New extension to fuzzy dynamic system and fuzzy fixed point results with an application

1.   Introduction

Dynamic process is a mightful formalistic apparatus for association with a large spectrum analysis of multistage decision making problems. Such problems appear and are congruent in essentially all human activities. Unfavourably, for explicit reasons, the analysis of fuzzy dynamic process is difficult. Fuzzy dynamic process are characteristic of all dynamic process where the variables associated are state and decision variables. Fuzzy dynamic iterative process is established as a process getting preprocessed inputs and having outputs that are furthermore defuzzified for realistic applications.

In the light of epistemic access, the term fuzzy sets appear as descriptions or perceptions of nonexistent underlying crisp values. As an example, it is noted that the temperature was high form but the numerical value is uncharted. This leads the way, to a number of classical problems which usually provide themselves to fuzzification fashions like Zadeh's generalization theorem ^[19].

In functional analysis, the field Banach fixed point theory originate as an imperative apparatus over the last some decades in non-linear sciences and engineering via behavioral science, economics, etc see (^{[4,6,7,8,11,12,14,16,20,21,23,24,25,26,29,31]}). To be unequivocal, while codifying an experiment mathematically, many number of researchers to interrogate the solvability of a functional equation in terms of differential equations, integral equations, or fractional differential equations. Such as the existence and uniqueness of a solution are often achieved by finding fixed point of a particular contraction mapping, (see more ^{[1,3,9,10,13,15,18,30]}). The three major structure in Banach fixed point theory are metric structure, topological structure, and discrete structure. These idea was extend by either generalized metric spaces into by modifying the structure of the contraction operators. However, Nadler ^[22] display the concept of Hausdorff metric discoursed the Banach fixed point theory for multi-valued mapping rather than single-valued mappings.

On the other hand, Alghamdi et al. ^[2] improved the idea of partial metric space to $b$-metric-like space. They produced interesting theorems of fixed point in the newly defined frame. Their concept was expedited by various researchers in many ways (see more ^[17,27,28]).

This article regards fuzzy dynamic process as fuzzy dynamic process on $b$ -metric-like space, specifically the mapping of set-valued (extended) fuzzy intervals endowed with the $b$-metric-like. From that point of view, a natural topic is convergence theorems via fuzzy dynamic process in the class of $b$ -metric-like space. Our view of convergence theorems in $b$ -metric-like space, then, disposes of fuzzy dynamic process entirely. Instead, we just adopt the standard setting of fuzzy dynamic process in $b$ -metric-like space which defines convergence theorems in generalized $\mathcal{F}$-contraction via expectations of fuzzy Suzuki Hardy Rogers type contraction operators. Subsequently, corollaries are originated from the main result. To explain the example in the main section, a table and diagram has been created that best illustrates the Fuzzy dynamic process to the readers. At the end, gives an application of our results in solving Hukuhara differentiability through the fuzzy initial valued problem and fuzzy functions. The pivotal role of Hukuhara differentiability in Fuzzy dynamic process is stated. At last, a summary of the article is described in the conclusion section.

2.   Preliminaries

Formally, an fuzzy set is defined as ^[32]:

A fuzzy set on $G$ is a mapping that assigns every value of $G$ to some element in $[0, 1]$. The family of all such mappings is expressed as $F(G)$. For a fuzzy set $A$ on $G$ and $< /p > < p > \mu < /p > < p > \in G$, the value $A(</p > < p > \mu < /p > < p > )$ is known as the membership grade of $< /p > < p > \mu < /p > < p >$ in $A$. The $\alpha -$level set of $A$ expressed as $\left[ A\right] _{\alpha }$ is given by

For a nonempty set $G$ and an ms $G^{\prime }$, a mapping $T:G\rightarrow F(G^{\prime })$ is a fuzzy mapping and is a fuzzy subset of $G\times G^{\prime }$ having the membership function $T(g)(g^{\prime })$. $T(g)(g^{\prime })$ describes the membership grade of $g^{\prime }$ in $T(g)$, while $[T\left(g\right) ]_{\alpha }$ states the $\alpha -$level set of $T(g)$, for more details see ^[5].

