A Contemporary Approach of Integral Khan-Type Multivalued Contractions with Generalized Dynamic Process and an Application

Abstract

The aim of this article is to investigate the relationship between integral-type contractions and the generalized dynamic process. The fixed-point results for multivalued mappings that satisfy both the integral Khan-type contraction and the integral $\theta$-contraction are established in a complete metric space. Furthermore, some corollaries are derived based on our main contribution. To demonstrate the novelty of our findings, several examples are provided. Finally, we look into whether nonlinear fractional differential equations have solutions utilizing the obtained results.

1. Introduction

The fixed point (FP) theory is a notable part of functional analysis and topology as the basis of the subject. In the realm of functional analysis and topology, the theory of FP holds a position of great significance. Serving as the foundation of these subjects, FP theory is fundamentally grounded in the concept of complete metric spaces (MS). This body of work has far-reaching applications and it has also been instrumental in addressing problems, not only within mathematics, but also in fields as diverse as engineering, computer science, and economics. In 1922, Banach laid the first and most basic brick of the FP theory by introducing the Banach contraction principle (BCP), and pointed out that the inequality $\varrho \left(\Gamma \xi ,\Gamma \mu \right)\le c\varrho \left(\xi ,\mu \right)$, $c\in \left[0,1\right)$ is a contraction for all $\xi ,\mu$ in a complete MS $(\Delta ,\varrho )$, whereas the mapping $\Gamma :\Delta \to \Delta$ has a unique fixed point (UFP) ${\xi }^{*}\in \Delta$, such that $\Gamma {\xi }^{*}={\xi }^{*}.$ Numerous results have appeared in the literature concerning the FPs of mappings that act as contractions over their entire domains. In 2002, Branciari [1] analyzed the existence of FPs for mappings defined on complete MS and focused on a general contraction of integral-type. For additional information, refer to [2,3]. Nadler [4] introduced a novel approach to FP theorems, contributing to the expansion of the BCP and incorporating multivalued mappings within complete MS. Subsequent to these contributions, several seminal works have been published, aiding researchers in deepening their understanding of the FP theory for multivalued mappings. In 1997, Berinde [5] explored the concept of a comparison function. Subsequently, in 2014, Latif et al. [6] introduced another notion of comparison functions and obtained new FP results for generalized $\left(\alpha ,\psi \right)$-Meir–Keeler contraction mappings. Liu et al. [7] introduced the notions of $\left(\psi ,\varphi \right)$-type contractions and $\left(\psi ,\varphi \right)$-type Suzuki contractions with establishing new FP theorems within the framework of complete MSs. Following this, Xiao et al. [8] examined a newly generalized multivalued Khan-type $\left(\psi ,\varphi \right)$-contraction and derived further FP theorems in complete MSs, as detailed in [9,10,11,12,13]. On the other hand, the dynamic process is a powerful formal tool for addressing a large-scale analysis of multistage decision-making problems. Such problems inherently arise and are pervasive across human activities. Unfortunately, the analysis of dynamic process, generalized dynamic process, and fuzzy dynamic process present considerable challenges. Klim and Wardowski [14] defined the concept of a dynamic process and developed an FP theorem for a specific case of nonlinear F-contraction set-valued maps in a complete MS. Subsequently, a generalized dynamic process of mappings was explored by Hussain et al. [15], who fulfilled coincidence and common FP results for generalized $(f,L)$-almost F-contraction mappings. Ali et al. [16] also introduced the notion of a fuzzy dynamic system, discussing new refinements of F-fuzzy Suzuki-type FP results for fuzzy operators and establishing the concept of the fuzzy dynamic system as an alternative to the Picard iterative sequence. In terms of applications, fractional differential equations have emerged in numerous branches of physics and mathematics, having a multitude of solutions that are documented in the literature. Therefore, fixed-point (FP) theory is an indispensable tool for advancing studies and calculations of solutions to nonlinear fractional differential equations (NFDEs). It also aids in finding solutions to initial value problems in both dynamic programming and mechanical engineering. Consequently, this mathematical branch is extremely potent for establishing the existence and uniqueness of solutions for a wide array of problems that are governed by nonlinear relations. Building upon this body of work, our research aims to study the relationship between generalized dynamic processes and integral-type contractions. We discuss new FP results for integral Khan-type multivalued $\left(\psi ,\varphi \right)$-contraction and integral multivalued $\theta$-contraction, along with the generalized dynamic process, in a complete MS.

2. Preliminaries

The following Theorem is very well known in the literature, and it was introduced by Banach as follows:

Theorem 1. 

Let $\left(\Delta ,\wp \right)$ be an MS and $\Gamma :\Delta \to \Delta$ be a contraction mapping; that is, for all $\xi ,\mu \in \Delta and$ $\wp \left(\Gamma \xi ,\Gamma \mu \right)\le c\wp \left(\xi ,\mu \right),$ where $c\in \left(0,1\right).$ Then, Γ has a UFP ${\xi }^{*}$ in $\Delta .$

Theorem 2 

([1]). Let $\left(\Delta ,\wp \right)$ be an MS and $\Gamma :\Delta \to \Delta$ be a mapping so that

$${\int }_0^{\wp \left(\Gamma \xi ,\Gamma \mu \right)}\phi \left(s\right)ds\le \beta {\int }_0^{\wp \left(\xi ,\mu \right)}\phi \left(s\right)ds,$$

for all $\xi ,\mu \in \Delta ,$ $\beta \in \left(0,1\right),$ and $\phi \in \mathsf{{\rm Y}},$ where Υ is the class of all functions $\phi :\left[0,\infty \right)\to \left[0,\infty \right)$, which is Lebesgue integrable and summable (i.e., with finite integral) on each compact subset of $\left[0,\infty \right)$, and ${\int }_0^{ϵ}\phi \left(s\right)ds>0$ for all $ϵ>0.$ Then, Γ has a UFP $a\in \Delta$, such that for each $\xi \in \Delta ,$ ${\displaystyle \underset{n\to \infty }{lim}}{\Gamma }^n\xi =a.$

Let $\left(\Delta ,\wp \right)$ be a given metric space. Then, we denote $CB\left(\Delta \right)$ as the family of all non-empty, closed, and bounded subsets of $\Delta$, and $K\left(\Delta \right)$ as the family of all non-empty and compact subsets of $\Delta .$

Definition 1 

([4]). Suppose that $\left(\Delta ,\wp \right)$ is an MS. The Pompeiu–Hausdorff metric $H_{\wp }:CB\left(\Delta \right)\times CB\left(\Delta \right)\to \left[0,\infty \right)$ induced by the distance ℘ is formulated as follows:

$$H_{\wp }\left(A,B\right)=max\left\{\underset{a_1\in A}{sup}\wp \left(a_1,B\right),\underset{a_2\in B}{sup}\wp \left(A,a_2\right)\right\},$$

for all $A,B\in CB\left(\Delta \right).$ In addition, $\left(CB\left(\Delta \right),H_{\wp }\right)$ is known as a Pompeiu–Hausdorff MS, where $\wp \left(a_1,B\right)=\underset{a_2\in B}{inf}\wp \left(a_1,a_2\right).$ Here, we say that an element $a\in \Delta$ is an FP of a multivalued map $\Gamma :\Delta \to CB\left(\Delta \right)$ if $a\in \Gamma a.$

In [5], Berinde introduced the concept of the comparison function, as follows:

Definition 2. 

Let Ψ denote the family of all comparison functions $\psi :\left(0,+\infty \right)\to \left(0,+\infty \right)$, so that

$\left({\psi }_1\right)$

  ψ is monotone increasing; that is, $t_1<t_2$ implies that $\psi \left(t_1\right)\le \psi \left(t_2\right);$

$\left({\psi }_2\right)$

 ${\displaystyle \underset{j\to \infty }{lim}}{\psi }^j\left(t\right)=0,$ for all $t>0$, where ${\psi }^j$ is the $j^{th}$ iterate of $\psi .$ Clearly, if $\psi \in \mathsf{\Psi }$, then $\psi \left(t\right)<t$ for each $t>0.$

Example 1. 

Let $\psi \left(t\right)=\alpha t,$ $0<\alpha <1,$ for all $t>0.$

• Then, for all $t_1,t_2>0$ with $t_1<t_2$ implies that $\alpha t_1\le \alpha t_2;$ that is, $\psi \left(t_1\right)\le \psi \left(t_2\right).$ Therefore, the condition $\left({\psi }_1\right)$ is satisfied.
• On the other hand, as ${\psi }^2\left(t\right)=\psi \left(\psi \left(t\right)\right)=\psi \left(\alpha t\right)={\alpha }^{2}t.$ Continuing in the same way, we obtain ${\psi }^j\left(t\right)={\alpha }^{j}t.$ Thus, ${\displaystyle \underset{j\to \infty }{lim}}{\psi }^j\left(t\right)=0,$ for all $t>0$ and $j\in \mathbb{N}.$
• Finally, $\psi \left(t\right)=\alpha t<t,$ for each $t>0.$ Hence, the condition $\left({\psi }_2\right)$ is satisfied, and then $\psi \in \mathsf{\Psi }.$

Definition 3 

([7]). Let Φ denote the family of all functions $\varphi :\left(0,+\infty \right)\to \left(0,+\infty \right)$, so that

$\left({\varphi }_1\right)$

 ϕ is non-decreasing;

$\left({\varphi }_2\right)$

 for each sequence $\left\{t_j\right\}\subset \left(0,\infty \right),$ ${\displaystyle \underset{j\to \infty }{lim}}\varphi \left(t_j\right)=0$⇔${\displaystyle \underset{j\to \infty }{lim}}{t}_j=0;$

$\left({\varphi }_3\right)$

 ϕ is continuous.

