Certain dynamic iterative scheme families and multi-valued fixed point results

1.   Introduction

In the frame of functional analysis, the study of metric fixed point theory (MFPT) untied a portal to a new area of pure and applied mathematics. It states sufficient conditions for the existence of a (unique) fixed point and also provides an iterative system by which we can make approximations to the fixed point and error bounds. This idea was explored and furthered by a good number of researchers (see ^[2,3,29,31]). It is widely held that MFPT originated in the year 1922 through the work of S. Banach ^[7] when he established the famous contraction mapping principle (CMP). Iteration systems are used in each branch of applied mathematics, and the criteria for convergence proofs and error estimates are very often produced by an application of the CMP and its variants. Moreover, E. Sperner (1928) proved the combinatorial geometric well-known lemma on the decomposition of a triangle which displays an important rule in the theory of CMP. These are the most important tool for proving tlre existence and uniqueiess of solutions to different mathematical models (differential, integral, ordinary and partial differential quations, variational inequalities). Some other fields are steady-state tempreture ditribution, chemical reactions, neutron transport theoty, economic, flow of fluids, optimal control theory, fractals, etc.

The classical fixed point theorems of Banach and Brouwer marked the development of the two most prominent and complementary facets of the theory, namely, the metric fixed point theory and the topological fixed point theory. The metric theory encompasses results and methods that involve properties of an essentially isometric nature. It originates with the concept of Picard successive approximations for establishing existence and uniqueness of solutions to nonlinear initial value problems of the 1st order and goes back as far as Cauchy, Liouville, Lipschitz, Peano, Fredholm, and most particularly, Emile Picard. The concept was investigated by extending metric spaces into its extensions. Kamran et al. ^[19] initiated the idea of an extended $b$-metric space, which is one of the most highlight extension of a $b$-metric space. After, in 2018, Mlaiki et al. ^[23] generalized the notion of an extended $b$-metric space to a controlled metric space. Many recent developments on metric structures and fixed point theory are investigated in ^[10,12,13] and also in the references therein. Later on, Nadler ^[24] used the idea of the Pompeiu-Hausdorff metric and gave the contraction theorem for set-valued maps instead of single-valued maps. In 2002, Branciari ^[9] introduced a well known contraction, known as the Branciari contraction. In 2012, Wardowski ^[32] initiated a new class of contractions, known as an $F$-contraction mapping and investigated the existence and uniqueness of fixed point results (see more ^{[14,15,22,26]}). Recently, many developments on fractional calculus and fixed point results based on (generalized) $F$ -contraction mappings are investigated in ^{[8,11,20,25,28,33]} and also in the references therein in the associated approach.

2.   Preliminaries

Let $N(\Psi)$ represent the family of all non-empty subsets of a non-empty set $\Psi$, and $C(\Psi)$ be the family of all non-empty closed subsets of $\Psi$. Let $Y:\Psi \rightarrow N\left(\Psi \right)$ be a set-valued mapping, and $\varepsilon _{0}\in \Psi$ be arbitrary and fixed. Define

Any element of $\check{D}\left(Y, \varepsilon _{0}\right)$ is named as dynamic iterative-scheme of $Y$ starting from point $\varepsilon _{0}$. The dynamic-iterative scheme $\left(\varepsilon _{j}\right) _{j\in \mathbb{N} \cup \left \{ 0\right \} }$ onward has the form $\left(\varepsilon _{j}\right)$ (see ^[17]).

Example 2.1. Let $\Psi = C\left(\left[ 0, 1\right] \right)$ be a Banach space with the norm $\left \Vert \varepsilon \right \Vert = \sup_{r\in \left[ 0, 1\right] }\left \vert \varepsilon \left(r\right) \right \vert$ where $\varepsilon \in \Psi$. Let $Y:\Psi \rightarrow 2^{\Psi }$ be so that for every $\varepsilon \in \Psi$, $Y\left(\varepsilon \right)$ is a collection of the function

that is,

and let $\varepsilon _{0}\left(u\right) = u,$ $u\in \left[ 0, 1\right],$ so $\left(\frac{1}{j!\left(j+1\right)!}r^{j+1}\right)$ is a dynamic process of $Y$ with starting point $\varepsilon _{0}.$ The mapping $Y:\Psi \rightarrow R$ is said to be $\check{D}\left(Y, \varepsilon _{0}\right)$ dynamic lower semi-continuous at $\varepsilon \in \Psi,$ if for each dynamic iterative-scheme $\left(\varepsilon _{j}\right) \in \check{D}\left(Y, \varepsilon _{0}\right)$ and for each subsequence $\left(\varepsilon _{j\left(i\right) }\right)$ of $\left(\varepsilon _{j}\right)$ converges to $\varepsilon$, we write $Y\left(\varepsilon \right) \leq \lim \inf_{i\rightarrow \infty }Y\left(\varepsilon _{j\left(i\right) }\right)$. In this case, $Y$ is $\check{D}\left(Y, \varepsilon _{0}\right)$-dynamic lower semi-continuous. If $Y$ is $\check{D}\left(Y, \varepsilon _{0}\right)$ dynamic lower semi-continuous at each $\varepsilon \in \Psi$, then $Y$ is known as lower semi-continuous. If for each sequence $\left(\varepsilon _{j}\right) \subset \Psi$ and $\varepsilon \in \Psi$ so that $\left(\varepsilon _{j}\right) \rightarrow \varepsilon$, we write $Y\left(\varepsilon \right) \leq \lim \inf_{i\rightarrow \infty }Y\left(\varepsilon \left(j\right) \right)$ (more see ^[4,17]).

Branciari ^[9] introduced the following concepts:

Definition 2.2. ^[9] Let $\left(\Psi, \delta \right)$ be a metric space and $Y:\Psi \rightarrow \Psi$ be so that

for all $\varepsilon _{1}, \varepsilon _{2}\in \Psi,$ where $\beta \in \left(0, 1\right)$, $\Phi :\kappa \rightarrow \kappa$ is a non-negative Lebesgue integrable mapping which is summable on each compact subset of $\kappa$ $\left[ \kappa = \left[ 0, +\infty \right) \right]$ and $\int_{0}^{\epsilon }\Phi \left(s\right) ds$ for all $\epsilon > 0.$ Then, $Y$ has fixed point.

Lemma 2.3. ^[21] Let $\left(\varepsilon _{i}\right) _{i\in \mathbb{N}}$ be a sequence so that $\lim_{i\rightarrow +\infty }\varepsilon_{i} = \varepsilon$. Then

Lemma 2.4. ^[21] Let $\left(\varepsilon _{i}\right) _{i\in \mathbb{N}}$ be a sequence. Then

For the last few years, contraction theorems have rapidly been evolving, not only in the metric frame, but also in many different extended spaces and the controlled metric space is one of them. In 2018, Mlaiki et al. ^[23] generalized the notion of an extended $b$-metric space to a controlled metric space. Alamgir et al.^[1] introduced the idea of a Hausdorff controlled metric and proved some well-known results in control metric spaces.

