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Approximate coincidence point and common fixed point results for a hybrid pair of mappings with constraints in partially ordered Menger PMspaces
Journal of Inequalities and Applications volume 2019, Article number: 254 (2019)
Abstract
We study an approximate coincidence point and a common fixed point problem for a hybrid pair of mappings with constraints in Menger PMspaces, and obtain some new results. We derive interesting consequences of the main results by using the properties of a Menger–Hausdorff metric, and analogous results based on graphs instead of partial orders can be similarly formulated. Moreover, we construct two examples to reveal that the main results are valid, and show that the main results can be used to explore the existence of solutions to a system of nonlinear integral equations.
Introduction
A statistical metric was defined by Menger and later revisited by other authors, which led further to the emergence of the definition of a probabilistic metric space [1]. Many efforts have been devoted to the study of fixed point and optimization problems in PMspaces since the formulation of PMspace theory [2,3,4,5,6,7,8]. Fixed point and related problems in the framework of different types of spaces equipped with a partial order have also been explored [9,10,11,12,13,14,15,16,17].
Let \((X, d, \preceq _{1}, \preceq _{2})\) be a partially ordered metric space, where ⪯_{1} and ⪯_{2} are two partial orders, and \(S,O,P,Q,R:X\rightarrow X\) be selfmappings. In [18], the authors raised the problem of seeking \(x\in X\) satisfying
In [18], the authors gave sufficient conditions for the existence of solutions to problem (1.1), in which the continuity of O, P, Q, and R is required. This requirement is weakened to the continuity of O and P (or Q and R) in [19], and the result in [18] is also extended under general contractive conditions by making use of a general class of functions. Recently, the main results of [18] and [19] were generalized to the framework of a Menger PMspace [20].
Let \((X, d, \preceq _{1}, \preceq _{2})\) be a partially ordered metric space, where ⪯_{1} and ⪯_{2} are two partial orders, \(S:X\rightarrow \mathcal{N}(X)\) be a setvalued mapping, where \(\mathcal{N}(X)\) denotes the collection of all nonempty subsets of X, and \(O,P,Q,R:X\rightarrow X\) be selfmappings. The authors in [21] investigated the following approximate fixed point problem for a setvalued mapping with constraints: seeking \(x\in X\) satisfying
The authors in [21] studied problem (1.2) and obtained an interesting result.
Let \((X,\mathfrak{F},\Delta ,\preceq _{1},\preceq _{2})\) be a partially ordered Menger PM space, where ⪯_{1} and ⪯_{2} are partial orders, \(S:X\rightarrow \mathcal{N}(X)\) be a setvalued mapping, and \(f,O,P,Q,R:X\rightarrow X\) be selfmappings. Now, consider the following two problems. The first one is to seek \(x\in X\) satisfying
The second one is to seek \(x\in X\) satisfying
In this paper, we investigate the abovementioned two problems, which are in fact an approximate coincidence point problem and a common fixed point problem for a hybrid pair of mappings (i.e., a singlevalued one and a setvalued one) with constraints in the framework of a partially ordered Menger PMspace, respectively. The rest of the paper is arranged as follows. In Sect. 2, we give some preliminaries. In Sect. 3, we provide an approximate coincidence point theorem and a common fixed point theorem for a hybrid pair of mappings in partially ordered Menger PMspaces and give two examples. In Sect. 4, we derive some consequent results of the theorems proved in Sect. 3. An application of the main results in discussing the solutions to a system of Volterra integral equations is presented in Sect. 5. Finally, we summarize the paper with some concluding remarks.
Preliminaries
A distribution function is a mapping \(F:\mathbb{R}\rightarrow \mathbb{R}^{+}\) satisfying nondecreasingness, leftcontinuity, \(\sup_{u\in \mathbb{R}}F(u)=1\), and \(\inf_{u\in \mathbb{R}}F(u)=0\). We denote by \(\mathfrak{D}\) the collection of all distribution functions and by γ the following special distribution function:
Definition 2.1
([3])
A triangular norm (a tnorm) is a mapping \(\Delta :[0,1]\times [0,1]\rightarrow [0,1]\) satisfying \(\Delta (x,1)=x\), \(\Delta (x,y)=\Delta (y,x)\), \(\Delta (x_{1},x_{2})\geq \Delta (y_{2},y_{2})\) for \(x_{1}\geq y_{1}\), \(x _{2}\geq y_{2}\) and \(\Delta (x,\Delta (y,z))=\Delta (\Delta (x,y),z)\).
Definition 2.2
([3])
A Menger probabilistic metric space (a Menger PMspace) is a triplet \((X, \mathfrak{F},\Delta )\), where X is a nonempty set, Δ is a tnorm, and \(\mathfrak{F}:X\times X \rightarrow \mathfrak{D}\) is a mapping satisfying (we rewrite \(\mathfrak{F}(a,b)\) as \(F_{a,b}\))
 (MPM1):

