(s,\varkappa \left(s\right))ds+C_0+C_{1}t.$$

From $\varkappa \left(0\right)=0,$ we get $C_0=0.$ Thus,

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

Now using $\varkappa \left(1\right)={\displaystyle \underset{0}{\overset{1}{\int }}}\varkappa \left(s\right)ds,$ we obtain

$$C_1=2\left[{\displaystyle \underset{0}{\overset{1}{\int }}}{\displaystyle \frac{{\left(t-s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}}g(s,\varkappa \left(s\right))-{\displaystyle \underset{0}{\overset{1}{\int }}}{\displaystyle \underset{0}{\overset{s}{\int }}}{\displaystyle \frac{{\left(s-\tau \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}}g(\tau ,\varkappa \left(\tau \right))d\tau ds\right].$$

Substituting this value into the expression for $\varkappa \left(t\right),$ we obtain the equivalent integral equation:

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

Therefore, the given boundary value problem is equivalent to the nonlinear integral equation:

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

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

$$G(t,s)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\left(t-s\right)}^{\alpha -1}+{\displaystyle \frac{2t}{\mathsf{\Gamma }\left(\alpha \right)}}\left({\displaystyle \frac{{\left(1-s\right)}^{\alpha }}{\alpha }}-{\left(1-s\right)}^{\alpha -1}\right),0\le s\le t\le 1,\\ {\displaystyle \frac{2t}{\mathsf{\Gamma }\left(\alpha \right)}}\left({\displaystyle \frac{{\left(1-s\right)}^{\alpha }}{\alpha }}-{\left(1-s\right)}^{\alpha -1}\right),0\le t\le s\le 1.\end{array}\right.$$

Based on the preceding analysis, we state the following theorem establishing the existence and uniqueness of the solution.

Theorem 4. 

Let $\mathsf{\Omega }=C\left(\right[0,1],\mathbb{R})$ be the space of real-valued continuous functions on $[0,1]$ equipped with the metric

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

Then d satisfies all the conditions of Definition 3 for any $\eta \ge 0$; hence, ($\mathsf{\Omega },d$) form a complete C-VSMS. Consider the closed subsets

$$A=\left\{\varkappa \in \mathsf{\Omega }:\varkappa \left(0\right)=0\right\}$$

and

$$B=\left\{\varkappa \in \mathsf{\Omega }:\varkappa \left(1\right)={\displaystyle \underset{0}{\overset{1}{\int }}}\varkappa \left(s\right)ds\right\}.$$

Define the integral operator

$$\left(\mathcal{H}\varkappa \right)\left(t\right)={\displaystyle \underset{0}{\overset{1}{\int }}}G(t,s)g(s,\varkappa \left(s\right))ds.$$

Assume that the following conditions hold. (i) The function $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous and satisfies the Lipschitz condition

$$\left|g\left(t,\varkappa \right)-g\left(t,\varsigma \right)\right|\le L\left|\varkappa -\varsigma \right|,\forall t\in [0,1],\forall \varkappa ,\varsigma \in \mathbb{C}$$

for some constant $L\ge 0.$ (ii) Let

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

and assume that $\varpi =LM<1.$ Then, the fractional boundary value problem (30) has a unique solution ${\varkappa }^{*}\in A\cap B$.

Proof. 

We first verify that

$$\mathcal{H}\left(A\right)\subseteq B,\mathcal{H}\left(B\right)\subseteq A.$$

For any $\varkappa \in A,$ we have

$$\begin{array}{ccc}\hfill \left(\mathcal{H}\varkappa \right)\left(1\right)& =& {\displaystyle \underset{0}{\overset{1}{\int }}}G\left(1,s\right)g\left(s,\varkappa \left(s\right)\right)ds\hfill \\ & =& {\displaystyle \underset{0}{\overset{1}{\int }}}{\displaystyle \underset{0}{\overset{1}{\int }}}G\left(s,\tau \right)g\left(\tau ,\varkappa \left(\tau \right)\right)d\tau ds\hfill \\ & =& {\displaystyle \underset{0}{\overset{1}{\int }}}\left(\mathcal{H}\varkappa \right)\left(s\right)ds,\hfill \end{array}$$