Definition 2.5. ^[23] A controlled-metric space on a non-empty set $\Psi$ is a function $\delta _{\varsigma }:\Psi \times \Psi \rightarrow R^{+}$ with $\varsigma :\Psi \times \Psi \rightarrow \left[ 1, \infty \right)$ so that $\forall$ $\varepsilon _{1}, \varepsilon _{2}, \varepsilon _{3}\in \Psi$

$\left(C_{1}\right) :$ if $\delta _{\varsigma }\left(\varepsilon _{1}, \varepsilon _{2}\right) = 0$ if and only if $\varepsilon _{1} = \varepsilon _{2};$

$\left(C_{2}\right) :$ $\delta _{\varsigma }\left(\varepsilon _{1}, \varepsilon _{2}\right) = \delta _{\varsigma }\left(\varepsilon _{2}, \varepsilon _{1}\right);$

$\left(C_{3}\right) :$ $\delta _{\varsigma }\left(\varepsilon _{1}, \varepsilon _{3}\right) \leq \varsigma \left(\varepsilon _{1}, \varepsilon _{2}\right) \delta _{\varsigma }\left(\varepsilon _{1}, \varepsilon _{2}\right) +\varsigma \left(\varepsilon _{2}, \varepsilon _{3}\right) \delta _{\varsigma }\left(\varepsilon _{2}, \varepsilon _{3}\right).$

The pair $\left(\Psi, \delta _{\varsigma }\right)$ is known as a controlled-metric space.

A sequence $\left \{ \varepsilon _{i}\right \}$ in controlled m.s $\Psi$ is convergent to some $\varepsilon \in \Psi$ if for every $\epsilon > 0,$ there is $I = I\left(\epsilon \right) \in \mathbb{N}$ so that $\delta _{\varsigma }\left(\varepsilon _{i}, \varepsilon \right) < \epsilon$ for all $i\geq I$ and have write $\lim_{i\rightarrow \infty }(\varepsilon _{i}) = \varepsilon$, a sequence $\left \{ \varepsilon _{i}\right \}$ in $\left(\Psi, \delta _{\varsigma }\right)$ is Cauchy if for every $\epsilon > 0,$ there is $I = I\left(\epsilon \right) \in \mathbb{N}$ so that $\delta _{\varsigma }\left(\varepsilon _{i}, \varepsilon _{i^{\prime }}\right) < \epsilon$ for all $i, i^{\prime }\geq I$ and a controlled-metric space $\left(\Psi, \delta _{\varsigma }\right)$ is complete if every Cauchy sequence in $\Psi$ converges.

Let $x\in \Psi$ and $\epsilon > 0,$ the open ball $B\left(x, \epsilon \right)$ is

The mapping $Y:\Psi \rightarrow \Psi$ is continuous at $x\in \Psi$ if for each $\epsilon > 0,$ there is $\alpha > 0$ so that

Owing to above proposition, we clearly say that if $Y$ is continuous at $x\in \Psi,$ then for $x_{i}\rightarrow x$, we have $Yx_{i}\rightarrow Yx$ as $i\rightarrow \infty.$

Further, Alamgir et al. ^[1] discussed some well-known results via the Hausdorff controlled metric. Define the Pompeiu-Hausdorff controlled metric $\hat{H}_{\varsigma }$ induced by $\delta _{\varsigma }$ on $CB(\Psi)$ as follows:

for each $\theta _{1}, \theta _{2}\in CB(\Psi)$, where $\check{D}\left(\varepsilon _{1}, \theta _{2}\right) = \inf_{\varepsilon _{2}\in \theta _{2}}\delta _{\varsigma }\left(\varepsilon _{1}, \varepsilon _{2}\right)$.

Lemma 2.6. ^[1] Let $\theta _{1}$ be a nonempty subsets of acontrolled-metric space $\left(\Psi, \delta \right)$, then

for $\varepsilon _{1}, \varepsilon _{2}\in \Psi$, where $\varsigma \left(\varepsilon _{2}, \theta _{1}\right) = \inf_{\varepsilon ^{\ast }\in \theta_{1}}\varsigma \left(\varepsilon _{2}, \varepsilon ^{\ast }\right)$ and $\delta _{\varsigma }\left(\varepsilon _{2}, \theta _{1}\right) = \inf_{\varepsilon ^{\ast }\in \theta _{1}}\delta _{\varsigma }\left(\varepsilon _{2}, \varepsilon ^{\ast }\right).$

Lemma 2.7. ^[1] Let $\theta _{1}, \theta _{2}\in CB\left(\Psi\right)$, then for all $\beta > 0$ and $\varepsilon _{2}\in \theta _{2}$, there is $\varepsilon _{1}\in \theta _{1}$ so that

Lemma 2.8. ^[1] Let $\theta _{1}$ and $\theta _{2}$ be nonemptysubsets of a controlled-metric space $\left(\Psi, \delta \right)$. If $\alpha \in \theta _{1}$, then

In 2012, Wardowski ^[32] initiated F-contractions and a related fixed point result was presented.

Definition 2.9. ^[32] $Y:\Psi \rightarrow \Psi$ is called an $F$ -contraction on a metric space $\left(\Psi, \delta \right)$, if there exist $F\in \nabla _{\mathcal{F}}$ and $\tau \in R^{+}/\left \{ 0\right \}$ so that $\delta \left(Y\varepsilon _{1}, Y\varepsilon _{2}\right) > 0$, implies

for each $\varepsilon _{1}, \varepsilon _{2}\in \Psi$, where $\nabla _{ \mathcal{F}}$ is the family of all functions $F:\left(0, +\infty \right) \longrightarrow \left(-\infty, +\infty \right)$, so that

$\left(\mathcal{F}_{i}\right)$ $\varepsilon _{1} < \varepsilon _{2}$ implies $F\left(\varepsilon _{1}\right) < F\left(\varepsilon _{2}\right)$ for all $\varepsilon _{1}, \varepsilon _{2}\in \left(0, +\infty \right);$

$\left(\mathcal{F}_{ii}\right)$ for each sequence $\left \{ \varepsilon \left(j\right) \right \}$ of positive real numbers,

$\left(\mathcal{F}_{iii}\right)$ there is $k\in \left(0, 1\right)$ such that $\lim_{c\rightarrow 0}\left(c\right) ^{k}F\left(c\right) = 0$. Then there is a unique fixed point of $Y$.

Example 2.10 As examples of $F$-contractions, one writes:

$\left(i\right) :F\left(\varepsilon \right) = \ln \left(\varepsilon \right);$

$\left(ii\right) :F\left(\varepsilon \right) = \ln \left(\varepsilon \right) +\varepsilon;$

$\left(iii\right) :F\left(\varepsilon \right) = -\frac{1}{\sqrt{\varepsilon }};$

$\left(iv\right) :F\left(\varepsilon \right) = \ln \left(\varepsilon ^{2}+\varepsilon \right).$

Owing to $\left(\mathcal{F}_{i}\right)$ and (2.1), each F-contraction $Y$ is a contractive mapping, and so each F-contraction mapping is continuous.

