Certain new iteration of hybrid operators with contractive $M$ -dynamic relations

1.   Introduction and preliminaries

After the Banach fixed point theorem, many interesting generalizations have been established by various authors ^{[1,3,4,5,6,8,9]}. The generalizations were presented either by changing the axioms of the metric space, or by modifying the contractive condition. However, a new debate was instigated when using the idea of the Pompeiu-Hausdorff metric. Namely, Nadler ^[20] discussed the Banach fixed point theorem for multivalued mappings. In continuation to this, Patle et al. ^[21] presented the idea of an $\widetilde{m}$-Pompeiu-Hausdorff metric, which was further promoted as an $\widetilde{m}$-metric, i.e., let $(\Omega, d_{m})$ be an $\widetilde{m}$-metric space. For $s_{1}\in \Omega$ and $\Phi _{1}\subseteq \Omega$,

Define the Pompeiu-Hausdorff metric $\mathcal{H}$ induced by $d_{m}$ on $CB(\Omega)$ as follows:

for all $\Phi _{1}, \Phi _{2}\in CB(\Omega)$, where

An element $s\in \Omega$ is known as a fixed point of a set-valued mapping $\rho$: $\Omega \rightarrow CB(\Omega)$ such that $s\in \rho \left(s\right)$. Furthermore, let $\rho _{1}$: $\Omega \rightarrow \Omega$ and $\rho _{2}$: $\Omega \rightarrow CB(\Omega)$, and a point $s\in \Omega$ is called a coincidence point of $\rho _{1}$ and $\rho _{2}$ if $\rho _{1}s\in \rho _{2}s.$ The set of all such elements is denoted by $\hat{C}\left(\rho _{1}, \rho _{2}\right)$. If for some element $s\in \Omega$, we have $s = \rho _{1}s\in \rho _{2}s$, then an element $s$ is called a common fixed point of $\rho _{1}$ and $\rho _{2}.$ A mapping $\rho$: $\Omega \rightarrow CB(\Omega)$ is known as continuous at point $c\in \Omega,$ if for any sequence $\left \{ s_{n}\right \}$ in $\Omega$ with

we have

This article is divided into three sections: Section 1 deals with the fundamental preliminaries and results that pertain to our main work. In section 2, we present some theorems dealing with Chatterjea type $F$ -contractions for hybrid operators based on an $M$-dynamic iterative process in the setting of $\widetilde{m}$-metric spaces. An example and some corollaries are developed as consequences of the obtained theorems. This portion also has graphs that best illustrate our results for the better understanding of readers. Section 3 gives an application of our results in finding a solution of a multistage system. The pivotal role of functional equations in a dynamic system related to a multistage process is stated. Another application is also stipulated discussing the solution of integral equations. At last, a summary of the article is described in the conclusion section. Throughout our work, denote by $N(\Omega)$, $CL(\Omega)$, $CB(\Omega)$ and $K(\Omega)$ the collections of all the following non-empty: subsets of $\Omega$, closed subsets of $\Omega$, bounded closed subsets of $\Omega$ and compact subsets of $\Omega$, respectively.

The multivalued version of the Banach fixed point theorem was initiated by Nadler ^[20], and some lemmas are useful for the rest: Let $\left(\Omega, d\right)$ be a complete metric space, and $\Gamma$: $\Omega \rightarrow CB(\Omega)$ is known as a Nadler contraction if there is $\delta \in \lbrack 0, 1)$ such that

Then, $\Gamma$ has a fixed point.

Lemma 1.1. ^[20] Let $\left(\Omega, d\right)$ be a metric space, $\beta \in CB(\Omega)$ and $s\in \Omega$. Then, for each $\epsilon > 0,$ there is $v\in \beta$ such that

Lemma 1.2. ^[25] Let $\left(\Omega, d\right)$ be a metric space and $\beta$, $\beta ^{\ast }\in CB(\Omega)$ with $\mathcal{H}(\beta, \beta ^{\ast }) > 0.$ Then, for every $h > 1$ and $s\in \beta$, there is $\nu = \nu (s)\in \beta ^{\ast }$ such that

Later, many papers dealing with multivalued mappings and hybrid operators appeared. For more details, see ^{[2,9,10,11,13,14,17,18,19,20,21,22,23,24]}. In 1995, Matthews ^[16] established the theory of a partial metric space and improved the Banach fixed point theorem in the context of partial metric spaces. After, based on the result of Matthews ^[16], Asadi et al. ^[8] introduced a new idea of an $\widetilde{m}$-metric space and studied its topological behavior. They also established fixed point results, which are generalizations of Banach and Kannan types fixed point theorems.

