Existence of solution of a system of non-linear differential inclusions with non-local, integral boundary conditions via fixed points of hybrid contractions

Abstract

In the present article, we have introduced the notions of γ-admissibility for the pair of q-ROF set-valued maps and admissible hybrid q-ROF $\mathcal{Z}$-contraction. Notions introduced in the article generalizes the existing concepts in fuzzy literature. Common fixed point result for a pair of γ-admissible q-ROF mappings in b-metric spaces utilizing the introduced contraction is presented. A nontrivial example to support the obtained results is also included. As an application, we have discussed the existence of solution of system of non-linear n-th order differential inclusions with non-local and integral boundary conditions.

1 Introduction

Fuzzy set theory has played a key role in the better interpretation of data and its utilization in real life. After Zadeh [1] introduced the concept of fuzzy sets, notions of intuitionistic fuzzy sets [2] and Pythagorean fuzzy sets [3] were initiated by Atanassov as generalizations of fuzzy sets. Intuitionistic fuzzy sets represent both grade of membership and non-membership of an element to a set with respect to a specific trait, while Pythagorean fuzzy sets enhance the selection space of element by squaring the sum of membership and non-membership of an element bounded by one. Yager, in 2016, introduced q-rung orthopair fuzzy sets (q-ROF sets), which increases selection space for an element more, as compared to Pythagorean fuzzy sets, by adding the q-th power of membership and non-membership of an element that is bounded by bounded by one. The phenomenon is well explained by Ali in [4]. The idea of q-ROF sets has vast application in engineering, agriculture, medical industry, image processing and decision making [5–9].

Significance of fixed point results originated with Banach contraction theorem [10], which guarantees the existence of unique fixed point for a single-valued mapping. Nadler [11] extended this idea for multivalued mappings. Later, various fixed point results for multivalued mappings utilizing various contractive conditions in a variety of metric spaces have been presented [12–18]. These results have vide applications in proving the existence of solutions of differential, integral and fractional equations and inclusions.

Following the introduction of Banach contraction, various contractive conditions are presented as its generalizations. In the same context, α-ψ contraction by Samet et al. [19] and $\mathcal{Z}$-contraction by Khojasteh [20] utilizing simulation functions, are presented. The idea of $\mathcal{Z}$-contraction is recently investigated by Rashid et al. in context of L-fuzzy sets and represented an application of their results in a graphic contraction [21]. They both discussed fixed point results utilizing the introduced contractions in complete metric spaces. The idea of α-ψ contraction is widely studied for a variety of mappings in context of different generalized metric spaces [22–24]. Authors have not only obtained fixed point results for α-ψ contraction, but also have presented application for the existence of solution of fractional order functional differential equations [25], existence of solution of integral equations inclusions [26], existence of solution of non-linear fractional differential equations inclusions [27] and many other mathematical structures.

The concept of fuzzy mapping, utilizing the concept of fuzzy sets, was first introduced by Weiss [28] and later Butnariu [29] and Heilpern [30] presented fixed point result utilizing the introduced mapping. Many fixed point results for fuzzy mappings, intuitionistic fuzzy mappings with various contractive conditions were inaugurated by mathematicians in fuzzy literature and their applications are also studied for surety of solutions of differential and integral equations and inclusions, for example [31–33]. Recently, the concepts of q-rung $(\alpha ,\beta )$-level cuts and q-ROF mapping are introduced by Rashid et al. in [34] based on the concept of q-ROF sets. They also proved fixed point result utilizing Suzuki-type contractive conditions and presented existence of solution of system of non-linear integral inclusions as an application.

Motivated by the study of α-ψ contraction and $\mathcal{Z}$-contraction, we have introduced γ-admissibility for the pair of q-ROF set-valued maps and admissible hybrid q-ROF $\mathcal{Z}$-contraction. Next, we have presented a common fixed point result for a pair of q-ROF set valued maps utilizing the introduced contraction in b-metric space. A nontrivial example illustrating our main result is also included. Depicting the significance of obtained results, existence of solution of system of non-linear n-th order differential inclusion with non-local and integral boundary conditions is discussed as an application.

2 Preliminaries

Definition 1

[35] Consider $X\neq \emptyset$ be a set and $b\geq 1$ a real number. A real valued function $d:X\times X\rightarrow [0,\infty )$ is a b-metric on X if the following conditions are fulfilled for all $x,y,z\in X$:

• (i) $d(x,y)=0$ if and only if $x=y$;

• (ii) $d(x,y)=d(y,x)$;

• (iii) $d(x,y)\leq b[d(x,z)+d(z,y)]$.

And $(X,d)$ is called a b-metric space.

Example 1

1. Consider $X=l_{p}(\mathbb{R})$ with $0< p<1$, where $l_{p}(\mathbb{R})=\{\{x_{n}\}\subset \mathbb{R}: \sum_{n=1}^{\infty}|x_{n}|^{p}< \infty \}$. Define $d:X\times X\rightarrow \mathbb{R_{+}}$ as:

$$\begin{aligned} d(x,y)= \Biggl(\sum_{n=1}^{\infty} \vert x_{n}-y_{n} \vert ^{p} \Biggr)^{1/p}, \end{aligned}$$

where $x=\{x_{n}\}$, $y=\{y_{n}\}$. Then $(X,d)$ is a b-metric space with $b=2^{1/p}$.

2. The space $L_{p}$ ($0< p<1$) of all real functions $x(t)$, $t\in [0,1]$ such that $\int _{0}^{1}|x(t)|^{p}\,dt<\infty$, is a b-metric space with $b=2^{1/p}$ if we take

$$\begin{aligned} d(x,y)= \biggl( \int _{0}^{1} \bigl\vert x(t)-y(t) \bigr\vert ^{p}\,dt \biggr)^{1/p}, \end{aligned}$$

for each $x,y\in L_{p}$.

Definition 2

[36] Let $(X,d)$ be a b-metric space. A sequence $\{x_{n}\}_{n\in \mathbb{N}}$ is called:

• (i) Cauchy if and only if for all $\varepsilon >0$ there exists $n(\varepsilon )\in \mathbb{N}$ such that $d(x_{n},x_{m})<\varepsilon$ for all $n,m\geq n(\varepsilon )$.

• (ii) Convergent in the case there is $x\in X$ such that for every $\varepsilon >0$ there is $n(\varepsilon )\in \mathbb{N}$ so that for every $n\geq n(\varepsilon )$, $d(x_{n},x)<\varepsilon$. This can be written as $\lim_{n\rightarrow \infty}(x_{n})=x$.

The b-metric space $(X,d)$ is complete if every Cauchy sequence is convergent.

Definition 3

[36] Let $(X,d)$ be a b-metric space and $A\subset X$. Then A is said to be:

• (i) Compact in case only if, for every sequence in A, there is a subsequence convergent to an element of A.

• (ii) Closed in case only if, for each sequence $\{x_{n}\}_{n\in \mathbb{N}}$ in A that converges to an element x, we have $x\in A$.

Definition 4

A nonempty set A in a b-metric space $(X,d)$ is proximal if for any $x\in X$, there is $k\in A$ such that $d(x,A)=d(x,k)$.

Consider the family of nonempty subsets of X, $CB(X)$ represents the set of all nonempty closed and bounded subsets of X, $P(X)$ represents the family of all nonempty proximal subsets of X and $K(X)$ represents the class of all nonempty closed and compact subsets of X.

Let $(X,d)$ be a b-metric space. For $A,B\in K(X)$, the function $H:K(X)\times K(X)\longrightarrow \mathbb{R}_{+}$, defined by

$$\begin{aligned} H(A,B)=\textstyle\begin{cases} \max \{\sup_{x\in A}d(x,B),\sup_{x\in B}d(x,A)\}, & \text{if it exists} \\ \infty , & \text{otherwise,} \end{cases}\displaystyle \end{aligned}$$

is called generalized Hausdorff b-metric on $K(X)$, where $d(x,A)=\inf_{y\in A}d(x,y)$.

Lemma 1

[11] Let $A,B\in CB(X)$ and $x\in A$, for any real number $l\geq 1$ there is $y\in B$ so that $d(x,y)\leq l.H(A,B)$. Also $d(x,B)\leq H(A,B)$.

In order to unify different types of contractions, Khojasteh et al. [20] recently introduced a family of auxiliary functions named simulation functions.

Definition 5

[20] A mapping $\rho :\mathbb{R_{+}}\times \mathbb{R_{+}}\rightarrow \mathbb{R}$ is called a simulation function if it meets the following criteria:

• (i) $\rho (0,0)=0$;

• (ii) $\rho (a,b)< b-a$ for all non-negative a, b;

• (iii) $\lim_{n\longrightarrow \infty}\sup \rho (a_{n},b_{n})<0$, if $\{a_{n}\}_{n\geq 1}$ and $\{b_{n}\}_{n\geq 1}$ are sequences from $(0,\infty )$ so that $\lim_{n\longrightarrow \infty}a_{n}=\lim_{n\longrightarrow \infty}b_{n}>0$.

We represent the collection of all simulation functions by $\mathcal{Z}$.

Example 2

1. If $\phi :\mathbb{R_{+}}\rightarrow [0,1)$ is a function such that $\limsup_{t\rightarrow r^{+}}\varphi (t)<1$ for all $r>0$, and we define

$$\begin{aligned} \rho _{A}(a,b)=b\varphi (b)-a\quad \text{for all } a,b\in \mathbb{R_{+}} \end{aligned}$$

2. If $\eta :\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ is an upper semicontinuous mapping such that $\eta (t)< t$ for all $t>0$ and $\eta (0)=0$, and we define

$$\begin{aligned} \rho _{B}(a,b)=\eta (b)-a \quad \text{for all } a,b\in \mathbb{R_{+}} \end{aligned}$$

3. Let $k\in \mathbb{R}$ be such that $k<1$ and let $\rho _{C}:\mathbb{R_{+}}\times \mathbb{R_{+}}\rightarrow \mathbb{R}$ be the function defined by:

$$\begin{aligned} \rho _{C}(a,b)=\textstyle\begin{cases} 2(b-a), & \text{if } b< a; \\ kb-a, & \text{otherwise.} \end{cases}\displaystyle \end{aligned}$$

Then, the functions $\rho _{A}$ and $\rho _{B}$ are simulation functions, but $\rho _{C}$ is not.