The concepts of generalized, multivalued Khan-type $\left(\psi ,\varphi \right)$-contraction and generalized multivalued $\theta$-Khan-contraction were shown in the work of [8], as follows:

Definition 4. 

Let $\left(\Delta ,\wp \right)$ be an MS. $\Gamma :\Delta \to CB\left(\Delta \right)$ is said to be a generalized multivalued Khan-type $\left(\psi ,\varphi \right)$-contraction, if there exist $\varphi \in \mathsf{\Phi }$ and $\psi \in \mathsf{\Psi }$, such that for all $\xi ,\mu \in \Delta ,$ we have

$\left(i\right)$

 If $max\left\{\wp \left(\xi ,\Gamma \mu \right),\wp \left(\mu ,\Gamma \xi \right)\right\}\ne 0,$ then $\Gamma \xi \ne \Gamma \mu$ and

$$\begin{array}{c}\hfill \varphi \left(H_{\wp }\left(\Gamma \xi ,\Gamma \mu \right)\right)\le \psi \left[\varphi \left(M\left(\xi ,\mu \right)\right)\right],\end{array}$$

where

$$M\left(\xi ,\mu \right)=\frac{\wp \left(\xi ,\Gamma \xi \right)\wp \left(\xi ,\Gamma \mu \right)+\wp \left(\mu ,\Gamma \mu \right)\wp \left(\mu ,\Gamma \xi \right)}{max\left\{\wp \left(\xi ,\Gamma \mu \right),\wp \left(\mu ,\Gamma \xi \right)\right\}}.$$

$\left(ii\right)$

 If $max\left\{\wp \left(\xi ,\Gamma \mu \right),\wp \left(\mu ,\Gamma \xi \right)\right\}=0,$ then $\Gamma \xi =\Gamma \mu .$

Theorem 3. 

Let $\left(\Delta ,\wp \right)$ be a complete MS and $\Gamma :\Delta \to K\left(\Delta \right)$ be a generalized multivalued Khan-type $\left(\psi ,\varphi \right)$-contraction. Then, Γ has an FP ${\xi }^{*}\in \Delta .$

Example 2. 

Let

• ${\varphi }_1\left(t\right)=t,$ for all $t>0;$
• ${\varphi }_2\left(t\right)=ln\theta \left(t\right),$ $\theta \in \mathsf{\Pi },$ for all $t>0;$
• ${\varphi }_3\left(t\right)=e^{\Re \left(t\right)},$ $\Re \in F,$ for all $t\ge 0.$
• It is easy to check that ${\varphi }_1\in \mathsf{\Phi }.$
• Now, let $t_1,t_2>0$, with $t_1<t_2.$ Since $\theta \in \mathsf{\Pi },$ then $\theta \left(t_1\right)\le \theta \left(t_2\right)$ implies that $ln\theta \left(t_1\right)\le ln\theta \left(t_2\right);$ that is, ${\varphi }_2\left(t_1\right)\le {\varphi }_2\left(t_2\right).$ Therefore, the condition $\left({\varphi }_1\right)$ is satisfied.
• Let $\left\{t_j\right\}$ be a sequence and ${\varphi }_2\left(t_j\right)=ln\theta \left(t_j\right)$ for all $t>0$ and $j\in \mathbb{N}.$ Since $\theta \in \mathsf{\Pi },$ we have

  $$\begin{array}{ccc}& & {\displaystyle \underset{j\to \infty }{lim}}\theta \left(t_j\right)=1\iff {\displaystyle \underset{j\to \infty }{lim}}{t}_j=0\hfill \\ & \Rightarrow & {\displaystyle \underset{j\to \infty }{lim}}ln\theta \left(t_j\right)=0\iff {\displaystyle \underset{j\to \infty }{lim}}{t}_j=0\hfill \\ & \Rightarrow & {\displaystyle \underset{j\to \infty }{lim}}{\varphi }_2\left(t_j\right)=0\iff {\displaystyle \underset{j\to \infty }{lim}}{t}_j=0.\hfill \end{array}$$

  Hence, the condition $\left({\varphi }_2\right)$ is satisfied. It is also easy to check that ${\varphi }_2$ is continuous, and then, we find that ${\varphi }_2\in \mathsf{\Phi }.$
• Likewise, let $t_1,t_2>0$, with $t_1<t_2.$ Since $\Re \in F,$ then $\Re \left(t_1\right)<\Re \left(t_2\right)$ implies that $e^{\Re \left(t_1\right)}\le e^{\Re \left(t_2\right)};$ that is, ${\varphi }_3\left(t_1\right)\le {\varphi }_3\left(t_2\right).$ Therefore, the condition $\left({\varphi }_1\right)$ is satisfied.
• Let $\left\{t_j\right\}$ be a sequence, and ${\varphi }_3\left(t_j\right)=e^{\Re \left(t_j\right)}$ for all $t>0$ and $j\in \mathbb{N}.$ Since $\Re \in F,$ we have

  $$\begin{array}{ccc}& & {\displaystyle \underset{j\to \infty }{lim}}\Re \left(t_j\right)=-\infty \iff {\displaystyle \underset{j\to \infty }{lim}}{t}_j=0\hfill \\ & \Rightarrow & {\displaystyle \underset{j\to \infty }{lim}}{e}^{\Re \left(t_j\right)}=0\iff {\displaystyle \underset{j\to \infty }{lim}}{t}_j=0\hfill \\ & \Rightarrow & {\displaystyle \underset{j\to \infty }{lim}}{\varphi }_3\left(t_j\right)=0\iff {\displaystyle \underset{j\to \infty }{lim}}{t}_j=0.\hfill \end{array}$$

  Hence, the condition $\left({\varphi }_2\right)$ is satisfied. It is also easy to check that ${\varphi }_3$ is continuous, and then, we find that ${\varphi }_3\in \mathsf{\Phi }.$

Definition 5 

([15]). A generalized dynamic process of mappings $f:\Delta \to \Delta$ and $\Gamma :\Delta \to CB\left(\Delta \right)$ is defined as:

$$\Xi \left(f,\Gamma ,{\xi }_0\right)=\left\{{\left\{{\xi }_j\right\}}_{j\in \mathbb{N}\cup \left\{0\right\}}:{\xi }_{j+1}=f{\xi }_j\in \Gamma {\xi }_{j-1},\mathrm{for}\mathrm{all}j\in \mathbb{N}\right\},$$

where ${\xi }_0\in \Delta$ is a starting point. In short, $\Xi \left(f,\Gamma ,{\xi }_0\right)$ stands for $f{\xi }_j.$ The sequence $\left\{{\xi }_j\right\}$ for which $\left(f{\xi }_j\right)$ is a generalized dynamic process is called the f iterative sequence of Γ, starting with ${\xi }_0.$

The following Lemmas are very useful in our study, and they are taken and proven in [17].

Lemma 1. 

Let ${\left\{{\xi }_j\right\}}_{j\in \mathbb{N}}$ be a non-negative sequence, such that $\underset{j\to \infty }{lim}{\xi }_j=\xi .$ Then,

$$\underset{j\to \infty }{lim}{\int }_0^{{\xi }_j}\phi \left(s\right)ds\le {\int }_0^{\xi }\phi \left(s\right)ds.$$

Lemma 2. 

Let ${\left\{{\xi }_j\right\}}_{j\in \mathbb{N}}$ be a non-negative sequence. Then,

$$\underset{j\to \infty }{lim}{\int }_0^{{\xi }_j}\phi \left(s\right)ds=0\iff \underset{j\to \infty }{lim}{\xi }_j=0.$$

Recently, Jleli and Samet [18] have established the following FP theorem.

Theorem 4. 

Let $\left(\Delta ,\wp \right)$ be a complete GMS and $\Gamma :\Delta \to \Delta$ be a given map. Suppose that there exist $\theta \in \mathsf{\Pi }$ and $k\in \left(0,1\right)$, such that for $\xi ,\mu \in \Delta ,$ we have

$$\wp \left(\Gamma \xi ,\Gamma \mu \right)\ne 0\Rightarrow \theta \left(\wp \left(\Gamma \xi ,\Gamma \mu \right)\right)\le {\left[\theta \left(\wp \left(\xi ,\mu \right)\right)\right]}^k,$$

where Π is denoted by the set of functions $\theta :\left(0,1\right)\to \left(0,1\right)$ satisfying the following conditions:

$\left({\mathsf{\Pi }}_1\right)$

 θ is non-decreasing;

$\left({\mathsf{\Pi }}_2\right)$

 for each sequence $\left\{t_j\right\}\subset \left(0,1\right),$ ${\displaystyle \underset{j\to \infty }{lim}}\theta \left(t_j\right)=1$⇔${\displaystyle \underset{j\to \infty }{lim}}{t}_j=0;$

$\left({\mathsf{\Pi }}_3\right)$

 there exist $r\in \left(0,1\right)$ and $\ell \in \left(0,1\right]$ such that ${\displaystyle \underset{j\to \infty }{lim}}\frac{\theta \left(t\right)-1}{t^r}=\ell ;$

$\left({\mathsf{\Pi }}_4\right)$

 θ is continuous.

