Stabilization of nonlinear hybrid stochastic time-delay neural networks with Lévy noise using discrete-time feedback control

1.   Introduction and preliminaries

If the image of a point $x$ under two single-valued mappings is $x$ itself, then $x$ is said to be a fixed point of these mappings. Banach ^[7] proved a meaningful result for contraction mappings. Due to its significance, several authors, like Acar et al. ^[3], Altun et al. ^[5], Aslantas et al. ^[6], Sahin et al. ^[27], Hussain et al. ^[17], Hammad et al. ^[14,15,16] and Ceng et al. ^[8,9,10,11] presented many related useful applications in fixed point theory. In ^[23,31], the authors showed a new iterative scheme for the solution of nonlinear mixed Volterra Fredholm type fractional delay integro-differential equations of different orders. Chistyakov ^[13] introduced the notion of a modular metric space. Mongkolkeha et al. ^[21] established some results in modular metric spaces for contraction mappings. Chaipunya et al. ^[12], Abdou et al. ^[2] and Alfuraidan et al. ^[4] showed fixed point results for multivalued mappings in modular metric spaces. Abdou et al. ^[1] proved fixed point theorems of pointwise contractions in modular metric spaces. Hussain et al. ^[19] discussed some fixed point theorems for generalized $F$-contractions in fuzzy metric and modular metric spaces. Later, Padcharoen et al. ^[22] introduced the concept of $\alpha$-type $F$-contractions in modular metric spaces and showed fixed point and periodic point results for such a contraction. Recently, Rasham et al. ^[26] introduced a modular-like metric space and proved results for families of mappings in such spaces. In this research work, we prove existence of fixed point results for a hybrid pair of multivalued maps fulfilling generalized rational type $F$-contractions, by using a weaker class of strictly increasing mappings $F$ rather than the class of mappings introduced by Wardowski ^[30].

Let us state the following preliminary concepts.

Definition 1.1. ^[26] Let $B$ be a non-empty set. A function $\upsilon :(0, \infty)\times B\times B\rightarrow \lbrack 0, \infty)$ is said to be a modular-like metric on $B,$ if for each $e, i, o\in B$ and $\upsilon (a, i, o) = \upsilon _{a}(i, o)$, the following hold:

$(i)$ $\upsilon _{a}(i, o) = \upsilon _{a}(o, i)$ for all $a > 0;$

$(ii)$ $\upsilon _{a}(i, o) = 0$ for all $a > 0$ implies $i = o;$

$(iii)$ $\upsilon _{l+n}(i, o)\leq \upsilon _{l}(i, e)+\upsilon _{n}(e, o)$ for all $l, n > 0$.

The pair $(B, \upsilon)$ is said to be a modular-like metric space. If we change $(ii)$ by "$\upsilon _{l}(i, o) = 0$ for each $l > 0$ iff $i = o$", then $(B, \upsilon)$ becomes a modular metric space. While, by changing $(ii)$ with "$\upsilon _{l}(i, o) = 0$ for some $l > 0$, such that $i = o$", we obtain a regular modular-like metric space. For $s\in B$ and $\varepsilon > 0,$ $\overline{C_{\upsilon _{l}}(s, \varepsilon)} = \{t\in B:|\upsilon _{l}(s, t)-\upsilon _{l}(t, t)|\leq \varepsilon \}$ is a closed ball in $(B, \upsilon).$

Example 1.2. Let $B = [0, \infty)\times \lbrack 0, \infty).$ Define $\upsilon :(0, \infty)\times B\times B\rightarrow \lbrack 0, \infty)$ as

The functions given in $(i)$ and $(ii)$ are examples of a modular-like metric on $B.$

Definition 1.3. ^[26] Let $(B, \upsilon)$ be a modular-like metric space.

$({\rm{i}})$ A sequence $(a_{n})_{n\in \mathbb{N} }$ in $B$ is said to be $\upsilon$-convergent to a point $a\in B$ for some $l > 0$ if $\underset{n\rightarrow +\infty }{\lim }\upsilon _{l}(a_{n}, a) = \upsilon _{l}(a, a)$.