One can easily find more interesting examples of simulation functions in [20, 37–40].

Definition 6

[20] Consider $(X,d)$ be a metric space and $T:X\rightarrow X$ a mapping. Then T is a $\mathcal{Z}$-contraction with respect to $\rho \in \mathcal{Z}$, if is satisfies:

$$\begin{aligned} \rho \bigl(d(T x,Ty),d(x,y)\bigr)\geq 0 \quad \text{for all } x,y\in X. \end{aligned}$$

Theorem 1

[20] On a complete metric space, every $\mathcal{Z}$-contraction has a unique fixed point.

Definition 7

[41] A q-rung orthopair fuzzy subset A of X, denoted as q-ROF set, is an orthopair,

$$\begin{aligned} A=(\mu _{A}, \nu _{A})_{q} \end{aligned}$$

or

$$\begin{aligned} A=\bigl\{ \bigl(x,\mu _{A}(x),\nu _{A}(x)\bigr)|x\in X\bigr\} , \end{aligned}$$

where, $\mu _{A},\nu _{A}:X\rightarrow [0,1]$ indicate the degree of membership (belongingness) and nonmembership (non-belongingness) of each element x to set A respectively, which fulfils

$$\begin{aligned} \bigl(\mu _{A}(x)\bigr)^{q}+\bigl(\nu _{A}(x) \bigr)^{q}\leq 1 \quad \text{for all } x\in X, q\geq 1. \end{aligned}$$

Definition 8

[34] Consider A a q-ROF set, i.e, $A\in F^{q}(X)$ (class of q-ROF subsets of X) and $x\in X$, then, for $\alpha \in [0,1]$, q-rung α-level set of A is defined as

$$\begin{aligned}{} [A]^{q}_{\alpha}=\bigl\{ x\in X:\bigl(\mu _{A}(x) \bigr)^{q}\geq \alpha \text{ and } \bigl( \nu _{A}(x) \bigr)^{q}\leq 1-\alpha \bigr\} . \end{aligned}$$

For $(\alpha ,\beta )\in (0,1]\times [0,1)$ with $\alpha +\beta \leq 1$, the q-rung $(\alpha ,\beta )$-level sets of A are defined as

$$\begin{aligned}{} [A]^{q}_{(\alpha ,\beta )}=\bigl\{ x\in X:\bigl(\mu _{A}(x) \bigr)^{q}\geq \alpha \text{ and } \bigl(\nu _{A}(x) \bigr)^{q}\leq \beta \bigr\} , \end{aligned}$$

and

$$\begin{aligned} A^{q}_{(\alpha ,\beta )}=\bigl\{ x\in X:\bigl(\mu _{A}(x) \bigr)^{q}>\alpha \text{ and } \bigl(\nu _{A}(x) \bigr)^{q}< \beta \bigr\} . \end{aligned}$$

Definition 9

[34] A q-ROF set A in a metric linear space V is said to be an approximate quantity if and only if $[A]^{q}_{(\alpha ,\beta )}$ is compact and convex in V for each $(\alpha ,\beta )\in (0,1]\times [0,1)$ with

$$\begin{aligned} \sup_{x\in V}\bigl(\mu _{A}(x)\bigr)^{q}=1 \quad \text{and}\quad \inf_{x\in V}\bigl( \nu _{A}(x) \bigr)^{q}=0. \end{aligned}$$

Collection of all approximate quantities in V is represented by $W(V)$.

For $A,B\in W(V)$ and $(\alpha ,\beta )\in (0,1]\times [0,1)$, define

$$\begin{aligned}& p_{(\alpha ,\beta )}^{q}(A,B) = d\bigl([A]^{q}_{(\alpha ,\beta )},[B]^{q}_{( \alpha ,\beta )} \bigr) \\& \hphantom{p_{(\alpha ,\beta )}^{q}(A,B)} = \inf_{x\in [A]^{q}_{(\alpha ,\beta )},y\in [B]^{q}_{(\alpha , \beta )}}d(x,y) \\& p^{q}(A,B) = \sup_{\alpha}\inf_{\beta} p_{(\alpha ,\beta )}(A,B) \\& D_{(\alpha ,\beta )}^{q}(A,B) = H\bigl([A]_{(\alpha ,\beta )}^{q},[B]_{( \alpha ,\beta )}^{q} \bigr) \\& D^{q}(A,B) = \sup_{\alpha}\inf_{\beta}D_{(\alpha ,\beta )}^{q}(A,B). \end{aligned}$$

Any crisp set A can be represented as a q-ROF set by considering the characteristic function of orthopairs

$$\begin{aligned}& \bigl(\mu _{A}(x)\bigr)^{q}=\textstyle\begin{cases} 1, & x\in A \\ 0, & x\notin A \end{cases}\displaystyle \\& \bigl(\nu _{A}(x)\bigr)^{q}=\textstyle\begin{cases} 0, & x\in A \\ 1, & x\notin A \end{cases}\displaystyle \end{aligned}$$

Definition 10

[34] Consider X an arbitrary set, Y be any metric space. A mapping $T:X\rightarrow F^{q}(Y)$ is known as q-rung orthopair fuzzy mapping (q-ROF mapping).

Definition 11

[34] Let X be a metric space. A point $x^{*}\in X$ is called a q-ROF fixed point of a q-ROF mapping $T:X\rightarrow F^{q}(X)$ if there exists $(\alpha ,\beta )\in (0,1]\times [0,1)$ with $\alpha +\beta \leq 1$ such that

$$\begin{aligned} x^{*}\in \bigl[T x^{*}\bigr]^{q}_{(\alpha ,\beta )}. \end{aligned}$$

Rus [42] first introduced the concept of a comparison function, which has since been extensively studied by a number of authors in an attempt to broaden the scope of contraction-type mappings.

Definition 12

Consider $\varphi :\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$, a non-decreasing function, is known as comparison function [42] if $\varphi ^{n}(t)\rightarrow 0$ as $n\rightarrow \infty$ for every $t\in \mathbb{R_{+}}$ and a b-comparison function [43] if for $b\geq 1$, $j\geq j_{0}$ and any $t\geq 0$, there exist $j_{0}\in \mathbb{N}$, $\lambda \in (0,1)$ and a convergent non-negative series $\sum_{n=1}^{\infty}x_{n}$ such that $b^{j+1}\varphi ^{j+1}(t)\leq \lambda b^{j}\varphi ^{j}(t)+x_{j}$, where $\varphi ^{n}$ is the nth iterate of φ.

Example 3

Consider $\varphi :\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ the functions defined by

• (i) $\varphi (t)=\gamma t$ for all $t\geq 0$, where $\gamma \in (0,1)$.

• (ii) $\varphi (t)=\frac{t}{t+1}$ for each $t\geq 0$.

are comparison functions.

Consider $(X,d)$ be a b-metric space with coefficient $b\geq 1$. Then $\varphi (t)=k(t)$; $t\in \mathbb{R_{+}}$ with $0< k<\frac{1}{b}$ is a b-comparison function. Let $\Omega _{b}$ represents the family of all functions $\varphi :\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ that satisfy, φ is b-comparison function, $\varphi (t)=t$ if and only if $t=0$ and φ is continuous.

Lemma 2

[42] A comparison function $\varphi :\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ has the characteristics that each iterate $\varphi ^{n}$, $n\in \mathbb{N}$ is also a comparison function and $\varphi (t)< t$ for all $t>0$.

Lemma 3

[42] The series $\sum_{j=0}^{\infty}b^{j}\varphi ^{j}(t)$ in b-comparison function $\varphi :\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ converges for every $t\in \mathbb{R_{+}}$.

Lemma 4

[44] For a b-metric space $(X,d)$, $A,B\in K(X)$ and $x,y\in X$, the following conditions always hold:

• (i) $d(x,B)\leq H(A,B)$ for each $x\in A$.

• (ii) $d(x,B)\leq d(x,b)$ for any $b\in B$.

• (iii) $d(x,A)\leq b[d(x,y)+d(y,A)]$.

• (iv) $d(x,A)=0$ if and only if $x\in A$.

• (v) $H(A,B)=0$ if and only if $A=B$.

• (vi) $H(A,B)=H(B,A)$.

• (vii) $H(A,B)\leq b[H(A,C)+H(C,B)$.

3 Main result

We initiate this section by discussing the idea of γ-admissibility of pair of q-ROF set-valued maps after getting inspired by the concept of β-admissibility raised by Samet [19].

Definition 13

For a metric space $(X,d)$, $\gamma :X\times X\rightarrow \mathbb{R_{+}}$ and q-ROF mappings S and T in $F^{q}(X)$, the ordered pair $(S,T)$ is γ-admissible if it satisfies the following conditions:

• (i) for each $x\in X$ and $y\in [Sx]^{q}_{(\alpha (x),\beta (x))}$, where $(\alpha (x),\beta (x))\in (0,1]\times [0,1)$ with $\gamma (x,y)\geq 1$, we have $\gamma (y,z)\geq 1$ for all $z\in [Ty]^{q}_{(\alpha (x),\beta (x))}\neq \phi$, where $(\alpha (y),\beta (y))\in (0,1]\times [0,1)$.

• (ii) for each $x\in X$ and $y\in [T x]^{q}_{(\alpha (x),\beta (x))}$ where $(\alpha (x),\beta (x))\in (0,1]\times [0,1)$ with $\gamma (x,y)\geq 1$, we have $\gamma (y,z)\geq 1$ for all $z\in [Sy]^{q}_{(\alpha (y),\beta (y))}\neq \phi$, where $(\alpha (y),\beta (y))\in (0,1]\times [0,1)$.

If $S=T$, then T is called γ-admissible. If $(S,T)$ is γ-admissible pair, then so is $(T,S)$.