Definition 2.1. ^[5] A point $g\in G$ is called a fuzzy fixed point of a fuzzy mapping $T:G\rightarrow F(G)$ if there is $\alpha \in (0, 1]$ such that $g\in \lbrack T\left(g\right) ]_{\alpha }.$

In the recent past, Wardowski ^[31] provided the term known as F-contraction and implemented on Banach fixed point theory. Which is the efficient generalization of Banach fixed point theory. Formally, an F-contraction is defined as follows ^[31]:

Definition 2.2. Let $\nabla _{\mathcal{F}}$ is the set of mappig $\mathcal{F} :R^{+}\longrightarrow R$ satisfying $\left({\rm{F}}_{i}\right) -\left({\rm{F}}_{iii}\right)$:

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

$\left({\rm{F}}_{ii}\right)$ For every sequence $\left \{ \mu _{\sigma }\right \}$ in $R^{+}$ such that

$\left({\rm{F}}_{iii}\right)$ There exist $k\in \left(0, 1\right)$ such that $\lim_{\mu \rightarrow 0}()\mu ^{k}\mathcal{F}(\mu) = 0$.

A mapping $T:G\rightarrow G$ is called an $\mathcal{F}$-contraction on a metric space $(G, d),$ if there is $\tau \in {\mathbb{R}}^{+}/\left \{ 0\right \}$ such that

After, we recall the following some basic idea of dynamic system:

Let $\xi :G\rightarrow C(G)$ be a mapping. A set

is called dynamic process $\check{D}\left(\xi, \mu_{0}\right)$ of $\mu$ with starting point $\mu_{0}$. Where $\mu_{0}\in G$ be arbitrary and fixed. In the light of $\check{D}\left(\xi, \mu_{0}\right),$ $\left(\mu_{a }\right) _{a \in N-\left \{0\right \} }$ onward has the form $\left(\mu_{a }\right)$ (see more ^[18]).

Further, the literature contains many generalizations of the idea of fixed point theory in metric spaces and its topological behavior. In particularly, Alghamdi et al. ^[2] designed the fashion of $b$-metric-like space as follows:

Definition 2.3. ^[2] Let $G$ be a $b$ -metric-like space with $G\neq \phi$ and $s\geq 1.$ A function $d:G\times G\rightarrow {\mathbb{R}}^{+}\cup \left \{ 0\right \}$ such that for every $\mu _{1}, \mu _{2}, \mu _{3}\in G$, the following conditions $\left(b_{i}\right),$ $\left(b_{ii}\right)$ and $\left(b_{iii}\right)$ hold true:

$\left(b_{i}\right)$ the condition: $d\left(\mu _{1}, \mu _{2}\right) = 0$ implies $\mu _{1} = \mu _{2};$

$\left(b_{ii}\right)$ the condition is hold true: $d\left(\mu _{1}, \mu _{2}\right) = d\left(\mu _{2}, \mu _{1}\right);$

$\left(b_{iii}\right)$ the condition is satisfied: $d\left(\mu _{1}, \mu _{3}\right) \leqq s[d\left(\mu _{1}, \mu _{2}\right) +d\left(\mu _{2}, \mu _{3}\right) ].$

The pair $\left(G{\rm{ }}, d\right)$ is known as a $b$ -metric-like space.

Example 2.4. Define $\left(G, d\right)$ with $s = 2$ by\newline

with

for all $\mu _{1}, \mu _{2}\in G = \{0, 1, 2\}.$ Then, $\left(G, d\right)$ is a $b$ -metric-like space$.$ Clearly, it is neither a $b$-metric nor a metric-like space, see more detail in ^[2].

Remark 2.5. Owing to above definition (2.3), every partial metric is a $b$ -metric-like space but converse may not hold true in general, see more ^[2]

Nadler ^[22], design the idea of Hausdorff metric and extended the Banach contraction theorem for multi-valued operators instead of single-valued operators. Hereinafter, we investigate the concept of Hausdorff $b$-metric-like as follows. Let $\left(G, \mu \right)$ be a $b$-metric-like space. For $\mu_{1}\in G$ and $L_{1}\subseteq G$, let $d _{b}\left(\mu_{1}, L_{2}\right) = \inf \left \{d \left(\mu_{1}, \mu_{2}\right) :\mu_{2}\in L_{2}\right \}.$ Define $\hat{H_{b}}:CB\left(G\right) \times CB\left(G\right) \rightarrow \left[ 0, +\infty \right)$ by

for each $L_{1}, L_{2}\in CB\left(G\right).$ Where $CB\left(G\right)$ denote the family of all non-empty closed and bounded-subsets of $G$ and $CL\left(G\right)$ the family of all non-empty closed-subsets of $G$.