$({\rm{ii}})$ A sequence $(a_{n})_{n\in \mathbb{N} }$ in $B$ is said to be an $\upsilon$-Cauchy sequence for some $l > 0$ if $\underset{n, m\rightarrow \infty }{\lim }\upsilon _{l}(a_{m}, a_{n})$ exists and is finite.

$({\rm{iii}})$ $B$ is called $\upsilon$-complete if each $\upsilon$-Cauchy sequence $(a_{n})_{n\in \mathbb{N} }$ in $B$ is $\upsilon$-convergent to some $a\in B$, that is,

$({\rm{iv}})$ If every sequence has a convergent subsequence, then $B$ is called compact.

Definition 1.4. ^[26] Let $(B, \upsilon)$ be a modular-like metric space and $U\subseteq B.$ An element $p_{0}$ in $U$ verifying

is called a best approximation in $U$ for $s\in B$. If each $s\in B$ possesses a best approximation in $U,$ then $U$ is called a proximinal set.

From now on, let $P(B)$ represent the set of proximinal compact subsets in $B.$

Example 1.5. Let $B = [0, \infty)$ and $\upsilon _{l}(s, r) = \frac{1}{w} (s+r)$ with $w > 0.$ Take $U =[7,8]$. Then for any $m\in B$,

So $7$ is a best approximation in $U$ for any $m\in B.$ Moreover, $[7,8]$ is a proximinal set.

Definition 1.6. ^[26] The mapping $H_{\upsilon _{l}}:P(B)\times P(B)\rightarrow \lbrack 0, \infty),$ given by

is known as an $\upsilon _{l}$- Hausdorff metric$.$ Note that $(P(B), H_{\upsilon _{l}})$ is named as an $\upsilon _{l}$- Hausdorff metric space.

Example 1.7. Let $B = [0, \infty)$ and $\upsilon _{l}(\theta, \vartheta) = \frac{1}{l}(\theta +\vartheta)$ with $l > 0.$ Taking $W = \left[5, 6\right]$ and $Q = \left[9, 10\right]$ we get $H_{\upsilon _{l}}(W, Q) = \frac{15}{l}.$

Definition 1.8. ^[26] Let $(X, \upsilon)$ be a modular-like metric space. $\upsilon$ is said to satisfy the $\triangle _{M}$-condition if $\underset{n, m\rightarrow \infty }{\lim }\upsilon _{p}(x_{n}, x_{m}) = 0$, where $p\in \mathbb{N}$ implies $\underset{n, m\rightarrow \infty }{\lim }\upsilon _{l}(x_{n}, x_{m}) = 0$, for some $l > 0$.

Definition 1.9. ^[28] Let $C\neq \Phi,$ $Y:C\rightarrow P(C)$ be a multivalued mapping, $E\subseteq C$ and $\alpha :C\times C\rightarrow \lbrack 0, +\infty)$ be a function. Then $Y$ is said to be $\alpha _{\ast }$-admissible on $E$ if $\alpha _{\ast }(Ye, Yz) = \inf \{\alpha (l, m):l\in Ye, m\in Yz\}\geq 1$, whenever $\alpha (e, z)\geq 1$ for all $e$, $z$ $\in E.$

Definition 1.10. ^[29] Let $B\neq \Phi,$ $Y:B\rightarrow P(B)$ be a multi-valued mapping, $R\subseteq B$ and $\alpha :B\times B\rightarrow \lbrack 0, \infty)$ be a function. Then $Y$ is said to be $\alpha _{\ast }$-dominated on $R$ if for all $v\in R, \ \alpha _{\ast }(v, Yv) = \inf \{\alpha (v, j):j\in Yv\}\geq 1.$

Definition 1.11. ^[30] Let $(C, d)$ be a metric space. A self mapping $H:C\rightarrow C$ is said to be a $Q$-contraction if for each $g, k\in C,$ there is $\tau > 0$ such that $d(Ca, Cg) > 0$ implies

where $Q:(0, \infty)\rightarrow \mathbb{R}$ satisfies the following:

(F1) For any $k\in (0, 1)$, $\underset{\sigma \rightarrow 0^{+}}{\lim }\sigma ^{k}Q(\sigma) = 0$;

(F2) For each $u, v > 0$ such that $u < v$, $Q(u) < Q(v)$;

(F3) $\underset{n\rightarrow +\infty }{\lim }\sigma _{n}$ $= 0$ if and only if $\underset{n\rightarrow +\infty }{\lim }Q(\sigma _{n}) = -\infty$ for every positive sequence $\{\sigma _{n}\}_{n = 1}^{\infty }$.

