- Research
- Open access
- Published:
Existence of solution of a system of non-linear differential inclusions with non-local, integral boundary conditions via fixed points of hybrid contractions
Boundary Value Problems volume 2024, Article number: 90 (2024)
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:
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
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
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
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
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:
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:
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,
or
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
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
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
and
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
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
Any crisp set A can be represented as a q-ROF set by considering the characteristic function of orthopairs
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
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:
for all \(x,y\in X\), where
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:
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
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 \(\gamma (x_{0},x_{1})\geq 1\)
-
(a)
-
(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:
Now, from Lemma 1, we have
Now, if \(x_{0}=x_{1}\) and \(S=T\), then:
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:
Similarly,
Now solving for \(A(x_{n-1},x_{n})\) and \(B(x_{n-1}, x_{n})\), \(n\in \mathbb{N} \), in general, we have:
It is clear that, \(( \frac{f+g}{2} ) ^{h}\leq \frac{f^{h}+g^{h}}{2} \), so
Using equations (2) and (3) in (1), we have:
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:
A contradiction. So,
Similarly,
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,
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
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
Now by using the triangle inequality in X, we show that \(\wp \in [T \wp ]^{q}_{(\alpha (\wp ),\beta (\wp ))}\). So
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,
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
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.
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:
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
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
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
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.
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,
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
and
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
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
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:
Assume four arbitrary mappings \(U,V,Y,Z:X\rightarrow [ 0,1 ] \) such that:
and
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
and
Now, let \(z(t)\in [ Ex ] _{(\alpha ,\beta )_{Ex}}^{q}\) and \(\omega (t)\in [ Gx ] _{(\alpha ,\beta )_{Gx}}^{q}\) such that
Now,
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.
Data Availability
No datasets were generated or analysed during the current study.
References
Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)
Atanassov, K.T., Stoeva, S.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)
Atanassov, K.T.: A second type of intuitionistic fuzzy sets. BUSEFAL 56, 66–70 (1993)
Ali, M.I.: Another view on q-rung orthopair fuzzy sets. Int. J. Intell. Syst. 33(11), 2139–2153 (2018)
Zhu, H., Zhao, J., Li, H.: q-ROF-SIR methods and their applications to multiple attribute decision making. Int. J. Mach. Learn. Cybern., 1–13 (2022)
Naz, S., Akram, M., Sattar, A., Al-Shamiri, M.M.A.: 2-tuple linguistic q-rung orthopair fuzzy CODAS approach and its application in arc welding robot selection. AIMS Math. 7(9), 17529–17569 (2022)
Deveci, M., Pamucar, D., Gokasar, I., Koppen, M., Gupta, B.B.: Personal mobility in metaverse with autonomous vehicles using Q-rung orthopair fuzzy sets based OPA-RAFSI model. IEEE Trans. Intell. Transp. Syst.. (2022)
Ashraf, S., Rehman, N., Hussain, A., AlSalman, H., Gumaei, A.H.: q-rung orthopair fuzzy rough Einstein aggregation information-based EDAS method: applications in robotic agrifarming. Comput. Intell. Neurosci. 2021 (2021)
Qiyas, M., Abdullah, S., Khan, N., Naeem, M., Khan, F., Liu, Y.: Case study for hospital-based post-acute care-cerebrovascular disease using sine hyperbolic q-rung orthopair fuzzy Dombi aggregation operators. Expert Syst. Appl. 215, 119224 (2023)
Banach, S.: On operations in abstract sets and their application to integral equations. Fundam. Math. 3(1), 133–181 (1922)
Nadler, S.B. Jr: (1969). Multi-valued contraction mappings
Shatanawi, W., Shatnawi, T.A.M.: New fixed point results in controlled metric type spaces based on new contractive conditions. AIMS Math. 8(4), 9314–9330 (2023). https://doi.org/10.3934/math.2023468
Rezazgui, A.-Z., Ahmad Tallafha, A., Shatanawi, W.: Common fixed point results via \(A\nu -\alpha \)-contractions with a pair and two pairs of self-mappings in the frame of an extended quasi b-metric space. AIMS Math. 8(3), 7225–7241 (2023). https://doi.org/10.3934/math.2023363
