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Common fixedpoint results of fuzzy mappings and applications on stochastic Volterra integral equations
Journal of Inequalities and Applications volume 2022, Article number: 110 (2022)
Abstract
The objective of the present research is to establish and prove some new common fuzzy fixedpoint theorems for fuzzy setvalued mappings involving Θcontractions in a complete metric space. For this purpose, a novel integraltype contraction condition is applied to obtain these results. In this way, several useful and classical results have been generalized. Moreover, a concrete example is created to furnish our results. An application to stochastic Volterra integral equations has been given to enhance the validity of our results.
Introduction
The fuzzy logics were created using a group structure with ambiguous knowledge. Due to the flexibility of FSs in dealing with unreliability, this is even better for humanistic logic based on authentic reality and limitless knowledge. This notion is unquestionably a basic aspect of classical sets. Another important feature of this information is that it enables evaluation of the negative and positive elements of incorrect notions. Fuzzy mathematics is an area of mathematics that deals with FS theory. Zadeh [32] in 1965 proposed FSs to demonstrate knowledge/analysis with nonstatistical uncertainty. Many developments and generalizations in FS theory have been made in the last few years; for further information, please see [19, 31, 33–38] and references therein. Across chemistry, biology, technology, mathematical analysis, machine intelligence, mechanical theory, and several other subjects, FS theory has a wide range of applications. In the study of mathematical analysis, FP results offer optimum conditions for simulating the solutions of linear and nonlinear operator equations. In 1922, Banach [14] proposed and demonstrated a theorem that guaranteed the existence and uniqueness of a FP in a CMS X of the selfmap f on X with contractive condition \(d(f\mu , f\nu )\leq \alpha \,d(\mu , \nu )\), where \(\alpha \in (0,1) \). This result is known as Banach’s FP theorem. By introducing the concept of fuzzy contraction mappings in association with the \(d_{\infty} \)metric for FSs, Heilpern [22] provided a fuzzy extension of the Banach [14] and Nadler [29] FP theorems. Following this conclusion, several authors (e.g., [4, 7–13, 25, 26]) generalized it and investigated the presence of (common) FPs of fuzzy approximate quantityvalued mappings meeting contractive class conditions on metriclinear spaces.
Branciari [16] introduced FPs of mappings that satisfy integraltype contractive conditions. Namely, given a MS \((X, d)\), Branciari considered a selfmapping T on X satisfying the contractive criteria of the form
for all \(x,y \in X\), where \(\lambda \in (0,1)\) and \(\Delta : [0,\infty ) \rightarrow [0,\infty )\) is a Lebesgue integrable function and is summable on every compact subset of \([0,\infty )\) and satisfies \(\int _{0}^{\epsilon}\Delta (t)\,dt > 0\), for all \(\epsilon > 0\).
This paper is organized as follows: In Sect. 2, some fundamental notions are reviewed, including FM, fuzzy FP, the Hausdorff metric, Θcontraction, etc. In Sect. 3, the existence of common fuzzy FPs of fuzzy functions for Θcontractions in connection with integraltype contractions are established. Moreover, a significant example is constructed for the validity of the results. Section 4 gives an application of our research work. In Sect. 5, some concluding remarks and future directions are given.
Preliminaries
This section recalls some fundamental notions, like fuzzy set (FS), fuzzy mapping (FM), fuzzy fixed point (FFP), fuzzy coincidence point, the Hausdorff metric, Θcontraction, etc. Let \((X,d)\) be a metric space (MS). Let \(\operatorname{CB}(X)\) be the collection of all closed and bounded subsets of X.
Definition 2.1
([15])
For \(M,N\in \operatorname{CB}(X) \), take
The Hausdorff metric H on \(\operatorname{CB} ( X ) \) induced by d is defined by
Let Π be the class of functions \(\Delta : [0,\infty ) \rightarrow [0,\infty )\) so that:

(i)
Δ is Lebesgue integrable and summable on each compact subset of \([0,\infty )\);

(ii)
\(\int _{0}^{\tau}\Delta (\upsilon )\,d\upsilon >0\), for each \(\tau > 0\).