Then, Γ has a UFP.

In [19], Jleli et al. have established the following extension of Theorem 4.

Theorem 5. 

Let $\left(\Delta ,\wp \right)$ be a complete GMS and $\Gamma :\Delta \to \Delta$ be a given map. Suppose that there exist $\theta \in \mathsf{\Pi }$ and $k\in \left(0,1\right)$ such that

$$\wp \left(\Gamma \xi ,\Gamma \mu \right)\ne 0\Rightarrow \theta \left(\wp \left(\Gamma \xi ,\Gamma \mu \right)\right)\le {\left[\theta \left(M\left(\xi ,\mu \right)\right)\right]}^k,$$

for $\xi ,\mu \in \Delta ,$ where

$$M\left(\xi ,\mu \right)=\{\wp \left(\xi ,\mu \right),\wp \left(\xi ,\Gamma \xi \right),\wp \left(\mu ,\Gamma \mu \right)\}.$$

Then, Γ has a UFP.

Definition 6 

([20]). Let $\Re :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be a mapping satisfying the following conditions:

$\left({\Re }_1\right)$

 ℜ is strictly increasing, i.e., $\forall \alpha ,\beta \in {\mathbb{R}}^{+}$, so that $\alpha <\beta$ then $\Re \left(\alpha \right)<\Re \left(\beta \right);$

$\left({\Re }_2\right)$

 for any sequence ${\left\{{\alpha }_j\right\}}_{j=1}^{\infty }$ of positive real numbers, ${\displaystyle \underset{j\to \infty }{lim}}{\alpha }_j=0\iff {\displaystyle \underset{j\to \infty }{lim}}\Re \left({\alpha }_j\right)=-\infty ;$

$\left({\Re }_3\right)$

 there exists $k\in \left(0,1\right)$ such that ${\displaystyle \underset{j\to \infty }{lim}}{\alpha }^k\Re \left({\alpha }_j\right)=0.$

• A mapping $\Gamma :\Delta \to \Delta$ is said to be an ℜ-contraction if there exists $\tau >0$ such that for all $\xi ,\mu \in \Delta$

  $$\begin{array}{c}\hfill \wp \left(\Gamma \xi ,\Gamma \mu \right)>0\Rightarrow \tau +\Re \left(\wp \left(\Gamma \xi ,\Gamma \mu \right)\right)\le \Re \left(\wp \left(\xi ,\mu \right)\right).\end{array}$$

  Define F as the set of functions ℜ satisfying $\left({\Re }_1\right)-\left({\Re }_3\right).$

3. Main Results

In this portion, we study the existence of FPs for integral Khan-type multivalued $\left(\psi ,\varphi \right)$-contractions and integral multivalued $\theta$-contractions with respect to the generalized dynamic process $\Xi \left(f,\Gamma ,{\xi }_0\right)$ in complete MSs.

Definition 7. 

Let f be a self-mapping on an MS $\left(\Delta ,\wp \right)$. A mapping $\Gamma :\Delta \to CB\left(\Delta \right)$ is said to be an integral Khan-type multivalued $\left(\psi ,\varphi \right)$-contraction with respect to a generalized dynamic process $\Xi \left(f,\Gamma ,{\xi }_0\right);$ ${\xi }_0\in \Delta$ if there exist $\varphi \in \mathsf{\Phi },$ $\psi \in \mathsf{\Psi }$ and $\phi \in \mathsf{{\rm Y}}$ so that for all ${\xi }_j,{\xi }_{j+1}\in \Xi \left(f,\Gamma ,{\xi }_0\right)$ and $j\in \mathbb{N}.$

$\left(i\right)$

 If $max\left\{\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left(f{\xi }_{j+1},\Gamma {\xi }_j\right)\right\}\ne 0,$ then $\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right)\ne 0$ and

$$\begin{array}{c}\hfill \wp \left(f{\xi }_j,f{\xi }_{j+1}\right)>0\Rightarrow \varphi \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right],\end{array}$$

(1)

where

$$M\left({\xi }_{j-1},{\xi }_j\right)=\frac{\wp \left(f{\xi }_{j-1},\Gamma {\xi }_{j-1}\right).\wp \left(f{\xi }_{j-1},\Gamma {\xi }_j\right)+\wp \left(f{\xi }_j,\Gamma {\xi }_j\right).\wp \left(f{\xi }_j,\Gamma {\xi }_{j-1}\right)}{max\{\wp \left(f{\xi }_{j-1},\Gamma {\xi }_j\right),\wp \left(f{\xi }_j,\Gamma {\xi }_{j-1}\right)\}}.$$

(2)

(ii)

 If $max\left\{\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left(f{\xi }_{j+1},\Gamma {\xi }_j\right)\right\}=0,$ then $\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right)=0$.

Lemma 3. 

Let $\left(\Delta ,\wp \right)$ be a complete MS and $\Gamma :\Delta \to \Delta$ be a self-mapping. Then, the following assertions are equivalent:

$\left(i\right)$

  Γ is an integral θ-contraction with $\theta \in \mathsf{\Pi };$

$\left(ii\right)$

  Γ is an integral $\left(\psi ,\varphi \right)$-contraction with $\psi \in \mathsf{\Psi },$ and $\varphi \in \mathsf{\Phi }.$

Proof. 

$\left(i\right)$⇒$\left(ii\right)$ If $\Gamma$ is an integral $\theta$-contraction, then there exist $\theta \in \mathsf{\Pi },$ $\phi \in \mathsf{{\rm Y}}$, and k ∈$\left(0,1\right)$, such that for all ${\xi }_j,{\xi }_{j+1}\in \Delta ,$ we have

$$\begin{array}{c}\hfill \wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)\ne 0\Rightarrow \theta \left({\int }_0^{\wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le {\left[\theta \left({\int }_0^{\wp \left({\xi }_j,{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\right]}^k.\end{array}$$

Using $\varphi \left(t\right)=ln\theta \left(t\right)$ and $\psi \left(t\right)=kt,$ then it is easy to verify that $\Gamma$ is an integral $\left(\psi ,\varphi \right)$-contraction and

$$\begin{array}{ccc}& & \theta \left({\int }_0^{\wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le {\left[\theta \left({\int }_0^{\wp \left({\xi }_j,{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\right]}^k\hfill \\ & \iff & ln\theta \left({\int }_0^{\wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le kln\left[\theta \left({\int }_0^{\wp \left({\xi }_j,{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\right]\hfill \\ & \iff & \varphi \left({\int }_0^{\wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{\wp \left({\xi }_j,{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\right].\hfill \end{array}$$

Hence, $\Gamma$ is an integral $\left(\psi ,\varphi \right)$-contraction.

• $\left(ii\right)$⇒$\left(i\right)$ If $\Gamma$ is an integral $\left(\psi ,\varphi \right)$-contraction, then there exist $\psi \in \mathsf{\Psi },$ $\varphi \in \mathsf{\Phi }$, and $\phi \in \mathsf{{\rm Y}}$, such that for all ${\xi }_j,{\xi }_{j+1}\in \Delta ,$ we have

  $$\begin{array}{c}\hfill \wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)\ne 0\Rightarrow \varphi \left({\int }_0^{\wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{\wp \left({\xi }_j,{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\right].\end{array}$$

  Using $\theta \left(t\right)=e^{\varphi \left(t\right)}$ and $\psi \left(t\right)=kt,$ then it is easy to verify that $\Gamma$ is an integral $\theta$-contraction and

  $$\begin{array}{ccc}& & \varphi \left({\int }_0^{\wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{\wp \left({\xi }_j,{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\right]\hfill \\ & \iff & \theta \left({\int }_0^{\wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le {\left[\theta \left({\int }_0^{\wp \left({\xi }_j,{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\right]}^k.\hfill \end{array}$$

Hence, $\Gamma$ is an integral $\theta$-contraction. □

Lemma 4. 

Let $\left(\Delta ,\wp \right)$ be a complete MS and $\Gamma :\Delta \to CB\left(\Delta \right)$ be a multivalued mapping. Then, the following assertions are equivalent:

$\left(i\right)$

 Γ is an integral multivalued θ-contraction with $\theta \in \mathsf{\Pi };$

$\left(ii\right)$

 Γ is an integral multivalued $\left(\psi ,\varphi \right)$-contraction with $\psi \in \mathsf{\Psi }$ and $\varphi \in \mathsf{\Phi }.$

Proof. 

The proof immediately follows the same steps as the proof of Lemma 3. □

Now, we state and prove the first main result.

Theorem 6. 

Presume that $\left(\Delta ,\wp \right)$ is a complete MS and that $f:\Delta \to \Delta$ is a nonlinear self-mapping. Let $\Gamma :\Delta \to CB\left(\Delta \right)$ be an integral Khan-type multivalued $\left(\psi ,\varphi \right)$-contraction with respect to the generalized dynamic process $\Xi \left(f,\Gamma ,{\xi }_0\right).$ Then, Γ and f possess a CFP ${\xi }^{*}$ so that ${\displaystyle \underset{j\to \infty }{lim}}f{\xi }_j={\xi }^{*}\in \Gamma {\xi }^{*}.$

Proof. 