so $\mathcal{H}\varkappa \in B.$ Hence, $\mathcal{H}\left(A\right)\subseteq B.$ Moreover, for any $\varkappa \in B,$

$$\left(\mathcal{H}\varkappa \right)\left(0\right)={\displaystyle \underset{0}{\overset{1}{\int }}}G\left(0,s\right)g\left(s,\varkappa \left(s\right)\right)ds=0,$$

so $\mathcal{H}\varkappa \in A.$ Hence, $\mathcal{H}\left(B\right)\subseteq A.$ For $\varkappa ,\varsigma \in C[0,1],$ we have

$$\begin{array}{ccc}\hfill \left|\left(\mathcal{H}\varkappa \right)\left(t\right)-\left(\mathcal{H}\varsigma \right)\left(t\right)\right|& =& \left|{\displaystyle \underset{0}{\overset{1}{\int }}}G\left(t,s\right)g\left(s,\varkappa \left(s\right)\right)ds-{\displaystyle \underset{0}{\overset{1}{\int }}}G\left(t,s\right)g\left(s,\varsigma \left(s\right)\right)ds\right|\hfill \\ & \le & {\displaystyle \underset{0}{\overset{1}{\int }}}\left|G\left(t,s\right)\right|g\left(s,\varkappa \left(s\right)\right)ds-g\left(s,\varsigma \left(s\right)\right)ds.\hfill \end{array}$$

By the assumption (i), we have

$$\left|\left(\mathcal{H}\varkappa \right)\left(t\right)-\left(\mathcal{H}\varsigma \right)\left(t\right)\right|\le L{∥\varkappa -\varsigma ∥}_{\infty }{\displaystyle \underset{0}{\overset{1}{\int }}}\left|G(t,s)\right|ds.$$

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

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

By hypothesis (ii), we have

$$\varpi =LM<1.$$

Thus,

$${∥\mathcal{H}\varkappa -\mathcal{H}\varsigma ∥}_{\infty }\le \varpi {∥\varkappa -\varsigma ∥}_{\infty }.$$

Now, by definition of d

$$d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )={∥\mathcal{H}\varkappa -\mathcal{H}\varsigma ∥}_{\infty }\left(1+i\right)\le \varpi {∥\varkappa -\varsigma ∥}_{\infty }\left(1+i\right)=\varpi d(\varkappa ,\varsigma ).$$

Thus, all the conditions of Corollary 1 are satisfied and the operator has unique ${\varkappa }^{*}\in C[0,1]$ which is a solution of the boundary-value problem (30). □

Example 6. 

Consider the following nonlinear Caputo fractional differential equation with nonlocal boundary conditions

$$\left\{\begin{array}{c}{D}_t^{\alpha }\left(\varkappa \left(t\right)\right)={\displaystyle \frac{1}{4}}\varkappa \left(t\right)+t,t\in (0,1)\\ \varkappa \left(0\right)=0,\varkappa \left(1\right)={\displaystyle \underset{0}{\overset{1}{\int }}}\varkappa \left(s\right)ds,\end{array}\right.$$

(31)

where $\alpha ={\displaystyle \frac{3}{2}}$ (so $1<\alpha <2$). This problem is equivalent to the Fredholm integral equation

$$\varkappa \left(t\right)={\displaystyle \underset{0}{\overset{1}{\int }}}G(t,s)\left({\displaystyle \frac{1}{4}}\varkappa \left(s\right)+s\right)ds,$$

where $G(t,s)$ denotes the corresponding Green’s function. Let

$$\mathsf{\Omega }=C\left(\right[0,1],\mathbb{R})$$

be the space of real-valued continuous functions. Define

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

which satisfies all the condition of Definition 3 with any $\eta \ge 0.$ Then ($\mathsf{\Omega },d$) is a complete C-VSMS. Following the cyclic structure of Corollary 1, let

$$A=\left\{\varkappa \in \mathsf{\Omega }:\varkappa \left(0\right)=0\right\}$$

and

$$B=\left\{\varkappa \in \mathsf{\Omega }:\varkappa \left(1\right)={\displaystyle \underset{0}{\overset{1}{\int }}}\varkappa \left(s\right)ds\right\}.$$