Definition 2.6. ^[5] Let $L_{1}, L_{2}\in V(G)$, $\alpha \in (0, 1]$. Then $d_{\alpha }(L_{1}, L_{2}) = inf_{g\in L_{1\alpha }, g^{\prime }\in L_{2\alpha }}d(g, g^{\prime })$,

where $\hat{H}_{b^{l}}$ is the Hausdorff distance.

Lemma 2.7. Let $L_{1}$ and $L_{2}$ be nonempty proximal subsets of a $b$ -MLS $\left(G, d\right)$. If $g\in L_{1}$, then

Lemma 2.8. Let $(G, d)$ be a $b$ -metric-like space. For all $L_{1}, L_{2}\in CB(G)$ and for any $g\in L_{1}$ such that $d(g, L_{2}) = d(g, g^{\prime }),$ where $g^{\prime }\in L_{2}.$ Then, $\hat{H}_{b^{l}}\left(L_{1}, L_{2}\right) \geq d\left(g, g^{\prime }\right).$

In the following, the concept of fuzzy dynamic process as a generalization of dynamic process, and some elementary facts about these concepts are discussed.

3.   Fuzzy dynamic process:$\check{D}\left(\left[ T \mu \right] _{ \alpha }, \mu _{0}\right)$

In this section, first we deal with some new aspects of the fuzzy dynamic process as follows:

Definition 3.1. Let $T:G\rightarrow F(G)$ be a fuzzy mapping. If there is $\alpha \in (0, 1]$, and let $\mu _{0}\in G$ be arbitrary and fixed such that

Every membership value of $\check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ is called a fuzzy dynamic process of $T$ starting point $\mu _{0}$. The fuzzy dynamic process $(\mu _{j})_{j\in {\mathbb{N}}\cup \left \{ 0\right \} }$ onward is written as $\left(\mu _{j}\right)$.

Example 3.2. Let $G = C\left(\left[ 0, 1\right] \right)$ be a Banach space with norm $\left \Vert \mu \right \Vert = \sup_{r\in \left[ 0, 1\right] }\left \vert \mu \left(r\right) \right \vert$ for $\mu \in G$. Let $T:G\rightarrow F(G)$ be a fuzzy mapping. If there is $\alpha \in (0, 1]$ such that for every $\mu \in G$, $\left[ T\mu \right] _{\alpha }$ is a set of the function

that is,

and let $\mu _{0}\left(\delta \right) = \delta,$ $\delta \in \left[ 0, 1 \right]$. Then the iterative sequence

is a fuzzy dynamic process of mapping $T$ with starting point $\mu _{0}.$ The mapping $T:G\rightarrow F\left(\mathbb{R}\right)$ is said to be $\check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ fuzzy dynamic lower semi-continuous at $\mu \in G,$ if for every fuzzy dynamic process $(\mu _{j})\in D(T, \mu _{0})$ and for every subsequence $(\mu _{j\left(i\right) })$ of $(\mu _{j})$ convergent to $\mu$

In this case, $T$ is fuzzy dynamic lower semi-continuous $\check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$. If $T$ is fuzzy dynamic lower semi-continuous $\check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ at each $\mu \in G$, then $T$ is known as lower semi-continuous. For every sequence $(\mu _{j})\subset G$ and $\mu \in G$ such that $(\mu _{j})\rightarrow \mu$, we have $\left[ T\mu \right] _{\alpha }\leq \lim \inf_{i\rightarrow +\infty }\left[ T\mu \left(j\right) \right] _{\alpha }.$

Example 3.3. Let $G = {\mathbb{R}}^{+}\cup \left \{ 0\right \}$. Define $T:G\rightarrow F\left(G\right)$ by

all $\mu \in G,$ there is $\alpha \left(\mu \right) = 1$ such that $\left[ T\mu \right] _{\alpha \left(\mu \right) } = \left[ 0, \frac{\mu }{2} \right].$ Apply the following iterative procedure to generate a sequence $\{ \mu _{n}\}$ of fuzzy sets is given by (see Table 1 and Figure 1)

Where $\mu _{0} = 2$ is intial point and $h = \frac{1}{2}$.

We obtain,

is a fuzzy dynamic process of $T$ starting at point $\mu _{0} = 2.$

Further, in the following we develop fuzzy fixed point theorems with respect to fuzzy dynamic process $\check{D} \left(\left[ T \mu \right] _{ \alpha }, \mu _{0}\right)$ as follows.