As ${\xi }_0\in \Xi \left(f,\Gamma ,{\xi }_0\right),$ if there is $j_0\in \mathbb{N}$, such that ${\xi }_{j_0}={\xi }_{j_0+1},$ then the existence of an FP is concluded. We build a sequence in integral Khan-type multivalued $\left(\psi ,\varphi \right)$-contraction with respect to the generalized dynamic process $\Xi \left(f,\Gamma ,{\xi }_0\right)$, as follows.

$$\Xi \left(f,\Gamma ,{\xi }_0\right)=\left\{{\left\{{\xi }_j\right\}}_{j\in \mathbb{N}\cup \left\{0\right\}}:{\xi }_{j+1}=f{\xi }_j\in \Gamma {\xi }_{j-1},\forall j\in \mathbb{N}\right\}.$$

Now, we consider ${\xi }_j\ne {\xi }_{j+1},$ then $\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right)=\wp \left({\xi }_{j+1},{\xi }_{j+2}\right)>0$ for every $j\in \mathbb{N}.$ Utilizing (1), we have

$$\begin{array}{ccc}\hfill \varphi \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)& \le & \psi \left[\varphi \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right]\hfill \\ & =& \psi \left[\varphi \left({\int }_0^{\frac{\wp \left(f{\xi }_{j-1},\Gamma {\xi }_{j-1}\right).\wp \left(f{\xi }_{j-1},\Gamma {\xi }_j\right)+\wp \left(f{\xi }_j,\Gamma {\xi }_j\right).\wp \left(f{\xi }_j,\Gamma {\xi }_{j-1}\right)}{max\{\wp \left(f{\xi }_{j-1},\Gamma {\xi }_j\right),\wp \left(f{\xi }_j,\Gamma {\xi }_{j-1}\right)\}}}\phi \left(s\right)ds\right)\right]\hfill \\ & =& \psi \left[\varphi \left({\int }_0^{\wp \left(f{\xi }_{j-1},f{\xi }_j\right)}\phi \left(s\right)ds\right)\right]\hfill \\ & <& \varphi \left({\int }_0^{\wp \left(f{\xi }_{j-1},f{\xi }_j\right)}\phi \left(s\right)ds\right).\hfill \end{array}$$

(3)

From (3) and $\left({\varphi }_1\right),$ we obtain

$${\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds<{\int }_0^{\wp \left(f{\xi }_{j-1},f{\xi }_j\right)}\phi \left(s\right)ds.$$

Consequently, the sequence $\left\{{\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right\}$ is decreasing and bounded from below. Therefore, there is a constant $\varrho >0$ such that $\underset{j\to \infty }{lim}{\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds=\varrho .$ Assume that $\varrho >0.$ Thus, from (3), we have

$$\begin{array}{c}\hfill \varphi \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le {\psi }^j\left[\varphi \left({\int }_0^{\wp \left(f{\xi }_0,f{\xi }_1\right)}\phi \left(s\right)ds\right)\right].\end{array}$$

(4)

Taking the limit as $j\to \infty$ in (4), and from properties of $\psi ,$ we obtain

$$0\le \underset{j\to \infty }{lim}\varphi \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le \underset{j\to \infty }{lim}{\psi }^j\left[\varphi \left({\int }_0^{\wp \left(f{\xi }_0,f{\xi }_1\right)}\phi \left(s\right)ds\right)\right]=0,$$

hence,

$$\underset{j\to \infty }{lim}\varphi \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)=0.$$

From $\left({\varphi }_2\right),$ we obtain

$$\underset{j\to \infty }{lim}{\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds=0.$$

From Lemma 2, we obtain

$$\underset{j\to \infty }{lim}\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)=\underset{j\to \infty }{lim}\wp \left({\xi }_{j+1},{\xi }_{j+2}\right)=0.$$

(5)

Next, we prove that $\left\{{\xi }_n\right\}$ is a Cauchy sequence in $\left(\Delta ,\wp \right).$ Suppose the contrary. Then, there exists $ϵ>0$ such that for each positive integer $p,$ there are positive integers $n\left(p\right),m\left(p\right)\in \mathbb{N}$ with $n\left(p\right)>m\left(p\right)>p$ satisfying

$$\begin{array}{c}\hfill \wp \left(f{\xi }_{n\left(p\right)},f{\xi }_{m\left(p\right)}\right)\ge ϵ.\end{array}$$

(6)

Let $m\left(p\right)$ be the smallest integer and satisfying (6), so that

$$\begin{array}{c}\hfill \wp \left(f{\xi }_{n\left(p\right)},f{\xi }_{m\left(p\right)-1}\right)<ϵ,\end{array}$$

(7)

for all $p\in \mathbb{N}.$ Using the triangle inequality in (6), we obtain

$$\begin{array}{c}\hfill ϵ\le \wp \left(f{\xi }_{n\left(p\right)},f{\xi }_{m\left(p\right)}\right)\le \wp \left(f{\xi }_{n\left(p\right)},f{\xi }_{m\left(p\right)+1}\right)+\wp \left(f{\xi }_{m\left(p\right)+1},f{\xi }_{m\left(p\right)}\right).\end{array}$$

(8)

Letting $p\to \infty$ in (8), using (5) and (7), we obtain

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim}\wp \left(f{\xi }_{n\left(p\right)},f{\xi }_{m\left(p\right)+1}\right)=ϵ.\end{array}$$

(9)

Thus, there exists $p_1\in \mathbb{N},$ so that

$$\begin{array}{c}\hfill \wp \left(f{\xi }_{n\left(p\right)},f{\xi }_{m\left(p\right)+1}\right)>\frac{ϵ}{2},\end{array}$$

for all $p>p_1.$ This implies that

$$\begin{array}{c}\hfill max\left\{\wp \left(f{\xi }_{n\left(p\right)},f{\xi }_{m\left(p\right)+1}\right),\wp \left(f{\xi }_{n\left(p\right)+1},f{\xi }_{m\left(p\right)}\right)\right\}>\frac{ϵ}{2},\end{array}$$

(10)

for all $p>p_1.$ Applying the triangle inequality again in (6), we have

$$\begin{array}{c}\hfill ϵ\le \wp \left(f{\xi }_{n\left(p\right)},f{\xi }_{m\left(p\right)}\right)\le \wp \left(f{\xi }_{n\left(p\right)},f{\xi }_{n\left(p\right)+1}\right)+\wp \left(f{\xi }_{n\left(p\right)+1},f{\xi }_{m\left(p\right)+1}\right)+\wp \left(f{\xi }_{m\left(p\right)+1},f{\xi }_{m\left(p\right)}\right).\end{array}$$

Taking the limit as $p\to \infty$ and utilizing (5), we obtain

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim}\wp \left(f{\xi }_{n\left(p\right)+1},f{\xi }_{m\left(p\right)+1}\right)\ge ϵ.\end{array}$$

Thus, there exists $p_2\in \mathbb{N},$ so that

$$\begin{array}{c}\hfill \wp \left(f{\xi }_{n\left(p\right)+1},f{\xi }_{m\left(p\right)+1}\right)>\frac{ϵ}{2},\end{array}$$

(11)

for all $p>p_2.$ Now, from (11), $\left({\varphi }_1\right)$, and (1), we have

$$\begin{array}{ccc}& \varphi & \left({\int }_0^{\frac{ϵ}{2}}\phi \left(s\right)ds\right)\hfill \\ & \le & \varphi \left({\int }_0^{\wp \left(f{\xi }_{n\left(p\right)+1},f{\xi }_{m\left(p\right)+1}\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{M\left({\xi }_{n\left(p\right)+1},{\xi }_{m\left(p\right)+1}\right)}\phi \left(s\right)ds\right)\right]\hfill \\ & <& \varphi \left({\int }_0^{\frac{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right).\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)+\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right).\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)}{max\left\{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right),\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right)\right\}}}\phi \left(s\right)ds\right).\hfill \end{array}$$

(12)

On the other hand, from (10), we find that

$$\begin{array}{ccc}\hfill 0& \le & \frac{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right).\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)+\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right).\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)}{max\left\{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right),\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right)\right\}}\hfill \\ & \le & \wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right)+\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right).\hfill \end{array}$$

Let $p\to \infty$ in the above inequality, and by taking (5) into account, we obtain

$$\underset{p\to \infty }{lim}\frac{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right).\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)+\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right).\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)}{max\left\{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right),\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right)\right\}}=0.$$

So, there exists $p_3\in \mathbb{N}$, such that

$$\begin{array}{cc}\frac{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right).\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)+\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right).\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)}{max\left\{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right),\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right)\right\}}& <\frac{ϵ}{2},\end{array}$$