Definition 14

Let $(X,d)$ be a b-metric space and $S,T:X\rightarrow F^{q}(X)$ are q-ROF set-valued maps. An ordered pair $(S,T)$ is called pairwise admissible hybrid q-ROF $\mathcal{Z}$-contraction with respect to $\rho \in \mathcal{Z}$, if there exists $(\alpha ,\beta )\in (0,1]\times [0,1)$, a function $\gamma :X\times X\rightarrow \mathbb{R_{+}}$ and a b-comparison function φ such that it satisfies following conditions:

$$0\le \left\{\begin{array}{c}{\rho }_1(\gamma (x,y)H({[Sx]}_{(\alpha (x),\beta (x))}^q,{[Ty]}_{(\alpha (y),\beta (y))}^q),\phi (A(x,y)))\\ +{\rho }_2(\gamma (x,y)H({[Sx]}_{(\alpha (x),\beta (x))}^q,{[Ty]}_{(\alpha (y),\beta (y))}^q),\phi (B(x,y)))\end{array}\right\}$$

(3.1)

for all $x,y\in X$, where

$$\begin{aligned}& A(x,y) = k_{1}\bigl(d(x,y)\bigr)+k_{2}\bigl(d\bigl(x,[T x]^{q}_{(\alpha (x),\beta (x))}\bigr)\bigr)+k_{3}\bigl(d \bigl(y,[Sy]^{q}_{( \alpha (y),\beta (y))}\bigr)\bigr) \\& \hphantom{A(x,y) =}{} +k_{4} \biggl( \frac{d(y,[Sy]^{q}_{(\alpha (y),\beta (y))})(1+d(x,[T x]^{q}_{(\alpha (x),\beta (x))}))}{1+d(x,y)} \biggr) \\& \hphantom{A(x,y) =}{} +k_{5} \biggl( \frac{d(y,[T x]^{q}_{(\alpha (x),\beta (x))})(1+d(x,[Sy]^{q}_{(\alpha (y),\beta (y))}))}{1+d(x,y)} \biggr), \\& B(x,y) = \bigl(d(x,y)\bigr)^{k_{1}}.\bigl(d\bigl(x,[T x]^{q}_{(\alpha (x),\beta (x))}\bigr)\bigr)^{k_{2}}.\bigl(d \bigl(y,[Sy]^{q}_{( \alpha (y),\beta (y))}\bigr)\bigr)^{k_{3}} \\& \hphantom{B(x,y) =}{} \cdot \biggl( \frac{d(y,[Sy]^{q}_{(\alpha (y),\beta (y))})(1+d(x,[T x]^{q}_{(\alpha (x),\beta (x))}))}{1+d(x,y)} \biggr)^{k_{4}} \\& \hphantom{B(x,y) =}{} \cdot \biggl( \frac{d(x,[Sy]^{q}_{(\alpha (y),\beta (y))})+d(y,[T x]^{q}_{(\alpha (x),\beta (x))})}{2b} \biggr)^{k_{5}}, \end{aligned}$$

where $(\alpha (x),\beta (x))$ and $(\alpha (y),\beta (y))\in (0,1]\times [0,1)$ with $b\geq 1$ and $k_{i}\geq 0$ such that $\sum^{5}_{i=1}k_{i}=1$.

If $(S,T)$ is pairwise admissible hybrid q-ROF $\mathcal{Z}$-contraction, then so is $(T,S)$.

Remark 1

If $(S,T)$ is pairwise admissible hybrid q-ROF $\mathcal{Z}$-contraction, then by second axiom of definition 5, we can easily formulate:

$$\begin{aligned} 2\gamma ( x,y ) H \bigl( [ Sx ] _{ ( \alpha (x),\beta (x) ) }^{q}, [ Ty ] _{ ( \alpha (y),\beta (y) ) }^{q} \bigr) \leq \varphi \bigl( A(x,y ) \bigr)+\varphi \bigl( B(x,y ) \bigr) \end{aligned}$$

(3.2)

Definition 15

Let $(X,d)$ be a b-metric space and $S,T:X\rightarrow F^{q}(X)$ are q-ROF set-valued maps. The ordered pair $(S,T)$ is H-continuous at $\wp \in X$, if for any sequence $\{x_{n}\}_{n\geq 1}$ in X,

• (i) $\lim_{n\rightarrow \infty}d(x_{n},\wp )=0$ implies $\lim_{n\rightarrow \infty}H([Sx_{n}]^{q}_{(\alpha (x_{n}),\beta (x_{n}))},[T \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))})=0$, where $(\alpha (x_{n}),\beta (x_{n}))\in (0,1]\times [0,1)$ and $(\alpha (\wp ),\beta (\wp ))\in (0,1]\times [0,1)$,

• (ii) $\lim_{n\rightarrow \infty}d(x_{n},\wp )=0$ implies $\lim_{n\rightarrow \infty}H([T x_{n}]^{q}_{(\alpha (x_{n}),\beta (x_{n}))},[S \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))})=0$, where $(\alpha (x_{n}),\beta (x_{n}))$ and $(\alpha (\wp ),\beta (\wp ))\in (0,1]\times [0,1)$.

If $(S,T)$ is H-continuous, then $(T,S)$ is also H-continuous.

If $(S,T)$ is continuous at each point of X, then the pair $(S,T)$ is H-continuous.

Let $F^{q}_{S}(X)$ be a subset of $F^{q}(X)$ defined by

$$\begin{aligned} F^{q}_{S}(X)=\bigl\{ A\in F^{q}(X):[A]^{q}_{(\alpha ,\beta )} \in K(X), \text{where } (\alpha ,\beta )\in \bigl(0,1]\times [0,1\bigr) \text{ and } q\geq 1 \bigr\} . \end{aligned}$$

Theorem 2

Consider $(X,d)$ a complete b-metric space and $S,T:X\rightarrow F^{q}_{S}(X)$ be pairwise admissible hybrid q-ROF $\mathcal{Z}$-contraction with respect to $\rho \in \mathcal{Z}$. Assume that:

• (i) $(S,T)$ is a γ-admissible pair;

• (ii) there exists $x_{0}\in X$ and

  • (a) $x_{1}\in [Sx_{0}]^{q}_{(\alpha (x_{0}),\beta (x_{0}))}$ such that $\gamma (x_{0},x_{1})\geq 1$

  • (b) $x_{1}\in [T x_{0}]_{(\alpha (x_{0}),\beta (x_{0}))}$ such that $\gamma (x_{0},x_{1})\geq 1$

• (iii) $(S,T)$ is H-continuous.

• (iv) The sets $[Sx]^{q}_{(\alpha (x),\beta (x))}$ and $[T x]^{q}_{(\alpha (x),\beta (x))}$ are proximal for every $x\in X$.

Then S and T have at least one common q-ROF fixed point in X.

Proof

Starting with $x_{0}\in X$ and since $[ Sx_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}\neq \emptyset$, there exists $x_{1}\in X$ such that $x_{1}\in [ Sx_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}$. Similarly, for $x_{1}$ we have $x_{2}\in X$ such that $x_{2}\in [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q}$. So, in general, $x_{2n+1}\in [ Sx_{2n} ] _{ ( \alpha (x_{2n}),\beta (x_{2n}) ) }^{q}$ and $x_{2n+2}\in [ T x_{2n+1} ] _{ ( \alpha (x_{2n+1}),\beta (x_{2n+1}) ) }^{q}$. Now from condition (3.1), we have:

$$\begin{aligned}& 0 \leq \rho _{1} \bigl( \gamma ( x_{0},x_{1} ) H \bigl( [ Sx_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}, [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} \bigr) ,\varphi \bigl( A(x_{0},x_{1} ) \bigr) \bigr) \\& \hphantom{0 \leq} {}+\rho _{2} \bigl( \gamma ( x_{0},x_{1} ) H \bigl( [ Sx_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}, [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} \bigr) , \varphi \bigl( B(x_{0},x_{1} ) \bigr) \bigr) \\& \hphantom{0 } \leq \varphi \bigl( A(x_{0},x_{1} ) \bigr)-\gamma ( x_{0},x_{1} ) H \bigl( [ Sx_{0} ] _{ ( \alpha (x_{0}), \beta (x_{0}) ) }^{q}, [ T x_{1} ] _{ ( \alpha (x_{1}), \beta (x_{1}) ) }^{q} \bigr) +\varphi \bigl( B(x_{0},x_{1} ) \bigr) \\& \hphantom{0 \leq} {}-\gamma ( x_{0},x_{1} ) H \bigl( [ Sx_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}, [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} \bigr) , \\& 2\gamma ( x_{0},x_{1} ) H \bigl( [ Sx_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}, [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} \bigr) \leq \varphi \bigl( A(x_{0},x_{1} ) \bigr)+\varphi \bigl( B(x_{0},x_{1} ) \bigr) \end{aligned}$$

Now, from Lemma 1, we have

$$\begin{aligned} d ( x_{1},x_{2} ) &\leq 2\gamma ( x_{0},x_{1} ) H \bigl( [ Sx_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}, [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} \bigr) \\ & \leq \varphi \bigl( A(x_{0},x_{1} ) \bigr)+\varphi \bigl( B(x_{0},x_{1} ) \bigr). \end{aligned}$$

(1)

Now, if $x_{0}=x_{1}$ and $S=T$, then:

$$\begin{aligned}& A(x_{1},x_{1}) = k_{1}d ( x_{1},x_{1} ) +k_{2}d \bigl( x_{1}, [ Sx_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} \bigr) +k_{3}d \bigl( x_{1}, [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} \bigr) \\& \hphantom{A(x_{1},x_{1}) =} {}+k_{4} \biggl[ \frac{d ( x_{1}, [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} ) ( 1+d ( x_{1}, [ Sx_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} ) ) }{1+d ( x_{1},x_{1} ) } \biggr] \\& \hphantom{A(x_{1},x_{1}) =} {}+k_{5} \biggl[ \frac{d ( x_{1}, [ Sx_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} ) ( 1+d ( x_{1}, [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} ) ) }{1+d ( x_{1},x_{1} ) } \biggr] , \\& A(x_{1},x_{1}) = k_{2}d ( x_{1},x_{1} ) +k_{3}d \bigl( x_{1}, [ Sx_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} \bigr) \\& \hphantom{A(x_{1},x_{1}) =} {} +k_{4} \biggl[ \frac{d ( x_{1}, [ Sx_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} ) ( 1+d ( x_{1},x_{0} ) ) }{1+d ( x_{1},x_{1} ) } \biggr] \\& \hphantom{A(x_{1},x_{1}) =} {}+k_{5} \biggl[ \frac{d ( x_{1},x_{0} ) ( 1+d ( x_{1}, [ Sx_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} ) ) }{1+d ( x_{1},x_{1} ) } \biggr] \\& \hphantom{A(x_{1},x_{1}) }= k_{4}d ( x_{1},x_{1} ) +k_{5}d ( x_{1},x_{0} ) =0 \end{aligned}$$