4.   Fuzzy fixed point theorems with respect to fuzzy dynamic process:$\check{D} \left(\left[ T \mu \right] _{ \alpha }, \mu _{0}\right)$

Now, we start with the following main definition:

Definition 4.1. Let $\left(G, d\right)$ be a $b$ -metric-like space with $s\geq 1$. A mapping $T:G\rightarrow F\left(G\right)$ is called a $F$-fuzzy Suzuki-Hardy-Rogers (abbr., F-FSHR) type contraction with respect to $\check{ D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ and $\alpha :G\rightarrow (0, 1]$ such that $\left[ T\mu _{i}\right] _{\alpha \left(i\right) }$ are nonempty closed subsets of $G$ if for some $\mathcal{F}\in \nabla _{\digamma }$ and $\tau :(0, +\infty)\rightarrow (0, +\infty)$ such that

we have

where

for all $\mu _{i}\in \check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$, $\hat{H}_{b}(\left[ T\mu _{i}\right] _{\alpha \left(i\right) }, \left[ T\mu _{i+1}\right] _{\alpha \left(i+1\right) }) > 0$, where $e_{1}, e_{2}, e_{3}, e_{4}, e_{5}\in \left[ 0, 1 \right]$ such that $e_{1}+e_{2}+e_{3}+e_{4}+e_{5} = 1$ and $1-e_{3}-e_{5} > 0$.

Remark 4.2. To continue with our results, the behavior of self distance in $b$-metric-like space is defined by

Additionally, we assume that $\mu _{i}\in \check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ satisfying fuzzy dynamic process for below condition:

for all $i\in \mathbb{N}$. If for the investigated process that does not satisfy (4.2), there is some $i_{0}\in \mathbb{N}$ such that

and

then we get $\mu _{i_{0-1}} = \mu _{i_{0}}\in \left[ T\mu _{i_{0-1}}\right] _{\alpha \left(i_{0-1}\right) }$ which implies the existence of fuzzy fixed point. In the light of this consideration, fuzzy dynamic process satisfying (4.2) does not depreciate a generality of our analysis.

Now, we proceed to our main result:

Theorem 4.3. Let $(G, d, s)$ be a complete $b$ -metric-like space. Let $T:G\rightarrow \mu _{\alpha }(G)$ be an F-FSHR type contraction with respect to $\mu _{i}$. Assume that the following holds:

$\left(i\right)$ There is a fuzzy dynamic iterative process $\mu _{i}\in \check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ such that for each $l\geq 0$ $\lim \inf_{k\rightarrow l^{+}}\tau \left(k\right) > 0;$

$\left(ii\right)$ A mapping $G\ni \mu _{i}\longmapsto d_{b}(\mu _{i}, \left[ T\mu _{i}\right] _{\alpha \left(i\right) })$ is fuzzy dynamic lower semi-continuous $\check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$;

$\left(iii\right)$ If, in addition, $\mathcal{F}$is super-additive, i.e., for $\mu_{1}, \mu_{2}, \xi_{1}, \xi_{2}\in R^{+}$ we have

Then $T$ has a fuzzy fixed point.

Proof. Choose an arbitrary point $\mu _{0}\in G$. In veiw of fuzzy dynamic iterative process, we have

In case that there is $i_{0}\in \mathbb{N}$ such that $\mu _{i_{0}} = \mu _{i_{0+1}}$, then our proof of Theorem (4.3) go ahead as follows. If we let $\mu _{i}\neq \mu _{i+1}$ for all $i\in \mathbb{N}$, then we have

From (4.1) and in the light of Lemma (2.8), we have

Now, we survey to the following inequality

for all $i\in \mathbb{N}$. Suppose, on the contrary, there is $i_{0}\in \mathbb{N}$ such that $d(\mu _{i_{0}+1}, \left[ T\mu _{i_{0}+1}\right] _{\alpha \left(i_{0}+1\right) })\geq d(\mu _{i_{0}}, \left[ T\mu _{i_{0}} \right] _{\alpha \left(i_{0}\right) })$. By (4.4) and Lemma (2.8), we have

Owing to the above hypothesis, this, in turn, yields:

Since $\mathcal{F}$ is super-additive, we can write

From this, By given condition $e_{1}+e_{2}+e_{3}+e_{4}+e_{5} = 1$, we have

a contradiction. Hence (4.5) holds true. In the light of above hypothesis, Therefore $d_{b}\left(\mu _{i}, \left[ T\mu _{i}\right] _{\alpha \left(i\right) }\right)$ is a decreasing sequence with respect to real number and it is bounded from below. Suppose that there is $\Psi \geq 0$ such that

We now to prove that $\Psi = 0$. Suppose, based on contrary that $\Psi > 0.$ Then, for every $\varepsilon > 0$, there is a natural number $j$ such that

By $\left(\mathcal{F}_{i}\right)$,

Also, by applying (4.3), we have

Since F-FSHR type contraction with respect to $\check{D}(T, \mu _{0})$, we have

Due to the above hypothesis, this, in turn, yields:

This implies

Since

By appealing to above observation, we obtain

Continuing these fashion, we obtain

Upon setting $i\rightarrow +\infty$, we have$\ $

Also, in veiw of $\left(\mathcal{F}_{ii}\right),$ we get

So, there is $i_{1}\in {\mathbb{N}}$ such that $d_{b}(\mu _{j+i}, \left[ T\mu _{j+i}\right] _{\alpha \left(j+i\right) }) < \Psi$ for all $i > i_{1}$, which is a contradiction with repect to $\Psi$. Therefore, we have

Now, we show that

Let us assume on the contrary that, for every $\varepsilon > 0$ there are sequences $\gamma (i)$ and $\delta (i)$ in ${\mathbb{N}}$ such that

for all $i\in {\mathbb{N}}$. So, we have

By (4.12), $\exists$ $i_{2}\in {\mathbb{N}}$ such that

for all $i > i_{2}$, which together with (4.15) yields

In view of $(\mathcal{F}_{i}),$ we can write

From (4.14) and (4.16), we write

Applying the triangle inequality, we find that

Next, if we setting to the limit $i\rightarrow +\infty$ in (4.19) and make use of (4.12), then,

Also, there is $i_{3}\in {\mathbb{N}}$ such that

for all $i > i_{3}$, that is, $d\left(\mu _{\gamma _{(i)+1}}, \mu _{\delta _{(i)+1}}\right) > 0 > 0$ for $i > i_{3}.$ Further, from (4.1) and Lemma (2.8), we can write

for all $i > \max \left \{ i_{1}, i_{2}\right \}.$ In view of (4.16)–(4.18), inequaility (4.20) yields

for all $i > \max \left \{ i_{1}, i_{2}\right \}$. Taking the limit $i\rightarrow +\infty$ in (4.21), we get

which by vertue of $(\mathcal{F}_{ii}),$ implies that $\lim_{i\rightarrow +\infty }d\left(\mu _{\gamma _{(i)+1}}, \mu _{\delta _{(i)+1}}\right) = 0$. In the light of (4.19), we can write $\lim_{i\rightarrow +\infty }d(\mu _{\gamma _{(i)}}, \mu _{\delta _{(i)}}) = 0,$ which contradicts. Hence (4.13) holds true. Hence $\{ \mu _{i}\}$ is a Cauchy sequence in $G.$ Since $G$ is a complete $b$ -metric-like space, there is a point $c\in G$ such that

Now, we show futher the following inequatlity

Assume on the contrary that $\exists$ $i_{0}\in {\mathbb{N}}$ such that

Then from (4.5) and (4.24), we have

a contradiction. Thus (4.23) holds true. So, we can write

or

Now, let us now examine the following cases:

Case 1. Assume that (4.25) holds true. From (4.25), we have

By (4.12) and (4.22), there is $i_{4}\in {\mathbb{N}}$ such that for some $\varepsilon _{1} > 0$

From (4.27) and (4.28), we have

for all $i > i_{4}.$ Taking the limit as $i\rightarrow +\infty$ in (4.29), we find that $\lim_{i\rightarrow +\infty }\mathcal{F}\left(d_{b}\left(\mu _{i+1}, \left[ T\left(c\right) \right] _{\alpha \left(c\right) }\right) \right) = -\infty$. By means of $(\mathcal{F}_{ii}),$ we have

On the other hand, we see that

Further, in the light of above hypothesis with respect to $G\ni c\longmapsto d_{b}(c, \left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) })$ is $\check{D}(T, \mu _{0})$-fuzzy dynamic lower semi-continuous, we have

Also, the closedness of $\left[ T\left(c\right) \right] _{\alpha \left(c\right) }$ implies that $c\in \left[ T\left(c\right) \right] _{\alpha \left(c\right) }$ which means that $c$ is a fuzzy fixed point of $T.$