Let $\digamma$ denote the set of mappings such that (F1)–(F3) hold.

Lemma 1.12. ^[26] Let $({\rm{£}}, \upsilon)$ be amodular-like metric space. Let $(P({\rm{£}}), H_{\upsilon _{l}})$ be aHausdorff $\upsilon _{l}-$metric-like space. Then, for all $b\in U$ and foreach $U, Y\in P({\rm{£}}),$ there is $b_{a}\in Y$ such that $H_{\upsilon_{l}}(U, Y)\geq \upsilon _{l}(a, b_{a})$.

Example 1.13. ^[24] Let $W = \mathbb{R}.$ Consider $\alpha :W\times W\rightarrow \lbrack 0, \infty)$ as

Define $L, N:W\rightarrow P(W)$ by

The $\alpha _{\ast }$-dominated property for $L$ and $N$ holds. Note that $L$ and $N$ are not $\alpha _{\ast }$-admissible.

2.   Main results

Let $({\rm{£}}, \upsilon)$ be a modular-like metric space$,$ $\delta _{0}\in {\rm{£}}$, and $R, C:{\rm{£}} \rightarrow$ $P({\rm{£}})$ be two multifunctions on ${\rm{£}}$. For $\delta _{1}\in R\delta _{0}$ with $\upsilon _{1}(\delta _{0}, R\delta _{0}) = \upsilon _{1}(\delta _{0}, \delta _{1})$, take $\delta _{2}\in C\delta _{1}$ such that $\upsilon _{1}(\delta _{1}, C\delta _{1}) = \upsilon _{1}(\delta _{1}, \delta _{2}).$ Choose $\delta _{3}\in R\delta _{2}$ such that $\upsilon _{1}(\delta _{2}, R\delta _{2}) = \upsilon _{1}(\delta _{2}, \delta _{3}).$ In this way, we get a sequence $\{CR(\delta _{n})\}$ in ${\rm{£}},$ where

for all $n\in \mathbb{N\cup \{}0\mathbb{\}}.$ Note that $\upsilon _{1}(\delta _{2n}, R\delta _{2n}) = \upsilon _{1}(\delta _{2n}, \delta _{2n+1})$ and $\upsilon _{1}(\delta _{2n+1}, C\delta _{2n+1}) = \upsilon _{1}(\delta _{2n+1}, \delta _{2n+2}).$ $\{CR(\delta _{n})\}$ is said to be a sequence in ${\rm{£}}$ generated by $\delta _{0}.$ If $R = C,$ then we denote $\{{\rm{£}} R(\delta _{n})\}$ instead of $\{CR(\delta _{n})\}.$

Theorem 2.1. Let $({\rm{£}}, \upsilon)$ be a completemodular-like metric space. Suppose that $\upsilon$ is regular and verifiesthe $\bigtriangleup _{M}$-condition. Let $\delta _{0}\in$ ${\rm{£}},$ $\alpha :{\rm{£}} \times {\rm{£}} \rightarrow \lbrack 0, \infty)$ and $R, C:{\rm{£}} \rightarrow$ $P({\rm{£}})$ be $\alpha _{\ast }$-dominatedmultifunctions on ${\rm{£}}.$ Assume there are $\tau > 0$ and $Q\in \digamma$such that

where $t, \delta \in \{CR(\delta _{n})\},$ $\alpha (t, \delta)\geq 1$ or $\alpha (\delta, t)\geq 1,$ and $H_{\upsilon _{1}}(Rt, C\delta) > 0$. Then the sequence $\{CR(\delta _{n})\}$ generated by $\delta _{0}$ convergesto $e\in {\rm{£}}$ and for each $n\in \mathbb{N},$ $\alpha (\delta _{n}, \delta _{n+1})\geq 1.$ Furthermore, if $e$ satisfies (2.1), $\alpha (\delta _{n}, e)\geq 1$ and $\alpha (e, \delta _{n})\geq 1$ forall integers $n\geq 0$, then $R$ and $C$ have a common fixed point $e$ in ${\rm{£}}.$