(13)

for all $p>p_3.$ Fusing (12) and (13), we obtain

$$\begin{array}{ccc}& \varphi & \left({\int }_0^{\frac{ϵ}{2}}\phi \left(s\right)ds\right)\hfill \\ & <& \varphi \left({\int }_0^{\frac{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right).\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)+\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right).\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right)}{max\left\{\wp \left(f{\xi }_{n\left(p\right)+1},\Gamma {\xi }_{m\left(p\right)+1}\right),\wp \left(f{\xi }_{m\left(p\right)+1},\Gamma {\xi }_{n\left(p\right)+1}\right)\right\}}}\phi \left(s\right)ds\right)\hfill \\ & <& \varphi \left({\int }_0^{\frac{ϵ}{2}}\phi \left(s\right)ds\right),\hfill \end{array}$$

for all $p>max\{p_1,p_2,p_3\},$ which is a contradiction. Hence, $\left\{{\xi }_n\right\}$ is a Cauchy sequence in a complete MS $\left(\Delta ,\wp \right).$ So, there exists ${\xi }^{*}\in \Delta$, such that $\underset{n\to \infty }{lim}f{\xi }_n={\xi }^{*},$ that is, $\underset{n\to \infty }{lim}\wp \left({\xi }_n,{\xi }^{*}\right)=0.$ By virtue of (1), we infer that

$$\begin{array}{ccc}\hfill \varphi \left({\int }_0^{\wp \left({\xi }^{*},f{\xi }^{*}\right)}\phi \left(s\right)ds\right)& =& \underset{n\to \infty }{lim}\varphi \left({\int }_0^{\wp \left(f{\xi }_{n+1},f{\xi }^{*}\right)}\phi \left(s\right)ds\right)\hfill \\ & \le & \underset{n\to \infty }{lim}\psi \left[\varphi \left({\int }_0^{M\left({\xi }_n,{\xi }^{*}\right)}\phi \left(s\right)ds\right)\right]\hfill \\ & <& \underset{n\to \infty }{lim}\varphi \left({\int }_0^{\frac{\wp \left(f{\xi }_n,\Gamma {\xi }_n\right).\wp \left(f{\xi }_n,\Gamma {\xi }^{*}\right)+\wp \left(f{\xi }^{*},\Gamma {\xi }^{*}\right).\wp \left(f{\xi }^{*},\Gamma {\xi }_n\right)}{max\{\wp \left(f{\xi }_n,\Gamma {\xi }^{*}\right),\wp \left(f{\xi }^{*},\Gamma {\xi }_n\right)\}}}\phi \left(s\right)ds\right).\hfill \end{array}$$

(14)

From (14), together with $\left({\varphi }_1\right)$ and $\left({\varphi }_3\right),$ we deduce that

$${\int }_0^{\wp \left({\xi }^{*},f{\xi }^{*}\right)}\phi \left(s\right)ds=0.$$

From Lemma 2, we conclude that $\wp \left({\xi }^{*},f{\xi }^{*}\right)=0,$ that is, ${\xi }^{*}=f{\xi }^{*}\in \Gamma {\xi }^{*}.$ Therefore, the mappings f and $\Gamma$ have a common FP ${\xi }^{*}$. □

Example 3. 

Let $\Delta =[0,\infty )$ with the usual metric $\wp \left(\xi ,\mu \right)=\left|\xi -\mu \right|,$ then $\left(\Delta ,\wp \right)$ is a complete MS. Consider the two mappings $f:\Delta \to \Delta$ by $f\xi =\frac{\xi }{4}$ and $\Gamma :\Delta \to CB\left(\Delta \right)$ by $\Gamma \xi =\left[0,\frac{4\xi }{3}\right].$ The sequence $\left\{{\xi }_j\right\}$ can be defined as ${\xi }_j={\xi }_0{\ell }^{j-1}$ for all $j\in \mathbb{N}$ with ${\xi }_0=4$ and $\ell =\frac{1}{4}$, so that ${\xi }_1=4$ and

$${\xi }_2=1=f{\xi }_1\in \Gamma {\xi }_0=\left[0,\frac{16}{3}\right],{\xi }_3=\frac{1}{4}=f{\xi }_2\in \Gamma {\xi }_1=\left[0,\frac{16}{3}\right];$$

$${\xi }_4=\frac{1}{16}=f{\xi }_3\in \Gamma {\xi }_2=\left[0,\frac{4}{3}\right],{\xi }_5=\frac{1}{64}=f{\xi }_4\in \Gamma {\xi }_3=\left[0,\frac{1}{3}\right].$$

Continuing with the same scenario, we deduce that

$$\Xi \left(f,\Gamma ,4\right)=\{1,\frac{1}{4},\frac{1}{16},\frac{1}{64},\dots \}$$

is a generalized dynamic process of f and Γ with the starting point ${\xi }_0=4.$ Define the functions $\phi ,\varphi ,\psi :\left(0,\infty \right)\to \left(0,\infty \right)$ using $\phi \left(t\right)=\frac{{16}^t}{3},$ $\varphi \left(t\right)=4t$ and $\psi \left(t\right)=\frac{2t}{ln16}.$ For all ${\xi }_j,{\xi }_{j+1}\in \Xi (f,\Gamma ,4)$ and $\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)>0,$ we have

$$\begin{array}{ccc}\hfill \varphi \left({\int }_0^{\wp \left({\xi }_{j+1},{\xi }_{j+2}\right)}\phi \left(s\right)ds\right)& =& \varphi \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)=\varphi \left({\int }_0^{\frac{1}{4}|{\xi }_j-{\xi }_{j+1}|}\phi \left(s\right)ds\right)\hfill \\ & =& 4{\int }_0^{\frac{1}{4}|{\xi }_j-{\xi }_{j+1}|}\frac{{16}^s}{3}ds=4\left(\frac{{16}^{\frac{1}{4}|{\xi }_j-{\xi }_{j+1}|}}{3ln16}-\frac{1}{3ln16}\right)\hfill \\ & \le & \frac{2}{ln16}\left[4\left(\frac{{16}^{|{\xi }_j-{\xi }_{j+1}|}}{3ln16}-\frac{1}{3ln16}\right)\right]\hfill \\ & =& \frac{2}{ln16}\left[4\left({\int }_0^{\wp \left({\xi }_j,{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\right]\hfill \\ & \le & \psi \left[\varphi \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right].\hfill \end{array}$$

We can choose any two arbitrary points ${\xi }_1=1$ and ${\xi }_2=\frac{1}{4}$ of $\Xi \left(f,\Gamma ,4\right)$ to obtain the following

$$\begin{array}{ccc}\hfill \varphi \left({\int }_0^{\wp \left(f{\xi }_1,f{\xi }_2\right)}\phi \left(s\right)ds\right)& =& \varphi \left({\int }_0^{\wp \left(f1,f\frac{1}{4}\right)}\phi \left(s\right)ds\right)\hfill \\ & =& 4\left(\frac{{16}^{\frac{3}{16}}}{3ln16}-\frac{1}{3ln16}\right)\approx 0.33.\hfill \end{array}$$

Thereafter, we have

$$M(1,\frac{1}{4})=\frac{\wp \left(f1,\Gamma 1\right).\wp \left(f1,\Gamma \frac{1}{4}\right)+\wp \left(f\frac{1}{4},\Gamma \frac{1}{4}\right).\wp \left(f\frac{1}{4},\Gamma 1\right)}{max\{\wp \left(f1,\Gamma \frac{1}{4}\right),\wp \left(f\frac{1}{4},\Gamma 1\right)\}}=\frac{17}{64},$$

so that

$$\psi \left[\varphi \left({\int }_0^{\frac{17}{64}}\frac{{16}^s}{3}ds\right)\right]=\frac{2}{ln16}\left[4\left(\frac{{16}^{\frac{17}{64}}}{3ln16}-\frac{1}{3ln16}\right)\right]\approx 0.38.$$

Therefore, we conclude that

$$\begin{array}{c}\hfill \varphi \left({\int }_0^{\wp \left(f1,f\frac{1}{4}\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{M\left(1,\frac{1}{4}\right)}\phi \left(s\right)ds\right)\right].\end{array}$$

Hence, the contractive condition of Theorem 6 is satisfied and Γ is an integral Khan-type multivalued $(\psi ,\varphi )$-contraction with respect to the generalized dynamic process $\Xi \left(f,\Gamma ,4\right),$ so that 0 is a CFP of f and Γ as the form $0=f0\in \Gamma 0.$

• Otherwise, we have

  $$\begin{array}{c}\hfill \varphi \left({\int }_0^{\wp \left({\xi }_{j+1},{\xi }_{j+2}\right)}\phi \left(s\right)ds\right)>\psi \left[\varphi \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right].\end{array}$$

  For instance, if ${\xi }_1=0$ and ${\xi }_2=2,$ then we obtain

  $$\begin{array}{ccc}\hfill \varphi \left({\int }_0^{\wp \left(f{\xi }_1,f{\xi }_2\right)}\phi \left(s\right)ds\right)& =& \varphi \left({\int }_0^{\wp \left(f0,f2\right)}\phi \left(s\right)ds\right)\hfill \\ & =& 4\left(\frac{{16}^{\frac{1}{2}}}{3ln16}-\frac{1}{3ln16}\right)\approx 1.44.\hfill \end{array}$$

  Thereafter, we have

  $$M(0,2)=\frac{\wp \left(f0,\Gamma 0\right).\wp \left(f0,\Gamma 2\right)+\wp \left(f2,\Gamma 2\right).\wp \left(f2,\Gamma 0\right)}{max\{\wp \left(f0,\Gamma 2\right),\wp \left(f2,\Gamma 0\right)\}}=\frac{1}{2}.$$

  So that

  $$\psi \left[\varphi \left({\int }_0^{\frac{1}{2}}\frac{{16}^s}{3}ds\right)\right]=\frac{2}{ln16}\left[4\left(\frac{{16}^{\frac{1}{2}}}{3ln16}-\frac{1}{3ln16}\right)\right]\approx 1.04.$$

  Therefore, we conclude that

  $$\begin{array}{c}\hfill \varphi \left({\int }_0^{\wp \left(f0,f2\right)}\phi \left(s\right)ds\right)>\psi \left[\varphi \left({\int }_0^{M\left(0,2\right)}\phi \left(s\right)ds\right)\right].\end{array}$$

  That is, the contractive condition of Theorem 6 is not satisfied for some ${\xi }_j,{\xi }_{j+1}\in \Delta$.