Definition 2.2
([32])
Let X be a nonempty set. A fuzzy set P in X is characterized by a membership (characteristic) function \(f_{P} (x)\) that associates with each point in X, a real number in \([0, 1]\). Let \(\mathcal{F}(X)\) be the family of all FSs in X. If P is a FS and \(x\in X\), then the functional values \(P(x) \) are called the grade of membership of x in P.
Definition 2.3
([22])
The αlevel set of a FS P in X, denoted by \([ P ] _{\alpha }\), is defined by
Here, N̅ denotes the closure of N. For a subset P of X, the characteristic function of P is denoted by \(\chi _{P}\).
A FS P in a metriclinear space V is said to be an approximate quantity if and only if \([ P ] _{\alpha}\) is compact and convex in V for each \(\alpha \in (0,1] \) and \(\sup_{x\in V}P(x)=1\).
Some subcollections of \(\mathcal{F}_{L}(X)\) and \(\mathcal{F}_{L}(V)\) are defined as follows:
For \(P,B\in \mathcal{F}(X)\), \(P\subset B \) means \(P(x)\leq B(x)\) for each \(x\in X\). If there is \(\alpha \in (0,1]\) so that \([ P ] _{\alpha }, [ B ] _{\alpha }\in \operatorname{CB} ( X ) \), then define
Let \(\alpha \in (0,1]\). If \([ P ] _{\alpha }, [ B ] _{\alpha }\in \operatorname{CB} ( X ) \), define \(p,d_{\infty}:\mathfrak{C} ( X ) \times \mathfrak{C} ( X ) \rightarrow \mathbb{R}\) (induced by the Hausdorff metric H) by
Definition 2.4
([10])
For an arbitrary set W and a metric space X, a FM is a function F from W into \(\mathcal{F}(X)\). A fuzzy mapping F is a fuzzy subset on \(W\times X \) with membership function \(F(x)(y)\). The functionalvalue \(F(x)(y) \) is the grade of membership of y in \(F(x)\). The family of all mappings from W into \(\mathcal{F}(X) \) is denoted by \(( \mathcal{F}(X) ) ^{W}\).
Definition 2.5
([10])
An αfuzzy FP of FM S defined on a MS \((X, d)\) is an element \(u\in X\) so that \(u\in [Su]_{\beta}\), for some \(\beta \in (0,1]\).
Definition 2.6
([10])
A common αfuzzy FP of two FMs \(F, T:W\rightarrow \mathcal{F}(X)\), is a point \(u\in W\) if \(u\in [Fu]_{\alpha}\cap [Tu]_{\alpha}\).
Definition 2.7
([24])
Consider a mapping Θ from \((0,\infty )\) to \((1,\infty )\) so that:
 (\(\Theta _{1}\)):

Θ is nondecreasing;
 (\(\Theta _{2}\)):

for any sequence \(\{\gamma _{n}\}\subset (0,\infty )\), \(\lim_{n\rightarrow \infty} \Theta (\gamma _{n})=1\) iff
$$ \lim_{n\rightarrow \infty}\gamma _{n}=0^{+}; $$  (\(\Theta _{3}\)):

there are \(u \in (0, 1)\) and \(0 < l < \infty \) so that \(\lim_{\gamma \rightarrow 0^{+}} \frac{\Theta (\gamma )1}{\gamma ^{u}}=l\).
A function \(T:X\rightarrow X\) is known as a Θcontraction if there are Θ that satisfies \((\Theta _{1})(\Theta _{3})\) and a number k between 0 and 1 so that for all \(x, y\in X\),
Theorem 2.1
([24])
Let \((X, d)\) be a CMS and \(T:X\rightarrow X\) be a Θcontraction, then T has a unique FP.
In 2017, Hancer et al. [21] added the following general condition \((\Theta _{4})\):
 (\(\Theta _{4}\)):

\(\Theta (\inf J) = \inf \Theta (J)\), where \(J\subset (0,\infty )\) with \(\inf (J) > 0\).