Proof. Consider a sequence $\{CR(\delta _{n})\}.$ Obviously, $\delta _{n}\in {\rm{£}}$ for each integer $n\geq 0$. If $j$ is odd, then $j = 2\grave{ \imath}+1$ for some $\grave{\imath}\in \mathbb{N}$. By definition of $\alpha _{\ast }$-dominated mappings, one has $\alpha _{\ast }(\delta _{2\grave{\imath}}, R\delta _{2\grave{\imath}})\geq 1$ and $\alpha _{\ast }(\delta _{2\grave{\imath}+1}, C\delta _{2\grave{\imath} +1})\geq 1.$ Since $\alpha _{\ast }(\delta _{2\grave{\imath}}, R\delta _{2\grave{ \imath}})\geq 1,$ one gets $\inf \{\alpha (\delta _{2\grave{\imath}}, b):b\in R\delta _{2\grave{\imath}}\}\geq 1.$ Also, $\delta _{2\grave{\imath}+1}\in R\delta _{2\grave{\imath}}$ and so $\alpha (\delta _{2\grave{\imath}}, \delta _{2 \grave{\imath}+1})\geq 1.$ Moreover, $\delta _{2\grave{\imath}+2}\in C\delta _{2\grave{\imath}+1}$ and so $\alpha (\delta _{2\grave{\imath}+1}, \delta _{2 \grave{\imath}+2})\geq 1.$ In view of Lemma 1.12, we have

This implies

Now, if

then from (2.2), we have

which is a contradiction. Therefore,

for all $\grave{\imath}\geq 0.$ Hence, from (2.2), we have

Similarly, we have

for all $\grave{\imath}\geq 0.$ By (2.3) and (2.4), we have

Repeating these steps, we get

Similarly, we have

By (2.5) and (2.6), we obtain

Letting $n\rightarrow \infty$ in (2.7), one obtains

Since $Q\in \digamma,$

Due to (F1) of $\digamma$, there is $k\in \left(0, 1\right)$ such that

By (2.7), for all $n\in \mathbb{N},$ we obtain

Using (2.8), (2.9) and taking $n\rightarrow \infty$ in (2.10), we have

By (2.11), there is $n_{1}\in \mathbb{N}$ such that $n(\upsilon _{1}(\delta _{n}, \delta _{n+1}))^{k}\leq 1$ for all $n\geq n_{1}$, or

Letting $p > 0$ and $m = n+p >$ $n > n_{1},$ we get

Since $k\in \left(0, 1\right),$ $\frac{1}{k} > 1$ and so the series $\sum_{j = 1}^{\infty }\frac{1}{j^{\frac{1}{k}}}$ converges. Thus,

Since $\upsilon$ satisfies the $\triangle _{M}$-condition, we have

Hence $\{CR(\delta _{n})\}$ is Cauchy in the regular complete modular-like metric space $({\rm{£}}, \upsilon)$ and so there is $e\in {\rm{£}}$ such that $\{CR(\delta _{n})\}\rightarrow e$ as $n\rightarrow \infty$ and thus

Now, by Lemma 1.12, one obtains

Now, there exists $\delta _{2n+1}\in R\delta _{2n}$ such that $\upsilon _{1}(\delta _{2n}, R\delta _{2n}) = \upsilon _{1}(\delta _{2n}, \delta _{2n+1})$. From assumption, $\alpha (\delta _{n}, e)\geq 1$. Assume that $\upsilon _{1}(e, Ce) > 0$. Then there is an integer $p > 0$ such that $\upsilon _{1}(\delta _{2n+1}, Ce) > 0$ for $n\geq p$. Now, if $H_{\upsilon _{1}}(R\delta _{2n}, Ce) > 0,$ then by (2.1), we have