Our goal is to introduce a new concept of non-linear ($F$, $F_{H}$)-dynamic-iterative scheme for Branciari Ćirić type-contractions and establish some related multi-valued fixed point results on controlled-metric spaces. Finally, we give concrete examples, an application and some open questions.

3.   A family of $F$-dynamic-iterative scheme: $\check{D}\left(Y, \varepsilon _{0}\right)$

First, we introduce the following definition.

Definition 3.1. Let $\left(\Psi, \delta _{\varsigma }\right)$ be a controlled-metric space and $Y:\Psi \rightarrow CB\left(\Psi \right)$ be a set valued Branciari Ćirić type contraction based on $F$ -dynamic-iterative scheme $\check{D}\left(Y, \varepsilon _{0}\right).$ If there are $F\in \nabla _{\mathcal{F}}$, $\tau :\left(0, +\infty \right) \rightarrow \left(0, +\infty \right)$ a non-constant function and $\Phi :\kappa \rightarrow \kappa$ a non-negative Lebesgue integrable mapping which is summable on each compact subset of $\kappa$ so that

where

for all $i\in \mathbb{N}$, $\varepsilon _{i}\in \check{D}\left(Y, \varepsilon _{0}\right)$ and for every given $\epsilon > 0$ so that $\int_{0}^{\epsilon }\Phi \left(s\right) \delta s > 0$.

Remark 3.2. In continuing way of our results, we consider only the dynamic iterative scheme $\varepsilon _{i}\in \check{D}\left(Y, \varepsilon _{0}\right)$ that verifies the following criteria:

When the process does not verify (3.2), then there is $i_{0}\in \mathbb{N}$ so that

and

Then we get $\varepsilon _{i_{0-1}} = \varepsilon _{i_{0}}\in Y\varepsilon _{i_{0-1}}$ which ensures the existence of a fixed point. In view of this consideration of dynamic iterative scheme satisfying (3.2), it does not depreciate a generality of our analysis.

Remark 3.3. Upon setting, clearly $Y$ is a contraction mapping with respect to $F$-dynamic iterative scheme $\check{D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right)$ and in the light of $\Phi \left(s\right) \equiv 1,$ we easily conclude that it is an $F$-contraction.

Theorem 3.4. Let $\left(\Psi, \delta _{\varsigma }\right)$ be a completecontrolled-metric space and $Y:\Psi \rightarrow CB\left(\Psi \right)$ be aset valued Branciari Ćirić type contraction with respect to $F$-dynamic-iterative scheme $\check{D}_{_{\varsigma }}\left(Y, \varepsilon_{0}\right)$. Assume that:

(D1) : There is a $F$-dynamic iterative scheme $\varepsilon _{i}\in \check{D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right)$ such that

(D2): A mapping $\Psi \ni \varepsilon _{i}\longmapsto \delta_{\varsigma }\left(\varepsilon _{i}, Y\varepsilon _{i}\right)$ is $\check{D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right)$-$F$-dynamic lowersemi-continuous.

Then $Y$ has a fixed point.

Proof. Choose $\varepsilon _{0}\in \Psi$ to be an arbitrary point. In view of $\varepsilon _{i}\in \check{D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right)$, we design the $F$-dynamic iterative scheme by the following family:

In case, there is $i_{0}\in \mathbb{N}$ so that $\varepsilon _{i_{0}}\in Y\varepsilon _{i_{0}}$, then $\varepsilon _{i_{0}}$ is a fixed point of $Y$ is clear. Therefore, if we let $\varepsilon _{i}\notin Y\varepsilon _{i}$ then $\check{ D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right) > 0$ for every $i\in \mathbb{N}$. Since $Y\varepsilon _{i}$ is compact, by (3.1) and Lemma 2.6, one writes

Moreover, since $Y\varepsilon _{i}$ is compact, we obtain $\varepsilon _{i+1}\in Y\varepsilon _{i}$ so that $\delta _{\varsigma }\left(\varepsilon _{i}, \varepsilon _{i+1}\right) = \check{D}_{_{\varsigma }}\left(\varepsilon _{i}, Y\varepsilon _{i}\right).$ Using (3.3), we have

Thus, the sequence $\left \{ \delta _{\varsigma }\left(\varepsilon _{i}, \varepsilon _{i+1}\right) \right \}$ is decreasing and hence it is convergent. Now, we show that $\lim_{i\rightarrow \infty }\delta _{\varsigma }\left(\varepsilon _{i}, \varepsilon _{i+1}\right) = 0.$ From, (D1) there is $\sigma > 0$ and $i_{0}\in \mathbb{N}$ so that $\tau \left(\delta _{\varsigma }\left(\varepsilon _{i-1}, \varepsilon _{i}\right) \right) > \sigma$ for all $i > i_{0}.$ Hence, we see that

Let us set $\varpi_{i} = \int_{0}^{\delta _{\varsigma }\left(\varepsilon _{i}, \varepsilon _{i+1}\right) }\Phi \left(s\right) \delta s > 0$ for $i = 0, 1, 2, \cdots,$ and from (3.4), we obtain $\lim_{i\rightarrow \infty }F\left(\varpi_{i}\right) = -\infty$. Using ($F_{ii}$), we get

In view of ($F_{iii}$), there is $\alpha \in \left(0, 1\right)$ so that

By (3.4), the following holds for all $i > i_{0},$

Taking limit as $i\rightarrow \infty$ in (3.7) and using (3.6), we have

Due to (3.8), there is $i_{1}\in \mathbb{N}$ so that $i\left[ \varpi_{i}\right] ^{\alpha }\leq 1$ for all $i\geq i_{1},$ we have

Now, in order to show that $\left \{ \varepsilon _{i}\right \}$ is a Cauchy sequence, we consider $j_{1},$ $j_{2}\in \mathbb{N}$ so that $j_{1} > j_{2}\geq i_{1}.$ From (3.9) and by the metric condition, we have

Owing to (3.10) and in view of the convergence of series $\sum_{l = j_{1}}^{\infty }\frac{1}{l^{\frac{1}{\alpha }}}$, we get $\int_{0}^{\delta _{\varsigma }\left(\varepsilon _{j_{1}}, \varepsilon _{j_{2}}\right) }\Phi \left(s\right) \delta s\rightarrow 0$. Hence, $\left \{ \varepsilon _{i}\right \}$ is Cauchy in $\left(\Psi, \delta _{\varsigma }\right).$ Further, for the completeness of $\Psi$ there is $\varepsilon ^{\ast }\in \Psi$ so that $\lim_{i\rightarrow \infty }\varepsilon _{i} = \varepsilon ^{\ast }.$ Since $Y$ is compact, we have $Y\varepsilon _{i}\rightarrow Y\varepsilon ^{\ast }$ and by Lemma 2.6, one writes