Clearly, both A and B are nonempty closed subsets of $\mathsf{\Omega }.$ Define the operator $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$ by

$$\left(\mathcal{H}\varkappa \right)\left(t\right)={\displaystyle \underset{0}{\overset{1}{\int }}}G(t,s)\left({\displaystyle \frac{1}{4}}\varkappa \left(s\right)+s\right)ds.$$

For any $\varkappa \in A,$ we have $\left(\mathcal{H}\varkappa \right)\left(0\right)=0$ and

$$\begin{array}{ccc}\hfill \left(\mathcal{H}\varkappa \right)\left(1\right)& =& {\displaystyle \underset{0}{\overset{1}{\int }}}G(1,s)g\left(s,\varkappa \left(s\right)\right)ds\hfill \\ & =& {\displaystyle \underset{0}{\overset{1}{\int }}}{\displaystyle \underset{0}{\overset{1}{\int }}}G(s,\tau )g\left(\tau ,\varkappa \left(\tau \right)\right)d\tau ds\hfill \\ & =& {\displaystyle \underset{0}{\overset{1}{\int }}}\left(\mathcal{H}\varkappa \right)\left(s\right),\hfill \end{array}$$

so $\mathcal{H}\varkappa \in B.$ Hence, $\mathcal{H}\left(A\right)\subseteq B.$ Similarly, for any $\varkappa \in B,$

$$\left(\mathcal{H}\varkappa \right)\left(0\right)=0,$$

so $\mathcal{H}\varkappa \in A$. Thus, $\mathcal{H}\left(B\right)\subseteq A.$ For any $t\in [0,1]$ and any $\varkappa ,\varsigma \in \mathbb{R},$ we have

$$\begin{array}{ccc}\hfill |g\left(t,\varkappa \right)-g\left(t,\varsigma \right)|& =& |{\displaystyle \frac{1}{4}}\varkappa -{\displaystyle \frac{1}{4}}\varsigma |\hfill \\ & =& {\displaystyle \frac{1}{4}}|\varkappa -\varsigma |.\hfill \end{array}$$

Thus, condition (i) holds with Lipschitz constant $L={\displaystyle \frac{1}{4}}.$ For $\alpha ={\displaystyle \frac{3}{2}},$ we have

$$\mathsf{\Gamma }\left({\displaystyle \frac{3}{2}}\right)={\displaystyle \frac{\sqrt{\pi }}{2}}\approx 0.8862.$$

A direct calculation (or standard estimate from the literature) yields

$${\displaystyle \underset{0}{\overset{1}{\int }}}\left|G\left(t,s\right)\right|ds\le 2.$$

Thus,

$$M=\underset{t\in [0,1]}{sup}{\displaystyle \underset{0}{\overset{1}{\int }}}\left|G\left(t,s\right)\right|ds=2.$$

And

$$\varpi =LM={\displaystyle \frac{1}{4}}\times 2={\displaystyle \frac{1}{2}}<1.$$

Thus, condition (ii) is satisfied. Now for any $\varkappa \in A$ and $\varsigma \in B.$ Then,

$$\begin{array}{ccc}\hfill \left|\left(\mathcal{H}\varkappa \right)\left(t\right)-\left(\mathcal{H}\varsigma \right)\left(t\right)\right|& \le & {\displaystyle \underset{0}{\overset{1}{\int }}}|G\left(t,s\right)|·{\displaystyle \frac{1}{4}}|\varkappa \left(s\right)-\varsigma \left(s\right)|ds\hfill \\ & =& {\displaystyle \frac{1}{4}}{\displaystyle \underset{0}{\overset{1}{\int }}}|G\left(t,s\right)|·|\varkappa \left(s\right)-\varsigma \left(s\right)|ds\hfill \\ & \le & {\displaystyle \frac{1}{4}}{∥\varkappa -\varsigma ∥}_{\infty }{\displaystyle \underset{0}{\overset{1}{\int }}}|G\left(t,s\right)|ds.\hfill \end{array}$$