The set of all continuous functions Θ satisfying \((\Theta _{1})(\Theta _{4})\) is denoted by Ξ.
For further study on Θcontractions, please see [2, 5, 23, 27, 30].
Lemma 2.1
([29])
For a MS X, let M and N be nonempty and belong to \(\operatorname{CB}(X)\). If \(m\in M\), then \(d ( m,N ) \leqslant H ( M,N ) \).
Lemma 2.2
([22])
For a complete metriclinear space \((V,d_{v}) \) and a FM \(T:V\longrightarrow \mathcal{W}(V)\), let \(x_{0}\in V\). Then, there is \(x_{1}\in X\) such that \(\chi _{\{x_{1}\}}\subset T(x_{0})\).
Lemma 2.3
([28])
If \(\{\rho _{n}\}\) is a sequence in \([0,\infty )\) and \(\varphi \in \Phi \), then \(\lim_{n\rightarrow \infty}\int _{0}^{\rho _{n}}\varphi (\upsilon )\,d\upsilon =0\) if and only if \(\rho _{n}\rightarrow 0\) as \(n\rightarrow \infty \).
Integraltype fuzzy fixedpoint theorems
In this chapter, an integraltype contraction condition is used to establish some common fuzzy FPs of FSvalued mappings involving Θcontractions in a MS.
Theorem 3.1
Let \((\Upsilon , d)\) be a CMS and \(\Phi , \Psi :\Upsilon \rightarrow \mathcal{F}(\Upsilon )\) be two FMs. Suppose for each \(\mu \in \Upsilon \), there exist \(\alpha _{\Phi (\mu )}\), \(\alpha _{\Psi (\mu )}\in (0,1]\) such that \([\Phi \mu ]_{\alpha _{\Phi (\mu )}}\) and \([\Psi \mu ]_{\alpha _{\Psi (\mu )}}\) are nonempty, and belong to \(\operatorname{CB}(\Upsilon )\). Assume that there are \(\Theta \in \Xi \), \(\Delta \in \Pi \) and \(k\in (0,1)\) such that
for all \(\mu , \nu \in \Upsilon \) with \(H ([\Phi \mu ]_{\alpha _{\Phi (\mu )}}, [\Psi \nu ]_{\alpha _{ \Psi (\nu )}} ) > 0\). Then, there is some \(z\in \Upsilon \) such that \(z\in [\Phi z]_{\alpha _{\Phi (z)}}\cap [\Psi z]_{\alpha _{\Psi (z)}}\).
Proof
Let \(\mu _{0}\in \Upsilon \) be arbitrary. By hypothesis, there is \(\alpha _{\Phi (\mu _{0})}\in (0,1]\) so that \([\Phi \mu _{0}]_{\alpha _{\Phi (\mu _{0})}}\) is a nonempty, bounded, and closed subset of ϒ. Take \(\alpha _{\Phi (\mu _{0})}=\alpha _{1}\). Let \(\mu _{1}\in [\Phi \mu _{0}]_{\alpha _{\Phi (\mu _{0})}}\). For this \(\mu _{1}\), there is \(\alpha _{\Psi (\mu _{1})}\in (0,1]\) so that \([\Psi \mu _{1}]_{\alpha _{\Psi (\mu _{1})}}\) is a nonempty, bounded, and closed subset of ϒ. Due to Lemma 2.1,
From \((\Theta _{1})\), (2), and (3), we obtain
From \((\Theta _{4})\), we have
Thus,
Now, from (4), there is \(\mu _{2}\in [\Psi \mu _{1}]_{\alpha _{\Psi (\mu _{1})}}\) such that
For this \(\mu _{2}\) there is \(\alpha _{\Phi (\mu _{2})}\in (0,1]\) such that \([\Phi \mu _{2}]_{\alpha _{\Phi (\mu _{2})}}\) is a nonempty, bounded, and closed subset of ϒ. Due to Lemma 2.1,
From \((\Theta _{1})\), (2) and (6), we obtain
From \((\Theta _{4})\), we have
Thus,
Now, from (7), there is \(\mu _{3}\in [\Phi \mu _{2}]_{\alpha _{\Phi (\mu _{2})}}\) so that
Continuing this process, we generate a sequence \(\{\mu _{n}\}\) in ϒ so that
and
with
and
Combining (9) and (10), one writes
which further implies that
Since \(\Theta \in \Xi \), we have at the limit \(n\rightarrow \infty \),
Thus,
by \((\Theta _{2})\). In view of \((\Theta _{3})\), there are \(q\in (0,1)\) and \(l\in (0,\infty ]\) so that
Case 1. Let \(l < \infty \) and \(\frac{l}{2} = C > 0\). Hence, there is \(n_{0}\in \mathbb{N}\) so that for all \(n > n_{0}\),
That is,
Then,
where \(D = \frac{1}{C}\).