\(F_{a,b}(w)=\gamma (w)\) for all \(w\in \mathbb{R}\) if and only if \(a=b\);
 (MPM2):

\(F_{a,b}(w)=F_{b,a}(w)\) for all \(w\in \mathbb{R}\);
 (MPM3):

\(F_{a,b}(w+v)\geq \Delta (F_{a,c}(w),F_{c,b}(v))\) for all \(a,b,c\in X\) and \(w,v\geq 0\).
Note that if \(\sup_{0< u<1}\Delta (u,u)=1\), then \((X, \mathfrak{F}, \Delta )\) is a Hausdorff topological space in the \((\varepsilon ,\lambda )\)topology τ, i.e., the family of sets \(\{U_{a}(\varepsilon ,\lambda ):\varepsilon >0,\lambda \in (0,1]\}\) (\(a \in X\)) is a basis of neighborhoods of a point a in X for τ, where \(U_{a}(\varepsilon ,\lambda )=\{b\in X: F_{a,b}(\varepsilon )>1\lambda \}\). The concepts of τconvergence of a sequence, τCauchy sequence in \((X,\mathfrak{F}, \Delta )\), and the τcompleteness of \((X,\mathfrak{F}, \Delta )\) can thus be introduced with respect to this topology. For more details, please refer to [3].
Let \((X,d)\) be a metric space, \(CL(X)\) be the collection of all nonempty closed subsets of X, and H be the Hausdorff metric which is defined by
for any \(\varTheta ,\varXi \in CL(X)\), where \(d(a,\varTheta )=\inf_{b\in \varTheta }d(a,b)\).
Let \((X, \mathfrak{F}, \Delta )\) be a Menger PMspace, \(\mathcal{N}(X)\) be the collection of all nonempty subsets of X, and \(\mathcal{CL}(X)\) be the collection of all nonempty τclosed subsets of X. For any \(\varTheta ,\varXi \in \mathcal{N}(X)\), define
where \(\tilde{\mathfrak{F}}\) is called the Menger–Hausdorff metric.
Remark 2.1
([3])

(1)
Let \((X,d)\) be a metric space. Define
$$ \mathfrak{F}(a,b) (u)=F_{a,b}(u)=\gamma \bigl(ud(a,b) \bigr) \quad \text{for all } a,b\in X \text{ and } u\in \mathbb{R}. $$(2.1)Then \((X,\mathfrak{F},\Delta _{\min })\) is a Menger PMspace, where \(\Delta _{\min }\) is the tnorm defined by \(\Delta _{\min }(x,y)= \min \{x,y\}\) for all \(x,y\in [0,1]\). Furthermore, the completeness of \((X,d)\) implies the τcompleteness of \((X,\mathfrak{F}, \Delta _{\min })\);

(2)
Let \(H(\cdot ,\cdot )\) be the Hausdorff metric, and define
$$ \begin{aligned}[b] \tilde{\mathfrak{F}}(\varTheta ,\varXi ) (u)&= \tilde{F}_{\varTheta ,\varXi }(u)\\&= \gamma \bigl(uH(\varTheta ,\varXi )\bigr) \quad \text{for all } \varTheta ,\varXi \in \mathcal{CL}(X) \text{ and } u\in \mathbb{R}.\end{aligned} $$(2.2)Then \(\tilde{\mathfrak{F}}\) is the Menger–Hausdorff metric induced by \(\mathfrak{F}\). Moreover, the τcompleteness of \((\mathcal{CL}(X), \tilde{\mathfrak{F}},\Delta )\) follows from the τcompleteness of \((X,\mathfrak{F},\Delta )\) provided that \(\Delta \geq \Delta _{m}\), where \(\Delta _{m}(x,y)=\max \{x+y1,0\}\) for all \(x,y\in [0,1]\).
Lemma 2.1
([3])
Let \((X,\mathfrak{F},\Delta )\) be a Menger PMspace. For any \(\varTheta ,\varXi \in \mathcal{CL}(X)\) and any \(a,b\in X\), the following statements hold:

(i)
For any \(a\in \varTheta \), \(F_{a,\varXi }(u)\geq \tilde{F}_{\varTheta , \varXi }(u)\) for all \(u\geq 0\);

(ii)
\(F_{a,\varTheta }(w+v)\geq \Delta (F_{a,b}(w),F_{y,\varTheta }(v))\) for all \(w,v\geq 0\);

(iii)
\(F_{a,\varTheta }(w+v)\geq \Delta (F_{a,\varXi }(w),\tilde{F}_{\varTheta ,\varXi }(v))\) for all \(w,v\geq 0\).
The concept of Fregularity was given in [20] as follows.
Definition 2.3
([20])
Let \((X, \mathfrak{F}, \Delta , \preceq )\) be a partially ordered Menger PMspace. We say that ⪯ is Fregular, if for sequences \(\{c_{n}\},\{d _{n}\}\subset X\) and \((c,d)\in X\times X\), we have
For a hybrid pair of mappings \((f,S)\), where \(f:X\rightarrow X\) is a selfmapping and \(S:X\rightarrow \mathcal{N}(X)\) is a setvalued one, and \(\xi \in (0,1)\), we introduce the following quantity:
Note that \(J_{\xi }^{x}(f,S)\neq \emptyset \) for arbitrary \(x\in X\). Moreover, for selfmappings M, N and a partial order ⪯ on X, if \(Mfy\preceq Nfy\) for some \(fy\in J_{\xi }^{x}(f,S)\), then we write \(MSx\preceq NSx\).
The following definition generalizes the corresponding one in [21] in two aspects. On the one hand, the interpretation of \(QSx\preceq _{2} RSx\) relies on the set \(J_{\xi }^{x}(f,S)\) which involves the probabilistic metric. On the other hand, the quantity \(J_{\xi }^{x}(f,S)\) is defined with respect to two mappings instead of one.
Definition 2.4
Let X be a nonempty set equipped with partial orders ⪯_{1} and ⪯_{2}. Let \(f,O,P,Q,R:X \rightarrow X\) be selfmappings, \(S:X\rightarrow \mathcal{N}(X)\) be a setvalued one, and \(\xi \in (0,1)\). We say that the hybrid pair of mappings \((f,S)\) is ξ\((O,P,Q,R,\preceq _{1},\preceq _{2})\)stable if
For a selfmapping \(f:X\rightarrow X\) and a setvalued mapping \(S:X\rightarrow \mathcal{N}(X)\) defined on X, if \(fw\in Sw\) for \(w\in X\), then we say that w is a coincidence point of f and S, and if \(fw\in \overline{Sw}\), then we say that w is an approximate coincidence point of f and S. Furthermore, if \(w\in X\) satisfies \(w=fw\in Sw\), then we say that w is a common fixed point of f and S. We denote by \(C(f,S)\) the set of coincidence points of f and S and by \(F(f,S)\) the set of common fixed points of the two mappings.
Definition 2.5
Let \((X, \mathfrak{F}, \Delta )\) be a Menger PMspace, \(f:X\rightarrow X\) be a selfmapping, and \(S:X\rightarrow \mathcal{N}(X)\) be a setvalued one. S is called τclosed with respect to f if \(G(S)_{f}\) is a τclosed subset of \((X\times X, F^{*})\), where
for all \((a,b),(\mu ,\nu )\in X\times X\) and \(u>0\), and \(G(S)_{f}=\{(a,fb):a \in X, fb\in Sa\}\).
For the sake of brevity, S̅ shall denote the setvalued mapping satisfying \(\overline{S}a=\overline{Sa}\) for all \(a\in X\).
Main results
In this section, we shall present and prove the main results of this paper. We first list some assumptions.
 \((A_{1})\) :