Now, we present some basic definitions and results as follows:

Definition 1.3. ^[8] An $\widetilde{m}$-metric space on a non-empty set $\Omega$ is a function $d_{m}$: $\Omega \times \Omega \rightarrow R^{+}\cup \left \{ 0\right \}$, such that for all $s_{1}, s_{2}, s_{3}\in \Omega$, the following conditions hold:

$\left(m_{\imath }\right) \; s_{1} = s_{2}$ if and only if $d_{m}\left(s_{1}, s_{1}\right) = d_{m}\left(s_{2}, s_{2}\right) = d_{m}\left(s_{1}, s_{2}\right)$;

$\left(m_{\imath \imath }\right) \; m_{s_{1}s_{2}}\leq d_{m}\left(s_{1}, s_{2}\right)$;

$\left(m_{\imath \imath \imath }\right) \; d_{m}\left(s_{1}, s_{2}\right) = d_{m}\left(s_{2}, s_{1}\right)$;

$\left(m_{\imath v}\right) \; d_{m}\left(s_{1}, s_{2}\right) -m_{s_{1}s_{2}}\leq \lbrack \left(d_{m}\left(s_{1}, s_{3}\right) -m_{s_{1}s3}\right) +(d_{m}\left(s_{3}, s_{2}\right) -m_{s_{3}s_{2}})]$.

The pair $\left(\Omega, d_{m}\right)$ is known as an $\widetilde{m}$ -metric space. Note that herein we define $m_{s_{1}s_{2}}$ and $M_{s_{1}s_{2}}$ by

Remark 1.4. ^[8] Every partial metric is an $\widetilde{m}$-metric, but the converse may not be hold true. For example, the following setting is an $\widetilde{m}$-metric, but it is not a partial metric. Let $\Omega = \left \{ 1, 2, 3\right \}$ and take $d_{m}\left(1, 1\right) = 1, \; d_{m}\left(2, 2\right) = 9, \; d_{m}\left(3, 3\right) = 5, \; d_{m}\left(1, 2\right) = d_{m}\left(2, 1\right) = 10, \; d_{m}\left(1, 3\right) = d_{m}\left(3, 1\right) = 7$ and $d_{m}\left(2, 3\right) = d_{m}\left(3, 2\right) = 7.$

Now, we discuss some basic concepts: convergence, Cauchyness, completeness and open balls of the $\widetilde{m}$-metric space $\left(\Omega \text{ }, d_{m}\right)$.

A sequence $\left \{ s\left(\imath \right) \right \}$ in $\left(\Omega \text{ }, d_{m}\right)$ is known as $\widetilde{m}$-convergent to $s\in \Omega$ iff

and an $\widetilde{m}$-Cauchy sequence if $\lim_{\imath, j\rightarrow \infty }\left(d_{m}(s_{\imath }, s_{j})-m_{s_{\imath }s_{j}}\right)$ exists and is finite. $\left(\Omega, d_{m}\right)$ is $\widetilde{m}$-complete if every $\widetilde{m}$-Cauchy sequence is $\widetilde{m}$-convergent to an element $s\in \Omega,$ and then each $\widetilde{m}$-metric space on $\Omega$ generates a $T_{0}$ topology denoted by $\bf{ \pmb{\mathsf{ τ}} }\left(\widetilde{m} \right)$. The base of topology $\bf{ \pmb{\mathsf{ τ}} }\left(\widetilde{m}\right)$ on $\Omega$ is the class of open $b^{m}$-balls:

where

for all $s_{1}\in \Omega$ and $r > 0$.

Lemma 1.5. ^[8] Let $\left \{ s\left(j\right) \right \}$ be a sequence in an $\widetilde{m}$-metric space. Let there be $\ell ^{m}\in \lbrack 0, 1)$ so that $d_{m}(s_{\imath +1}, s_{\imath })\leq \ell _{m}d^{m}(s_{\imath }, s_{\imath -1})$ for every $\imath \geq 1$. Then,

(L1) $\lim_{n\rightarrow \infty }d_{m}(s_{\imath }, s_{\imath -1}) = 0;$

(L2) $\lim_{n\rightarrow \infty }d_{m}(s_{\imath }, s_{\imath }) = 0;$

(L3) $\lim_{n\rightarrow \infty }m_{s_{j}s_{\imath }} = 0;$

(L4) $s_{\imath }$ is an $\widetilde{m}$-Cauchy sequence.