Case 2. Suppose \(l = \infty \). Let \(C>0\) be a real. Easily, there is \(n_{0}\in \mathbb{N}\) so that
for all \(n > n_{0}\). This implies that
where \(D = \frac{1}{C}\). In both cases, there are \(D > 0\) and \(n_{0} \in \mathbb{N}\) so that for all \(n > n_{0}\),
Now, we have
As \(n\rightarrow \infty \), the above inequality yields that
Hence, there is an integer \(n_{1}\) so that for all \(n > n_{1}\),
This implies that
for all \(n > n_{1}\). Hence,
for all \(n > n_{1}\). Now, to prove that \(\{\mu _{n}\}\) is a Cauchy sequence, suppose \(m,n\in \mathbb{N}\) such that \(m > n > n_{1}\). We have
Since \(0< q<1\), the series \(\sum_{i=n}^{\infty}\int _{0}^{\frac{1}{i^{1/q}}} \Delta (t)\,dt\) converges. When \(n,m\rightarrow \infty \), we obtain \(d(\mu _{n}, \mu _{m}) \rightarrow 0\). Hence, \(\{\mu _{n}\}\) is a Cauchy sequence in \((\Upsilon , d)\). Since ϒ is complete, there is \(z\in \Upsilon \) so that \(\lim_{n\rightarrow \infty}\mu _{n}\rightarrow z\). Now, we will show that \(z\in [\Psi z]_{\alpha _{\Psi (z)}}\). On the contrary, suppose that \(z\notin [\Psi z]_{\alpha _{\Psi (z)}}\), then there are \(p \in \mathbb{N}\) and a sequence \(\{\mu _{n_{t}}\}\) of \(\{\mu _{n}\}\) such that \(d(\mu _{n_{t+1}}, [\Psi z]_{\alpha _{\Psi (z)}} ) > 0\) \(\forall n_{t} \geq p\). By using \((\Theta _{1})\) and Lemma 2.1, we have
Now, from (2) and (19), we have
Letting \(t\rightarrow \infty \), then by using the continuity of Θ, the above inequality implies that
That is,
Hence, \(z\in [\Psi z]_{\alpha _{\Psi (z)}}\). Similarly, \(z\in [\Phi z]_{\alpha _{\Phi (z)}}\). Thus, \(z\in [\Phi z]_{\alpha _{\Phi (z)}}\cap [\Psi z]_{\alpha _{\Psi (z)}} \). □
Example 3.1
Let \(\Upsilon = [0,\infty )\) and define \(d: \Upsilon \times \Upsilon \rightarrow \mathbb{R}_{+}\) by
Define two mappings \(\Phi , \Psi : \Upsilon \rightarrow \mathcal{F}(\Upsilon )\) for \(\alpha \in [0,1 ]\) as
and
The αlevel sets are
Consider \(\Theta (t) = 2^{\sqrt[k]{t}}\). Then, there is some \(k=\frac{1}{\sqrt{3}}\in (0,1 )\) such that
for all \(\mu , \nu \in \Upsilon \) with \(H ([\Phi \mu ]_{\alpha _{\Phi (\mu )}}, [\Psi \nu ]_{\alpha _{ \Psi (\nu )}} ) > 0\). Hence, Theorem 3.1 can be applied to find \(0\in \Upsilon \) such that \(0\in [\Phi 0]_{\alpha} \cap [\Psi 0]_{\alpha}\).