If there exists a sequence \(\{a_{n}\}\subset X\), such that \(\{fa_{n}\}\) is τconvergent to fa for \(a\in X\), \(fa_{n} \in Sa_{n1}\) for all \(n\in \mathbb{N}\), and \(\lim_{n\rightarrow \infty }F_{fa_{n},Sa_{n}}(u)=1\) for all \(u>0\), then \(F_{fa,Sa}(u)=1\) for all \(u>0\).
 \((A_{2})\) :

The following implication holds for any \(b\in X\):
$$ fb\notin \overline{Sb}\quad \Rightarrow \quad \sup_{a\in X}\Delta \bigl(F_{fa,fb}(u),F _{fa,Sa}(u)\bigr)< 1 \quad \text{for all } u>0. $$  \((A_{3})\) :

S̅ is a τclosed setvalued mapping with respect to f.
The following two theorems are the main results of this paper.
Theorem 3.1
Let \((X,\mathfrak{F},\Delta ,\preceq _{1}, \preceq _{2})\) be a τcomplete partially ordered Menger PMspace, where ⪯_{1} and ⪯_{2} are partial orders on X and Δ is a continuous tnorm, \(f,O,P,Q,R:X\rightarrow X\) be selfmappings satisfying that \(f(X)\) is τclosed and O, P, Q, R are τcontinuous, and \(S:X\rightarrow \mathcal{N}(X)\) be a setvalued mapping. Furthermore, suppose that one of assumptions \((A_{1})\)–\((A _{3})\) holds, and the following hypotheses hold for some \(\xi \in (0,1)\):

(i)
⪯_{1} and ⪯_{2} are Fregular;

(ii)
\(Ofx_{0}\preceq _{1} Pfx_{0}\) for some \(x_{0}\in X\);

(iii)
\((f,S)\) is ξ\((O,P,Q,R,\preceq _{1},\preceq _{2})\)stable;

(iv)
\((f,S)\) is ξ\((Q,R,O,P,\preceq _{2},\preceq _{1})\)stable;