Taking supremum over t, we obtain

$$\begin{array}{ccc}\hfill {∥\mathcal{H}\varkappa -\mathcal{H}\varsigma ∥}_{\infty }& \le & {\displaystyle \frac{1}{4}}{∥\varkappa -\varsigma ∥}_{\infty }·\left(\underset{t\in [0,1]}{sup}{\displaystyle \underset{0}{\overset{1}{\int }}}|G(t,s)|ds\right)\hfill \\ & \le & {\displaystyle \frac{1}{4}}·2·{∥\varkappa -\varsigma ∥}_{\infty }\hfill \\ & =& {\displaystyle \frac{1}{2}}{∥\varkappa -\varsigma ∥}_{\infty }.\hfill \end{array}$$

In view of the definition of $d,$ we have

$$\begin{array}{ccc}\hfill d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )& =& {∥\mathcal{H}\varkappa -\mathcal{H}\varsigma ∥}_{\infty }\left(1+i\right)⪯{\displaystyle \frac{1}{2}}{∥\varkappa -\varsigma ∥}_{\infty }\left(1+i\right)\hfill \\ & =& \varpi d(\varkappa ,\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ).\hfill \end{array}$$

All hypotheses of Corollary 1 are satisfied and the operator has unique ${\varkappa }^{*}\in A\cap B,$ which is the unique solution of the fractional boundary value problem (31).

Example 7. 

Consider the following nonlinear Caputo fractional differential equation with nonlocal boundary conditions

$$\left\{\begin{array}{c}{D}_t^{\alpha }\left(\varkappa \left(t\right)\right)={\displaystyle \frac{1}{8}}sin\left(\varkappa \left(t\right)\right)+{\displaystyle \frac{t^2}{1+t^2}},t\in (0,1)\\ \varkappa \left(0\right)=0,\varkappa \left(1\right)={\displaystyle \underset{0}{\overset{1}{\int }}}\varkappa \left(s\right)ds,\end{array}\right.$$

(32)

where $\alpha ={\displaystyle \frac{3}{2}}$ (so $1<\alpha <2$), and the nonlinear function is

$$g\left(t,\varkappa \right)={\displaystyle \frac{1}{8}}sin\left(\varkappa \left(t\right)\right)+{\displaystyle \frac{t^2}{1+t^2}}.$$

This problem is equivalent to the Fredholm integral equation

$$\begin{array}{ccc}\hfill \varkappa \left(t\right)& =& \varkappa \left(0\right)+{\displaystyle \underset{0}{\overset{1}{\int }}}G(t,s)\left({\displaystyle \frac{1}{8}}sin\left(\varkappa \left(s\right)\right)+{\displaystyle \frac{s^2}{1+s^2}}\right)ds\hfill \\ & =& {\displaystyle \underset{0}{\overset{1}{\int }}}G(t,s)\left({\displaystyle \frac{1}{8}}sin\left(\varkappa \left(s\right)\right)+{\displaystyle \frac{s^2}{1+s^2}}\right)ds,\hfill \end{array}$$

where $G(t,s)$ is the Green’s function for the given boundary conditions. Let $\mathsf{\Omega }=C\left(\right[0,1],\mathbb{R})$ be the space of real-valued continuous functions. Define

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

which satisfies all the condition of Definition 3 with any $\eta \ge 0.$ Thenm ($\mathsf{\Omega },d$) is a complete C-VSMS. Following the cyclic structure of Corollary 1, let

$$A=\left\{\varkappa \in \mathsf{\Omega }:\varkappa \left(0\right)=0\right\}$$

and

$$B=\left\{\varkappa \in \mathsf{\Omega }:\varkappa \left(1\right)={\displaystyle \underset{0}{\overset{1}{\int }}}\varkappa \left(s\right)ds\right\}.$$