Case 2. Assume that (4.26) holds true. From (4.26), we can write

From (4.12) and (4.22), there is $i_{5}\in {\mathbb{N}}$ such that for some $\varepsilon _{2} > 0$

Now, from (4.30) and (4.31), we have

for all $i > i_{5}.$ Taking the limit as $i\rightarrow +\infty$ in (4.32), we see that $\lim_{i\rightarrow +\infty }\mathcal{F}\left(d_{b}\left(\mu _{i+2}, \left[ T\left(c\right) \right] _{\alpha \left(c\right) }\right) \right) = -\infty$. By means of $(\mathcal{F}_{ii}),$ we have

Consequently,

Further, in view of above fashion with respect to $G\ni c\longmapsto d_{b}(c, \left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) })$ is $\check{D}(T, \mu _{0})$-fuzzy dynamic lower semi-continuous, we have

Also, the closedness of $\left[ T\left(c\right) \right] _{\alpha \left(c\right) },$ which implies that $c\in \left[ T\left(c\right) \right] _{\alpha \left(c\right) }$. Hence, $c$ is a fuzzy fixed point of $T.$

Corollary 4.4. Let $\left(G, d\right)$ be a $b$ -metric-like space with $s\geq 1$. Assume that $T:G\rightarrow \mu (G)$ is a F-fuzzy Suzuki-Kannan (abbr., F-FSK) type contraction with respect to fuzzy dynamic system $\check{D} \left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ and $\alpha :G\rightarrow \lbrack 0, 1]$ such that $\left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) }$ are nonempty closed subsets of $G.$ Assume that for some $\mathcal{F}\in \nabla _{\digamma }$ and $\tau :(0, +\infty)\rightarrow (0, +\infty)$ such that

we have

where

for all $\mu _{i}\in \check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$, $\hat{H}_{b}\left(\left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) }, \left[ T\left(\mu _{i+1}\right) \right] _{\alpha \left(i+1\right) }\right) > 0$, where $e_{2}, e_{3}\in \left[ 0, 1\right]$ such that $e_{1}+e_{2} = 1$. Assume that $\left(i\right)$–$\left(iii\right)$ are satisfied. Then $T$ has a fuzzy fixed point.

Corollary 4.5. Let $\left(G, d\right)$ be a $b$ -metric-like space with $s\geq 1$. Assume that $T:G\rightarrow \mu (G)$ is a F-fuzzy Suzuki-Chatterjea (abbr., F-FSC) type contraction with respect to fuzzy dynamic system $\check{D} \left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ and $\alpha :G\rightarrow \lbrack 0, 1]$ such that $\left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) }$ are nonempty closed subsets of $G.$ Assume that for some $\mathcal{F}\in \nabla _{\digamma }$ and $\tau :(0, +\infty)\rightarrow (0, +\infty)$ such that

we have

where

for all $\mu _{i}\in \check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$, $\hat{H}_{b}\left(\left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) }, \left[ T\left(\mu _{i+1}\right) \right] _{\alpha \left(i+1\right) }\right) > 0$, where $e_{4}, e_{5}\in \lbrack 0, \frac{1}{2})$. Assume that $\left(i\right)$–$\left(iii\right)$ are satisfied. Then $T$ has a fuzzy fixed point.

Corollary 4.6. Let $\left(G, d\right)$ be a $b$ -metric-like space with $s\geq 1$. Assume that $T:G\rightarrow \mu (G)$ is a F-fuzzy Suzuki-Banach (abbr., F-FSB) type contraction with respect to fuzzy dynamic system $\check{D} \left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ and $\alpha :G\rightarrow \lbrack 0, 1]$ such that $\left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) }$ are nonempty closed subsets of $G.$ Assume that for some $\mathcal{F}\in \nabla _{\digamma }$ and $\tau :(0, +\infty)\rightarrow (0, +\infty)$ such that

we have

for all $\mu _{i}\in \check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$, $\hat{H}_{b}\left(\left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) }, \left[ T\left(\mu _{i+1}\right) \right] _{\alpha \left(i+1\right) }\right) > 0$, where $e_{1}\in \lbrack 0, 1)$. Assume that $\left(i\right)$ and $\left(ii\right)$ are satisfied. Then $T$ has a fuzzy fixed point.