Herein, we start by recognizing some basic approaches of a dynamic iterative process in the context of $\widetilde{m}$-metric spaces. Consider a function $\Psi$: $\Omega \rightarrow CB(\Omega)$ such that

for each $j\geq 1.$ The term $\check{D}(\Psi, s_{0})$ is an $M$-dynamic iterative process of the mapping $\Psi$ having the starting point $s_{0}.$ The $M$-dynamic iterative process. $\check{D}(\Psi, s_{0})$ is usually written as $s_{j}$, and let $\Psi _{1}$: $\Omega \rightarrow \Omega$ and $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ be such that

for every $j\geq 1.$ The set $\check{D}(\Psi _{1}, \Psi _{2}, s_{0})$ is said to be an $M$-dynamic iterative process of hybrid $\left(\Psi _{1}, \Psi _{2}\right)$ having the starting point $s_{0}$. The $M$-dynamic iterative process. $\check{D}(\Psi _{1}, \Psi _{2}, s_{0})$ is usually written as $\Psi _{1}\left(s_{j}\right)$. For more details, see ^[7,12,15].

Example 1.6. Let $\Omega = [0, \infty)$. Define $\Psi _{1}$: $\Omega \rightarrow \Omega$ and $\Psi _{2}$: $\Omega \rightarrow CB\left(\Omega \right)$ by $\Psi _{1}\left(s\right) = \frac{s}{2}, \; \Psi _{2}\left(s\right) = [0, \frac{s}{2}]$, respectively. A sequence $\{s_{i}\}$ is given as $s_{i} = s_{0}g^{i-1}$ for all $n\in N$ with $s_{0} = 2$ and $g = \frac{1}{2}$. Then, we have Table 1.

In light of the above table, one asserts that

is an $M$-dynamic iterative process of hybrid operator $\Psi _{1}$ and $\Psi _{2}$ starting from $\varepsilon _{0} = 2.$

In 2012, Wardowski ^[26] introduced the concept of $F$-contractions as follows:

Definition 1.7. ^[26] Let $K$: $(0, \infty)\longrightarrow (-\infty, +\infty)$ be such that:

$\left(K_{\imath }\right) \; K$ is strictly increasing, for every $s_{1}, s_{2}\in (0, \infty)$ such that $s_{1} < s_{2}$ implies $K\left(s_{1}\right) < K\left(s_{2}\right).$

$\left(K_{\imath \imath }\right)$ For each positive sequence $\left \{ s\left(j\right) \right \}$,

$\left(K_{\imath \imath \imath }\right)$ There is $k\in \left(0, 1\right)$ such that $\lim_{c\rightarrow 0}c^{k}K(c) = 0$.

Herein, $\nabla$ is the set of functions verifying $\left(K_{\imath }\right)$–$\left(K_{\imath \imath \imath }\right)$.

Let $(\Omega, d)$ be a metric space. $\Gamma _{2}$: $\Omega \rightarrow \Omega$ is known as a $F$-contraction if there is $\tau > 0$ such that

for each $s_{1}, s_{2}\in \Omega.$

Definition 1.8. ^[7] Consider $\Gamma _{1}$: $\Omega \rightarrow \Omega$ and $\Gamma _{2}$: $\Omega \rightarrow CB(\Omega)$. The pair $(\Gamma _{1}, \Gamma _{2})$ is called weakly compatible if $\Gamma _{1}\left(n\right) = \Gamma _{2}\left(n\right)$ for some $n\in \Omega$, and then $\Gamma _{1}\Gamma _{2}\left(n\right) = \Gamma _{2}\Gamma _{1}\left(n\right).$

In this manuscript, we will generalize the following result with respect to $M$-dynamic iterative process $\check{D}(\Psi _{1}, \Psi _{2}, s_{0})$ for hybrid operators: Let $\left(\Omega, d_{m}\right)$ be a complete $\widetilde{m}$-metric space and let $\Gamma$: $\Omega \rightarrow CB(\Omega)$ be a multivalued mapping. Suppose there is $\ell _{m}\in (0, \frac{1}{2})$ such that

for all $s, v\in \Omega.$ Then, $f$ possesses a fixed point ^[21]. In particular, we will consider Chatterjea type contractions based on $M$-dynamic iterative process: $\check{D}(\Psi _{1}, \Psi _{2}, s_{0})$ and its graphical interpretations. The main approach is to find new common fixed points for hybrid operators.

2.   Main results

First, in order to give our new generalized definition, we have the following:

Definition 2.1. Let $\left(\Omega, d_{m}\right)$ be an $\widetilde{m}$ -metric space and let $\Psi _{1}$: $\Omega \rightarrow \Omega.$ The mapping $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is called a set-valued $F$-Chatterjea type contraction with respect to the $M$-dynamic iterative process: $\check{D}(\Psi _{1}, \Psi _{2}, s_{0})$, where $s_{0}\in \Omega$, if there are $K\in \nabla$ and $\tau$: $(0, \infty)\rightarrow (0, \infty)$ such that

where

for all $s_{\imath }, s_{\imath +1}\in \check{D}(\Psi _{1}, \Psi _{2}, s_{0})$, $d_{m}\left(\Psi _{1}\left(s_{\imath }\right), \Psi _{1}\left(s_{\imath +1}\right) \right) > 0$ and $\ell _{m}\in \left[ 0, \frac{1}{2}\right)$.