Both A and B are nonempty closed subsets of $\mathsf{\Omega }.$ Define the operator $\mathcal{H}:\mathsf{\Omega }\to \mathsf{\Omega }$ by

$$\left(\mathcal{H}\varkappa \right)\left(t\right)={\displaystyle \underset{0}{\overset{1}{\int }}}G(t,s)\left({\displaystyle \frac{1}{8}}sin\left(\varkappa \left(s\right)\right)+{\displaystyle \frac{s^2}{1+s^2}}\right)ds.$$

For any $\varkappa \in A,$ we have $\left(\mathcal{H}\varkappa \right)\left(0\right)=0.$ Also,

$$\begin{array}{ccc}\hfill \left(\mathcal{H}\varkappa \right)\left(1\right)& =& {\displaystyle \underset{0}{\overset{1}{\int }}}G(1,s)g\left(s,\varkappa \left(s\right)\right)ds\hfill \\ & =& {\displaystyle \underset{0}{\overset{1}{\int }}}{\displaystyle \underset{0}{\overset{1}{\int }}}G(s,\tau )g\left(\tau ,\varkappa \left(\tau \right)\right)d\tau ds\hfill \\ & =& {\displaystyle \underset{0}{\overset{1}{\int }}}\left(\mathcal{H}\varkappa \right)\left(s\right),\hfill \end{array}$$

so $\mathcal{H}\varkappa \in B.$ Hence, $\mathcal{H}\left(A\right)\subseteq B.$ Similarly, for any $\varkappa \in B,$

$$\left(\mathcal{H}\varkappa \right)\left(0\right)=0,$$

so $\mathcal{H}\varkappa \in A$. Thus, $\mathcal{H}\left(B\right)\subseteq A.$ For any $t\in [0,1]$ and any $\varkappa ,\varsigma \in \mathbb{R},$ we have

$$\begin{array}{ccc}\hfill |g\left(t,\varkappa \right)-g\left(t,\varsigma \right)|& =& |{\displaystyle \frac{1}{8}}sin\left(\varkappa \right)-{\displaystyle \frac{1}{8}}sin\left(\varsigma \right)|\hfill \\ & \le & {\displaystyle \frac{1}{8}}|\varkappa -\varsigma |,\hfill \end{array}$$

since $\left|sin\left(\varkappa \right)-sin\left(\varsigma \right)\right|\le \left|\varkappa -\varsigma \right|.$ Thus, condition (i) holds with Lipschitz constant $L={\displaystyle \frac{1}{8}}.$ For $\alpha ={\displaystyle \frac{3}{2}},$ we have

$$\mathsf{\Gamma }\left({\displaystyle \frac{3}{2}}\right)={\displaystyle \frac{\sqrt{\pi }}{2}}\approx 0.8862.$$

A straightforward computation shows that

$${\displaystyle \underset{0}{\overset{1}{\int }}}\left|G\left(t,s\right)\right|ds\le 2.$$

Thus,

$$M=\underset{t\in [0,1]}{sup}{\displaystyle \underset{0}{\overset{1}{\int }}}\left|G\left(t,s\right)\right|ds=2.$$

Now,

$$\varpi =LM={\displaystyle \frac{1}{8}}\times 2={\displaystyle \frac{1}{4}}<1.$$

Thus, condition (ii) is satisfied. Now for any $\varkappa \in A$ and $\varsigma \in B.$ Then,

$$\begin{array}{ccc}\hfill \left|\left(\mathcal{H}\varkappa \right)\left(t\right)-\left(\mathcal{H}\varsigma \right)\left(t\right)\right|& \le & {\displaystyle \underset{0}{\overset{1}{\int }}}|G\left(t,s\right)|·{\displaystyle \frac{1}{8}}|sin\left(\varkappa \left(s\right)\right)-sin\left(\varsigma \left(s\right)\right)|ds\hfill \\ & \le & {\displaystyle \frac{1}{8}}{\displaystyle \underset{0}{\overset{1}{\int }}}|G\left(t,s\right)|·|\varkappa \left(s\right)-\varsigma \left(s\right)|ds\hfill \\ & \le & {\displaystyle \frac{1}{8}}{∥\varkappa -\varsigma ∥}_{\infty }{\displaystyle \underset{0}{\overset{1}{\int }}}|G\left(t,s\right)|ds.\hfill \end{array}$$