Joshi, M., Tomar, A., Abdeljawad, T.: On fixed points, their geometry and application to satellite web coupling problem in S-metric spaces. AIMS Math. 8(2), 4407–4441 (2023). https://doi.org/10.3934/math.2023220
Razani, A.: An existence theorem for ordinary differential equation in Menger probabilistic metric space. Miskolc Math. Notes 15(2), 711–716 (2014)
Razani, A.: A fixed point theorem in the Menger probabilistic metric space. N.Z. J. Math. 35(1), 109–114 (2006)
Chen, L., Xia, X., Zhao, Y., Liu, X.: Common fixed point theorems for two mappings in complete b-metric spaces. Fractal Fract. 6(2), 103 (2022)
Razani, A., Safari, F.: An elliptic type inclusion problem on the Heisenberg Lie group. Math. Slovaca 73(4), 957–968 (2023)
Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for \(\alpha -\psi \)-contractive type mappings. Nonlinear Anal., Theory Methods Appl. 75(4), 2154–2165 (2012)
Khojasteh, F., Shukla, S., Radenovic, S.: A new approach to the study of fixed point theory for simulation functions. Filomat 29(6), 1189–1194 (2015)
Rashid, M., Azam, A., Dar, F., Ali, F., Al-Kadhi, M.A.: A comprehensive study on advancement in hybrid contraction and graphical analysis of fuzzy fixed points with application. Mathematics 11(21), 4489 (2023)
Rashid, M., Shahid, L.: Some Generalized Contractive Mappings in Generalized Spaces. Punjab Univ. J. Math. 51(8) (2020)
Karapinar, E.: \(\alpha -\psi \)-Geraghty contraction type mappings and some related fixed point results. Filomat 28(1), 37–48 (2014)
Asl, J.H., Rezapour, S., Shahzad, N.: On fixed points of \(\alpha -\psi \)-contractive multifunctions. Fixed Point Theory Appl. 2012, 212 (2012)
Budhia, L., Aydi, H., Ansari, A.H., Gopal, D.: Some new fixed point results in rectangular metric spaces with an application to fractional-order functional differential equations. Nonlinear Anal., Model. Control 25(4), 580–597 (2020)
Salimi, P., Latif, A., Hussain, N.: Modified \(\alpha -\psi \)-contractive mappings with applications. Fixed Point Theory Appl. 2013(1), 151 (2013)
Karapinar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric spaces with an application to nonlinear fractional differential equations. Mathematics 7(5), 444 (2019)
Weiss, M.D.: Fixed points, separation, and induced topologies for fuzzy sets. J. Math. Anal. Appl. 50(1), 142–150 (1975)
Butnariu, D.: Fixed points for fuzzy mappings. Fuzzy Sets Syst. 7(2), 191–207 (1982)
Heilpern, S.: Fuzzy mappings and fixed point theorem. J. Math. Anal. Appl. 83(2), 566–569 (1981)
Azam, A., Tabassum, R., Rashid, M.: Coincidence and fixed point theorems of intuitionistic fuzzy mappings with applications. J. Math. Anal. 8(4), 56–77 (2017)
Shagari, M.S., Fulatan, I.A.: A role of fuzzy set-valued maps in integral inclusions. Casp. J. Math. Sci. 11(1) (2022)
Alansari, M., Shagari, M.: Solutions of fractional differential inclusions and stationary points of intuitionistic fuzzy-set-valued maps. Symmetry 15(8), 1535 (2023)
Rashid, M., Shahid, L., Agarwal, R.P., Hussain, A., Al-Sulami, H.: q-ROF mappings and Suzuki type common fixed point results in b-metric spaces with application. J. Inequal. Appl. 2022(1), 155 (2022)
Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1(1), 5–11 (1993)
Boriceanu, M.: Fixed Point Theory for Multivalued Generalized Contractions on a Set with two b-Metrics. Stud. Univ. Babeş–Bolyai, Math. 3 (2009)
Roldan-Lopez-de-Hierro, A.F., Karapinar, E., Roldan-Lopez-de-Hierro, C., Martinez-Moreno, J.: Coincidence point theorems on metric spaces via simulation functions. J. Comput. Appl. Math. 275, 345–355 (2015)
Alharbi, A.S., Alsulami, H.H., Karapinar, E.: On the power of simulation and admissible functions in metric fixed point theory. J. Funct. Spaces 2017 (2017)
Alqahtani, B., Fulga, A., Karapinar, E.: Fixed point results on d-symmetric quasi-metric space via simulation function with an application to Ulam stability. Mathematics 6(10), 208 (2018)
Aydi, H., Felhi, A., Karapinar, E.R.D.A.L., Alojail, F.A.: Fixed points on quasi-metric spaces via simulation functions and consequences. J. Math. Anal. 9(2), 10–24 (2018)
Yager, R.R.: Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst. 25(5), 1222–1230 (2016)
Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)
Berinde, V.: Generalized contractions in quasimetric spaces. In: Seminar on Fixed Point Theory, vol. 3, pp. 3–9 (1993)
Singh, S.L., Prasad, B.: Some coincidence theorems and stability of iterative procedures. Comput. Math. Appl. 55(11), 2512–2520 (2008)
Acknowledgements
The authors I. Ayoob and N. Mlaiki would like to thank the Prince Sultan University for paying the publication fees for this work through TAS LAB.
Funding
This research work did not receive any external funding.
Author information
Authors and Affiliations
Contributions
M.R., L.S., F.D., I.A. and N.M. wrote the main manuscript text. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Rashid, M., Shahid, L., Dar, F. et al. Existence of solution of a system of non-linear differential inclusions with non-local, integral boundary conditions via fixed points of hybrid contractions. Bound Value Probl 2024, 90 (2024). https://doi.org/10.1186/s13661-024-01902-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-024-01902-x