Corollary 4.7. Let $\left(G, d\right)$ be a $b$ -metric-like space with $s\geq 1$. Assume that $T:G\rightarrow \mu (G)$ is a F-fuzzy Banach (abbr., F-FB) type contraction with respect to fuzzy dynamic system $\check{D} \left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$ and $\alpha :G\rightarrow \lbrack 0, 1]$ such that $\left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) }$ are nonempty closed subsets of $G.$ Assume that for some $\mathcal{F}\in \nabla _{\digamma }$ and $\tau :(0, +\infty)\rightarrow (0, +\infty)$ such that

for all $\mu _{i}\in \check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right)$, $\hat{H}_{b}\left(\left[ T\left(\mu _{i}\right) \right] _{\alpha \left(i\right) }, \left[ T\left(\mu _{i+1}\right) \right] _{\alpha \left(i+1\right) }\right) > 0$, where $e_{1}\in \lbrack 0, 1)$. Assume that $\left(i\right)$ and $\left(ii\right)$ are satisfied. Then $T$ has a fuzzy fixed point.

Example 4.8. Let $G = {\mathbb{R}}^{+}\cup \left \{ 0\right \}$ and $d:G\times G\rightarrow {\mathbb{R}}^{+}\cup \left \{ 0\right \}$ be a function defined by

Clearly, $\left(d, G\right)$ is a complete $b$-metric-like space with $s = \frac{4}{3}$. Define a fuzzy mapping $T:G\rightarrow F(G)$ by

Define $\mathcal{F}:{\mathbb{R}}^{+}\rightarrow {\mathbb{R}}$ and $\tau :{ \mathbb{R}}^{+}\rightarrow {\mathbb{R}}^{+}$ by $\mathcal{F}(\mu) = \ln (\mu)$ and

For all $\mu \in \check{D}\left(\left[ T\mu \right] _{\alpha }, \mu _{0}\right),$ there is $\alpha \left(\mu \right) = 1$ such that $\left[ T\mu \right] _{\alpha \left(\mu \right) } = \left[ 0, \frac{\mu }{2}\right].$ Then we have

setting $e_{2} = e_{3} = e_{4} = e_{4} = 0$ and $e_{1} = 1$, we obtain

Hence all the required possible hypothesis of Corollary 4.6 are satisfied, Thus $T$ has a fuzzy fixed point.

5.   An application to Hukuhara fuzzy differentiability problem

Fuzzy differential equations and fuzzy integral equations have always been of key importance in dynamical programming and engineering problems. Therefore, various authors used different techniques for solving an fuzzy differential equations and fuzzy integral equations. Among those, Hukuhara differentiability for fuzzy valued function is the most celebrated problem. This section renders solution of a fuzzy differential equations. For this we explore Hukuhara differentiability for fuzzy functions and fuzzy initial valued problem in the setting of $b$ -metric-like space.

Definition 5.1. A function $g: {\mathbb{R}} \rightarrow \left[ 0, 1\right]$ is called a fuzzy real number if

$\left(i\right)$ $g$ is normal, i.e., there is $\mu _{0}\in {\mathbb{R}}$ in such a way that $g\left(\mu _{0}\right) = 1$;

$\left(ii\right)$ $ga$ is fuzzy convex, i.e., $g\left(\beta \left(\mu _{1}\right) +\left(1-\beta \right) \mu _{2}\right) \geq \min \left \{ g\left(\mu _{1}), g\left(\mu _{2}\right) \right) \right \}$, $0\leq \beta \leq 1$, for all $\mu _{1}, \mu _{2}\in {\mathbb{R}}$;

$\left(iii\right)$ $g$ is upper semi-continuous;

$\left(iiii\right)$ $\left[ g\right] ^{0} = cl\left \{ \mu \in R:g\left(\mu \right) > 0\right \}$ is compact.