Similarly, $B(x_{1},x_{1})=0$, so we get $2\gamma ( x_{0},x_{1} ) H ( [ Sx_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}, [ T x_{1} ] _{ ( \alpha (x_{1}),\beta (x_{1}) ) }^{q} ) \leq 0$, which implies $x_{1}\in [ Sx_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}$ and $x_{1}\in [ T x_{0} ] _{ ( \alpha (x_{0}),\beta (x_{0}) ) }^{q}$, i.e., $x_{1}$ is a common fixed point of S and T. Now if $x_{0}\neq x_{1}$ and $S\neq T$, then from (1), we have:

$$ d ( x_{1},x_{2} ) \leq \varphi \bigl( A(x_{0},x_{1} ) \bigr)+\varphi \bigl( B(x_{0},x_{1} ) \bigr). $$

Similarly,

$$ d ( x_{2},x_{3} ) \leq \varphi \bigl( A(x_{1},x_{2} ) \bigr)+\varphi \bigl( B(x_{1},x_{2} ) \bigr). $$

Now solving for $A(x_{n-1},x_{n})$ and $B(x_{n-1}, x_{n})$, $n\in \mathbb{N}$, in general, we have:

$$\begin{aligned}& A(x_{n-1},x_{n}) = k_{1}d ( x_{n-1},x_{n} ) +k_{2}d \bigl( x_{n-1}, [ Sx_{n-1} ] _{ ( \alpha (x_{n-1}),\beta (x_{n-1}) ) }^{q} \bigr) +k_{3}d \bigl( x_{n}, [ T x_{n} ] _{ ( \alpha (x_{n}),\beta (x_{n}) ) }^{q} \bigr) \\& \hphantom{A(x_{n-1},x_{n}) =} {}+k_{4} \biggl[ \frac{d ( x_{n}, [ T x_{n} ] _{ ( \alpha (x_{n}),\beta (x_{n}) ) }^{q} ) + ( 1+d ( x_{n-1}, [ Sx_{n-1} ] _{ ( \alpha (x_{n-1}),\beta (x_{n-1}) ) }^{q} ) ) }{1+d ( x_{n-1},x_{n} ) } \biggr] \\& \hphantom{A(x_{n-1},x_{n}) =} {}+k_{5} \biggl[ \frac{d ( x_{n}, [ Sx_{n-1} ] _{ ( \alpha (x_{n-1}),\beta (x_{n-1}) ) }^{q} ) ( 1+d ( x_{n-1}, [ T x_{n} ] _{ ( \alpha (x_{n}),\beta (x_{n}) ) }^{q} ) ) }{1+d ( x_{n-1},x_{n} ) } \biggr] \\& \hphantom{A(x_{n-1},x_{n}) }= k_{1}d ( x_{n-1},x_{n} ) +k_{2}d ( x_{n-1},x_{n} ) +k_{3}d ( x_{n},x_{n+1} ) \\& \hphantom{B(x_{n-1},x_{n}) =}{} +k_{4} \biggl[ \frac{d ( x_{n},x_{n+1} ) ( 1+d ( x_{n-1},x_{n} ) ) }{1+d ( x_{n-1},x_{n} ) } \biggr] \\& \hphantom{A(x_{n-1},x_{n}) =} {}+k_{5} \biggl[ \frac{d ( x_{n},x_{n} ) ( 1+d ( x_{n-1},x_{n+1} ) ) }{1+d ( x_{n-1},x_{n} ) } \biggr] \\& A(x_{n-1},x_{n})= ( k_{1}+k_{2} ) d ( x_{n-1},x_{n} ) + ( k_{3}+k_{4} ) d ( x_{n},x_{n+1} ) , \\& B(x_{n-1},x_{n}) = \bigl( d ( x_{n-1},x_{n} ) \bigr) ^{k_{1}}. \bigl( d \bigl( x_{n-1}, [ Sx_{n-1} ] _{ ( \alpha (x_{n-1}), \beta (x_{n-1}) ) }^{q} \bigr) \bigr) ^{k_{2}}. \bigl( d \bigl( x_{n}, [ T x_{n} ] _{ ( \alpha (x_{n}),\beta (x_{n}) ) }^{q} \bigr) \bigr) ^{k_{3}} \\& \hphantom{B(x_{n-1},x_{n}) =}{} \cdot \biggl[ \frac{d ( x_{n}, [ T x_{n} ] _{ ( \alpha (x_{n}),\beta (x_{n}) ) }^{q} ) ( 1+d ( x_{n-1}, [ Sx_{n-1} ] _{ ( \alpha (x_{n-1}),\beta (x_{n-1}) ) }^{q} ) ) }{1+d(x_{n-1},x_{n})} \biggr] ^{k_{4}} \\& \hphantom{B(x_{n-1},x_{n}) =}{} \cdot \biggl[ \frac{d ( x_{n-1}, [ T x_{n} ] _{ ( \alpha (x_{n}),\beta (x_{n}) ) }^{q} ) +d ( x_{n}, [ Sx_{n-1} ] _{ ( \alpha (x_{n-1}),\beta (x_{n-1}) ) }^{q} ) }{2b} \biggr] ^{k_{5}} \\& \hphantom{B(x_{n-1},x_{n}) =}{} \cdot \biggl[ \frac{d ( x_{n-1}, [ T x_{n} ] _{ ( \alpha (x_{n}),\beta (x_{n}) ) }^{q} ) +d ( x_{n}, [ Sx_{n-1} ] _{ ( \alpha (x_{n-1}),\beta (x_{n-1}) ) }^{q} ) }{2b} \biggr] ^{k_{5}} \\& \hphantom{B(x_{n-1},x_{n})}= \bigl( d ( x_{n-1},x_{n} ) \bigr) ^{k_{1}}. \bigl( d ( x_{n-1},x_{n} ) \bigr) ^{k_{2}}. \bigl( d ( x_{n},x_{n+1} ) \bigr) ^{k_{3}}. \bigl( d ( x_{n},x_{n+1} ) \bigr) ^{k_{4}} \\& \hphantom{B(x_{n-1},x_{n}) =}{} \cdot \biggl[ \frac{d ( x_{n-1},x_{n+1} ) +d ( x_{n},x_{n} ) }{2b} \biggr] ^{k_{5}} \end{aligned}$$

(2)

It is clear that, $( \frac{f+g}{2} ) ^{h}\leq \frac{f^{h}+g^{h}}{2}$, so

$$\begin{aligned}& B(x_{n-1},x_{n})\leq \bigl( d ( x_{n-1},x_{n} ) \bigr) ^{k_{1}+k_{2}}. \bigl( d ( x_{n},x_{n+1} ) \bigr) ^{k_{3}+k_{4}}. \biggl[ b \frac{d ( x_{n-1},x_{n} ) ^{k_{5}}+ ( d ( x_{n},x_{n+1} ) ) ^{k_{5}}}{2b} \biggr] \\& B(x_{n-1},x_{n})\leq \bigl( d ( x_{n-1},x_{n} ) \bigr) ^{k_{1}+k_{2}}. \bigl( d ( x_{n},x_{n+1} ) \bigr) ^{k_{3}+k_{4}}. \biggl[ \frac{d ( x_{n-1},x_{n} ) ^{k_{5}}}{2}+ \frac{ ( d ( x_{n},x_{n+1} ) ) ^{k_{5}}}{2} \biggr] \end{aligned}$$

(3)

Using equations (2) and (3) in (1), we have:

$$\begin{aligned} d ( x_{1},x_{2} ) \leq &\varphi \bigl[ ( k_{1}+k_{2} ) d ( x_{0},x_{1} ) + ( k_{3}+k_{4} ) d ( x_{1},x_{2} ) \bigr] \\ &{}+\varphi \biggl[ \bigl( d ( x_{0},x_{1} ) \bigr) ^{k_{1}+k_{2}}. \bigl( d ( x_{1},x_{2} ) \bigr) ^{k_{3}+k_{4}}. \biggl[ \frac{d ( x_{0},x_{1} ) ^{k_{5}}}{2}+ \frac{ ( d ( x_{1},x_{2} ) ) ^{k_{5}}}{2} \biggr] \biggr] \end{aligned}$$

Suppose $d ( x_{0},x_{1} ) \leq d ( x_{1},x_{2} )$. Since φ is non-decreasing and $k_{1}+k_{2}+k_{3}+k_{4}+k_{5}\leq 1$, we have:

$$\begin{aligned}& d ( x_{1},x_{2} ) \leq \varphi \bigl[ ( k_{1}+k_{2} ) d ( x_{1},x_{2} ) + ( k_{3}+k_{4} ) d ( x_{1},x_{2} ) \bigr] +\varphi \bigl[ \bigl( d ( x_{0},x_{1} ) \bigr) ^{k_{1}+k_{2}k_{3}+k_{4}+k_{5}} \bigr] \\& \hphantom{d ( x_{1},x_{2} )}\leq \varphi \bigl[ d ( x_{1},x_{2} ) \bigr] + \varphi \bigl[ d ( x_{1},x_{2} ) \bigr] =j_{1}d ( x_{1},x_{2} ) +j_{1}d ( x_{1},x_{2} ) , \\& d ( x_{1},x_{2} ) \leq ( j_{1}+j_{2} ) d ( x_{1},x_{2} ) < d ( x_{1},x_{2} ) , \quad \text{where }0< j_{1},j_{2}< \frac{1}{b}. \end{aligned}$$

A contradiction. So,

$$ d ( x_{1},x_{2} ) \leq \varphi \bigl[ d ( x_{0},x_{1} ) \bigr] +\varphi \bigl[ d ( x_{0},x_{1} ) \bigr] =2\varphi \bigl[ d ( x_{0},x_{1} ) \bigr] . $$

Similarly,

$$\begin{aligned} d ( x_{1},x_{2} ) \leq& \varphi \bigl[ d ( x_{1},x_{2} ) \bigr] +\varphi \bigl[ d ( x_{1},x_{2} ) \bigr] \\ =&2\varphi \bigl[ d ( x_{1},x_{2} ) \bigr] \leq 2\varphi \bigl( 2\varphi \bigl( d ( x_{0},x_{1} ) \bigr) \bigr) =2^{2}\varphi ^{2} \bigl( d ( x_{0},x_{1} ) \bigr) . \end{aligned}$$

In general, $d ( x_{n},x_{n+1} ) \leq 2^{n}\varphi ^{n} ( d ( x_{0},x_{1} ) )$.