Corollary 3.1
Let \((\Upsilon , d)\) be a CMS and \(\Phi , \Psi :\Upsilon \rightarrow \mathcal{F}(\Upsilon )\) be two fuzzy maps. Suppose for each \(\mu \in \Upsilon \), there exist \(\alpha _{\Phi (\mu )}\), \(\alpha _{\Psi (\mu )}\in (0,1]\) such that \([\Phi \mu ]_{\alpha _{\Phi (\mu )}}\) and \([\Psi \mu ]_{\alpha _{\Psi (\mu )}}\) are nonempty, and belong to \(\operatorname{CB}(\Upsilon )\). Assume that there are \(\Theta \in \Xi \) and \(k\in (0,1)\) so that
for all \(\mu , \nu \in \Upsilon \) with \(H ([\Phi \mu ]_{\alpha _{\Phi (\mu )}}, [\Psi \nu ]_{\alpha _{ \Psi (\nu )}} ) > 0\). Then, there is \(z\in \Upsilon \) so that \(z\in [\Phi z]_{\alpha _{\Phi (z)}}\cap [\Psi z]_{\alpha _{\Psi (z)}}\).
Proof
By letting \(\Delta (t)\equiv 1\) in Theorem 3.1, we will obtain the required result. □
Theorem 3.2
Let \((\Upsilon , d)\) be a CMS and \(\Phi :\Upsilon \rightarrow \mathcal{F}(\Upsilon )\) be a FM. Suppose for each \(\mu \in \Upsilon \), there is \(\alpha _{\Phi (\mu )}\in (0,1]\) such that \([\Phi \mu ]_{\alpha _{\Phi (\mu )}}\) is nonempty, and belongs to \(\operatorname{CB}(\Upsilon )\). If there are \(\Theta \in \Xi \), \(\Delta \in \Pi \) and \(k\in (0,1)\) so that for all \(\mu , \nu \in \Upsilon \),
with \(H ([\Phi \mu ]_{\alpha _{\Phi (\mu )}}, [\Phi \nu ]_{\alpha _{ \Phi (\nu )}} ) > 0\), then there is \(z\in \Upsilon \) so that \(z\in [\Phi z]_{\alpha _{\Phi (z)}}\).
Corollary 3.2
Let \((\Upsilon , d)\) be a CMS and \(\Phi :\Upsilon \rightarrow \mathcal{F}(\Upsilon )\) be a FM. Suppose for each \(\mu \in \Upsilon \), there are \(\alpha _{\Phi (\mu )}\in (0,1]\) such that \([\Phi \mu ]_{\alpha _{\Phi (\mu )}}\) is nonempty, and belong to \(\operatorname{CB}(\Upsilon )\). If there are \(\Theta \in \Xi \) and \(k\in (0,1)\) so that
for all \(\mu , \nu \in \Upsilon \) with \(H ([\Phi \mu ]_{\alpha _{\Phi (\mu )}}, [\Phi \nu ]_{\alpha _{ \Phi (\nu )}} ) > 0\), then there is \(z\in \Upsilon \) so that \(z\in [\Phi z]_{\alpha _{\Phi (z)}}\).
Proof
Put \(\Delta (t)=1\) in Theorem 3.2 to obtain the required result. □
Now, we will establish common FP results.
Theorem 3.3
Let \((\Upsilon , d)\) be a CMS and \(A, B:\Upsilon \rightarrow \mathcal{CB}(\Upsilon )\) be two multivalued maps. Assume that there are \(\Theta \in \Xi \), \(\Delta \in \Pi \) and \(k\in (0,1)\) so that for all \(\mu , \nu \in \Upsilon \),
with \(H (A\mu , B\nu ) > 0\). Then, there is some \(z\in \Upsilon \) such that \(z\in Az\cap Bz\).