Letting $n\rightarrow \infty$ and using (2.13), we get

Since $Q$ is strictly increasing, (2.14) implies

This is a contradiction. Hence $\upsilon _{1}(e, Ce) = 0$ and so $e\in Ce.$

Similarly, we can show that $\upsilon _{1}(e, Re) = 0$, that is, $e\in Re.$ Hence $e$ is a common fixed point of both mappings $R$ and $C$ in ${\rm{£}}.$

Corollary 2.2. Let $({\rm{£}}, \upsilon)$ be a completemodular-like metric space. Suppose that $\upsilon$ is regular and verifiesthe $\bigtriangleup _{M}$-condition. Let $\alpha :{\rm{£}} \times {\rm{£}} \rightarrow \lbrack 0, \infty)$ and $R, C:{\rm{£}} \rightarrow$ $P({\rm{£}})$be $\alpha _{\ast }$-dominated multifunctions on ${\rm{£}}.$ Assume thereare $\tau > 0$ and $Q\in \digamma$ such that

where $t, \delta \in {\rm{£}},$ $\alpha (t, \delta)\geq 1$ or $\alpha(\delta, t)\geq 1,$ and $H_{\upsilon _{1}}(Rt, C\delta) > 0$. Then there exists a sequence $\{\delta _{n}\}$ in ${\rm{£}}$ converging to $e\in {\rm{£}}$ and for each $n\in \mathbb{N},$ $\alpha (\delta _{n}, \delta _{n+1})\geq 1.$ Also, if $\alpha (\delta_{n}, e)\geq 1$ and $\alpha (e, \delta _{n})\geq 1$ for all integers $n\geq 0$, then $R$ and $C$ have a common fixed point $e$ in ${\rm{£}}.$

Example 2.3. Let ${\rm{£}} = \mathbb{R} _{+}\cup \{0\}$. Take $\upsilon _{2}(r, m) = r+m$ and $\upsilon _{1}(e, t) = \frac{ 1}{2}(e+t)$ for all $e, t\in {\rm{£}}.$ Define $R, C:{\rm{£}} \rightarrow$ $P({\rm{£}})$ by

and

Suppose that $v_{0} = 1$. Then $\upsilon _{1}(v_{0}, Rv_{0}) = \upsilon _{1}(1, R1) = \upsilon _{1}(1, \frac{1}{3})$ and so $v_{1} = \frac{1}{3}.$ Now, $\upsilon _{1}(v_{1}, Cv_{1}) = \upsilon _{1}(\frac{1}{3}, C\frac{1}{3}) = \upsilon _{1}(\frac{1}{3}, \frac{1}{12})$ and thus $v_{2} = \frac{1}{12}.$ Now, $\upsilon _{1}(v_{2}, Rv_{2}) = \upsilon _{1}(\frac{1}{12}, R\frac{1}{12}) = \upsilon _{1}(\frac{1}{12}, \frac{1}{36})$ and so $v_{3} = \frac{1}{36}.$ Continuing in this way, we have $\{CR(v_{n})\} = \{1, \frac{1}{3}, \frac{1}{12}, \frac{1}{36}, \frac{1 }{144}, \cdots \}.$ Define $\alpha :{\rm{£}} \times {\rm{£}} \rightarrow \lbrack 0, \infty)$ as

Let $v, y\in \{CR(v_{n})\}$ with $\alpha (v, y)\geq 1$. Then

Also,

If $Q(t) = \ln t$ and $\tau = \ln (1.2),$ then we have

Hence all the conditions in Theorem 2.1 hold and so $R$ and $C$ possess a common fixed point.

Note that

and

If $v = 2$ and $y = 3$, then we have

Also

Now,

This implies that

So the condition (2.1) does not hold on the whole space. Hence Corollary 2.2 and the other existing results in modular metric spaces cannot be applied to ensure the existence of a common fixed point. However, Theorem 2.1 is valid here.

Taking $R = C$ in Theorem 2.1, we may state the following corollary.