(v)
\(F_{fy,Sy}(u)\geq F_{fx,fy}(\frac{u}{\eta })\) for all \(x\in X\) and \(fy\in J_{\xi }^{x}(f,S)\) with \((Ofx\preceq _{1} Pfx\textit{ and }Qfy\preceq _{2} Rfy)\) or \((Ofy\preceq _{1} Pfy\textit{ and }Qfx\preceq _{2} Rfx)\) and \(u>0\), where \(\eta \in (0,\xi )\).
Then there exists at least one solution to problem (1.3).
Proof
By condition (ii), \(Ofx_{0}\preceq _{1} Pfx_{0}\) for some \(x_{0}\in X\). Using (iii) and Definition 2.4, we get \(QTx_{0} \preceq _{2} RTx_{0}\), which implies that there exists \(fx_{1}\in J _{\xi }^{x_{0}}(f,S)\) such that \(Qfx_{1}\preceq _{2} Rfx_{1}\). Utilizing (iv) and Definition 2.4, we get \(OTx_{1}\preceq _{1} PTx_{1}\), which implies that there exists \(fx_{2}\in J_{\xi }^{x_{1}}(f,S)\) such that \(Ofx_{2}\preceq _{2} Pfx_{2}\). Hence, we can inductively construct a sequence \(\{x_{n}\}\subset X\) satisfying
and
Noting that \(fx_{n+1}\in J_{\xi }^{x_{n}}(f,S)\), we get
It immediately follows from (3.2) and (3.3) that
It is easy to prove from (3.4) that \(\{fx_{n}\}\) is a τCauchy sequence. Therefore, by the τcompleteness of \((X, \mathfrak{F}, \Delta )\), there exists \(u\in X\) such that \(fx_{n}\stackrel{\tau }{\rightarrow } a(n\rightarrow \infty )\). By the τclosedness of \(f(X)\), we have \(a=fb\) for some \(b\in X\), i.e., \(fx_{n}\stackrel{\tau }{\rightarrow } fb(n \rightarrow \infty )\).
Since O, P, Q, R are τcontinuous, by (3.1) and (i), we obtain
and
Now, if there exists a subsequence \(\{x_{n_{l}}\}\) of \(\{x_{n}\}\) satisfying that \(Sx_{n_{l}}=Sb\) for all \(l\in \mathbb{N}\). Note that
Thus, it follows from \(fx_{n_{l}}\in Sb\) and \(fx_{n}\stackrel{ \tau }{\rightarrow } fb(n\rightarrow \infty )\) that \(fb\in \overline{Sb}\). Suppose that this is not true, then \(Sx_{n}\neq Sb\) for all \(n\in \mathbb{N}\). Using (3.2) and the fact that \(fx_{n+1}\in J_{\xi }^{x_{n}}(f,S)\), we get
Hence, we get \(\lim_{n\rightarrow \infty }F_{fx_{n},Sx_{n}}(u)=1\) for all \(u>0\) by letting \(n\rightarrow \infty \). Now, we proceed the proof by considering the following three cases.
Case 1. Suppose that assumption \((A_{1})\) holds. By the inductive construction of \(\{x_{n}\}\) and \((A_{1})\), we have \(F_{fb,Sb}(u)=1\) for all \(u>0\). So \(fb\in \overline{Sb}\).
Case 2. Suppose that assumption \((A_{2})\) holds. Now, suppose \(fb\notin \overline{Sb}\). Then we have
which cannot be true. Therefore, \(fb\in \overline{Sb}\).
Case 3. Suppose that assumption \((A_{3})\) holds. Noting that Δ is continuous, by Definition 2.5, we obtain
It follows from the above assertion that \((b,b)\in G(\overline{S})_{f}\), that is,
Combining (3.5), (3.6), and (3.7), we conclude that b is a solution to problem (1.3). □
We shall further study the existence of solutions to problem (1.4).
Theorem 3.2
If the range of the mapping S is assumed to be \(\mathcal{CL}(X)\) rather than \(\mathcal{N}(X)\), and O, P (or Q, R) are τcontinuous, while the other conditions are the same as the ones in Theorem 3.1, then there exists \(b\in X\) such that \(fb\in Sb\) with constraints \(Ofb\preceq _{1} Pfb\) and \(Qfb\preceq _{2} Rfb\). Furthermore, if we assume that \(ffx=fx\) for \(x\in C(f,S)\), then there exists at least one solution to problem (1.4).
Proof
We first assume that O, P are τcontinuous. Then, following the proof in Theorem 3.1, we can get Eqs. (3.5) and (3.7). Since Sb is τclosed now, we get
Now, we prove (3.6) holds. In fact, by (iv) and (3.5), we obtain
which implies that there exists \(fy\in J_{\xi }^{b}(f,S)\) such that \(Qy\preceq _{2} Ry\). Noting that \(fb\in Sb\), we can easily verify that
Therefore, we deduce that (3.6) holds. If Q and R are τcontinuous, the conclusion can be proved by using the same method. □