Note that, for $\alpha \in (0, 1],$

expresses $\alpha -cut$ of the fuzzy set $g.$ For $g\in P^{1}$, where $P^{1}$ represents the family of fuzzy real numbers, one can write $\left[ g\right] ^{\alpha }\in C_{c}\left({\mathbb{R}}\right) for all \alpha \in \left[ 0, 1\right]$, where $C_{c}\left(R\right)$ denotes the set of all compact and convex subsets of $\mathbb{R}$. The supremum on $P^{1}$ endowed with the $b$-metric-like is defined by

for all $g_{1}, g_{2}\in P^{1},$ $g_{1, s_{1}}^{\alpha }-g_{2, s_{1}}^{\alpha } = diam\left(\left[ g\right] \right).$ Consider the continuous fuzzy function defined on $\left[ 0, \Gamma \right]$, for $\Gamma > 0$ as $C\left(\left[ 0, \Gamma \right], P^{1}\right)$ endowed with the complete $b$ -metric-like with respect to $b$-metric-like as:

for all $g_{1}, g_{2}\in C^{1}\left(\left[ 0, \Gamma \right], P^{1}\right).$ Consider the fuzzy initial valued problem:

where $g^{\prime }$ is the Hukuhara differentiability and $f$ is the fuzzy function, i.e., $f:I\times P^{1}\rightarrow P^{1}$ is continuous. Denote the set of all continuous fuzzy functions $f:I\rightarrow P^{1}$ which have continuous derivatives by $C^{1}\left(I, P^{1}\right).$ A family $\mu \in C^{1}\left(I, P^{1}\right)$ is a solution of fuzzy initial valued problem (5.1) if and only if

where (5.2) is called a fuzzy Volterra integral equation.

Theorem 5.2. Let $f:I\times P^{1}\rightarrow P^{1}$ be a continuous function such that

for $g, g^{\prime }\in P^{1}$, In addition, assume that $\tau :(0, +\infty)\rightarrow (0, +\infty)$ such that

where $g < g^{\prime } for all \mu \in I$ and $g, g^{\prime }\in P^{1}.$ Then the FIVP (5.1) has a fuzzy solution with respect to $C^{1}\left(I, P^{1}\right).$

Proof. Let $\tau :(0, +\infty)\rightarrow (0, +\infty)$ and the family $C^{1}\left(I, P^{1}\right)$ endow with the $b$ -metric-like as:

for all $g, g^{\prime }\in C^{1}\left(I, P^{1}\right).$ Let $S:G\rightarrow (0, 1]$. Due to (5.2) for $g\in G,$ one can write

Assume that $g < g$. Then we have

This implies $Y_{g}\left(\mu \right) \neq Y_{g^{\prime }}\left(\mu \right)$. Assume a fuzzy mapping $T:G\rightarrow P^{G}$ is defined by

Owing to $\alpha \left(g\right) = S\left(g\right)$ and $\alpha \left(g^{\prime }\right) = S\left(g^{\prime }\right),$ we have

and on the same fashion, we have

Therefore,

Then, in view of above hypothesis we have:,

By appealing to the above fashion, we obtain

or equivalently,

Owing to logarithms, we have

Owing to the above speculation, this, in turn, yields:

Due to $\mathcal{F}$-contraction, with the setting $\mathcal{F}\left(\mu \right) = \ln \mu, for all \mu \in C^{1}\left(I, P^{1}\right)$, we have

It follows that there is $c\in C^{1}\left(I, P^{1}\right)$ such that $c\in \left[ Tc\right] _{\alpha \left(c\right) }.$ Hence all the possible hypothesis of Corollary 4.7 are satisfied and consequently fuzzy initial valued problem (5.1) has a fuzzy solution $c\in C^{1}\left(I, P^{1}\right)$ in $C^{1}\left(I, P^{1}\right).$

6.   Concluding conclusions and observations

The article regards with new approach of fuzzy dynamic process on $b$ -metric-like space, specifically the mapping of set-valued (extended) fuzzy intervals endowed with the $b$ -metric-like. After we just adopt the standard setting of fuzzy dynamic process in $b$ -metric-like space which defines convergence theorems in generalized $\mathcal{F}$-contraction via expectations of fuzzy Suzuki-type contraction mappings. Subsequently, corollaries are originated from the main result. To explain the example in the main section, a graphically interpretation has been created that best illustrates the fuzzy dynamic process to the readers. At the end, gives an application of our results in solving Hukuhara differentiability through the fuzzy initial valued problem and fuzzy functions. The pivotal role of Hukuhara differentiability in fuzzy dynamic process is stated. In future, this methodology can be inspected intuitionistic fuzzy and picture fuzzy sets the fuzzy dynamic process for a hybrid pair of mappings can be examined.

Acknowledgements

All the authors have equal contribution, read and approved the final manuscript. The authors S. Subhi, N. Mlaiki and W. Shatanawi would like to thank Prince Sultan University for paying the publication fees for this work through TAS LAB.

Conflict of interest

The authors declare that they have no conflicts of interests.