Remark 2.2. Observe that we will only consider the $M$-dynamic iterative process. $s_{\imath }\in \check{D}(\Psi _{1}, \Psi _{2}, s_{0})$ that satisfies the following:

for each integer $\imath \geq 2$. If the process does not satisfy (2.2), then there exists some $\imath _{0}\in N$ such that

imply that

which implies the existence of a common fixed point.

Our main result is given as follows:

Theorem 2.3. Let $\left(\Omega, d_{m}\right)$ be a complete $\widetilde{m}$-metric space and $\Psi _{1}$: $\Omega \rightarrow \Omega$. Assume $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is a set-valued F-Chatterjea type contraction with respect to the $M$-dynamic iterative process: $\check{D} (\Psi _{1}, \Psi _{2}, s_{0})$. Then, the hybrid pair $(\Psi _{1}, \Psi _{2})$ possesses a common fixed point, say, $c\in \Omega$.

Proof. Let $s_{0}\in \Omega$ be an arbitrary element. In the presence of an $M$ -dynamic iterative process, we obtain the following:

If $s_{\imath _{0}} = s_{\imath _{0+1}}$ for some $\imath _{0}\in N$, the proof is done. Now, let us take for $\imath \in \mathbb{N}$,

If we let $s_{\imath }\neq s_{\imath +1}$ for every $\imath \in \mathbb{N}$, then owing to (2.1),

Next, we have to examine the following:

for all $\imath \in \mathbb{N}$.

Assume, on the contrary, that there exists $\sigma \in \mathbb{N}$ such that $\zeta (\sigma +1)\geq \zeta (\sigma)$. By (2.4), one writes

Using $(\Psi _{\imath })$, one writes

That is,

This implies that

Take

Since $\ell _{m}\in \lbrack 0, \frac{1}{2})$, we have $0\leq \vartheta _{m} < 1$. Therefore, $\zeta (\sigma +1) < \zeta (\sigma)$, which is a contradiction. Hence, (2.5) holds. Thus, the real sequence

is decreasing, so there is $\Upsilon \geq 0$ so that

Next, we have to prove that $\Upsilon = 0$. Suppose, on the contrary, that $\Upsilon > 0$, and for every $\varepsilon > 0$, there is $v\in N$ such that

Using $(K_{\imath })$, we have

Since $K$ is a strictly increasing-mapping w.r.t. an $M$-dynamic iterative process, one writes

By $(\Psi _{\imath }$), we have

That is,

Hence,

Again, $(\Psi _{\imath })$ and $0\leq \vartheta _{m} = \frac{\ell _{m}}{1-\ell _{m}} < 1$. Therefore,

for all $v\in N$.

Next, by given hypothesis on $\tau$ there is $\kappa > 0$ and $v\in N$ such that

for all $v > v_{0}$. Thus, we obtain for all $v > v_{0}$ with setting of $\delta = \frac{1}{1-\ell _{m}}$the following inequalities:

Letting $\imath \rightarrow \infty$ along with $\left(\Psi _{ii}\right)$, we have $\ \lim_{\imath \rightarrow \infty }\Psi (\zeta (v+\imath)) = -\infty$ in such a way that

Then, there exists $\imath _{1}\in N$ such that $\zeta (v+\imath) < \Gamma$ for all $\imath > \imath _{1}$. This contradicts the definition of $\Gamma$. Thus,

From $(K_{\imath \imath \imath })$, there is $k\in \left(0, 1\right)$ such that

By (2.11), the following holds:

Taking the limit as $\imath \rightarrow \infty$ in (2.14), we have

From (2.14), there is $\imath _{1}\in N$ such that $n\left[ \zeta \left(\imath \right) \right] ^{k}\leq 1$ for all $\imath \geq \imath _{1}$. We have

Next is to prove that $\left \{ s_{\imath }\right \}$ is an $\widetilde{m}$ -Cauchy sequence. For this, we consider $j_{1}, \; j_{2}\in \mathbb{N}$ such that $j_{1} > j_{2}\geq \imath _{1}.$ Using the triangular inequality of $\left(m_{iv}\right)$ and from (2.15), we have

Due to the convergence of the series $\sum_{l = j_{1}}^{\infty }\frac{1}{l^{\frac{1}{k} }},$ letting $\imath \rightarrow \infty$, we get

Hence, $\left \{ s_{\imath }\right \}$ is $\widetilde{m}$-Cauchy in $\left(\Omega, d_{m}\right).$ Owing to the completeness of $\Omega,$ we have $s_{\imath }\rightarrow s$ as $\imath \rightarrow \infty$ for some $s\in \Omega.$ So, we have

and

Due to $(L_{2})$, we obtain $d_{m}\left(s_{\imath }, s_{\imath }\right) \rightarrow 0$ as $\imath \rightarrow \infty$, so

and

By (2.16)–(2.18), we obtain

and

Hence,

for all $\imath \in \mathbb{N}$.