So, $\check{D}_{_{\varsigma }}\left(\varepsilon ^{\ast }, Y\varepsilon ^{\ast }\right) = 0$ and $\varepsilon ^{\ast }\in Y\varepsilon ^{\ast }.$ Now, by right continuity of $F$ we examine $\varepsilon ^{\ast }\in Y\varepsilon ^{\ast }.$ Suppose on the contrary, $\varepsilon ^{\ast }\notin Y\varepsilon ^{\ast }$ then there are $i_{0}\in \mathbb{N}$ and a subsequence $\left \{ \varepsilon _{i_{k}}\right \}$ of $\left \{ \varepsilon _{i}\right \}$ so that $\check{D}_{_{\varsigma }}\left(\varepsilon _{i_{k}+1}, Y\varepsilon ^{\ast }\right) > 0$ for each $i_{k}\geq i_{0}$ [Otherwise, there is $i_{1}\in \mathbb{N}$ so that $\varepsilon _{i}\in Y\varepsilon ^{\ast }$ for every $i\geq i_{1}$, which yields $\varepsilon ^{\ast }\in Y\varepsilon ^{\ast }$, it is a contradiction]. Since $\check{D} _{_{\varsigma }}\left(\varepsilon _{i_{k}+1}, Y\varepsilon ^{\ast }\right) > 0$ for each $i_{k}\geq i_{0}$, one writes

Taking a limit as $k\rightarrow \infty$ in (3.12),

which is a contradiction. Thus, since $\Psi \ni \varepsilon _{i}\longmapsto \delta _{\varsigma }(\varepsilon _{i}, Y\varepsilon _{i})$ is $\check{D} _{_{\varsigma }}\left(Y, \varepsilon _{0}\right)$-$F$-dynamic lower semi-continuous, we have

The closedness of $Y\varepsilon ^{\ast }$ implies that $\varepsilon ^{\ast }\in Y\varepsilon ^{\ast }$ which means that $\varepsilon ^{\ast }$ has a fixed point of $Y.$

Some direct consequences of Theorem 3.4 are as follows:

Remark 3.5. In light of Theorem 3.4, we derive the following contractive condition:

where

for all $i\in \mathbb{N}$, $\varepsilon _{i}\in \check{D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right)$ and $\hat{H}_{\varsigma }\left(Y\varepsilon _{i}, Y\varepsilon _{i+1}\right) > 0.$ Then, $Y$ has a fixed point.

Corollary 3.6. Let $\left(\Psi, \delta _{\varsigma }\right)$ be a completecontrolled-metric space and $Y:\Psi \rightarrow CB(\Psi)$ be a set-valuedBranciari Ćirić type contraction based on $F$-dynamic-iterativescheme $\check{D}\left(Y, \varepsilon _{0}\right).$ Suppose for some $F\in\nabla _{\mathcal{F}}$, $\tau :\left(0, +\infty \right) \rightarrow \left(0, +\infty \right)$ a non-constant function and $\Phi :\kappa \rightarrow \kappa$ a non-negative Lebesgue integrable mapping which is summable oneach compact subset of $\kappa$ so that

where

for all $i\in \mathbb{N}$, $\varepsilon _{i}\in \check{D}_{_{\varsigma}}\left(Y, \varepsilon _{0}\right),$ $\delta _{\varsigma }\left(Y\varepsilon _{i}, Y\varepsilon _{i+1}\right) > 0$ and for each given $\epsilon > 0$ such that $\int_{0}^{\epsilon }\Phi \left(s\right) \delta s > 0$. Assume that (3.4) and (3.4) are satisfied. Then $Y$ has a fixedpoint.

Remark 3.7. In view of Corollary 3.6, we state the following contractive condition:

where

$(i)$ $\Delta _{1}\left(\varepsilon _{i-1}, \varepsilon _{i}\right) = \delta _{\varsigma }\left(\varepsilon _{i-1}, \varepsilon _{i}\right);$

$(ii)$ $\Delta _{2}\left(\varepsilon _{i-1}, \varepsilon _{i}\right) = \max \left \{ \delta _{\varsigma }\left(\varepsilon _{i-1}, \varepsilon _{i}\right), \check{D}_{_{\varsigma }}\left(\varepsilon _{i-1}, Y\varepsilon _{i-1}\right), \check{D}_{_{\varsigma }}\left(\varepsilon _{i}, Y\varepsilon _{i}\right) \right \};$

$(iii)$ $\Delta _{3}\left(\varepsilon _{i-1}, \varepsilon _{i}\right) = \max \left \{ \delta _{\varsigma }\left(\varepsilon _{i-1}, \varepsilon _{i}\right), \frac{\check{D}_{_{\varsigma }}\left(\varepsilon _{i-1}, Y\varepsilon _{i-1}\right), \check{D}_{_{\varsigma }}\left(\varepsilon _{i}, Y\varepsilon _{i}\right) }{2}, \frac{\check{D}_{_{\varsigma }}\left(\varepsilon _{i-1}, Y\varepsilon _{i}\right) +\check{D}_{_{\varsigma }}\left(\varepsilon _{i}, Y\varepsilon _{i-1}\right) }{2}\right \}$

for all $i\in \mathbb{N}$, $\varepsilon _{i}\in \check{D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right).$ Then $Y$ has a fixed point.

Corollary 3.8. Let $\left(\Psi, \delta _{\varsigma }\right)$ be a completecontrolled-metric space and $Y:\Psi \rightarrow CB(\Psi)$ be a set-valuedBranciari Ćirić type contraction based on $F$-dynamic-iterativescheme $\check{D}\left(Y, \varepsilon _{0}\right).$ If for some $F\in\nabla _{\mathcal{F}}$, $\tau _{j}:(0, +\infty)\rightarrow (0, +\infty),$ $j = 1, 2$ a non-constant function and $\Phi :\kappa \rightarrow \kappa$ is anon-negative Lebesgue integrable mapping which is summable on each compactsubset of $\kappa$ so that one of the following holds:

$\left(G_{1}\right)$ $\delta _{\varsigma }\left(Y\varepsilon_{i}, Y\varepsilon _{i+1}\right) > 0$ $\Rightarrow \tau _{j = 1}\left(\Delta\left(\varepsilon _{i-1}, \varepsilon _{i}\right) \right) -\frac{1}{\int_{0}^{\hat{H}_{\varsigma }\left(Y\varepsilon _{i}, Y\varepsilon_{i+1}\right) }\Phi \left(s\right) \delta s}\leq -\frac{1}{\int_{0}^{\Delta\left(\varepsilon _{i-1}, \varepsilon _{i}\right) }\Phi \left(s\right)\delta s};$

$\left(G_{2}\right)$ $\delta _{\varsigma }\left(Y\varepsilon_{i}, Y\varepsilon _{i+1}\right) > 0$ $\Rightarrow \tau _{j = 2}\left(\Delta\left(\varepsilon _{i-1}, \varepsilon _{i}\right) \right) +\frac{1}{1-\exp\int_{0}^{\hat{H}_{\varsigma }\left(Y\varepsilon _{i}, Y\varepsilon_{i+1}\right) }\Phi \left(s\right) \delta s}\leq \frac{1}{1-\exp\int_{0}^{\Delta \left(\varepsilon _{i-1}, \varepsilon _{i}\right) }\Phi\left(s\right) \delta s},$

where

for all $i\in \mathbb{N}$, $\varepsilon _{i}\in \check{D}_{_{\varsigma}}\left(Y, \varepsilon _{0}\right)$ and for each given $\epsilon > 0$ sothat $\int_{0}^{\epsilon }\Phi \left(s\right) \delta s > 0$. Assume that (D1)and (D2) are satisfied. Then $Y$ has a fixed point.