Let $m,n\in \mathbb{N}$ with $m>n$, then,

$$\begin{aligned} d(x_{n},x_{m})\leq {}&hd(x_{n},x_{n+1})+h^{2}d(x_{n+1},x_{n+2})+ \cdots +h^{m-n}d(x_{m-1},x_{m}) \\ \leq {}&h2^{n}\varphi ^{n}\bigl(d(x_{0},x_{1}) \bigr)+h^{2}2^{n+1}\varphi ^{n+1} \bigl(d(x_{0},x_{1})\bigr)+ \cdots +h^{m-n}2^{m-1} \varphi ^{m-1}\bigl(d(x_{0},x_{1})\bigr) \\ ={}&h^{n-n+1}2^{n}\varphi ^{n}\bigl(d(x_{0},x_{1}) \bigr)+h^{n-n+2}2^{n+1}\varphi ^{n+1} \bigl(d(x_{0},x_{1})\bigr) \\ & {}+\cdots +h^{m-n+1-1}2^{m-1}\varphi ^{m-1} \bigl(d(x_{0},x_{1})\bigr) \\ ={}&\frac{1}{h^{n-1}}\bigl[h^{n}2^{n}\varphi ^{n} \bigl(d(x_{0},x_{1})\bigr)+h^{n+1}2^{n+1} \varphi ^{n+1}\bigl(d(x_{0},x_{1})\bigr) \\ & {}+\cdots +h^{m-1}2^{m-1}\varphi ^{m-1} \bigl(d(x_{0},x_{1})\bigr)\bigr] \\ ={}&\frac{1}{h^{n-1}}\sum^{m-1}_{i=n}h^{i}2^{i} \varphi ^{i}\bigl(d(x_{0},x_{1})\bigr) \\ \leq {}&\frac{1}{h^{n-1}}\sum^{\infty}_{i=0}h^{i} \varphi ^{i}\bigl(d(x_{0},x_{1})\bigr). \end{aligned}$$

(3.3)

Since φ is a b-comparison function, it follows that the series $\sum^{\infty}_{i=0}h^{i}\varphi ^{i}(d(x_{0},x_{1}))$ is convergent. Setting $S_{k}=\sum^{k}_{i=1}h^{i}\varphi ^{i}(d(x_{0},x_{1}))$, (3.3) becomes

$$\begin{aligned} d(x_{n},x_{m})\leq \frac{1}{h^{n-1}}(S_{m-1}-S_{n-1}). \end{aligned}$$

(3.4)

We achieve $d(x_{n},x_{m})\rightarrow 0$ by applying $\lim_{n,m\rightarrow \infty}$ in (3.4), proving that $\{x_{n}\}$ is a Cauchy sequence in X and completeness of X implies there exists $u\in X$ such that

$$\begin{aligned} \lim_{n\rightarrow \infty}d(x_{n},u)=0. \end{aligned}$$

(3.5)

Now by using the triangle inequality in X, we show that $\wp \in [T \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))}$. So

$$\begin{aligned} d\bigl(\wp ,[T \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))}\bigr)\leq {}& h\bigl(d(\wp ,x_{2n+1})+d\bigl(x_{2n+1},[T \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))} \bigr)\bigr) \\ \leq{} &hd(\wp ,x_{2n+1})+hH\bigl([Sx_{2n}]^{q}_{(\alpha (x_{2n}),\beta (x_{2n}))},[T \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))}\bigr). \end{aligned}$$

(3.6)

Since $(S,T)$ pair is H-continuous, applying limit as $n\rightarrow \infty$ in (3.6), and using (3.5), we get $d(\wp ,[T \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))})=0$, which indicates that $\wp \in [T \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))}$. Similarly, we can also demonstrate that $\wp \in [S\wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))}$. Thus,

$$\begin{aligned} \wp \in [T \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))}\cap [S\wp ]^{q}_{( \alpha (\wp ),\beta (\wp ))}, \end{aligned}$$

that is, ℘ is common q-ROF fixed point of S and T mappings. □

Example 4

Consider $X=[0,1]$ and $d(x,y)=|x-y|^{2}$ for all $x,y\in X$. Then $(X,d,b=2)$ is a complete b-metric space. Consider two q-ROF set-valued maps $S,T:X\rightarrow F^{q}(X)$. For each $x\in X$, Sx and Tx are q-ROF sets defined as:

If $x=1$:

$$\begin{aligned}& \mu _{T(1)}(t)=\textstyle\begin{cases} (\frac{1}{4})^{\frac{1}{q}}, & \text{if } t\neq 1 \\ 1, & \text{if }t=1 \end{cases}\displaystyle \\& \nu _{T(1)}(t)=\textstyle\begin{cases} (\frac{2}{3})^{\frac{1}{q}}, & \text{if } t\neq 1 \\ 0, & \text{if } t=1, \end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned}& \mu _{S(1)}(t)=\textstyle\begin{cases} (\frac{5}{18})^{\frac{1}{q}}, & \text{if } t\neq 1 \\ 1, & \text{if } t=1 \end{cases}\displaystyle \\& \nu _{S(1)}(t)=\textstyle\begin{cases} (\frac{9}{20})^{\frac{1}{q}}, & \text{if } 0\leq t\leq \frac{1}{2} \\ 0, & \text{if } \frac{1}{2}< t\leq 1. \end{cases}\displaystyle \end{aligned}$$

If $x\neq 1$:

$$\begin{aligned}& \mu _{T x}(t)=\textstyle\begin{cases} (\alpha )^{\frac{1}{q}}, & \text{if } 0\leq t\leq \frac{x^{2}}{2} \\ (\frac{2\alpha}{5q})^{\frac{1}{q}}, & \text{if } \frac{x^{2}}{2}< t \leq \frac{x}{2} \\ (\frac{\alpha}{3q})^{\frac{1}{q}}, & \text{if } \frac{x}{2}< t\leq 1 \end{cases}\displaystyle \\& \nu _{T x}(t)=\textstyle\begin{cases} 0, & \text{if } 0< t\leq \frac{x^{2}}{4} \\ (\beta ^{2})^{\frac{1}{q}}, & \text{if } \frac{x^{2}}{4}< t\leq \frac{x^{2}}{2} \\ (\beta )^{\frac{1}{q}}, & \text{if } \frac{x^{2}}{2}< t\leq 1, \end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned}& \mu _{Sx}(t)=\textstyle\begin{cases} (\alpha )^{\frac{1}{q}}, & \text{if } 0\leq t\leq \frac{x^{2}}{2} \\ (\frac{\alpha}{4q})^{\frac{1}{q}}, & \text{if } \frac{x^{2}}{2}< t \leq x \\ (\frac{2\alpha}{9q})^{\frac{1}{q}}, & \text{if } x< t\leq 1 \end{cases}\displaystyle \\& \nu _{Sx}(t)=\textstyle\begin{cases} 0, & \text{if } 0\leq t\leq \frac{x^{2}}{3} \\ (\beta ^{3})^{\frac{1}{q}}, & \text{if } \frac{x^{2}}{3}< t\leq \frac{x^{2}}{2} \\ (\beta ^{2})^{\frac{1}{q}}, & \text{if } \frac{x^{2}}{2}< t\leq 1. \end{cases}\displaystyle \end{aligned}$$

Taking $\alpha =\frac{3}{5}$ and $\beta =\frac{1}{5}$, we obtain

$$\begin{aligned}& [T x]^{q}_{(\alpha ,\beta )}=\textstyle\begin{cases} {[0,\frac{x^{2}}{2}]}, & \text{if } x\neq 1 \\ \{1\}, & \text{if } x=1 \end{cases}\displaystyle \end{aligned}$$

(3.7)

$$\begin{aligned}& [Sx]^{q}_{(\alpha ,\beta )}=\textstyle\begin{cases} {[0,\frac{x^{2}}{2}]}, & \text{if } x\neq 1 \\ \{1\}, & \text{if } x=1. \end{cases}\displaystyle \end{aligned}$$

(3.8)

Certainly, Sx and $T x\in F^{q}_{S}(X)$ for each $x\in X$. Define $\gamma :X\times X\rightarrow \mathbb{R_{+}}$ and $\varphi :\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ by

$$\begin{aligned} \gamma (x,y)=\textstyle\begin{cases} 8, & \text{if } x=y=1 \\ \frac{1}{15 }, & \text{if } x,y\in \{2,3\} \text{ and } x\neq y \\ 0, & \text{elsewhere, } \end{cases}\displaystyle \end{aligned}$$

and $\varphi (t)=\frac{t}{4}$ for all $t>0$. Let $\rho _{1}(w,v)=\frac{1}{18}v-w$ and $\rho _{2}(w,v)=\frac{3}{5}v-w$ for $w,v\in \mathbb{R_{+}}$. Obviously $\rho _{1},\rho _{2}\in \mathcal{Z}$ and $\varphi \in \Omega _{b}$. Now we verify condition (3.1) under the following cases:

Case 1:

If $x=y=1$, $[Sx]^{q}_{(\alpha ,\beta )}=[Ty]^{q}_{(\alpha ,\beta )}=\{1\}$ and therefore $H([Sx]^{q}_{(\alpha ,\beta )},[Ty]^{q}_{(\alpha ,\beta )})=0$ for all $x,y\in X$. Putting these values into (3.1) yields

$$\begin{aligned}& \rho _{1} \bigl(\gamma (x,y)H\bigl([Sx]_{(\alpha ,\beta )}^{q}, [Ty]_{( \alpha ,\beta )}^{q}\bigr),\varphi \bigl(A(x,y)\bigr) \bigr) \\& \qquad {}+ \rho _{2} \bigl(\gamma (x,y)H\bigl([Sx]_{(\alpha ,\beta )}^{q}, [Ty]_{( \alpha ,\beta )}^{q}\bigr),\varphi \bigl(B(x,y)\bigr) \bigr) \\& \quad = \rho _{1}\bigl(0, \varphi \bigl(A(x,y)\bigr)\bigr) +\rho _{2} \bigl(0,\varphi \bigl(B(x,y)\bigr)\bigr) \\& \quad = \frac{1}{18}\varphi \bigl(A(x,y)\bigr) +\frac{3}{5}\varphi \bigl(B(x,y)\bigr)\geq 0. \end{aligned}$$

Case 2:

If $x,y\in \{\frac{2}{3},\frac{1}{5}\}$ and $x\neq y$, then letting $x=\frac{2}{3}$ and $y=\frac{1}{5}$, we have $[Sx]^{q}_{(\alpha ,\beta )}=[0,\frac{2}{9}]$, and $[Ty]^{q}_{(\alpha ,\beta )}=[0, \frac{1}{50}]$

$$\begin{aligned} A \biggl(\frac{2}{3},\frac{1}{5} \biggr) =& k_{1} d \biggl(\frac{2}{3}, \frac{1}{5} \biggr) +k_{2}d \biggl( \frac{2}{3}, \biggl[S\frac{2}{3} \biggr]_{(\alpha ,\beta )}^{q} \biggr) +k_{3} d \biggl(\frac{1}{5}, \biggl[T\frac{1}{5} \biggr]_{(\alpha ,\beta )}^{q} \biggr) \\ &{} +k_{4} \biggl( \frac{d(\frac{1}{5},[T\frac{1}{5}]_{(\alpha ,\beta )}^{q})(1+d(\frac{2}{3}, [S\frac{2}{3}]_{(\alpha ,\beta )}^{q}))}{1+d(\frac{2}{3},\frac{1}{5})} \biggr) \\ &{} +k_{5} \biggl( \frac{d(\frac{1}{5},[S\frac{2}{3}]_{(\alpha ,\beta )}^{q})(1+d(\frac{2}{3}, [T\frac{1}{5}]_{(\alpha ,\beta )}^{q}))}{1+d(\frac{2}{3},\frac{1}{5})} \biggr) \\ =& k_{1} \biggl(\frac{49}{225} \biggr)+k_{2} \biggl( \frac{16}{81} \biggr) + k_{3} \biggl(\frac{81}{2500} \biggr)+k_{4} \biggl( \frac{873}{27400} \biggr) +k_{5}(0) . \end{aligned}$$

Similarly,

$$\begin{aligned} B \biggl(\frac{2}{3},\frac{1}{5} \biggr) =& \biggl( \frac{49}{225} \biggr)^{k_{1}}. \biggl(\frac{16}{81} \biggr) ^{k_{2}}. \biggl( \frac{81}{2500} \biggr)^{k_{3}}. \biggl( \frac{873}{27400} \biggr) ^{k_{4}}. \biggl(\frac{9409}{90000} \biggr)^{k_{5}}. \end{aligned}$$

By setting $k_{1}=k_{2}=k_{3}=0$ and $k_{4}=k_{5}=\frac{1}{2}$, we get

$$\begin{aligned}& \rho _{1} \biggl(\frac{1}{15} \biggl(\frac{8281}{202500} \biggr), \varphi \biggl(A \biggl(\frac{2}{3},\frac{1}{5} \biggr) \biggr) \biggr) +\rho _{2} \biggl(\frac{1}{15} \biggl(\frac{8281}{202500} \biggr), \varphi \biggl(B \biggl(\frac{2}{3}, \frac{1}{5} \biggr) \biggr) \biggr) \\& \quad \geq 0. \end{aligned}$$

Case 3:

If $x,y\in X\setminus \{\frac{2}{3},\frac{1}{5}\}$, then $\gamma (x,y)=0$. Hence from (3.1), we obtain

$$\begin{aligned} \rho _{1} \bigl(0,\varphi \bigl(A(x,y)\bigr) \bigr)+\rho _{2} \bigl(0,\varphi \bigl(B(x,y)\bigr) \bigr) =\frac{1}{18} \varphi \bigl(A(x,y)\bigr)+\frac{3}{5}\varphi \bigl(B(x,y)\bigr) \geq 0 \end{aligned}$$

Moreover, it is clear that $(S,T)$ pair is γ-admissible. The sets $[Sx]^{q}_{(\alpha ,\beta )}$ and $[T x]^{q}_{(\alpha ,\beta )}$ are proximal and H-continuous for each $x\in X$. Since all the hypotheses of Theorem 2 are satisfied, we can observe from the Fig. 1 that S and T have many common q-ROF fixed points in X.

Figure 1

Graphical illustration of (3.7) and (3.8), representing infinitely many fixed points

4 Consequences

The following result can easily be obtained by considering a single mapping in the main result.

Corollary 1

Consider $(X,d)$ a complete b-metric space and $T:X\rightarrow F^{q}(X)$ a q-ROF set-valued map, which satisfies:

$$0\le \left\{\begin{array}{c}{\rho }_1(\gamma (x,y)H({[Tx]}_{(\alpha (x),\beta (x))}^q,{[Ty]}_{(\alpha (y),\beta (y))}^q),\phi (A(x,y)))\\ +{\rho }_2(\gamma (x,y)H({[Tx]}_{(\alpha (x),\beta (x))}^q,{[Ty]}_{(\alpha (y),\beta (y))}^q),\phi (B(x,y)))\end{array}\right\}$$

(4.1)

for all $x,y\in X$, where $\varphi \in \Omega _{b}$, $\rho \in \mathcal{Z}$ and $\gamma :X\times X\rightarrow \mathbb{R_{+}}$ is a function. Assume also that:

• (i) T is a γ-admissible;

• (ii) There exists $x_{0}\in X$ and $x_{1}\in [T x_{0}]^{q}_{(\alpha (x_{0}),\beta (x_{0}))}$ such that $\gamma (x_{0},x_{1})\geq 1$:

• (iii) T is H-continuous.

• (iv) $[T x]^{q}_{(\alpha (x),\beta (x))}$ is proximal for each $x\in X$.

Then, T has at least one q-ROF fixed point in X.

The following result generalizes the Nadler’s result [11] for multivalued mappings.

Corollary 2

Consider a complete b-metric space and suppose $T:X\rightarrow F^{q}(X)$ is a q-ROF mapping, that satisfies

$$ H \bigl( [ T x ] _{ ( \alpha (x),\beta (x) ) }^{q}, [ Ty ] _{ ( \alpha (y),\beta (y) ) }^{q} \bigr) \leq \lambda d(x,y) $$

for all $x,y\in X$, $\lambda \in (0,1)$, $\alpha ,\beta \in (0,1]\times {}[ 0,1)$.

Moreover, assume that:

• (i) T is H-continuous.

• (ii) $[ T x ] _{ ( \alpha (x),\beta (x) ) }^{q}$ is proximal for each $x\in X$.

Then, T has a q-ROF fixed point in X.

The following result can easily be accomplished by considering an intuitionistic fuzzy mapping in the main result.

Corollary 3

Consider a complete b-metric space and suppose $T:X\rightarrow F^{I}(X)$ be an intuitionistic fuzzy mapping that satisfies

$$0\le \left\{\begin{array}{c}{\rho }_1(\gamma (x,y)H({[Tx]}_{(\alpha (x),\beta (x))},{[Ty]}_{(\alpha (y),\beta (y))}),\phi (A(x,y)))\\ +{\rho }_2(\gamma (x,y)H({[Tx]}_{(\alpha (x),\beta (x))},{[Ty]}_{(\alpha (y),\beta (y))}),\phi (B(x,y)))\end{array}\right\}$$

for all $x,y\in X$, where $\varphi \in \Omega _{b}$, $\rho \in \mathcal{Z}$ and $\gamma :X\times X\rightarrow \mathbb{R_{+}}$ is a function. Assume also that:

• (i) T is a γ-admissible;

• (ii) There exists $x_{0}\in X$ and $x_{1}\in [T x_{0}]^{q}_{(\alpha (x_{0}),\beta (x_{0}))}$ such that $\gamma (x_{0},x_{1})\geq 1$:

• (iii) T is H-continuous.

• (iv) $[T x]^{q}_{(\alpha (x),\beta (x))}$ is proximal for each $x\in X$.

Then, T has at least one fixed point in X.

The following result can easily be accomplished by considering a fuzzy mapping in the main result.

Corollary 4

Consider a complete b-metric Sspace and suppose $T:X\rightarrow F(X)$ be a fuzzy mapping that satisfies

$$0\le \left\{\begin{array}{c}{\rho }_1(\gamma (x,y)H({[Tx]}_{\alpha (x)},{[Ty]}_{\alpha (y)}),\phi (A(x,y)))\\ +{\rho }_2(\gamma (x,y)H({[Tx]}_{\alpha (x)},{[Ty]}_{\alpha (y)}),\phi (B(x,y)))\end{array}\right\}$$

for all $x,y\in X$, where $\varphi \in \Omega _{b}$, $\rho \in \mathcal{Z}$ and $\gamma :X\times X\rightarrow \mathbb{R_{+}}$ is a function. Assume also that:

• (i) T is a γ-admissible.

• (ii) There exists $x_{0}\in X$ and $x_{1}\in [T x_{0}]^{q}_{\alpha (x_{0})}$ such that $\gamma (x_{0},x_{1})\geq 1$.

• (iii) T is H-continuous.

• (iv) $[T x]^{q}_{\alpha (x)}$ is proximal for each $x\in X$.

Then, T has at least one fixed point in X.

5 Application

We will use Theorem 2 to show the existence of common solution of a system of non-linear n-th order differential inclusion with non-local and integral boundary conditions.

$$ \left . \textstyle\begin{array}{l} u^{n}(t)\in F_{1}(t,u(t))\quad \text{a.e }t\in [ 0,1 ] \\ \wp (0)=u_{0}+g(u),\qquad u^{\prime }(0)=u^{\prime \prime }(0)=\cdots=u^{n-2}(0)=0 \\ \delta _{1}u(1)+\delta _{2}u^{\prime }(1)=\sum_{i=1}^{m} \kappa _{i}\int _{0}^{\zeta _{i}}u^{\prime }(s)\,ds \quad 0< \zeta _{i}< 1 \\ u^{n}(t)\in F_{2}(t,u(t))\quad \text{a.e }t\in [ 0,1 ] \\ u(0)=u_{0}+g(u),\qquad u^{\prime }(0)=u^{\prime \prime }(0)=\cdots=u^{n-2}(0)=0 \\ \delta _{1}u(1)+\delta _{2}u^{\prime }(1)=\sum_{i=1}^{m} \kappa _{i}\int _{0}^{\zeta _{i}}u^{\prime }(s)\,ds \quad 0< \zeta _{i}< 1\end{array}\displaystyle \right \} $$

(4)

where $F_{1},F_{2}: [ 0,1 ] \times \mathbb{R} \longrightarrow P ( \mathbb{R} )$ are multivalued maps and $P ( \mathbb{R} )$ is the family of all subsets of $\mathbb{R}$. $g:C ( [ 0,1 ] ,\mathbb{R} ) \rightarrow \mathbb{R}$ is continuous function, $\delta _{1}$, $\delta _{2}$, $\kappa _{i}$, $\zeta _{i}$ ($i=1,2,\ldots,m$) are real constants.