Taking the limit as $\imath \rightarrow \infty$ in the above equation and using (2.18), (2.21) and $(L_{2})$, we have

This yields

We shall show that $d_{m}\left(s, \Psi _{1}(s)\right) = 0.$ Due to $(m_{\imath v})$, we have

for all $\imath \in \mathbb{N}.$ Taking the superior limit as $\imath \rightarrow \infty$ in (2.25) and using (2.16)–(2.18) and (2.24), we have

We get that

By (2.1), we have

That is,

From (2.18), (2.26) and (2.27), we have

Owing to $(m_{\imath })$, we get

so we have $s = \Psi _{1}(s)\in \Psi _{2}\left(s\right).$ Next, let $s^{\ast }$ be another fixed point of $\Psi _{1}$ such that

which is a contradiction. Hence, the hybrid pair $(\Psi _{1}, \Psi _{2})$ has a common fixed point. □

Further, some corollaries are developed as consequences of the obtained theorems of our endeavor.

Corollary 2.4. Let $\left(\Omega, d_{m}\right)$ be a complete $\widetilde{m}$-metric space and $\Psi _{1}$: $\Omega \rightarrow \Omega$. Assume $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is a set-valued $F$-Kannan type contraction, that is,

where

with respect to $M$-dynamic iterative process. $\check{D}(\Psi _{1}, \Psi _{2}, s_{0}), \; d_{m}(\Psi _{1}\left(s_{\imath }\right), \Psi _{1}\left(s_{\imath +1}\right)) > 0$ and $\ell _{m}\in \left[ 0, \frac{1}{2}\right)$. Then, the hybrid pair $(\Psi _{1}, \Psi _{2})$ possesses a common fixed point.

Corollary 2.5. Let $\left(\Omega, d_{m}\right)$ be a complete $\widetilde{m}$-metric space and $\Psi _{1}$: $\Omega \rightarrow \Omega$. Assume $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is a set-valued $F$-Banach contraction, and

with respect to $M$-dynamic iterative process $\check{D}(\Psi _{1}, \Psi _{2}, s_{0})$, $d_{m}\left(\Psi _{1}\left(s_{\imath }\right), \Psi _{1}\left(s_{\imath +1}\right) \right) > 0$ and $\ell _{m}\in (0, 1)$. Then, the pair $(\Psi _{1}, \Psi _{2})$ possesses a common fixed point.

Corollary 2.6. Let $\left(\Omega, d_{m}\right)$ be a complete $\widetilde{m}$ -metric space and $\Psi _{1}$: $\Omega \rightarrow \Omega$. Assume $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is a set-valued F-Banach contractive. If there is $\Delta$: $\kappa \rightarrow \kappa$, a non-negative Lebesgue integrable operator which is summable on each compact subset of $\kappa$ such that

for all $s_{\imath }\in \check{D}(\Psi _{1}, \Psi _{2}, s_{0})$ and for all given $\epsilon > 0$ so that $\int_{0}^{\epsilon }\Delta \left(s\right) \delta s > 0$, then the pair $(\Psi _{1}, \Psi _{2})$ possesses a common fixed point$.$

Theorem 2.7. Let $\left(\Omega, d_{m}\right)$ be an $\widetilde{m}$ -metric space and let $\Psi _{1}$: $\Omega \rightarrow \Omega$ be continuous. The mapping $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is called a set-valued $F$-Chatterjea type contraction with respect to the $M$ -dynamic iterative process. $\check{D}(\Psi _{1}, \Psi _{2}, s_{0})$, where $s_{0}\in \Omega$, if $K\in \nabla$, $\Psi$ is right continuous at $\mathcal{H}\left(\Psi _{2}\left(s_{1}\right), \Psi _{2}\left(s_{2}\right) \right)$, and $\tau$: $(0, \infty)\rightarrow (0, \infty)$ such that

where

for all $s_{1}, s_{2}\in \check{D}(\Psi _{1}, \Psi _{2}, s_{0})$, $\mathcal{H} \left(\Psi _{2}\left(s_{1}\right), \Psi _{2}\left(s_{2}\right) \right) > 0$ and $\ell _{m}\in \left[ 0, \frac{1}{2}\right)$. Then, the hybrid pair $(\Psi _{1}, \Psi _{2})$ has a common fixed point, say, $c\in \Omega$.