Corollary 2.4. Let $({\rm{£}}, \upsilon)$ be a completemodular-like metric space. Suppose $\upsilon$ is regular and the $\bigtriangleup _{M}$-condition holds. Let $\delta _{0}\in {\rm{£}},$ $\alpha:{\rm{£}} \times {\rm{£}} \rightarrow \lbrack 0, \infty)$ and $R:{\rm{£}} \rightarrow$ $P({\rm{£}})$ be a $\alpha _{\ast }$-dominated set-valuedfunction on ${\rm{£}}.$ Assume there are $\tau > 0$ and $Q\in \digamma$ suchthat

where $t, \delta \in \{{\rm{£}} R(\delta _{n})\},$ $\alpha (t, \delta)\geq1,$ and $H_{\upsilon _{1}}(Rt, R\delta) > 0.$ Then, the sequence $\{{\rm{£}} R(\delta _{n})\}$ generated by $\delta _{0}$ converges to $e\in {\rm{£}}$and for each integer $n\geq 0,$ $\alpha (\delta _{n}, \delta _{n+1})\geq 1.$Also, if $e$ satisfies (2.15) and either $\alpha (\delta _{n}, e)\geq 1$ or $\alpha (e, \delta _{n})\geq 1$ for all integers $n\geq 0$, then $R$ has afixed point $e$ in ${\rm{£}}$.

3.   Applications to graph theory

Jachymski ^[20] initiated the graph concept in fixed point theory. Hussain et al. ^[18] gave new results for graphic contractions. Recently, Younis et al. ^[32] discussed a significant result on the graphical structure of extended $b$-metric spaces and Shoaib et al. ^[29] established some results on graph dominated set-valued mappings in the setting of $b$-metric like spaces. Further results on graph theory can be seen in ^[24,25,28].

Definition 3.1. ^[29] Let $A$ be a non-empty set and $\Upsilon = (\mathcal{V}(\Upsilon), \mathcal{L}(\Upsilon))$ be a graph with $\mathcal{V }(\Upsilon) = A$. A mapping $P$ from $A$ into $P(A)$ is said to be multi-graph dominated on $A$ if for each $\imath \in A$, we have $(\imath, \jmath)\in \mathcal{L}(\Upsilon),$ where $\jmath \in Pa$.

Theorem 3.2. Let $(U, \upsilon)$ be a complete modular-likemetric space endowed with a graph $\Upsilon$ and $\delta _{0}\in R$ satisfying the following:

(i) $R$ and $C$ are multi-graph dominated functions on $\{CR(\delta _{n})\}$;

(ii) There are $\tau > 0$ and $Q\in \digamma$ such that

where $w, h\in \{CR(\delta _{n})\}$, $(w_{, }h)\in \mathcal{L}(\Upsilon)$or $(h, w)\in \mathcal{L}(\Upsilon),$ and $H_{\upsilon _{1}}(Rw, Ch) > 0.$Suppose that the regularity of $R$ and the $\bigtriangleup _{M}$-conditionare verified. Then $(\delta _{n}, \delta _{n+1})\in \mathcal{L}(\Upsilon)$and $\{CR(\delta _{n})\}\rightarrow \delta ^{\ast }.$ Also, if $\delta^{\ast }$ satisfies (3.1), $(\delta _{n}, \delta ^{\ast })\in \mathcal{L}(\Upsilon)$ and $(\delta ^{\ast }, \delta _{n})\in \mathcal{L}(\Upsilon)$for all integers $n\geq 0$, then $R$ and $C$ have a common fixed point in $U.$

Proof. Define $\alpha :U\times U\rightarrow \lbrack 0, \infty)$ as $\alpha (w, h) = 1$ if $w\in U$ and $(w, h)\in \mathcal{L}(\Upsilon)$, and $\alpha (w, h) = 0,$ otherwise. The graph domination on $U$ yields that $(w, h)\in \mathcal{L}(\Upsilon)$ for all $h\in Rw$ and $(w, h)\in \mathcal{L} (\Upsilon)$ for each $h\in Cw$. So $\alpha (w, h) = 1$ for all $h\in Rw$ and $\alpha (w, h) = 1$ for each $h\in Cw.$ Thus $\inf \{\alpha (w, h):h\in Rw\} = 1$ and $\inf \{\alpha (w, h):h\in Cw\} = 1.$ Hence $\alpha _{\ast }(w, Rw) = 1$ and $\alpha _{\ast }(w, Cw) = 1$ for any $w\in R.$ So $R$ and $C$ are $\alpha _{\ast }$-dominated on $U.$ Furthermore,