We next prove the second part of the theorem. From (3.8), we get \(b\in C(f,S)\) and so \(a=fb=ffb=fa\in Sb\). Since \(J_{\xi }^{b}(f,S)=\{fb \}\), we have \(fa\in J_{\xi }^{b}(f,S)\). Noting that (3.5) and (3.6) hold, it follows from condition (v) of Theorem 3.1 that
Therefore, we have \(fa\in Sa\), and thus \(a=fa\in Sa\). Also, by (3.5) and (3.6), we have \(Oa\preceq _{1} Pa\) and \(Qa\preceq _{1} Ra\). Hence, a is a solution to problem (1.4).
Now, we give two examples to show that the above two theorems are valid.
Example 3.1
Let \(X=\{2+\frac{1}{4^{n}}:n\in \mathbb{N}\} \cup \{2,3\}\), and the following partial order ⪯ is imposed on X: \(x\preceq y\) iff \(x\leq y\) for \(x,y\in X\). Define \(f,O,P,Q,R:X \rightarrow X\) and \(S:X\rightarrow \mathcal{CL}(X)\) as follows:
and
Let \(\mathfrak{F}\) be defined by (2.1), where \(d(x,y)= \vert xy \vert \). Then \((X, \mathfrak{F}, \Delta _{\min })\) is τcomplete. Direct calculation shows that
for all \(u>0\), and so for any \(\xi \in (\frac{1}{4},1)\), we have \(J_{\xi }^{2+\frac{1}{4^{n}}}(f,S)=\{2+\frac{1}{4^{n+1}}\}\) for all \(n\in \mathbb{N}\), \(J_{\xi }^{2}(f,S)=\{2\}\) and \(J_{\xi }^{3}(f,S)= \{\frac{33}{16}\}\). Moreover, for each \(x\in X\) and \(fy\in J_{\xi } ^{x}(f,S)\), the inequality \(F_{fy,Sy}(u)\geq F_{fx,fy}(\frac{u}{ \eta })\) holds for \(\eta =\frac{1}{4}\in (0,\xi )\).
Noting that when \(Ofx\leq Pfx\), we have \(x\in \{2+\frac{1}{4^{n}}, n\in \mathbb{N}\}\), and there exists \(fy\in Sx\) such that \(F_{fx,Sx}(u)\leq F_{fx,fy}(\frac{u}{\xi })\) for all \(\xi \in ( \frac{1}{4},1)\) and \(u>0\), and \(Qfy\leq Rfy\), we observe that \((f,S)\) is ξ\((O,P,Q,R,\preceq ,\preceq )\)stable, where \(\xi \in (\frac{1}{4},1)\). Similarly, \((f,S)\) is ξ\((Q,R,O,P, \preceq ,\preceq )\)stable with \(\xi \in (\frac{1}{4},1)\). Besides, it is easy to check that assumption \((A_{1})\) holds. By Theorem 3.1, it is claimed that \(C(f,S)\neq \emptyset \). In this example, \(C(f,S)=\{2,3 \}\). Furthermore, it is obvious that \(ff2=f2\) and \(ff3\neq f3\), so by Theorem 3.2 \(F(f,S)\neq \emptyset \). Here, \(F(f,S)=\{2\}\).
Example 3.2
Let \(X=[0,\frac{\pi }{2}]\), and the partial order ⪯ is defined in the same way as in Example 3.1. Define \(O,P,Q,R:X\rightarrow X\) and \(S:X\rightarrow \mathcal{CL}(X)\) as follows:
and
Let \(\mathfrak{F}\) be defined as the same one in Example 3.1. Then \((X, \mathfrak{F}, \Delta _{\min })\) is τcomplete. By direct calculation, we get
for all \(u>0\), and so for all \(\xi \in (\frac{1}{2},1)\), we have \(J_{\xi }^{x}(f,S)=[(\frac{1}{2}\frac{1}{4\xi })x,\frac{x}{4}]\) for \(x\in [0,\frac{\pi }{2})\) and \(J_{\xi }^{\frac{\pi }{2}}(f,S)=\{\frac{ \pi }{2}\}\). Moreover, for \(x\in X\) and \(fy\in J_{\xi }^{x}(f,S)\), the inequality \(F_{fy,Sy}(u)\geq F_{fx,fy}(\frac{u}{\eta })\) holds for \(\eta =\frac{1}{2}\in (0,\xi )\) and \(u>0\).
We can easily check that \((f,S)\) are ξ\((O,P,Q,R,\preceq ,\preceq )\)stable and ξ\((Q,R,O,P,\preceq ,\preceq )\)stable for \(\xi \in (\frac{1}{2},1)\). Besides, it is obvious that assumption \((A_{1})\) holds. It can be seen that all the hypotheses of Theorem 3.1 and Theorem 3.2 hold. So we claim that \(F(f,S)\neq \emptyset \). In this example, \(F(f,S)=\{0,\frac{\pi }{2}\}\).
Consequent results
In this section, we give some results as consequences of Theorem 3.1 and Theorem 3.2. First, if we use another contraction condition using Menger–Hausdorff metric, we can get the following result which can be viewed as a corollary of Theorem 3.1.
Corollary 4.1
Let \((X,\mathfrak{F},\Delta ,\preceq _{1}, \preceq _{2})\) be a τcomplete partially ordered Menger PMspace, where ⪯_{1} and ⪯_{2} are partial orders on X and Δ is a continuous tnorm, \(f,O,P,Q,R:X\rightarrow X\) be selfmappings satisfying \(f(X)\) is τclosed and O, P, Q, R are τcontinuous, and \(S:X\rightarrow \mathcal{N}(X)\) be a setvalued mapping. Furthermore, suppose that the following hypotheses hold for some \(\xi \in (0,1)\):