Proof. Choose $s_{0}\in \Omega$ as an arbitrary point. Based on the $M$-dynamic iterative process, we have $\Psi _{1}\left(s_{1}\right) \in \Psi _{2}\left(s_{0}\right)$. Assume that $\mathcal{H}\left(\Psi _{2}\left(s_{0}\right), \Psi _{2}\left(s_{1}\right) \right) > 0.$ Since, by hypothesis, $\Psi$ is right continuous at $\mathcal{H}\left(\Psi _{2}\left(s_{0}\right), \Psi _{2}\left(s_{1}\right) \right)$, there is $h > 1$ such that

Since $\Psi _{1}\left(s_{1}\right) \in \Psi _{2}\left(s_{0}\right)$, we derive $d_{m}(\Psi _{1}\left(s_{1}\right), \Psi _{2}\left(s_{1}\right))\leq \mathcal{H}\left(\Psi _{2}\left(s_{0}\right), \Psi _{2}\left(s_{1}\right) \right)$, and thus there is $s^{\ast }\in \Psi _{2}\left(s_{1}\right)$ such that

Setting an element $s_{2}\in \Omega$ such that $\Psi _{1}\left(s_{2}\right) = s^{\ast }$, (2.29) becomes

In the case that

$\Psi _{1}\left(s_{1}\right) \in \Psi _{2}\left(s_{1}\right)$. Hence, the proof is complete. Assume that

Since $\Psi$ is strictly increasing, we have

Owing to (2.31) and applying Theorem (2.3), we easily derive that the pair $(\Psi _{1}, \Psi _{2})$ has a common fixed point. □

Corollary 2.8. Let $\left(\Omega, d_{m}\right)$ be a complete $\widetilde{m}$-metric space and $\Psi _{1}$: $\Omega \rightarrow \Omega$. Assume $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is a set-valued $F$-Kannan type contraction, that is,

where

with respect to $M$-dynamic iterative process. $\check{D}(\Psi _{1}, \Psi _{2}, s_{0}), \; \mathcal{H}(\Psi _{2}\left(s_{1}\right), \Psi _{2}\left(s_{2}\right)) > 0$ and $\ell _{m}\in \left[ 0, \frac{1}{2}\right)$. Then, the hybrid pair $(\Psi _{1}, \Psi _{2})$ possesses a common fixed point.

Corollary 2.9. Let $\left(\Omega, d_{m}\right)$ be a complete $\widetilde{m}$-metric space and $\Psi _{1}$: $\Omega \rightarrow \Omega$. Assume $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is a set-valued F-Banach contraction, and

with respect to $M$-dynamic iterative process $\check{D}(\Psi _{1}, \Psi _{2}, s_{0})$, $\mathcal{H}\left(\Psi _{2}\left(s_{1}\right), \Psi _{2}\left(s_{2}\right) \right) > 0$ and $\ell _{m}\in (0, 1)$. Then, the pair $(\Psi _{1}, \Psi _{2})$ possesses a common fixed point.

Corollary 2.10. Let $\left(\Omega, d_{m}\right)$ be a complete $\widetilde{m}$ -metric space and $\Psi _{1}$: $\Omega \rightarrow \Omega$. Assume $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is a set-valued F-Banach contractive. If there is $\Delta$: $\kappa \rightarrow \kappa$ a non-negative Lebesgue integrable operator which is summable on each compact subset of $\kappa$ such that

for all $s_{\imath }\in \check{D}(\Psi _{1}, \Psi _{2}, s_{0})$ and for all given $\epsilon > 0$ so that $\int_{0}^{\epsilon }\Delta \left(s\right) \delta s > 0$, then the pair $(\Psi _{1}, \Psi _{2})$ possesses a common fixed point$.$

Corollary 2.11. Let $\left(\Omega, d_{m}\right)$ be a complete $\widetilde{m}$ -metric space, and $\Psi _{1}$: $\Omega \rightarrow \Omega$ is continuous. Assume $\Psi _{2}$: $\Omega \rightarrow CB(\Omega)$ is a set-valued F-Banach contractive. If there is $\Delta$: $\kappa \rightarrow \kappa$ a non-negative Lebesgue integrable operator which is summable on each compact subset of $\kappa$ such that

for all $s_{\imath }\in \check{D}(\Psi _{1}, \Psi _{2}, s_{0})$ and for all given $\epsilon > 0$ so that $\int_{0}^{\epsilon }\Delta \left(s\right) \delta s > 0$, then the pair $(\Psi _{1}, \Psi _{2})$ possesses a common fixed point$.$

Next, we present an example to show the benefits of our endeavor.

Example 2.12. Let $\Omega = [0, \infty)$, and let $d_{m}:\Omega \times \Omega \rightarrow \lbrack 0, \infty)$ be defined by

Clearly, $\left(d_{m}, \Omega \right)$ is an $\widetilde{m}$-complete metric space. The mappings $\Psi _{1}$: $\Omega \rightarrow \Omega$, $\Psi _{2}$: $\Omega \rightarrow CB\left(\Omega \right)$, $\Psi$: $(0, \infty)\rightarrow \left(-\infty, +\infty \right)$ and $\tau$: $(0, \infty)\rightarrow (0, \infty)$ are given as

and

for $\alpha \in(0, \infty)$.