where $w, h\in U\cap \{CR(\delta _{n})\},$ $\alpha (w, h)\geq 1$ and $H_{\upsilon _{1}}(Rw, Ch) > 0.$ Also, (ii) is fulfilled. Due to Theorem 2.1, $\{CR(\delta _{n})\}$ is a sequence in $U$ and $\{CR(\delta _{n})\}\rightarrow \delta ^{\ast }\in U.$ Here, $\delta _{n}, \delta ^{\ast }\in U$ and either $(\delta _{n}, \delta ^{\ast })\in \mathcal{L}(\Upsilon)$ or $(\delta ^{\ast }, \delta _{n})\in \mathcal{L}(\Upsilon)$ yields that either $\alpha (\delta _{n}, \delta ^{\ast })\geq 1$ or $\alpha (\delta ^{\ast }, \delta _{n})\geq 1.$ So all the hypotheses of Theorem 2.1 hold. Thus $\delta ^{\ast }$ is a common fixed point of $R$ and $C$ in $U$ and $\upsilon _{1}(\delta ^{\ast }, \delta ^{\ast }) = 0.$

4.   On single-valued mappings

In this section, some corollaries related to single-valued mappings in modular-like metric space are derived. Let $({\rm{£}}, \upsilon)$ be a modular-like metric space$,$ $\delta _{0}\in {\rm{£}}$ and $R, C:{\rm{£}} \rightarrow {\rm{£}}$ be a pair of mappings. Let $\delta _{1} = R\delta _{0}$, $\delta _{2} = C\delta _{1}$, $\delta _{3} = R\delta _{2}.$ Consider a sequence $\{\delta _{n}\}$ in ${\rm{£}}$ such that $\delta _{2n+1} = R\delta _{2n}$ and $\delta _{2n+2} = C\delta _{2n+1},$ for integers $n\geq 0$. We represent this type of iteration by $\{CR(\delta _{n})\}.$ $\{CR(\delta _{n})\}$ is a sequence in ${\rm{£}}$ generated by $\delta _{0}.$ If $R = C,$ then we use $\{ {\rm{£}} R(\delta _{n})\}$ instead of $\{CR(\delta _{n})\}.$

Theorem 4.1. Let $({\rm{£}}, \upsilon)$ be a completemodular-like metric space. Suppose that the regularity of $\upsilon$ andthe $\bigtriangleup _{M}$-condition hold. Take $r > 0,$ $\delta _{0}\in{\rm{£}},$ $\alpha :{\rm{£}} \times {\rm{£}} \rightarrow \lbrack 0, \infty)$and let $R, C:{\rm{£}} \rightarrow {\rm{£}}$ be $\alpha _{\ast }$-dominatedmultifunctions on ${\rm{£}}.$ Then there are $\tau > 0$ and $Q\in \digamma$such that

where $t, \delta \in \{CR(\delta _{n})\},$ $\alpha (t, \delta)\geq 1,$ or $\alpha (\delta, t)\geq 1,$ and $\upsilon _{1}(Rt, C\delta) > 0.$ Then $\alpha (\delta _{n}, \delta _{n+1})\geq 1$ for all integers $n\geq 0$and $\{CR(\delta _{n})\}\rightarrow h\in {\rm{£}}.$ Also, if $h$ verifies(4.1), $\alpha (\delta _{n}, h)\geq 1$ and $\alpha (h, \delta _{n})\geq 1$ forall integers $n\geq 0$, then $R$ and $C$ admit a common fixed point $h$ in ${\rm{£}}.$

Proof. The proof is similar to the proof of Theorem 2.1.

Letting $R = C$ in Theorem 4.1, we have the following corollary.