(i)
⪯_{1} and ⪯_{2} are Fregular;

(ii)
\(Ofx_{0}\preceq _{1} Pfx_{0}\) for some \(x_{0}\in X\);

(iii)
\((f,S)\) is ξ\((O,P,Q,R,\preceq _{1},\preceq _{2})\)stable;

(iv)
\((f,S)\) is ξ\((Q,R,O,P,\preceq _{2},\preceq _{1})\)stable;

(v)
\(\tilde{F}_{Sx,Sy}(u)\geq F_{fx,fy}(\frac{u}{\eta })\) for all \(x,y\in X\) and \(u>0\), where \(\eta \in (0,\xi )\).
Then there exists at least one solution to problem (1.3).
Proof
First, by (v) and Lemma 2.1 (i), for all \(x\in X\) and \(fy\in J_{\xi }^{x}(f,S)\) and \(u>0\), we have
Thus, conditions (i)–(v) of Theorem 3.1 are satisfied. We next show that assumption \((A_{1})\) holds. In fact, combining the definition of Menger–Hausdorff metric and Lemma 2.1 (ii) and (iii), for all \(x,y\in X\) and \(u>0\), we get
Taking \(y=x_{n}\) yields that
We deduce that assumption \((A_{1})\) holds by letting \(n\rightarrow \infty \). Thus, it follows from Theorem 3.1 that the conclusion holds. □
Corollary 4.2
If the range of the mapping S is assumed to be \(\mathcal{CL}(X)\) rather than \(\mathcal{N}(X)\), and O, P (or Q, R) are τcontinuous, while the other conditions are the same as the ones in Corollary 4.1, then there exist \(b\in X\) such that \(fb\in Sb\) with constraints \(Ofb\preceq _{1} Pfb\) and \(Qfb\preceq _{2} Rfb\). Furthermore, if we assume that \(ffx=fx\) for \(x\in C(f,S)\), then there exists at least one solution to problem (1.4).
We can also derive some other consequent results by posing restrictions on the mappings or the partial orders from Corollary 4.1 and Corollary 4.2. For the sake of brevity, we omit them here.
Remark 4.1
Example 3.1 and Example 3.2 cannot be used to illustrate Corollary 4.1, since in Example 3.1,
and in Example 3.2, it holds that \(\tilde{F}_{S(0),S(\frac{\pi }{2})}(u)= \gamma (u\frac{\pi }{2})=F_{f(0),f(\frac{\pi }{2})}(u)\) for all \(u>0\).
By considering a graph instead of a partial order, one can establish analogous results as above. We first recall the concept of a directed graph.
Definition 4.1
([22])
An ordered pair \((X,E)\), where X is a nonempty set and \(E\subset X\times X\) is a binary relation, is called a directed graph G.
We next introduce the following definitions based on graphs.
Definition 4.2
Let \((X, \mathfrak{F}, \Delta , G)\) be a graphbased Menger PMspace, where \(G=(X,E)\) is a graph. The graph G is called \(F_{G}\)regular, if for sequences \(\{c_{n}\}, \{d_{n}\}\subset X\) and \((c,d)\in X\times X\), we have
Similarly, if there exists \(fy\in J_{\xi }^{x}(f,S)\) such that \((My,Ny)\in E\), where \(\xi \in (0,1)\), then we write \((MSx,NSx)\in E\).
Definition 4.3
Let X be a nonempty set with graphs \(G_{1}=(X,E_{1})\) and \(G_{2}=(X,E_{2})\) imposed on it, \(f,O,P,Q,R:X \rightarrow X\) be selfmappings, and \(S:X\rightarrow \mathcal{N}(X)\) be a setvalued mapping. The hybrid pair of mappings \((f,S)\) is called ξ\((O,P,Q,R,G_{1},G_{2})\)graphstable if
Based on the above definitions, we can prove the following two theorems using the same method in Theorem 3.1 and Theorem 3.2, and also derive corresponding corollaries.
Theorem 4.1
Let \((X, \mathfrak{F}, \Delta , G_{1}, G_{2})\) be a τcomplete graphbased Menger PMspace, where \(G_{1}\) and \(G_{2}\) are two graphs and Δ is a continuous tnorm, \(f,O,P,Q,R:X\rightarrow X\) be selfmappings satisfying \(f(X)\) is τclosed and O, P, Q, R are τcontinuous, and \(S:X\rightarrow \mathcal{N}(X)\) be a setvalued mapping. Suppose that one of assumptions \((A_{1})\)–\((A_{3})\) holds, and the following hypotheses hold for some \(\xi \in (0,1)\):