Definition 8. 

Let f be a self-mapping on an MS $\left(\Delta ,\wp \right)$. A mapping $\Gamma :\Delta \to CB\left(\Delta \right)$ is said to be an integral Khan-type multivalued θ-contraction with respect to the generalized dynamic process $\Xi \left(f,\Gamma ,{\xi }_0\right),$ ${\xi }_0\in \Delta$ if there exist $k\in \left(0,1\right),$ $\theta \in \mathsf{\Pi }$ and $\phi \in \mathsf{{\rm Y}}$ so that for all ${\xi }_j,{\xi }_{j+1}\in \Xi \left(f,\Gamma ,{\xi }_0\right)$ and $j\in \mathbb{N}.$

$\left(i\right)$

 If $max\left\{\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left(f{\xi }_{j+1},\Gamma {\xi }_j\right)\right\}\ne 0,$ then $\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right)\ne 0$ and

$$\begin{array}{c}\hfill \wp \left(f{\xi }_j,f{\xi }_{j+1}\right)>0\Rightarrow \theta \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le {\left[\theta \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right]}^k,\end{array}$$

(15)

where $M\left({\xi }_{j-1},{\xi }_j\right)$ is defined as in (1).

(ii)

 If $max\left\{\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left(f{\xi }_{j+1},\Gamma {\xi }_j\right)\right\}=0,$ then $\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right)=0$.

Now, we state and prove the second main result.

Theorem 7. 

Presume that $\left(\Delta ,\wp \right)$ is a complete MS and that $f:\Delta \to \Delta$ is a nonlinear self-mapping. Let $\Gamma :\Delta \to CB\left(\Delta \right)$ be an integral Khan-type multivalued θ-contraction with respect to the generalized dynamic process $\Xi \left(f,\Gamma ,{\xi }_0\right).$ Then, Γ and f possess a CFP ${\xi }^{*}$ so that $\underset{j\to \infty }{lim}f{\xi }_j={\xi }^{*}\in \Gamma {\xi }^{*}.$

Proof. 

As ${\xi }_0\in \Xi \left(f,\Gamma ,{\xi }_0\right),$ if there is $j_0\in \mathbb{N}$ such that ${\xi }_{j_0}={\xi }_{j_0+1},$ then the existence of an FP is concluded. We build a sequence in integral Khan-type multivalued $\theta$-contraction with respect to the generalized dynamic process $\Xi \left(f,\Gamma ,{\xi }_0\right)$ as in the following

$$\Xi \left(f,\Gamma ,{\xi }_0\right)=\left\{{\left\{{\xi }_j\right\}}_{j\in \mathbb{N}\cup \left\{0\right\}}:{\xi }_{j+1}=f{\xi }_j\in \Gamma {\xi }_{j-1},\forall j\in \mathbb{N}\right\}.$$

Now, we consider ${\xi }_j\ne {\xi }_{j+1},$ then $\wp \left(f{\xi }_j,\Gamma {\xi }_{j+1}\right)=\wp \left({\xi }_{j+1},{\xi }_{j+2}\right)>0$ for every $j\in \mathbb{N}.$ Utilizing (15), we have

$$\begin{array}{c}\hfill \theta \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le {\left[\theta \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right]}^k.\end{array}$$

(16)

Inserting ln in both sides of (16) and from Lemma 4 with considering that $\psi \left(t\right)=kt$ and $\varphi \left(t\right)=ln\theta \left(t\right),$ we obtain

$$\begin{array}{c}\hfill \varphi \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{\wp \left(f{\xi }_{j-1},f{\xi }_j\right)}\phi \left(s\right)ds\right)\right].\end{array}$$

Hereafter, following the same steps as in the proof of Theorem 6, we can conclude that f and $\Gamma$ have a CFP ${\xi }^{*}$. □

Example 4. 

Let $\Delta =[0,\infty )$ with the usual metric $\wp \left(\xi ,\mu \right)=\left|\xi -\mu \right|,$ then $\left(\Delta ,\wp \right)$ is a complete MS, and let $\theta :\left(0,\infty \right)\to \left(1,\infty \right)$ be defined as $\theta \left(t\right)=e^{\sqrt{t}};$ it is clear that $\theta \in \mathsf{\Pi }.$ Consider the two mappings $f:\Delta \to \Delta$ by $f\xi =\frac{\xi }{2}$ and $\Gamma :\Delta \to CB\left(\Delta \right)$ by $\Gamma \xi =\left[0,\frac{2\xi }{3}\right].$ The sequence $\left\{{\xi }_j\right\}$ can be defined as ${\xi }_j={\xi }_0{\ell }^{j-1}$ for all $j\in \mathbb{N}$ with ${\xi }_0=2$ and $\ell =\frac{1}{2}$, so that ${\xi }_1=2$ and

$${\xi }_2=1=f{\xi }_1\in \Gamma {\xi }_0=\left[0,\frac{4}{3}\right],{\xi }_3=\frac{1}{2}=f{\xi }_2\in \Gamma {\xi }_1=\left[0,\frac{4}{3}\right];$$

$${\xi }_4=\frac{1}{4}=f{\xi }_3\in \Gamma {\xi }_2=\left[0,\frac{2}{3}\right],{\xi }_5=\frac{1}{8}=f{\xi }_4\in \Gamma {\xi }_3=\left[0,\frac{1}{3}\right].$$

Continuing with the same scenario, we deduce that

$$\Xi \left(f,\Gamma ,2\right)=\{1,\frac{1}{2},\frac{1}{4},\frac{1}{8},...\}$$

is a generalized dynamic process of f and Γ with a starting point of ${\xi }_0=2.$ Define the function $\phi :\left(0,\infty \right)\to \left(0,\infty \right)$ by $\phi \left(t\right)=\frac{1}{3}.$ Now, for all ${\xi }_j,{\xi }_{j+1}\in \Xi (f,\Gamma ,2)$ and $\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)>0,$ we have

$$\begin{array}{ccc}\hfill \theta \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)& =& \theta \left({\int }_0^{\frac{1}{2}|{\xi }_j-{\xi }_{j+1}|}\frac{1}{3}ds\right)\hfill \\ & =& \theta \left(\frac{|{\xi }_j-{\xi }_{j+1}|}{6}\right)=e^{\frac{1}{\sqrt{2}}\sqrt{{\int }_0^{\wp \left({\xi }_j,{\xi }_{j+1}\right)}\phi \left(s\right)ds}}\hfill \\ & \le & {\left(e^{\sqrt{{\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds}}\right)}^{\frac{1}{\sqrt{2}}}\hfill \\ & =& {\left[\theta \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right]}^k.\hfill \end{array}$$

We can choose any two arbitrary points ${\xi }_1=1$ and ${\xi }_2=\frac{1}{2}$ of $\Xi \left(f,\Gamma ,2\right)$ to obtain the following

$$\begin{array}{ccc}\hfill \theta \left({\int }_0^{\wp \left(f{\xi }_1,f{\xi }_2\right)}\phi \left(s\right)ds\right)& =& \theta \left({\int }_0^{\wp \left(f1,f\frac{1}{2}\right)}\frac{1}{3}ds\right)\hfill \\ & =& \theta \left({\int }_0^{\wp \left(\frac{1}{2},\frac{1}{4}\right)}\frac{1}{3}ds\right)\approx 1.33.\hfill \end{array}$$

Thereafter, we have

$$M(1,\frac{1}{2})=\frac{\wp \left(f1,\Gamma 1\right).\wp \left(f1,\Gamma \frac{1}{2}\right)+\wp \left(f\frac{1}{2},\Gamma \frac{1}{2}\right).\wp \left(f\frac{1}{2},\Gamma 1\right)}{max\{\wp \left(f1,\Gamma \frac{1}{2}\right),\wp \left(f\frac{1}{2},\Gamma 1\right)\}}=\frac{5}{8}.$$

So that for $k=\frac{1}{\sqrt{2}},$ we obtain

$${\left[\theta \left({\int }_0^{M(1,\frac{1}{2})}\phi \left(s\right)ds\right)\right]}^k={\left[\theta \left({\int }_0^{\frac{5}{8}}\frac{1}{3}ds\right)\right]}^k={\left[\theta \left(\frac{5}{24}\right)\right]}^{\frac{1}{\sqrt{2}}}\approx 1.38.$$

Therefore, we conclude that

$$\begin{array}{c}\hfill \theta \left({\int }_0^{\wp \left(f1,f\frac{1}{2}\right)}\phi \left(s\right)ds\right)\le {\left[\theta \left({\int }_0^{M(1,\frac{1}{2})}\phi \left(s\right)ds\right)\right]}^k.\end{array}$$

Hence, the contractive condition of Theorem 7 is satisfied and Γ is an integral Khan-type multivalued θ-contraction with respect to the generalized dynamic process $\Xi \left(f,\Gamma ,2\right),$ so that 0 is a CFP of f and Γ as the form $0=f0\in \Gamma 0.$

• Otherwise, we have

  $$\begin{array}{c}\hfill \theta \left({\int }_0^{\wp \left(f{\xi }_j,f{\xi }_{j+1}\right)}\phi \left(s\right)ds\right)>{\left[\theta \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right]}^k.\end{array}$$