Proof. The proof directly proceed from Corollary 3.8 based on the functions $F\left(\varepsilon \right) = -\frac{1}{\varepsilon }$ and $F\left(\varepsilon \right) = \frac{1}{1-\exp \left(\varepsilon \right) },$ which is also fulfilled for the family $\nabla _{\mathcal{F}}$, then the result follows.

Example 3.9. Let $\Psi = R^{+}\cup \left \{ 0\right \}$. Define the complete controlled-metric spaces $\left(\Psi, \delta _{\varsigma }\right)$ by

and $\varsigma :\Psi \times \Psi \rightarrow \left[ 1, \infty \right)$ as

Consider a mapping $Y:\Psi \rightarrow CB\left(\Psi \right)$ defined by $Y\varepsilon = \left[ 0, \frac{\varepsilon }{2}\right], \varepsilon > 0$ and $\tau$ a non-constant function, that is, $\tau :R^{+}\rightarrow R^{+}$ is of the form

Consider tghe dynamic iterative process $\check{D}(Y, \varepsilon _{0}):$ A sequence $\{ \varepsilon _{i}\}$ is defined by $\varepsilon _{i} = \varepsilon _{0}g^{i-1}$ for each $i\in \mathbb{N}$ with starting point $\varepsilon _{0} = 2$ and $g = \frac{1}{2}$ so that (see Table 1):

By continuing the above iterative process, one asserts that

is a F-dynamic iterative process of $Y$ starting from $\varepsilon _{0} = 2.$

For $\varepsilon _{i}\in \check{D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right)$ and $Y$ as a Branciari Ćirić type contraction mapping with respect to $F$-dynamic-iterative scheme $\check{D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right),$ we obtain $\hat{H}_{\varsigma }\left(Y\varepsilon _{i}, Y\varepsilon _{i+1}\right) = \frac{\left \vert \varepsilon _{i-1}-\varepsilon _{i}\right \vert }{2}$ and $\Delta \left(\varepsilon _{i-1}, \varepsilon _{i}\right) = \left \vert \varepsilon _{i-1}-\varepsilon _{i}\right \vert.$ Now, by contractive condition (3.1) upon setting of $F(\varepsilon) = \ln (\varepsilon)$ and $\Phi \left(s\right) = 1$ for all $s\in R$, we see that $\tau \left(h\right) \leq \Omega (\imath),$ where

Hence, all the required hypotheses of Theorem 3.4 are satisfied and consequently in view of Tables 1 and 2, and Figures 1 and 2, the required hypotheses of Theorem 4.4 regarding to $\tau \left(h\right) \leq \Omega \left(\imath \right)$, are satisfied for all possible values. Here, $0\in Y(0)$ is a fixed point of $Y$ for a Branciari Ćirić type contraction mapping with respect to $F$-dynamic-iterative scheme $\check{D}_{_{\varsigma }}\left(Y, \varepsilon _{0}\right)$.

4.   A Family of $F_{H}$-dynamic-iterative scheme: $\check{D}(\digamma, Y, \alpha _{0})$

Here, we give our second general definition.

Definition 4.1. Let $\digamma :\Psi \rightarrow \Psi$ and $Y:\Psi \rightarrow CB(\Psi)$ be so that

for each integer $j\geq 1.$ The set $\check{D}(\digamma, Y, \alpha _{0})$ is said to be a hybrid dynamic-iterative scheme of $\digamma$ and $Y$ having the starting point $\alpha _{0}$. The hybrid dynamic-iterative scheme $\check{D}(\digamma, Y, \alpha _{0})$ is shortly written as $\digamma \left(\alpha _{j}\right)$.

Definition 4.2. Let $\digamma :\Psi \rightarrow \Psi$ and $Y:\Psi \rightarrow CB\left(\Psi \right)$ be an hybrid Branciari Ćirić type contraction on the controlled-metric space $\left(\Psi, \delta _{\varsigma }\right)$ with respect to $F_{H}$-dynamic-iterative scheme $\check{D}\left(\digamma, Y, \varepsilon _{0}\right).$ Suppose there are $F_{H}\in \nabla _{ \mathcal{F}}$, $\tau :\left(0, +\infty \right) \rightarrow \left(0, +\infty \right)$ a non-constant function and $\Phi :\kappa \rightarrow \kappa$ a non-negative Lebesgue integrable mapping which is summable on each compact subset of $\kappa$ so that

where

for all $i\in \mathbb{N}$, $\varepsilon _{i}\in \check{D}\left(\digamma, Y, \varepsilon _{0}\right)$ and for each given $\epsilon > 0$ so that $\int_{0}^{\epsilon }\Phi \left(s\right) \delta s > 0$.

Remark 4.3. Via Remark 3.2, we consider only the $F_{H}$-dynamic iterative scheme $\varepsilon _{i}\in \check{D}\left(\digamma, Y, \varepsilon _{0}\right)$ that satisfying the following condition:

If the investigated process that does not satisfy (4.3), then there is some $i_{0}\in \mathbb{N}$ so that

and

then we get $\digamma \varepsilon _{i_{0-1}} = \digamma \varepsilon _{i_{0}}\in Y\varepsilon _{i_{0-1}}$ which implies the existence of common fixed point. Due to this consideration of $F_{H}$-dynamic iterative scheme that satisfies (4.3), it does not depreciate a generality of our approach. Moreover, owing to Example 3.2, we easily conclude that the hybrid pair $\left(\digamma, Y\right)$ with respect to $F_{H}$-dynamic iterative scheme $\check{D}\left(\digamma, Y, \varepsilon _{0}\right)$ is a contraction mapping.

Theorem 4.4. Let $\digamma :\Psi \rightarrow \Psi$ and $Y:\Psi \rightarrow CB\left(\Psi \right)$ be an hybrid Branciari Ćirić typecontraction on the controlled-metric space $\left(\Psi, \delta _{\varsigma}\right)$ with respect to $F_{H}$-dynamic-iterative scheme $\check{D}\left(\digamma, Y, \varepsilon _{0}\right)$. Assume $F_{H}\left(\digamma, Y\right) \neq \phi,$ where $F_{H}\left(\digamma, Y\right)$ provided that $\digamma (\Psi)$ is complete and $Y$ is a closed multivalued mapping suchthat

(D3) there is an $F_{H}$-dynamic iterative scheme $\varepsilon_{i}\in \check{D}\left(\digamma, Y, \varepsilon _{0}\right)$ such that suchthat

(D4) for some $\varepsilon \in F_{H}\left(\digamma, Y\right),$ $\digamma$ is $Y$-weakly commuting at $\varepsilon$ so that $\digamma^{2}\varepsilon = Y\digamma \varepsilon.$

Then the hybrid pair $\left(\digamma, Y\right)$ has a common fixed point.