Proof
Consider \(\alpha :\Upsilon \rightarrow (0,1]\). Let \(\Phi ,\Psi :\Upsilon \rightarrow \mathcal{F}(\Upsilon )\) be two fuzzy maps defined by
and
Then,
and
Thus, Theorem 3.1 can be applied to obtain \(z\in \Upsilon \) so that
□
Corollary 3.3
Let \((\Upsilon , d)\) be a CMS and \(A, B:\Upsilon \rightarrow \mathcal{CB}(\Upsilon )\) be two multivalued maps. If there are \(\Theta \in \Xi \) and \(k\in (0,1)\) so that
for all \(\mu , \nu \in \Upsilon \) with \(H (A\mu , B\nu ) > 0\), then there is some \(z\in \Upsilon \) so that \(z\in Az\cap Bz\).
Proof
By considering \(\Delta (t) = 1\) in Theorem 3.3, we will obtain the required result. □
Corollary 3.4
Let \((\Upsilon , d)\) be a CMS and \(A:\Upsilon \rightarrow \mathcal{CB}(\Upsilon )\) be a multivalued map. If there are \(\Theta \in \Xi \), \(\Delta \in \Pi \) and \(k\in (0,1)\) so that
for all \(\mu , \nu \in \Upsilon \) with \(H (A\mu , A\nu ) > 0\), then there is \(z\in \Upsilon \) so that \(z\in Az\cap Bz\).
Theorem 3.4
Let \((\Upsilon , d)\) be a complete metriclinear space and \(\Phi , \Psi :\Upsilon \rightarrow \mathcal{F}(\Upsilon )\) be two fuzzy maps. If there are \(\Theta \in \Xi \), \(\Delta \in \Pi \) and \(k\in (0,1)\) so that
for all \(\mu , \nu \in \Upsilon \) with \(d_{\infty} (\Phi (\mu ), \Psi (\nu ) ) > 0\), then there is \(z\in \Upsilon \) so that \(\{z\}\subset \Phi (z)\) and \(\{z\}\subset \Psi (z) \).
Proof
Consider \(\mu \in \Upsilon \), Lemma 2.2 implies that there is some \(\nu \in \Upsilon \) such that \(\nu \in [\Phi \mu ]_{1}\). Also, we can find \(w\in \Upsilon \) such that \(w\in [\Psi \mu ]_{1}\). Hence, for each \(\mu \in \Upsilon \), \([\Phi \mu ]_{\alpha (\mu )}\) and \([\Psi \mu ]_{\alpha (\mu )}\) are nonempty, and belong to \(\operatorname{CB}(\Upsilon )\). Since \(\alpha (\mu ) = \alpha (\nu ) = 1\), one writes
∀ \(\mu ,\nu \in \Upsilon \). Since Θ is nondecreasing, one obtains
for all \(\mu ,\nu \in \Upsilon \). This implies that
for all \(\mu ,\nu \in \Upsilon \). Now, since \([\Phi \mu ]_{1} \subseteq [\Phi \mu ]_{\alpha}\) for any \(\alpha \in (0,1]\), and so \(d (\mu , [\Phi \mu ]_{\alpha} ) \leq d (\mu , [ \Phi \mu ]_{1} )\), for every \(\alpha \in (0,1]\). Thus, we have \(p (\mu , \Phi (\mu ) ) \leq d (\mu , [\Phi \mu ]_{1} )\). Similarly, \(p (\mu , \Psi (\mu ) ) \leq d (\mu , [\Psi \mu ]_{1} )\).