  For instance, if ${\xi }_1=0$ and ${\xi }_2=2,$ then we obtain

  $$\begin{array}{ccc}\hfill \theta \left({\int }_0^{\wp \left(f{\xi }_1,f{\xi }_2\right)}\phi \left(s\right)ds\right)& =& \theta \left({\int }_0^{\wp \left(f0,f2\right)}\phi \left(s\right)ds\right)\hfill \\ & =& \theta \left({\int }_0^{\wp \left(0,1\right)}\frac{1}{3}ds\right)=e^{\frac{1}{\sqrt{3}}}\approx 1.8.\hfill \end{array}$$

  Thereafter, we have

  $$M(0,2)=\frac{\wp \left(f0,\Gamma 0\right).\wp \left(f0,\Gamma 2\right)+\wp \left(f2,\Gamma 2\right).\wp \left(f2,\Gamma 0\right)}{max\{\wp \left(f0,\Gamma 2\right),\wp \left(f2,\Gamma 0\right)\}}=1.$$

  So that for $k=\frac{1}{\sqrt{2}},$ we obtain

  $${\left[\theta \left({\int }_0^{M(0,2)}\phi \left(s\right)ds\right)\right]}^k={\left[\theta \left({\int }_0^1\frac{1}{3}ds\right)\right]}^{\frac{1}{\sqrt{2}}}=e^{\frac{1}{\sqrt{6}}}\approx 1.5.$$

  Therefore, we conclude that

  $$\begin{array}{c}\hfill \theta \left({\int }_0^{\wp \left(f0,f2\right)}\phi \left(s\right)ds\right)>{\left[\theta \left({\int }_0^{M\left(0,2\right)}\phi \left(s\right)ds\right)\right]}^k.\end{array}$$

  That is, the contractive condition of Theorem 7 is not satisfied for some ${\xi }_j,{\xi }_{j+1}\in \Delta$.

4. Corollaries

Corollary 1. 

Let $\Gamma :\Delta \to CB\left(\Delta \right)$ be a mapping on a MS $\left(\Delta ,\wp \right).$ Then, Γ is said to be an integral Khan-type multivalued $\left(\psi ,\varphi \right)$-contraction with respect to dynamic process $\Xi \left(\Gamma ,{\xi }_0\right),$ ${\xi }_0\in \Delta$ if there exist $\varphi \in \mathsf{\Phi },$ $\psi \in \mathsf{\Psi }$ and $\phi \in \mathsf{{\rm Y}}$ such that for all ${\xi }_j,{\xi }_{j+1}\in \Xi \left(\Gamma ,{\xi }_0\right)$ and $j\in \mathbb{N}.$

$\left(i\right)$

 If $max\left\{\wp \left({\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left({\xi }_{j+1},\Gamma {\xi }_j\right)\right\}\ne 0,$ then $\Gamma {\xi }_j\ne \Gamma {\xi }_{j+1}$ and

$$\begin{array}{c}\hfill H_{\wp }\left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)>0\Rightarrow \varphi \left({\int }_0^{H_{\wp }\left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right],\end{array}$$

(17)

where

$$M\left({\xi }_{j-1},{\xi }_j\right)=\frac{\wp \left({\xi }_{j-1},\Gamma {\xi }_{j-1}\right).\wp \left({\xi }_{j-1},\Gamma {\xi }_j\right)+\wp \left({\xi }_j,\Gamma {\xi }_j\right).\wp \left({\xi }_j,\Gamma {\xi }_{j-1}\right)}{max\{\wp \left({\xi }_{j-1},\Gamma {\xi }_j\right),\wp \left({\xi }_j,\Gamma {\xi }_{j-1}\right)\}}.$$

(18)

(ii)

 If $max\left\{\wp \left({\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left({\xi }_{j+1},\Gamma {\xi }_j\right)\right\}=0,$ then $\Gamma {\xi }_j=\Gamma {\xi }_{j+1}$.

• Hence, Γ has an FP ${\xi }^{*}$ such that ${\xi }^{*}\in \underset{j\to \infty }{lim}\Gamma {\xi }_j=\Gamma {\xi }^{*}.$

Corollary 2. 

Let $\Gamma :\Delta \to \Delta$ be a mapping on a MS $\left(\Delta ,\wp \right).$ Then, Γ is said to be an integral Khan-type $\left(\psi ,\varphi \right)$-contraction with respect to dynamic process $\Xi \left(\Gamma ,{\xi }_0\right),$ ${\xi }_0\in \Delta$ if there exist $\varphi \in \mathsf{\Phi },$ $\psi \in \mathsf{\Psi }$, and $\phi \in \mathsf{{\rm Y}}$ such that for all ${\xi }_j,{\xi }_{j+1}\in \Xi \left(\Gamma ,{\xi }_0\right)$ and $j\in \mathbb{N}.$

$\left(i\right)$

 If $max\left\{\wp \left({\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left({\xi }_{j+1},\Gamma {\xi }_j\right)\right\}\ne 0,$ then $\Gamma {\xi }_j\ne \Gamma {\xi }_{j+1}$ and

$$\begin{array}{c}\hfill \wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)>0\Rightarrow \varphi \left({\int }_0^{\wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right],\end{array}$$

(19)

where $M\left({\xi }_{j-1},{\xi }_j\right)$ is defined as in (18).

(ii)

 If $max\left\{\wp \left({\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left({\xi }_{j+1},\Gamma {\xi }_j\right)\right\}=0,$ then $\Gamma {\xi }_j=\Gamma {\xi }_{j+1}$.

• Hence, Γ has an FP ${\xi }^{*}$ such that ${\xi }^{*}=\underset{j\to \infty }{lim}\Gamma {\xi }_j=\Gamma {\xi }^{*}.$

Corollary 3. 

Let $\Gamma :\Delta \to CB\left(\Delta \right)$ be a mapping on a MS $\left(\Delta ,\wp \right).$ Then, Γ is said to be an integral Khan-type multivalued θ-contraction with respect to dynamic process $\Xi \left(\Gamma ,{\xi }_0\right),$ ${\xi }_0\in \Delta$ if there exist $k\in \left(0,1\right),$ $\theta \in \mathsf{\Pi }$, and $\phi \in \mathsf{{\rm Y}}$ such that for all ${\xi }_j,{\xi }_{j+1}\in \Xi \left(\Gamma ,{\xi }_0\right)$ and $j\in \mathbb{N}.$

$\left(i\right)$

 If $max\left\{\wp \left({\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left({\xi }_{j+1},\Gamma {\xi }_j\right)\right\}\ne 0,$ then $\Gamma {\xi }_j\ne \Gamma {\xi }_{j+1}$ and

$$\begin{array}{c}\hfill H_{\wp }\left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)>0\Rightarrow \theta \left({\int }_0^{H_{\wp }\left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le {\left[\theta \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right]}^k,\end{array}$$

(20)

where $M\left({\xi }_{j-1},{\xi }_j\right)$ is defined as in (18).

(ii)

 If $max\left\{\wp \left({\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left({\xi }_{j+1},\Gamma {\xi }_j\right)\right\}=0,$ then $\Gamma {\xi }_j=\Gamma {\xi }_{j+1}$.

• Hence, Γ has an FP ${\xi }^{*}$ such that ${\xi }^{*}\in \underset{j\to \infty }{lim}\Gamma {\xi }_j=\Gamma {\xi }^{*}.$

Corollary 4. 

Let $\Gamma :\Delta \to \Delta$ be a mapping on a MS $\left(\Delta ,\wp \right).$ Then, Γ is said to be an integral Khan-type θ-contraction with respect to dynamic process $\Xi \left(\Gamma ,{\xi }_0\right),$ ${\xi }_0\in \Delta$ if there exist $k\in \left(0,1\right),$ $\theta \in \mathsf{\Pi }$, and $\phi \in \mathsf{{\rm Y}}$ such that for all ${\xi }_j,{\xi }_{j+1}\in \Xi \left(\Gamma ,{\xi }_0\right)$ and $j\in \mathbb{N}.$

$\left(i\right)$

 If $max\left\{\wp \left({\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left({\xi }_{j+1},\Gamma {\xi }_j\right)\right\}\ne 0,$ then $\Gamma {\xi }_j\ne \Gamma {\xi }_{j+1}$ and

$$\begin{array}{c}\hfill \wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)>0\Rightarrow \theta \left({\int }_0^{\wp \left(\Gamma {\xi }_j,\Gamma {\xi }_{j+1}\right)}\phi \left(s\right)ds\right)\le {\left[\theta \left({\int }_0^{M\left({\xi }_{j-1},{\xi }_j\right)}\phi \left(s\right)ds\right)\right]}^k,\end{array}$$

(21)

where $M\left({\xi }_{j-1},{\xi }_j\right)$ is defined as in (18).

(ii)

 If $max\left\{\wp \left({\xi }_j,\Gamma {\xi }_{j+1}\right),\wp \left({\xi }_{j+1},\Gamma {\xi }_j\right)\right\}=0,$ then $\Gamma {\xi }_j=\Gamma {\xi }_{j+1}$.