Proof. Consider $\varepsilon _{0}\in \Psi$ to be an arbitrary point. In view of (4.1), we have

In case, if there is $i_{0}\in \mathbb{N}$ so that $\varepsilon _{i_{0}}\in \digamma \varepsilon _{i_{0}}$, then $\varepsilon _{i_{0}}$ is a fixed point of $\digamma$ is clear. Therefore, if we let $\varepsilon _{i}\notin \digamma \varepsilon _{i}$ then $\check{D}(\digamma, Y, \varepsilon _{0}) > 0$ for every $i\in \mathbb{N}$. Using (4.2), one writes

which implies

Based on (4.1) and (4.4), we have

for all $i\in \mathbb{N}.$ Due to ($F_{i}$), we obtain for some $i$,

which gives a contradiction. Thus, we get

Consequently,

for all $i\in \mathbb{N}.$ Thus, the sequence $\left \{ \delta _{\varsigma }\left(\varepsilon _{i}, \varepsilon _{i+1}\right) \right \}$ is decreasing and hence convergent. Now, we show that $\lim_{i\rightarrow \infty }\delta _{\varsigma }\left(\varepsilon _{i}, \varepsilon _{i+1}\right) = 0.$ From (3.4) there is $\sigma > 0$ and $i_{0}\in \mathbb{N}$, so that $\tau \left(\delta _{\varsigma }\left(\varepsilon _{i-1}, \varepsilon _{i}\right) \right) > \sigma$ for all $i > i_{0}.$ Thus, we have

Setting $\varpi_{i} = \int_{0}^{\delta _{\varsigma }\left(\digamma \varepsilon _{i}, \digamma \varepsilon _{i+1}\right) }\Phi \left(s\right) \delta s > 0$ for $i = 0, 1, 2, ...$ and from (4.6), we obtain $\lim_{i\rightarrow \infty }F\left(\varpi_{i}\right) = -\infty$. Using ($F_{ii}$) implies that

From ($F_{iii}$), there is $\alpha \in \left(0, 1\right)$ so that

By (4.6), the following holds for all $i > i_{0},$

Taking limit as $i\rightarrow \infty$ in (4.9) and using (4.8), we have

Let us perceive that from (4.10), there is $i_{1}\in \mathbb{N}$ so that $i \left[ \varpi_{i}\right] ^{\alpha }\leq 1$ for all $i\geq i_{1}.$ We have

Now, we will show that $\left \{ \varepsilon _{i}\right \}$ is a Cauchy sequence. For this mark, we consider $j_{1},$ $j_{2}\in \mathbb{N}$ so that $j_{1} > j_{2}\geq i_{1}.$ From (4.11),

which yields

Owing to (4.12) and in view of convergence of series $\sum_{l = j_{1}}^{\infty }\frac{1}{l^{\frac{1}{\alpha }}}$, we get $\int_{0}^{\delta _{\varsigma }\left(\digamma \varepsilon _{j_{1}}, \digamma \varepsilon _{j_{2}}\right) }\Phi \left(s\right) \delta s\rightarrow 0$. Hence, $\left \{ \digamma \varepsilon _{i}\right \}$ is Cauchy in $\digamma \left(\Psi \right).$ Further, for the completeness of $\Psi$ there is $\varepsilon ^{\ast }\in \digamma \Psi$ so that $\lim_{i\rightarrow \infty }\digamma \varepsilon _{i} = \varepsilon ^{\ast }.$ Now, we claim that $\digamma \varepsilon ^{\ast }\in Y\varepsilon ^{\ast }.$ So, $\delta _{\varsigma }\left(\digamma \varepsilon ^{\ast }, Y\varepsilon ^{\ast }\right) = 0$ and $\digamma \varepsilon ^{\ast }\in Y\varepsilon ^{\ast }.$ In case, $\digamma \varepsilon ^{\ast }\notin Y\varepsilon ^{\ast }$ then $\delta _{\varsigma }\left(\digamma \varepsilon ^{\ast }, Y\varepsilon ^{\ast }\right) > 0$ as $\digamma$ is compact. By ($F_{i}$) and Lemma 2.6, we see that

Suppose on the contrary, $\digamma \varepsilon ^{\ast }\notin Y\varepsilon ^{\ast }$ then there are an $i_{0}\in \mathbb{N}$ and a subsequence $\left \{ \varepsilon _{i_{k}}\right \}$ of $\left \{ \varepsilon _{i}\right \}$ so that $\delta _{\varsigma }\left(\digamma \varepsilon _{i_{k}+1}, Y\varepsilon ^{\ast }\right) > 0$ for each $i_{k}\geq i_{0}$ [Otherwise, there is $i_{1}\in \mathbb{N}$ so that $\digamma \varepsilon _{i}\in Y\varepsilon ^{\ast }$ for every $i\geq i_{1}$, which yields $\digamma \varepsilon ^{\ast }\in Y\varepsilon ^{\ast }$, a contradiction]. Since $\delta _{\varsigma }\left(\digamma \varepsilon _{i_{k}+1}, Y\varepsilon ^{\ast }\right) > 0$ for each $i_{k}\geq i_{0}$, by the contractive condition, one writes

Letting $k\rightarrow \infty$ in (4.14),

a contradiction. Thus, $\digamma \varepsilon ^{\ast }\in Y\varepsilon ^{\ast }$, which means that $\varepsilon ^{\ast }$ has a common fixed point of the hybrid pair $(\digamma, Y).$ Further, for some $\varepsilon \in F_{H}\left(\digamma, Y\right),$ $\digamma$ is $Y$-weakly commuting at $\varepsilon$ so that $\digamma \varepsilon ^{2} = \digamma \varepsilon.$ So we obtain $\digamma ^{2}\varepsilon \in Y\digamma \varepsilon$. In the light of given hypothesis, we see that $\digamma \varepsilon = \digamma ^{2}\varepsilon$ and hence $\digamma \varepsilon = \digamma ^{2}\varepsilon \in Y\digamma \varepsilon$, Consequently, $\digamma \varepsilon \in F_{H}\left(\digamma, Y\right)$.

Some direct consequences of Theorem 4.4 are given.

Remark 4.5. In the light of Theorem 3.4, we obtain the following contractive conditions:

(i) $\tau (\int_{0}^{\Delta \left(\varepsilon _{i-1}, \varepsilon _{i}\right) }\Phi \left(s\right) \delta s)+F_{H}(\int_{0}^{\delta _{\varsigma }\left(\digamma \varepsilon _{i+1}, \digamma \varepsilon _{i}\right) }\Phi \left(s\right) \delta s)\leq F_{H}(\int_{0}^{\Delta \left(\varepsilon _{i-1}, \varepsilon _{i}\right) }\Phi \left(s\right) \delta s);$

(ii) $2\tau (\int_{0}^{\Delta \left(\varepsilon _{i-1}, \varepsilon _{i}\right) }\Phi \left(s\right) \delta s)+F_{H}(\int_{0}^{\delta _{\varsigma }\left(\digamma \varepsilon _{i+1}, \digamma \varepsilon _{i}\right) }\Phi \left(s\right) \delta s)\leq F_{H}(\int_{0}^{\Delta \left(\varepsilon _{i-1}, \varepsilon _{i}\right) }\Phi \left(s\right) \delta s).$

Due to above fashion, we easily see that the hybrid pair $\left(\digamma, Y\right)$ has a common fixed point.