Corollary 4.2. Let $({\rm{£}}, \upsilon)$ be a complete modularlike metric space. Suppose that the regularity of $\upsilon$ and the $\bigtriangleup _{M}$-condition hold. Choose $\delta _{0}\in {\rm{£}},$ $\alpha :{\rm{£}} \times {\rm{£}} \rightarrow \lbrack 0, \infty)$ and let $R:{\rm{£}} \rightarrow {\rm{£}}$ be a single-valued function on ${\rm{£}}$.Then there are $\tau > 0$ and $Q\in \digamma$ such that

where $t, \delta \in \{{\rm{£}} R(\delta _{n})\},$ $\alpha (t, \delta)\geq1,$ or $\alpha (\delta, t)\geq 1,$ and $\upsilon _{1}(Rt, R\delta) > 0.$Then $\alpha (\delta _{n}, \delta _{n+1})\geq 1$ for all integers $n\geq 0$and $\{\delta _{n}\}\rightarrow h\in {\rm{£}}.$ Also, if (4.2) holds for $h$, $\alpha (\delta _{n}, h)\geq 1$ and $\alpha (h, \delta _{n})\geq 1$ forall integers $n\geq 0$, then $R$ has a fixed point $h.$

5.   Integral equations

In this section, we apply our work to solve integral equations.

Theorem 5.1. Let $({\rm{£}}, \upsilon)$ be a completemodular-like metric space. Suppose that the regularity of $\upsilon$ andthe $\bigtriangleup _{M}$-condition hold. Take $r > 0,$ $\delta _{0}\in{\rm{£}}$ and let $R, C:{\rm{£}} \rightarrow {\rm{£}}$ be $\alpha _{\ast }$-dominatedmultifunctions on ${\rm{£}}.$ Then there are $\tau > 0$and $Q\in \digamma$ such that

where $t, \delta \in \{CR(\delta _{n})\},$ and $\upsilon _{1}(Rt, C\delta) > 0.$Then $\{CR(\delta _{n})\}\rightarrow f\in {\rm{£}}.$ Also, if $f$ verifies (5.1), then $R$ and $C$ admit a unique common fixed point $f$ in ${\rm{£}}.$

Let $W = C([0, 1], \mathbb{R} _{+})$ be the family of continuous functions defined on $[0, 1]$. The following are two integral equations:

for all $e\in \lbrack 0, 1]$, where $H, G:[0, 1]\times \lbrack 0, 1]\times W\rightarrow \mathbb{R}.$ For $\delta \in C([0, 1], \mathbb{R} _{+}),$ define supremum norm as $\Vert \delta \Vert _{\eta } = \sup\limits_{s\in \lbrack 0, 1]}\{\left\vert \delta (s)\right\vert e^{-\tau s}\}$, and take $\tau > 0$ arbitrarily. For all $c, w\in C([0, 1], \mathbb{R} _{+}),$ define

It is clear that $(C([0, 1], \mathbb{R} _{+}), d_{\tau })$ is a complete modular-like metric space$.$ So we have the following result.

Theorem 5.2. Suppose that

(i) $H, G:[0, 1]\times \lbrack 0, 1]\times C([0, 1], \mathbb{R} _{+})\rightarrow \mathbb{R}$;

(ii) Define

Assume that there is $\tau > 0$ such that

for all $e, f\in \lbrack 0, 1]$ and $u, \delta \in C([0, 1], \mathbb{R} ^{+}),$ where

Then (5.2) and (5.3) possess a unique solution.

Proof. By (ⅱ),

This implies

Thus

All the conditions of Theorem 5.1 hold for $Q(f) = \frac{-1}{f}$ for $f$ $> 0$ and $\upsilon _{1}(f, \delta) = \frac{1}{2}\Vert f+\delta \Vert _{\tau }$. Hence both the integral Eqs $(5.2)$ and $\left(5.3\right)$ admit a unique common solution.

6.   Conclusions

In this article, we have achieved some new results for a pair of set-valued mappings verifying a generalized rational Wardowski type contraction. Dominated mappings are applied to obtain some fixed point theorems. Applications on integral equations and graph theory are given. Moreover, we investigate our results in a more better new framework. New results in ordered spaces, modular metric space, dislocated metric space, partial metric space, $b$-metric space and metric space can be obtained as corollaries of our results. One can further extend our results to fuzzy mappings, bipolar fuzzy mappings and fuzzy neutrosophic soft mappings.

Conflict of interest

The authors declare that we have no conflict of interest.