Taking supremum over t, we obtain

$$\begin{array}{ccc}\hfill {∥\mathcal{H}\varkappa -\mathcal{H}\varsigma ∥}_{\infty }& \le & {\displaystyle \frac{1}{8}}{∥\varkappa -\varsigma ∥}_{\infty }·\left(\underset{t\in [0,1]}{sup}{\displaystyle \underset{0}{\overset{1}{\int }}}|G(t,s)|ds\right)\hfill \\ & \le & {\displaystyle \frac{1}{8}}·2·{∥\varkappa -\varsigma ∥}_{\infty }\hfill \\ & =& {\displaystyle \frac{1}{4}}{∥\varkappa -\varsigma ∥}_{\infty }.\hfill \end{array}$$

Now, by definition of $d,$ we have

$$\begin{array}{ccc}\hfill d(\mathcal{H}\varkappa ,\mathcal{H}\varsigma )& =& {∥\mathcal{H}\varkappa -\mathcal{H}\varsigma ∥}_{\infty }\left(1+i\right)⪯{\displaystyle \frac{1}{4}}{∥\varkappa -\varsigma ∥}_{\infty }\left(1+i\right)\hfill \\ & =& \varpi d(\varkappa ,\varsigma )⪯\varpi \mathsf{\Xi }(\varkappa ,\varsigma ).\hfill \end{array}$$

All the assumptions of Corollary 1 are fulfilled; therefore, the operator admits a unique FP ${\varkappa }^{*}\in A\cap B,$ which yields the unique solution of problem (32).

6. Conclusions

In this work, we explored the framework of C-VSMSs and developed some CFP theorems under new cyclic and interpolative rational contractive conditions. The proposed contractions extend and unify a variety of existing findings in C-VMSs and significantly broaden the scope of FP theory in this setting.

To demonstrate the effectiveness and originality of the obtained results, a non-trivial illustrative example was presented, verifying the applicability of the established theorems. Moreover, as a concrete application, the developed FP results were employed to investigate the existence and uniqueness of solutions for a class of nonlinear fractional ordinary differential equations with nonlocal boundary conditions. This application confirms that the introduced cyclic framework is well suited for studying fractional models arising in applied sciences.

7. Open Problems and Future Directions

Motivated by the results obtained in this work, several interesting research directions remain open for further investigation:

(1) It would be worthwhile to introduce and study a more general framework of complex-valued perturbed SMSs, extending the current structure of C-VSMSs. In such a setting, one may investigate whether the established cyclic and interpolative rational contractive conditions can be adapted or further generalized.

(2) Another natural direction is to develop polynomial-type contraction conditions within the context of C-VSMSs (or their possible generalizations) and to establish corresponding FP results. This may lead to a wider class of nonlinear contractive mappings beyond rational and interpolative types.

(3) The study of multivalued mappings in the setting of C-VSMSs remains open. In particular, the existence and uniqueness of FPs for multivalued contractions under cyclic or interpolative conditions could provide significant extensions of the current results.

(4) From the application point of view, it would be interesting to investigate fractional differential inclusions in the framework of C-VSMSs. The development of appropriate FP techniques for multivalued operators may allow one to establish existence results for such problems under nonlocal or integral boundary conditions.

(5) Furthermore, exploring the applicability of the proposed framework to more complex models arising in applied sciences, such as systems involving memory effects or hybrid fractional dynamics, remains an open and promising direction.

These problems indicate that the theory of C-VSMSs has strong potential for further development, both from theoretical and applied perspectives.