Corollary 4.6. Let $\digamma :\Psi \rightarrow \Psi$ and $Y:\Psi \rightarrow CB\left(\Psi \right)$ be an hybrid Branciari Ćirić typecontraction on the controlled-metric space $\left(\Psi, \delta _{\varsigma}\right)$ with respect to $F_{H}$-dynamic-iterative scheme $\check{D}\left(\digamma, Y, \varepsilon _{0}\right).$ Suppose there are $F_{H}\in \nabla _{\mathcal{F}},$ $\tau _{j}:(0, +\infty)\rightarrow (0, +\infty),$ $j = \overline{1, 6}$ a non-constant function and $\Phi :\kappa \rightarrow \kappa$ a non-negative Lebesgue integrable mapping which is summable on eachcompact subset of $\kappa$ so that $\delta \left(\digamma \alpha_{i}, \digamma \alpha _{i+1}\right) > 0$ and one of the following holds:

$(G1):$ $\tau _{j = 1}(\int_{0}^{\Delta (\alpha _{i-1}, \alpha _{i})}\Phi\left(s\right) \delta s)-\frac{1}{\int_{0}^{\delta \left(\digamma \alpha_{i}, \digamma \alpha _{i+1}\right) }\Phi \left(s\right) \delta s}\leq -\frac{1}{\int_{0}^{\Delta (\alpha _{i-1}, \alpha _{i})}\Phi \left(s\right)\delta s};$

$(G2):$ $\tau _{j = 2}(\int_{0}^{\Delta (\alpha _{i-1}, \alpha _{i})}\Phi\left(s\right) \delta s)+\exp (\int_{0}^{\delta \left(\digamma \alpha_{i}, \digamma \alpha _{i+1}\right) }\Phi \left(s\right) \delta s)\leq(\int_{0}^{\Delta (\alpha _{i-1}, \alpha _{i})}\Phi \left(s\right) \delta s);$

$\left(G3\right) :$ $\tau _{j- = 3}(\int_{0}^{\Delta (\alpha _{i-1}, \alpha_{i})}\Phi \left(s\right) \delta s)+\frac{1}{1-\exp (\int_{0}^{\delta\left(\digamma \alpha _{i}, \digamma \alpha _{i+1}\right) }\Phi \left(s\right) \delta s)}\leq \frac{1}{1-\exp (\int_{0}^{\Delta (\alpha_{i-1}, \alpha _{i})}\Phi \left(s\right) \delta s)};$

$\left(G4\right) :$ $\tau _{j = 4}(\int_{0}^{\Delta (\alpha _{i-1}, \alpha_{i})}\Phi \left(s\right) \delta s)^{c}+(\int_{0}^{\delta \left(\digamma\alpha _{i}, \digamma \alpha _{i+1}\right) }\Phi \left(s\right) \delta s)^{c}\leq (\int_{0}^{\Delta (\alpha _{i-1}, \alpha _{i})}\Phi \left(s\right) \delta s)^{c},$ $c > 0;$

$\left(G5\right) :$ $\tau _{j- = 5}(\int_{0}^{\Delta (\alpha _{i-1}, \alpha_{i})}\Phi \left(s\right) \delta s)-\frac{1}{\int_{0}^{\delta \left(\digamma \alpha _{i}, \digamma \alpha _{i+1}\right) }\Phi \left(s\right)\delta s}$$+\int_{0}^{\delta \left(\digamma \alpha _{i}, \digamma \alpha_{i+1}\right) }\Phi \left(s\right) \delta s\leq -\frac{1}{\int_{0}^{\Delta(\alpha _{i-1}, \alpha _{i})}\Phi \left(s\right) \delta s}$ $+\int_{0}^{\Delta (\alpha _{i-1}, \alpha _{i})}\Phi \left(s\right) \delta s;$

$\left(G6\right) \tau _{j- = 6}(\int_{0}^{\Delta (\alpha _{i-1}, \alpha_{i})}\Phi \left(s\right) \delta s)+\int_{0}^{\delta \left(\digamma \alpha_{i}, \digamma \alpha _{i+1}\right) }$$\Phi \left(s\right) \delta s\exp\int_{0}^{\delta \left(\digamma \alpha _{i}, \digamma \alpha _{i+1}\right)}\Phi \left(s\right) \delta s\leq \int_{0}^{\Delta (\alpha _{i-1}, \alpha_{i})}\Phi \left(s\right)$ $\delta s\exp \int_{0}^{\delta \left(\digamma \alpha _{i}, \digamma \alpha_{i+1}\right) }\Phi \left(s\right) \delta s,$

where,

$\Delta (\alpha _{i-1}, \alpha _{i}) =$$\max \left \{ \delta (\digamma \alpha_{i-1}, \digamma \alpha _{i}), \delta (\digamma \alpha _{i-1}, Y\alpha_{i-1}), \delta (\digamma \alpha _{i}, Y\alpha _{i}), \frac{\delta (\digamma\alpha _{i-1}, Y\alpha _{i})+\delta (\digamma \alpha _{i}, Y\alpha _{i-1})}{2}\right \}$,

for all $i\in \mathbb{N}$, $\alpha _{i}\in \check{D}(\digamma, Y, \alpha_{0})$ and for each given $\epsilon > 0$ so that $\int_{0}^{\epsilon }\Phi\left(s\right) \delta s > 0$. Assume that (D3) and (D4) are satisfied. Thenthe hybrid pair $\left(\digamma, Y\right)$ has a common fixed point.

Proof. The proof follows directly from Corollary 4.6 based on the functions: $F\left(\alpha \right) = -\frac{1}{\alpha },$ $F\left(\alpha \right) = \exp \left(\alpha \right),$ $F\left(\alpha \right) = \frac{1}{1-\exp \left(\alpha \right) },$ $F\left(\alpha \right) = \alpha ^{c > 0},$ $F\left(\alpha \right) = -\frac{1}{\alpha }+\alpha$ and $F\left(\alpha \right) = \alpha.\exp \left(\alpha \right).$ Also, for the family $\nabla _{\mathcal{F}}$, the result follows.