• Hence, Γ has an FP ${\xi }^{*}$ such that ${\xi }^{*}=\underset{j\to \infty }{lim}\Gamma {\xi }_j=\Gamma {\xi }^{*}.$

5. An Application

Nonlinear fractional differential equations (NFDEs) have always been of key importance in dynamical programmings and engineering problems. A lot of authors have used FP techniques for solving NFDEs; for more, see [21,22,23,24,25]. In this framework of studying, we apply Corollary 2 to find the analytical solution for the following NFDE. In particular, we define the Liouville–Caputo FDE of function $\xi :\left[0,1\right)\to \mathbb{R}$ by

$$D^{\vartheta }\left(\xi \left(t\right)\right)=\frac{1}{\omega (j-\vartheta )}{\int }_0^t{\left(t-s\right)}^{j-\vartheta -1}{\xi }^{\left(j\right)}\left(s\right)ds,\left(j-1<\vartheta <j,j=\left[\vartheta \right]+1\right)$$

order $\vartheta >0,$ where $\left[\vartheta \right]$ denotes the integer part of the positive real number and $\omega$ is a gamma function. Now, we consider the following NFDE:

$$\left\{\begin{array}{cc}{D}^{\vartheta }\left(\xi \left(t\right)\right)+\Im (t,\xi \left(t\right)=0,t\in I=[0,1],& \vartheta <1\\ \xi \left(0\right)=\xi \left(1\right)=0\end{array}\right.,$$

(22)

where $\Im :I\times \mathbb{R}\to \mathbb{R}$ is a continuous function. So, the NFDE (22) is equivalent to the following integral equation (IE) $\forall t\in I:$

$$\xi \left(t\right)={\int }_0^1\mho (t,s)\Im (s,\xi \left(s\right))ds.$$

(23)

The Green’s function $\mho :I\times I\to \mathbb{R}$ associated with (23) is defined as

$$G(s,t)=\left\{\begin{array}{cc}{\left(t(1-s)\right)}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1},& 0\le s\le t\le 1\\ \frac{{\left(t(1-s)\right)}^{\alpha -1}}{\vartheta \left(\alpha \right)},& 0\le t\le s\le 1\end{array}\right..$$

(24)

Let $\Delta =C(I,\mathbb{R})$ be the space of all continuous functions defined on I equipped with $\wp :\Delta \times \Delta \to {\mathbb{R}}^{+}$, so that

$$\wp \left({\xi }_1,{\xi }_2\right)=\parallel {\xi }_1\left(t\right)-{\xi }_2{\left(t\right)\parallel }_{\infty }=\underset{t\in I}{sup}|{\xi }_1\left(t\right)-{\xi }_2\left(t\right)|,\forall {\xi }_1,{\xi }_2\in \Delta .$$

Distinctly, $\left(\Delta ,\wp \right)$ is a complete MS. Now, we consider the NFDE (22) under the two postulates below:

(1)

There exists a constant $\tau \ge 1$, such that for all $q_1,q_2\in \mathbb{R}$, and $t\in I,$ we have

$$\begin{array}{c}\hfill |\Im (t,q_1)-\Im (t,q_2)|\le \frac{e^{\tau }}{\tau }M(q_1,q_2),\end{array}$$

where

$$M(q_1,q_2)=\frac{|q_1-\Gamma q_1|.|q_1-\Gamma q_2|+|q_2-\Gamma q_2|.|q_2-\Gamma q_1|}{max\left\{\right|q_1-\Gamma q_2|,|q_2-\Gamma q_1\left|\right\}}.$$

(2)

There is $\tau \ge 1$ so that $\underset{t,s\in I}{max}{\int }_0^1\mho (t,s)ds\le \frac{1}{\tau }.$

Now, we begin with the main theorem in this section.

Theorem 8. 

We mention that the NFDE (22) has at least one solution under the postulates (1) and (2), if and only if the operator Γ has an FP.

Proof. 

Define the operator $\Gamma :\Delta \to \Delta$ using

$$\Gamma \xi \left(t\right)={\int }_0^1\mho (t,s)\Im (s,\xi \left(s\right))ds,$$

(25)

for all $t\in I.$ Clearly, the solution of the IE (23) is equivalent to finding an FP of the operator $\Gamma .$ According to the above assumptions, for all ${\xi }_1,{\xi }_2\in \Delta$, and $t\in I,$ we obtain

$$\begin{array}{ccc}\hfill |\Gamma {\xi }_1\left(t\right)-\Gamma {\xi }_2\left(t\right)|& =& |{\int }_0^1\mho (t,s)\Im (s,{\xi }_1\left(s\right))d-{\int }_0^1\mho (t,s)\Im (s,{\xi }_1\left(s\right))ds|\hfill \\ & \le & {\int }_0^1\left[|\mho (t,s)\Im (s,{\xi }_1\left(s\right))-\mho (t,s)\Im (s,{\xi }_2\left(s\right))|\right]ds\hfill \\ & \le & {\int }_0^1\mho (t,s)\frac{e^{\tau }}{\tau }M\left({\xi }_1,{\xi }_2\right)ds\hfill \\ & =& \left[\frac{e^{\tau }}{\tau }M\left({\xi }_1,{\xi }_2\right)\right]\left[{\int }_0^1\mho (t,s)ds\right].\hfill \end{array}$$

Consequently, we have

$$\begin{array}{c}\hfill \underset{t\in I}{max}|\Gamma {\xi }_1\left(t\right)-\Gamma {\xi }_2\left(t\right)|\le \left[\frac{e^{\tau }}{\tau }M\left({\xi }_1,{\xi }_2\right)\right]\left[\underset{t,s\in I}{max}{\int }_0^1\mho (t,s)ds\right],\end{array}$$

implying that

$$\begin{array}{c}\hfill \wp \left(\Gamma {\xi }_1,\Gamma {\xi }_2\right)\le \frac{1}{{\tau }^2{e}^{\tau }}M\left({\xi }_1,{\xi }_2\right).\end{array}$$

We choose the functions $\psi \left(t\right)=\frac{1}{{\tau }^2{e}^{\tau }}t\in \mathsf{\Psi }$ and $\varphi \left(t\right)=t\in \mathsf{\Phi }$, such that

$$\varphi \left(\wp \left(\Gamma {\xi }_1,\Gamma {\xi }_2\right)\right)\le \psi \left[\varphi \left(M\left({\xi }_1,{\xi }_2\right)\right)\right].$$

Furthermore, via a contraction condition in Corollary 2 with considering $\phi \left(s\right)=1$ for all $s\in \mathbb{R},$ we obtain

$$\varphi \left({\int }_0^{\wp \left(\Gamma {\xi }_1,\Gamma {\xi }_2\right)}\phi \left(s\right)ds\right)\le \psi \left[\varphi \left({\int }_0^{M\left({\xi }_1,{\xi }_2\right)}\phi \left(s\right)ds\right)\right].$$

Thus, all stipulations of Theorem 8 are held and the operator $\Gamma$ has an FP. That is, the Equation (22) has at least one solution. □

Example 5. 

Let’s consider the following NIE:

$$\xi \left(t\right)=\frac{e^{-\tau }}{12\tau }{\int }_0^1{\left(t(1-s)\right)}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1}\xi \left(s\right)ds,0\le s\le t\le 1.$$

(26)

Let $\Gamma :\Delta \to \Delta ,$ be defined by

$$\Gamma \xi \left(t\right)=\frac{e^{-\tau }}{12\tau }{\int }_0^1{\left(t(1-s)\right)}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1}\xi \left(s\right)ds,0\le s\le t\le 1.$$

By specifying $\mho \left(t,s\right)=\frac{1}{12}{\int }_0^1{\left(t(1-s)\right)}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1}$, and $\Im \left(s,\xi \left(s\right)\right)=\frac{e^{-\tau }}{\tau }\xi \left(s\right)$ in Theorem 8, we find that the NIE (26) has at least one solution in $\Delta .$ Now, we have

$$\begin{array}{ccc}\hfill |\Im \left(s,{\xi }_1\left(s\right)\right)-\Im \left(s,{\xi }_2\left(s\right)\right)|& =& |\frac{e^{-\tau }}{\tau }{\xi }_1\left(s\right)-\frac{e^{-\tau }}{\tau }{\xi }_2\left(s\right)|\hfill \\ & \le & \frac{2e^{-\tau }}{\tau }|{\xi }_1\left(s\right)-{\xi }_2\left(s\right)|\hfill \\ & =& \frac{2e^{-\tau }}{\tau }\wp \left({\xi }_1,{\xi }_2\right)\hfill \\ & \le & \frac{2e^{-\tau }}{\tau }M\left({\xi }_1,{\xi }_2\right).\hfill \end{array}$$

For $\tau =2,$ $\alpha =2$, and $0\le s\le t\le 1,$ we obtain

$$\begin{array}{c}\hfill \underset{t,s\in I}{max}{\int }_0^1\mho (t,s)ds=\underset{t,s\in I}{max}{\int }_0^1\frac{1}{12}\left[{\left(t(1-s)\right)}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1}\right]ds\le \frac{1}{2}.\end{array}$$

Therefore, all conditions of Theorem 8 are satisfied, so that the mapping $\Gamma$ has an FP that is a solution to the problem (26).

6. Conclusions

This study developed FP outcomes for multivalued mappings that satisfy both the integral Khan-type multivalued $\left(\psi ,\varphi \right)$-contraction and the integral multivalued $\theta$-contraction, along with a generalized dynamic process in a complete MS. We also presented several corollaries derived from our main findings. These main outcomes are substantiated by demonstrative examples and an application focusing on the existence and uniqueness of a solution to the NFDE, serving as a practical review of our core results. Moreover, this work introduced the notion of a dynamic process as a new avenue for establishing FP results for the said contractions. By utilizing a Khan-type contraction and a $\theta$-contraction, we illustrated the relationship between integral-type contractions and a generalized dynamic process.