Consider $L^{1} ( [ 0,1 ] ,\mathbb{R} )$ as the Banach space of measureable functions $u: [ 0,1 ] \rightarrow \mathbb{R}\mathbbm{,}$ which are Lebesgue integrable and normed by $\Vert u \Vert _{L^{1}}=\int _{0}^{1} \vert u(t) \vert \,dt$.

Let $F_{1},F_{2}: [ 0,1 ] \times \mathbb{R} \longrightarrow P ( \mathbb{R} )$, for each $u\in C^{n-1} ( [ 0,1 ] ,\mathbb{R} )$ define the set of selections of $F_{1}$ and $F_{2}$ by,

$$\begin{aligned}& S_{F_{1},u}= \bigl\{ f_{1}\in L^{1} \bigl( [ 0,1 ] ,\mathbb{R} \bigr) :f_{1}\in F_{1}\bigl(t,u(t) \bigr) \bigr\} \quad \text{a.e }t\in [ 0,1 ]\\& S_{F_{2},u}= \bigl\{ f_{2}\in L^{1} \bigl( [ 0,1 ] ,\mathbb{R} \bigr) :f_{2}\in F_{2}\bigl(t,u(t) \bigr) \bigr\} \quad \text{a.e }t\in [ 0,1 ] \end{aligned}$$

By the common solution of differential inclusions in (4), we mean $u\in C^{n-1} ( [ 0,1 ] ,\mathbb{R} )$ such that $u(0)=u_{0}+g(u)$, $u^{\prime }(0)=u^{ \prime \prime }(0)=\cdots=u^{n-2}(0)=0$ and $\delta _{1}u(1)+\delta _{2}u^{\prime }(1)=\sum_{i=1}^{m} \kappa _{i}\int _{0}^{\zeta _{i}}u^{\prime }(s)\,ds$ $0< \zeta _{i}<1$ and there exists functions $f_{1},f_{2}\in L^{1} ( [ 0,1 ] ,\mathbb{R} )$ such that $f_{1}\in F_{1}(t,u(t))$, $f_{2}\in F_{2}(t,u(t))$ and

$$\begin{aligned} u(t) =& \bigl( 1-\delta _{1}\Lambda t^{n-1} \bigr) \bigl[ u_{0}+g(u) \bigr] + \int _{0}^{1} \frac{ ( t-s ) ^{n-1}}{ ( n-1 ) !}f_{1}(s)\,ds \\ &{}+\Lambda t^{n-1} \Biggl\{ \sum_{i=1}^{m} \kappa _{i} \int _{0}^{\zeta _{i}} \frac{ ( \zeta _{i}-s ) ^{n-1}}{ ( n-1 ) !}f_{1}(s)\,ds- \delta _{1} \int _{0}^{1} \frac{ ( 1-s ) ^{n-1}}{ ( n-1 ) !}f_{1}(s)\,ds \\ &{}- \delta _{2} \int _{0}^{1} \frac{ ( 1-s ) ^{n-2}}{ ( n-2 ) !}f_{1}(s)\,ds \Biggr\} \end{aligned}$$

and

$$\begin{aligned} u(t) =& \bigl( 1-\delta _{1}\Lambda t^{n-1} \bigr) \bigl[ u_{0}+g(u) \bigr] + \int _{0}^{1} \frac{ ( t-s ) ^{n-1}}{ ( n-1 ) !}f_{2}(s)\,ds \\ &{}+\Lambda t^{n-1} \Biggl\{ \sum_{i=1}^{m} \kappa _{i} \int _{0}^{\zeta _{i}} \frac{ ( \zeta _{i}-s ) ^{n-1}}{ ( n-1 ) !}f_{2}(s)\,ds- \delta _{1} \int _{0}^{1} \frac{ ( 1-s ) ^{n-1}}{ ( n-1 ) !}f_{2}(s)\,ds \\ &{}- \delta _{2} \int _{0}^{1} \frac{ ( 1-s ) ^{n-2}}{ ( n-2 ) !}f_{2}(s)\,ds \Biggr\} , \end{aligned}$$

where $\Lambda = \frac{1}{\delta _{1}+ ( n-1 ) \delta _{2}-\sum_{i=1}^{m}\kappa _{i}\zeta _{i}^{n-1}}$.

Theorem 3

Consider the system of non-linear n-th order differential inclusions as in (4). Assume that the following conditions hold:

$( A1 )$ The operators $F_{1}(t,u(t))$ and $F_{2}(t,u(t))$ are continuous.

$(A2)$ Consider a continuous function $\gamma : ( 0,\infty ) \rightarrow {}[ 1,\infty )$ such that

$$l_{0} \big\vert g(z)-g(\omega ) \big\vert \leq \frac{B ( z(t),\omega (t) ) }{2\gamma (z(t),\omega (t))}\quad \textit{and}\quad p_{0} \big\vert f_{1u}(s)-f_{2v}(s) \big\vert \leq \frac{A(z(t),\omega (t))}{2\gamma (z(t),\omega (t))},$$

where $p_{0}=\frac{1}{n!}+ \vert \Lambda \vert ( \frac{\sum_{i=1}^{m}\kappa _{i}\zeta _{i}^{n-1}}{n!}+ \frac{ \vert \delta _{1} \vert }{n!}+ \frac{ \vert \delta _{2} \vert }{ ( n-1 ) !} )$, $l_{0}= \vert 1-\Lambda \delta _{1}t^{n-1} \vert$ and

$$\begin{aligned}& A\bigl(z(t),\omega (t)\bigr) \\& \quad = k_{1}d \bigl( z(t),\omega (t) \bigr) +k_{2}d \bigl( z(t), \bigl[ Ez(t) \bigr] _{ ( \alpha (z(t)),\beta (z(t)) ) }^{q} \bigr) +k_{3}d ( \omega (t), \bigl[ G\omega (t) \bigr] _{ ( \alpha ( \omega (t)),\beta (\omega (t) ) }^{q} \bigr) \\& \qquad {}+k_{4} \biggl[ \frac{d ( \omega (t), [ G\omega (t) ] _{ ( \alpha (\omega (t)),\beta (\omega (t) ) }^{q} ) ( 1+d ( z(t), [ Ez(t) ] _{ ( \alpha (z(t)),\beta (z(t)) ) }^{q} ) ) }{1+d ( z(t),\omega (t) ) } \biggr] \\& \qquad {}+k_{5} \biggl[ \frac{d ( \omega (t), [ Ez(t) ] _{ ( \alpha (z(t)),\beta (z(t)) ) }^{q} ) ( 1+d ( z(t), [ G\omega (t) ] _{ ( \alpha (\omega (t)),\beta (\omega (t) ) }^{q} ) ) }{1+d ( z(t),\omega (t) ) } \biggr] , \\& B\bigl(z(t),\omega (t)\bigr) \\& \quad = \bigl( d \bigl( z(t),\omega (t) \bigr) \bigr) ^{k_{1}}. \bigl( d \bigl( z(t), \bigl[ Ez(t) \bigr] _{ ( \alpha (z(t)),\beta (z(t)) ) }^{q} \bigr) \bigr) ^{k_{2}} \\& \qquad {}\cdot \bigl( d \bigl( \omega (t), \bigl[ G\omega (t) \bigr] _{ ( \alpha (\omega (t)),\beta (\omega (t) ) }^{q} \bigr) \bigr) ^{k_{3}} \\& \qquad {}\cdot \biggl[ \frac{d ( \omega (t), [ G\omega (t) ] _{ ( \alpha (\omega (t)),\beta (\omega (t) ) }^{q} ) ( 1+d ( z(t), [ Ez(t) ] _{ ( \alpha (z(t)),\beta (z(t)) ) }^{q} ) ) }{1+d ( z(t),\omega (t) ) } \biggr] ^{k_{4}} \\& \qquad {}\cdot \biggl[ \frac{d ( z(t), [ G\omega (t) ] _{ ( \alpha (\omega (t)),\beta (\omega (t) ) }^{q} ) +d ( \omega (t), [ Ez(t) ] _{ ( \alpha (z(t)),\beta (z(t)) ) }^{q} ) }{2b} \biggr] ^{k_{5}}, \end{aligned}$$

where $( \alpha (z(t)),\beta (z(t)) )$, $( \alpha (\omega (t)), \beta (\omega (t) ) \in (0,1]\times {}[ 0,1)$, $b\geq 1$, $k_{i} \geq 0$ and $\sum_{1=1}^{5}k_{i}=1$.

Then, there exists a common solution of system of differential inclusions in (4).