Moreover,
Due to Theorem 3.1, we obtain \(z\in \Upsilon \), so that \(z\in [\Phi z]_{1}\cap [\Psi z]_{1}\), i.e., \(\{z\}\subset \Phi (z) \text{and} \{z\}\subset \Psi (z)\). □
Here, we consider that Ψ̂ is the setvalued mapping induced from FM \(\Psi :\Upsilon \rightarrow \mathcal{F}(\Upsilon )\), i.e.,
Corollary 3.5
Let \((\Upsilon , d)\) be a CMS. Consider two fuzzy maps \(\Phi , \Psi :\Upsilon \rightarrow \mathcal{F}(\Upsilon )\) such that ∀ \(\mu \in \Upsilon \), \(\widehat{\Phi}(\mu )\), \(\widehat{\Psi}(\mu )\) are nonempty, and belong to \(\operatorname{CB}(\Upsilon )\). Suppose there are \(k\in (0,1)\), \(\Delta \in \Pi \) and \(\Theta \in \Xi \) such that
for all \(\mu ,\nu \in \Upsilon \) with \(H (\widehat{\Phi}(\mu ), \widehat{\Psi}(\nu ) ) > 0\). Then, there is a point \(u\in \Upsilon \) such that \(\Phi (u)(u) \geq \Phi (u)(\mu )\) and \(\Psi (u)(u) \geq \Psi (u)(\mu )\) \(\forall \mu \in \Upsilon \).
Proof
From Theorem 3.3, we obtain \(u\in \Upsilon \) such that \(u\in \widehat{\Phi}(u)\cap \widehat{\Psi}(u)\). Then, by Lemma 2.2, we have \(\Phi (u)(u) \geq \Phi (u)(\mu )\) and \(\Psi (u)(u) \geq \Psi (u)(\mu )\) \(\forall \mu \in \Upsilon \). □
Application to stochastic Volterra integral equations
Stochastic integral equations arise in nearly every field of science and engineering. In recent time, researchers are becoming more interested in developing and unifying the concepts of probability theory and functional analysis, thereby establishing a variety of methods for studying the existence of solutions of integrodifferential equations (e.g., see [1, 3, 6]). However, problems abound that can be solved more effectively by the use of FS techniques than by classical probabilitybased methods [18, 32]. In continuation of this development, in this section, we investigate the existence of a common solution of a system of stochastic Volterra integral equations by using the idea of fuzzy maps.
With respect to our main objective here, a note on notations is in order. The stochastic integral equations and the notations are recorded randomly from [3, 17, 20] as follows. Denote by \((\Omega , \mathfrak{A}, \mathfrak{P} )\), a probability measure space, where Ω is a nonempty set, \(\mathfrak{A}\) is a σalgebra of subsets of Ω, and \(\mathfrak{P}\) is a complete probability measure on \(\mathfrak{A}\). Let \(\mathbb{R}_{+}=[0, \infty )\). The space of all continuous and bounded functions on \(\mathbb{R}_{+}\) with values in \(L_{2}:=L_{2}(\Omega , \mathfrak{A}, \mathfrak{P})\) is represented by \(C:= C (\mathbb{R}_{+}, L_{2}(\Omega , \mathfrak{A}, \mathfrak{P}) )\). We shall study the existence condition for a solution of the following system of Volterra stochastic differential equations:
where \(t\geq 0\) and (i) ω is a point of Ω, (ii) \(h(t;\omega )\) is called the stochastic free term defined for \(t\geq 0\), (iii) \(\mu (t;\omega )\) is the unknown stochastic variable for each \(t\geq 0\), (iv) \(k_{1}\) and \(k_{2}\) are stochastic kernels defined for \(0\leq \zeta \leq t<\infty \), (v) \(f(t, \mu )\) is a scalar function defined for \(t\geq 0\). By a random solution \(\mu (t;\omega )\) of the stochastic integral equations (27) and (28), we mean a function \(\mu (t;\omega )\) that belongs to \(C (\mathbb{R}_{+}, L_{2} (\Omega , \mathfrak{A}, \mathfrak{P} ) )\) and satisfies the equations a.e.