A sequence $\{s_{n}\}$ can be defined by $s_{i} = s_{0}g^{i-1}$ for all $i\in N$ with $s_{0} = 2$ and $g = \frac{1}{2}$. Then,

is an $M$-dynamic iterative process of a hybrid pair of mappings $\left(\Psi _{1}, \Psi _{2}\right)$, starting at the point $s_{0} = 2.$ Next, we choose $s_{1} = 1$, $s_{2} = \frac{1}{4}$ and $\ell _{m} = \frac{1}{3}$. Take the $M$-Hausdorff induced by an $\widetilde{m}$-metric under an $M$-dynamic iterative process, defined by

such that

Equivalently, one writes

Owing to $F$-contraction, (2.29) is written as $\tau \left(\alpha \right) \leq \Xi \left(\imath \right)$, where

Hence, by Table 2 and Figure 1, the required hypotheses of Corollary (2.8), regarding $\tau \left(\alpha \right) \leq \Xi \left(\imath \right)$ are satisfied. Here, $0 = \Psi _{1}(0)\in \Psi _{2}(0)$ is a common fixed point of $\Psi _{1}$ and $\Psi _{2}.$

3.   Applications

Herein, this section deals with applications of our obtained results. Two important applications are given below:

3.1.   An application to multistage systems

A study of a decision space and a state space makes up two fundamental parts of $M$-dynamic iterative programming problems. A state space is a family of states regarding initial states, action states and transitional states. So, a state space is the collection of parameters that show different type states. A decision space is the set of possible actions that can be taken to solve the problem. These natures allow us to formulate many problems in mathematical optimization and computer programming. In particular, the problem of $M$-dynamic iterative programming problems related to multistage process reduces to the problem of solving functional equations.

for $\wp _{1}\in s$,

for $\wp _{1}\in s$, where

Suppose $X_{1}$ and $X_{2}$ are Banach spaces. $S\subset X_{1}$ is a state space, and $H\subset X_{2}$ is a decision space. For more, see ^[4]. Let $\sigma \in B\left(S\right)$ be an collection of all bounded real valued functions on $S,$ and defined by $||\sigma || = \sup_{\wp \in S}|\sigma \left(\wp \right) |$. $\left(B\left(s\right), ||.||\right)$ which endowed with $\widetilde{m}$-metric, we have

for all $\sigma, \vartheta \in B\left(S\right)$. Define $X_{1}, X_{2}$: $B\left(S\right) \rightarrow B\left(S\right)$ by

for all $\varpi \in B\left(S\right)$ and $\wp \in S.$ In addition, assume that $\tau$: $(0, \infty)\rightarrow (0, \infty)$ such that

for all $\sigma, \vartheta \in B\left(S\right)$, where $\wp \in S$ and $t\in D.$

Theorem 3.1. Let semi-continuous mappings $X_{1}, X_{2}$: $B\left(s\right) \rightarrow B\left(s\right)$ as defined in (3.4) and (3.5) such that:

$\left(\imath \right) \; D_{1}, \; D_{2}, \; \Psi _{1}$ and $\Psi _{2}$ are continuous and bounded.

$\left(\imath \imath \right)$ For all $\varpi \in B\left(S\right) \; \exists \; t\in B\left(S\right)$ so that

$\left(\imath \imath \imath \right)$ Assume there is $\varpi \in B\left(s\right)$ such that ($X_{1}$ and $X_{2}$ are called weakly compatible),

Then, the functional equations defined by (3.1) and (3.2) possess a bounded solution.

Proof. Let $\left(B\left(S\right), d_{m}\right)$ be a complete $\widetilde{m}$ -metric space, where $d_{m}\left(\sigma, \vartheta \right)$ is the $\widetilde{m}$-metric, as defined by (3.3). Consider an arbitrary $\kappa > 0$ and $\varpi _{1}$, $\varpi _{2}\in B\left(S\right)$, so there are $\wp \in s$ and $t_{1}, t_{2}\in D$ such that

Using (3.7) and (3.10), one writes

which implies that

In the same manner, by (3.8) and (3.9)$,$

This implies

for every $\wp \in s.$ It follows that

This leads to

As a result, all conditions of Corollary (2.8) are fulfilled. Hence, a fixed point $\varpi ^{\ast }\in B(s)$, that is, $\varpi ^{\ast }$ is a bounded solution of (3.4) and (3.5). □

3.2.   An application to integral equations

In this section, we consider the following integral equation:

$\wp \in \left[ \eta _{1}, \eta _{2}\right]$. Herein, $D_{\imath }$: $\left[ \eta _{1}, \eta _{2}\right] ^{2}\times R\rightarrow P^{cv}\left(R\right)$ (set of nonempty compact and convex subsets of $R$)$, \; \imath = 1, 2$, the operator $D_{\imath }\left(\wp, \wp ^{\ast }, \varpi _{\imath }\left(\wp ^{\ast }\right) \right)$ is lower semicontinuous for all $\varpi \in C(\left[ \eta _{1}, \eta _{2}\right], \mathbb{R})$, and $\sigma _{\imath }$: $\left[ \eta _{1}, \eta _{2}\right] \rightarrow R$ is continuous, $\imath = 1, 2.$ Let $X = C(\left[ \eta _{1}, \eta _{2}\right], R)$ be endowed with the $\widetilde{m}$-metric defined by

Define a set-valued operator $\Psi$: $C(\left[ \eta _{1}, \eta _{2}\right], R)\rightarrow CL(C(\left[ \eta _{1}, \eta _{2}\right], R))$ by

where $\wp \in \left[ \eta _{1}, \eta _{2}\right]$, $\imath = 1, 2.$ Let $\varpi _{\imath }\in C(\left[ \eta _{1}, \eta _{2}\right], R)$ and

Consider, for $D_{\imath }$: $\left[ \eta _{1}, \eta _{2}\right] ^{2}\rightarrow P^{cv}\left(R\right),$ by Michael's selection theorem, there is a continuous function $d_{\varpi \imath }{}_{m}$: $\left[ \eta _{1}, \eta _{2}\right] ^{2}\rightarrow R$ defined as follows: $d_{\varpi \imath }{}_{m}\left(\wp, \wp ^{\ast }\right) \in d_{\imath }{}_{m}(\wp, \wp ^{\ast })$, $\wp, \wp ^{\ast }\in \left[ \eta _{1}, \eta _{2}\right]$ and

Precisely, the operator $\Psi (\varpi _{\imath })\neq \emptyset$ is closed.

Theorem 3.2. Let

and the set-valued operator $\Psi, T$: $C(\left[ \eta _{1}, \eta _{2}\right], R)\rightarrow CL(C(\left[ \eta _{1}, \eta _{2}\right], R))$ as given by

$\imath = 1, 2$, be such that the following hold:

$\left(\mathit{\text{i}}\right)$ There exists a continuous and bounded function $\Omega :X\rightarrow R^{+}\cup \left \{ 0\right \}$ such that

for every $\wp, \wp ^{\ast }\in \left[ \eta _{1}, \eta _{2}\right]$, $\varpi _{1}, \varpi _{2}\in X$ and

for each $\wp ^{\ast }\in \left[ \eta _{1}, \eta _{2}\right].$

$\left(\mathit{\text{ii}}\right)$ There exists

such that

implies that

Then, the above Eqs (3.13) and (3.14) possess a bounded solution.

Proof. Let $\varpi _{1}, \varpi _{2}\in X$ be such that $\varkappa _{1}\in \Psi \left(\varpi _{1}\right)$. It follows that

such that

By$\left(\imath \right)$, there exists $\varpi _{2}\left(\wp, \wp ^{\ast }\right) \in d_{\varpi _{2}}{}_{m}(\wp, \wp ^{\ast })$ such that

Consider

$\wp, \wp ^{\ast }\in \left[ \eta _{1}, \eta _{2}\right].$ Since the operator is lower semi-continuous, there exists $\varpi _{2}$: $\left[ \eta _{1}, \eta _{2}\right] ^{2}\times R\rightarrow R$, such that $d_{\varpi _{2}}{}_{m}\left(\wp, \wp ^{\ast }\right) \in T(\wp, \wp ^{\ast })$, for all $\wp, \wp ^{\ast }\in \left[ \eta _{1}, \eta _{2}\right].$ Thus,

for all $\wp \in \left[ \eta _{1}, \eta _{2}\right]$.

Now, we have

Hence, we have

Thus,

This yields, with $l_{m} = 1$ and $\Psi \left(\ell \right) = \ln \left(\ell \right)$,

As a result, all conditions of Corollary (2.8) are fulfilled, so the system of integral equations has a bounded solution. □

4.   Conclusions

In this manuscript, we have identified fixed points of a hybrid operators using the tool of $M$-dynamic processes. A contraction of the $F$-Chatterjea type is examined within the category of $M$-metric spaces, accompanied by an illustrative example. The obtained results are illustrated with graphical structures, and some corollaries are acquired consequently. At last, we illustrate our results by some applications by solving multistage systems that are primarily useful in dynamic systems. Along with that, we ensure the existence of a solution of integral equations. In the future, these extended results can be furthered to acquire fixed point theorems for Reich and Hardy Rogers type $F$-contractions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that they have no competing interests.