Proof

It is well known that a space with metric defined by $d ( u,v ) =\sup_{t\in [ 0,1 ] } ( u(t)-v(t) ) ^{2}= \Vert u-v \Vert _{\infty }^{2}$ $\forall u,v\in C ( [ 0,1 ] ,\mathbb{R} )$ is a complete b-metric space with $s=2$. We define a pair of q-ROF mappings $E,G:X\rightarrow F_{S}^{q}(X)$ as follows:

$$\begin{array}{c}E:X\to F_S^q(X):E(x(\sigma ))={\omega }_x(\sigma )\hfill \\ =\left\{\begin{array}{c}h\in X:h(\sigma )=(1-{\delta }_1\mathrm{\Lambda }{t}^{n-1})[u_0+g(h)]\\ +{\int }_0^1\frac{{(t-s)}^{n-1}}{(n-1)!}{f}_1(s)ds\\ +\mathrm{\Lambda }{t}^{n-1}\left\{\begin{array}{c}{\sum }_{i=1}^m{\kappa }_i{\int }_0^{{\zeta }_i}\frac{{({\zeta }_i-s)}^{n-1}}{(n-1)!}{f}_1(s)ds-{\delta }_1{\int }_0^1\frac{{(1-s)}^{n-1}}{(n-1)!}{f}_1(s)ds\\ -{\delta }_2{\int }_0^1\frac{{(1-s)}^{n-2}}{(n-2)!}{f}_1(s)ds\end{array}\right\}\end{array}\right\}\hfill \\ G:X\to F_S^q(X):G(x(\sigma ))\hfill \\ ={\mathrm{\Omega }}_x(\sigma )=\left\{\begin{array}{c}h\in X:h(\sigma )=(1-{\delta }_1\mathrm{\Lambda }{t}^{n-1})[u_0+g(h)]\\ +{\int }_0^1\frac{{(t-s)}^{n-1}}{(n-1)!}{f}_2(s)ds\\ +\mathrm{\Lambda }{t}^{n-1}\left\{\begin{array}{c}{\sum }_{i=1}^m{\kappa }_i{\int }_0^{{\zeta }_i}\frac{{({\zeta }_i-s)}^{n-1}}{(n-1)!}{f}_2(s)ds-{\delta }_1{\int }_0^1\frac{{(1-s)}^{n-1}}{(n-1)!}{f}_2(s)ds\\ -{\delta }_2{\int }_0^1\frac{{(1-s)}^{n-2}}{(n-2)!}{f}_2(s)ds\end{array}\right\}\end{array}\right\}\hfill \end{array}$$

Assume four arbitrary mappings $U,V,Y,Z:X\rightarrow [ 0,1 ]$ such that:

$$\begin{aligned}& \mu _{Ex}(h)=\textstyle\begin{cases} ( U(x) ) ^{1/q} & \text{if }h(\sigma )\in \omega _{x}( \sigma ) \\ 0 & \text{otherwise}\end{cases}\displaystyle \\& \nu _{Ex}(h)=\textstyle\begin{cases} 0 & \text{if }h(\sigma )\in \omega _{x}(\sigma ) \\ ( V(x) ) ^{1/q} & \text{otherwise}\end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned}& \mu _{Gx}(h)=\textstyle\begin{cases} ( Y(x) ) ^{1/q} & \text{if }h(\sigma )\in \Omega _{x}( \sigma ) \\ 0 & \text{otherwise}\end{cases}\displaystyle \\& \nu _{Gx}(h)=\textstyle\begin{cases} 0 & \text{if }h(\sigma )\in \Omega _{x}(\sigma ) \\ ( Z(x) ) ^{1/q} & \text{otherwise}\end{cases}\displaystyle \end{aligned}$$

If we take $\alpha _{E}(x)=U(x)$, $\beta _{E}(x)=0$, $\alpha _{G}(x)=Y(x)$ and $\beta _{G}(x)=0$, then we have

$$ \bigcup_{x\in X} [ Ex ] _{(\alpha ,\beta )_{Ex}}^{q}=\bigcup_{x\in X} \bigl\{ h\in X:\mu _{Ex}(h)= \bigl( U(x) \bigr) ^{1/q} \text{ and } \nu _{Ex}(h)=0 \bigr\} = \bigcup_{x\in X} \bigl\{ \omega _{x}(\sigma ) \bigr\} $$

and

$$ \bigcup_{x\in X} [ Gx ] _{(\alpha ,\beta )_{Gx}}^{q}=\bigcup_{x\in X} \bigl\{ h\in X:\mu _{Gx}(h)= \bigl( Y(x) \bigr) ^{1/q} \text{ and } \nu _{Gx}(h)=0 \bigr\} = \bigcup_{x\in X} \bigl\{ \Omega _{x} ( \sigma ) \bigr\} $$

Now, let $z(t)\in [ Ex ] _{(\alpha ,\beta )_{Ex}}^{q}$ and $\omega (t)\in [ Gx ] _{(\alpha ,\beta )_{Gx}}^{q}$ such that

$$\begin{array}{c}z(t)=\left\{\begin{array}{c}(1-{\delta }_1\mathrm{\Lambda }{t}^{n-1})[u_0+g(z)]+{\int }_0^1\frac{{(t-s)}^{n-1}}{(n-1)!}{f}_1(s)ds\\ +\mathrm{\Lambda }{t}^{n-1}\{{\sum }_{i=1}^m{\kappa }_i{\int }_0^{{\zeta }_i}\frac{{({\zeta }_i-s)}^{n-1}}{(n-1)!}{f}_1(s)ds-{\delta }_1{\int }_0^1\frac{{(1-s)}^{n-1}}{(n-1)!}{f}_1(s)ds\\ -{\delta }_2{\int }_0^1\frac{{(1-s)}^{n-2}}{(n-2)!}{f}_1(s)ds\}\end{array}\right\}\hfill \\ \omega (t)=(1-{\delta }_1\mathrm{\Lambda }{t}^{n-1})[u_0+g(\omega )]+{\int }_0^1\frac{{(t-s)}^{n-1}}{(n-1)!}{f}_2(s)ds\hfill \\ \phantom{\omega (t)=}+\mathrm{\Lambda }{t}^{n-1}\{\sum _{i=1}^m{\kappa }_i{\int }_0^{{\zeta }_i}\frac{{({\zeta }_i-s)}^{n-1}}{(n-1)!}{f}_2(s)ds-{\delta }_1{\int }_0^1\frac{{(1-s)}^{n-1}}{(n-1)!}{f}_2(s)ds\hfill \\ \phantom{\omega (t)=}-{\delta }_2{\int }_0^1\frac{{(1-s)}^{n-2}}{(n-2)!}{f}_2(s)ds\}\hfill \end{array}$$

Now,

$$\begin{array}{c}|z(t)-\omega (t)|\hfill \\ =\left|\begin{array}{c}(1-{\delta }_1\mathrm{\Lambda }{t}^{n-1})[u_0+g(z)]+{\int }_0^1\frac{{(t-s)}^{n-1}}{(n-1)!}{f}_1(s)ds\\ +\mathrm{\Lambda }{t}^{n-1}\{{\sum }_{i=1}^m{\kappa }_i{\int }_0^{{\zeta }_i}\frac{{({\zeta }_i-s)}^{n-1}}{(n-1)!}{f}_1(s)ds-{\delta }_1{\int }_0^1\frac{{(1-s)}^{n-1}}{(n-1)!}{f}_1(s)ds\\ -{\delta }_2{\int }_0^1\frac{{(1-s)}^{n-2}}{(n-2)!}{f}_1(s)ds\}\end{array}\right|\hfill \\ -\left|\begin{array}{c}(1-{\delta }_1\mathrm{\Lambda }{t}^{n-1})[u_0+g(\omega )]-{\int }_0^1\frac{{(t-s)}^{n-1}}{(n-1)!}{f}_2(s)ds\\ -\mathrm{\Lambda }{t}^{n-1}\{{\sum }_{i=1}^m{\kappa }_i{\int }_0^{{\zeta }_i}\frac{{({\zeta }_i-s)}^{n-1}}{(n-1)!}{f}_2(s)ds-{\delta }_1{\int }_0^1\frac{{(1-s)}^{n-1}}{(n-1)!}{f}_2(s)ds\\ -{\delta }_2{\int }_0^1\frac{{(1-s)}^{n-2}}{(n-2)!}{f}_2(s)ds\}\end{array}\right|\hfill \\ \le (1-{\delta }_1\mathrm{\Lambda }{t}^{n-1})|g(z)-g(\omega )|+{\int }_0^1\frac{{(t-s)}^{n-1}}{(n-1)!}ds|f_1(s)-f_2(s)|\hfill \\ +\left|\mathrm{\Lambda }{t}^{n-1}\right|\left\{\begin{array}{c}{\sum }_{i=1}^m{\kappa }_i{\int }_0^{{\zeta }_i}\frac{{({\zeta }_i-s)}^{n-1}}{(n-1)!}ds|f_1(s)-f_2(s)|\\ +|{\delta }_1|{\int }_0^1\frac{{(1-s)}^{n-1}}{(n-1)!}ds|f_1(s)-f_2(s)|\\ +|{\delta }_2|{\int }_0^1\frac{{(1-s)}^{n-2}}{(n-2)!}ds|f_1(s)-f_2(s)|\end{array}\right\}\hfill \\ \le (1-{\delta }_1\mathrm{\Lambda }{t}^{n-1})|g(z)-g(\omega )|+\frac{1}{n!}\hfill \\ +|\mathrm{\Lambda }|(\frac{{\sum }_{i=1}^m{\kappa }_i{\zeta }_i^{n-1}}{n!}+\frac{|{\delta }_1|}{n!}+\frac{|{\delta }_2|}{(n-1)!})|f_1(s)-f_2(s)|\hfill \\ \le l_0|g(z)-g(\omega )|+p_0|f_{1u}(s)-f_{2v}(s)|\hfill \\ \le \frac{B(z(t),\omega (t))}{2\gamma (z(t),\omega (t))}+\frac{A(z(t),\omega (t))}{2\gamma (z(t),\omega (t))}\hfill \\ H({[Ex]}_{{(\alpha ,\beta )}_{Ex}}^q,{[Gx]}_{{(\alpha ,\beta )}_{Gx}}^q)\le \frac{B(z(t),\omega (t))}{2\gamma (z(t),\omega (t))}+\frac{A(z(t),\omega (t))}{2\gamma (z(t),\omega (t))}\hfill \\ 2\gamma (z(t),\omega (t))H({[Ex]}_{{(\alpha ,\beta )}_{Ex}}^q,{[Gx]}_{{(\alpha ,\beta )}_{Gx}}^q)\le A(z(t),\omega (t))+B(z(t),\omega (t)).\hfill \end{array}$$

Hence by Theorem 2 there exists a common solution of system of differential inclusions in (4). □

6 Conclusion

We conclude this manuscript by achieving the following significant goals:

• (1) We have introduced γ-admissibility for the pair of q-ROF setsvalued maps and admissible hybrid q-ROF $\mathcal{Z}$-contraction.

• (2) Common fixed point result for a pair of q-ROF set valued maps utilizing the introduced contraction in b-metric space is presented.

• (3) A nontrivial example illustrating our main result, along with graphical representation, is also included.

• (4) Depicting the significance of obtained results, existence of solution of system of non-linear n-th order differential inclusion with non-local and integral boundary conditions is discussed as an application.

• (5) Consequences section elaborates the contribution of new results being presented in literature.

Open problem

The main result can also be studied for other generalized metric spaces like Menegr probabilistic metric spaces and newly introduced q-ROF metric spaces. Application for the existence of solution of differential and integral inclusions can also be studied then by utilizing the new results in said spaces.