Theorem 4.1
Consider the system of Volterra stochastic integral equations (27) and (28). Assume that the following conditions hold:

(i)
\(f:C\longrightarrow C\), \(h:\mathbb{R}_{+}\longrightarrow L_{2}\), \(k_{1}, k_{2}:\mathbb{R}_{+}\times \mathbb{R}_{+}\times L_{2} \longrightarrow L_{2}\) are continuous;

(ii)
\(\f(t, \mu (t;\omega ))f(t, \nu (t;\omega ))\\leq \\mu (t;\omega ) \nu (t;\omega )\\), where \(\mu (t;\omega ), \nu (t;\omega )\in C\);

(iii)
There exist \(\eta > 1\), \(K>0\), \(\lambda \geq 0\) and a nondecreasing function
\(\Theta : (0, \infty )\longrightarrow (1, \infty )\) satisfying
$$ \int _{0}^{\Theta (\lambda K\\mu \nu \)}\varphi (t)\,dt\leq \int _{0}^{[ \Theta (\\mu \nu \)]^{\frac{1}{\eta}}}\varphi (t)\,dt,$$provided \(\lambda < \frac{1}{K}\) and \(\varphi \in \Phi \), where \(K:=\k_{1}(t, \zeta ,\omega )k_{2}(t,\zeta ,\omega )\\).
Then, there exists a common random solution of equations (27) and (28) in C.
Proof
Let \(\Upsilon = C:=C (\mathbb{R}_{+}, L_{2}(\Omega , \mathfrak{A}, \mathfrak{P}) )\) be endowed with the uniform norm \(\.\\). Then, \((\Upsilon , \.\)\) is a Banach space. Assume that \(k_{1}\), \(k_{2}\) are such that \(P_{\mu}, Q_{\mu}\in \Upsilon \), where
Consider two arbitrary mappings \(M, N: \Upsilon \longrightarrow (0, 1]\) and a pair of FMs \(F, G:\Upsilon \longrightarrow \mathcal{F}(\Upsilon )\) defined as
If we take \(\alpha _{F(\mu )}= M(\mu )\) and \(\alpha _{G(\mu )}= N(\mu )\), then we obtain
Therefore,
Since Θ is nondecreasing, we obtain using condition (ii),
Consequently,
Thus, for \(k= \frac{1}{\eta}\in (0,1)\), all the conditions of Theorem 4.1 are satisfied to obtain \(z\in \Upsilon \) such that \(z\in [Fz]_{\alpha _{F}(z)}\cap [Gz]_{\alpha _{G(z)}}\), which corresponds to a common random solution of equations (27) and (28). □
Conclusion
FP theory plays an essential role in mathematics and applied sciences, such as mathematical modeling, optimization, economic theories and many more disciplines. Vagueness is an immense module in the life of an individual. To handle uncertainty in reallife problems, FS theory achieved a great success and popularity. Due to fuzzy techniques, outstanding results in science and technology are obtained that added an awesome modification in solving dailylife problems. In this paper, modern fuzzy techniques are applied in obtaining common FPs of two mutivalued mappings defined on a CMS. For this purpose, an integraltype Θcontraction is applied. In this way, we have generalized many useful and practical results in the existing literature. The latest and classic results are presented as direct and indirect consequences of our results. A nontrivial and stimulating example is erected for embellishment of our main result. Moreover, to show the strength and importance of the research work, as an application we have investigated the existence of a common solution of a system of stochastic Volterra integral equations by using the idea of fuzzy maps.
Availability of data and materials
The data used to support the findings of this study are available from the corresponding author upon request.
Abbreviations
 FP:

fixed point
 CMS:

complete metric space
 FS:

fuzzy set
 FM:

fuzzy mapping
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SK and MSS: Conceptualization, Methodology, Supervision, and Writing draft preparation. HA: Investigation, Formal Analysis, Investigation, Review and Validation. AM and TA: Investigation, Funding Acquisition and Validation. All authors have read and agreed with the final version of the manuscript.
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Kanwal, S., Shagari, M.S., Aydi, H. et al. Common fixedpoint results of fuzzy mappings and applications on stochastic Volterra integral equations. J Inequal Appl 2022, 110 (2022). https://doi.org/10.1186/s13660022028492
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DOI: https://doi.org/10.1186/s13660022028492
MSC
 46S40
 47H10
 54H25
Keywords
 Fixed point
 Hausdorff metric
 Fuzzy mapping
 Integral equation