Example 4.7. Let $\Psi = R^{+}\cup \left \{ 0\right \}$. Define the complete controlled-metric space $\left(\Psi, \delta _{\varsigma }\right)$ by

and $\varsigma :\Psi \times \Psi \rightarrow \left[ 1, \infty \right)$ as

Let $\digamma :\Psi \rightarrow \Psi$ and $Y:\Psi \rightarrow CB(\Psi)$ defined by $\digamma \varepsilon = \frac{\varepsilon -1}{2}$ and

Let $\tau$ be a non-constant function, that is, $\tau :R^{+}\rightarrow R^{+}$ is of the form

Design a sequence $\{ \varepsilon _{i}\}$ by $\varepsilon _{i} = \varepsilon _{i-1}+1$ with $\varepsilon _{0} = 1$. Then the following estimates hold (see Table 3):

Continuing in this way,

is an $F_{H}$-dynamic-iterative scheme of $\digamma$ and $Y$ starting from the point $\varepsilon _{0} = 1.$

For $\varepsilon _{i}\in \check{D}(\digamma, Y, \varepsilon _{0})$ and the hybrid pair $\left(\digamma, Y\right)$ for Branciari Ćirić type contraction mappings with respect to $F_{H}$-dynamic-iterative scheme $\check{D}(\digamma, Y, \varepsilon _{0}),$ we see that $\hat{H}_{\varsigma }\left(\digamma \varepsilon _{i}, \digamma \varepsilon _{i+1}\right) = \frac{ \left \vert \varepsilon _{i-1}-\varepsilon _{i}\right \vert }{2}$ and $\Delta \left(\varepsilon _{i-1}, \varepsilon _{i}\right) = \left \vert \varepsilon _{i-1}-\varepsilon _{i}\right \vert.$ Now, in view of (4.2) with $F(\varepsilon) = \ln (\varepsilon)$ and $\Phi \left(s\right) = 1$ for $s\in R$, we have $\tau \left(h\right) \leq \Omega (\imath),$ where

Hence, all the required hypotheses of Theorem 4.4 are satisfied and consequently the hybrid pair $\left(\digamma, Y\right)$ for Branciari Ćirić type contraction mapping with respect to $F_{H}$ -dynamic-iterative scheme $\check{D}(\digamma, Y, \varepsilon _{0})$ has a common fixed point. Hence, by Tables 3 and 4, and Figures 3 and 4, the required hypotheses of Theorem 4.4, regarding to $\tau \left(h\right) \leq \Omega \left(\imath \right)$, are satisfied for all possible values. Here, $0 = \digamma (0)\in Y(0)$ is a common fixed point of $\digamma$ and $Y.$ Next, observe that for $h\geq 70$ then $\tau \left(h\right) \nleq \Omega \left(\imath \right)$. So, Theorem 4.4 can not be satisfied.

5.   An application

Many recent developments on fractional calculus and fixed point theory are investigated in ^[16,30], and also in the references therein.

Consider the Liouville-Caputo fractional differential equations viewed on order $\kappa$ $\left(D_{\left(c, \kappa \right) }\right)$ given as

where $i-1 < \gamma < i,$ $i = \left[ \omega \right] +1,$ $\omega \in C^{i}\left(\left[ 0, +\infty \right] \right)$, the collection $\left[ \gamma \right]$ corresponds to a positive real number and $\Gamma$ is the Gamma function. Let the complete controlled-metric space $\delta _{\varsigma }:C\left(I\right) \times C\left(I\right) \rightarrow R^{+}$ be given as

with setting $\varsigma \left(g_{1}, g_{2}\right) = \varsigma \left(g_{2}, g_{3}\right) = 2.$ Now, consider the following fashion of Liouville-Caputo fractional derivative

where $x\in \left(0, 1\right)$ and $\gamma \in \left(1, 2\right]$ with

where $I = \left[ 0, 1\right],$ $\Xi \in C\left(I, R\right)$ and $L:I\times R\rightarrow R$ is a continuous function. Take $P:\Psi \rightarrow \Psi$ as

for $v\in \Psi$ and $x\in \left[ 0, 1\right].$ Now, we state the main result.

Theorem 5.1. Suppose that $L$ is non-decreasing on its second variable andthere is $\tau > 0$ so that $g_{i-1}, g_{i}\in D_{_{\varsigma }}(\Upsilon, g_{0})$ and $x\in \left[ 0, 1\right]$ implies

where $\Omega = \frac{\left(2\gamma -1\right) \left(\Gamma \left(\gamma+1\right) \right) }{2\left(5\gamma +2\right) }$ and

Then Eqs (5.3) and (5.4) have at least one solution, i.e., say $g^{\ast }\in \Psi.$

Proof. For each $x\in I$, consider

Now, we have

This yields that

It implies that

Therefore,

Now, by contractive condition (2.1) with $\Phi \left(s\right) = 1$ for all $s\in R$ and $F\left(s\right) = -\frac{1}{\sqrt{s}}$, we have

for all $i\in \mathbb{N}$, $g_{i}\in \Psi$ and for each given $\epsilon > 0$ so that $\int_{0}^{\epsilon }\Phi \left(s\right) \delta s > 0$. Thus, all the required hypotheses of Theorem 3.4 are satisfied and we ensure that the Eqs (5.3) and (5.4) have at least one solution in $P.$

Theorem 5.2. Let $L:I\times R\rightarrow R$ be a continuous function, non-decreasing on second variable and there is $\tau > 0$ so that $g_{i-1}, g_{i}\in D(\Upsilon, Y, g_{0})$ and $x\in \left[ 0, 1\right]$ implies

where $\Omega = \frac{\left(2\gamma -1\right) \left(\Gamma \left(\gamma+1\right) \right) }{2\left(5\gamma +2\right) }$ and

In the light of Theorem 5.1 with $\Upsilon$ is $Y$-weakly commutingat $g$ so that $\Upsilon ^{2}g = Y\Upsilon g$, we conclude that Eqs (5.3) and (5.4) have at least one solution.

Example 5.3. Consider the Liouville-Caputo fractional differential equations based on order $\gamma$ $\left(D_{\left(c, \gamma \right) }\right)$

and its integral boundary valued problem:

where $\gamma = \frac{3}{2},$ $\vartheta = \frac{3}{4}$ and $L\left(x, v\left(x\right) \right) = \frac{1}{\left(x+3\right) ^{2}}\frac{\left \vert \Xi \left(x\right) \right \vert }{1+\left \vert \Xi \left(x\right) \right \vert }.$ So, the above setting is an example of Eqs (5.3) and (5.4). Hence, the Eqs (5.9) and (5.10) have at least one solution.

6.   Open problems

In this section, we pose some challenging questions for the readers.

Problem 1: Can Theorems 3.4 and 4.4 be proved without the condition ($F_{iii}$)?

Problem 2: Can Theorems 3.4 and 4.4 be proved by Semi- $F$-contraction and without the continuity of $F$-contraction?

7.   Conclusions

In our present investigation, we have introduced and systematically studied an extension of the developments concerning $F$- contractions that were proposed, in the year 2012 by Wardowski. We have fruitfully developed and generalized the notion of the $F$- contractions to the case of non-linear $F$ and $F_{H}$-dynamic-iterative scheme for Branciari Ćirić type-contractions and proved several multi-valued fixed point results on controlled-metric spaces. An approximations of the dynamic-iterative scheme instead of the conventional Picard sequence are also determined. The paper also includes a tangible example and a graphical interpretation that displays the motivation for such investigations. The work is completed by giving an application of the proposed non-linear $F$ and $F_{H}$ -dynamic-iterative scheme to the Liouville-Caputo fractional derivatives and fractional differential equations. In the future, these results can be furthered to acquire fixed point results for single and multi-valued mappings in the context of double controlled-metric space and triple controlled-metric spaces.

Acknowledgments

The authors Aiman Mukheimer and Kamal Abodayeh would like to thank Prince Sultan University for paying APC and for the support through the TAS research LAB.

Conflicts of interest

The authors declare that they have no competing interests.