(i)
\(G_{1}\) and \(G_{2}\) are \(F_{G}\)regular;

(ii)
\((Ofx_{0},Pfx_{0})\in G_{1}\) for some \(x_{0}\in X\);

(iii)
\((f,S)\) is ξ\((O,P,Q,R,G_{1},G_{2})\)graphstable;

(iv)
\((f,S)\) is ξ\((Q,R,O,P,G_{2},G_{1})\)graphstable;

(v)
\(F_{fy,Sy}(u)\geq F_{fx,fy}(\frac{u}{\eta })\) for all \(x\in X\) and \(fy\in J_{\xi }^{x}(f,S)\) with \(((Ofx,Pfx)\in E_{1}\textit{ and }(Qfy, Rfy)\in E_{2})\) or \(((Ofy,Pfy)\in E_{1}\textit{ and }(Qfx, Rfx)\in E_{2})\) and \(u>0\), where \(\eta \in (0,\xi )\).
Then there exists \(a\in X\) such that \(fa\in \overline{Sa}\), \((Ofa,Pfa)\in E_{1}\) and \((Qfa,Rfa)\in E_{2}\).
Theorem 4.2
If the range of the mapping S is assumed to be \(\mathcal{CL}(X)\) rather than \(\mathcal{N}(X)\), and O, P (or Q, R) are τcontinuous, while the other conditions are the same as the ones in Corollary 4.1, then there exists \(v\in X\) such that \(fv\in Sv\) with constraints \(Ofv\preceq _{1} Pfv\) and \(Qfv\preceq _{2} Rfv\). Furthermore, if we assume that \(ffx=fx\) for \(x\in C(f,S)\), then there exists at least one solution to problem (1.4).
An application
In this section, we utilize the main results in Sect. 3 to investigate the existence of solutions for a system of nonlinear integral equations.
Let \(X=C([p,q],\mathbb{R})\), where \(C([p,q],\mathbb{R})\) denotes the space of all continuous functions on \([p,q]\) with \(q>p>0\), and impose the following norm on X:
Then \((X, \Vert \cdot \Vert _{1})\) is a Banach space.
Consider another norm
Note that the two norms \(\Vert \cdot \Vert _{1}\) and \(\Vert \cdot \Vert _{2}\) are equivalent (see [23]), which implies that \((X, \Vert \cdot \Vert _{2})\) is also a Banach space.
Define \(\mathfrak{F}: X\times X\rightarrow \mathfrak{D}\) by
and impose on X the following partial order:
Now, we consider the problem of the existence of solutions for the following system of Volterra integral equations:
for all \(u\in [p,q]\), where \(q>p>0\), \(h,x_{i}\in X\) and \(\varPhi _{i} \in C([p,q]\times [p,q]\times X,\mathbb{R})\).
Theorem 5.1
Let \((X,\mathfrak{F},\Delta _{\min })\) be the Menger PMspace induced by the Banach space X, where \(\mathfrak{F}\) is defined by (5.2), and X is equipped with the partial order ⪯ defined by (5.3), \(\varPhi _{i}\in C([p,q]\times [p,q] \times X,\mathbb{R})\), \(i=1,2\), \(\varphi _{1}\) and \(\varphi _{2}\) are two functionals on X, and the following hypotheses hold:

(i)
\(\Vert \varPhi _{j} \Vert _{\infty }=\sup_{u,v\in [p,q],x\in C([p,q],\mathbb{R})} \vert \varPhi _{i}(u,v,x(v)) \vert < \infty \) for \(j\in \{1,2\}\);

(ii)
\(\varphi _{1}(fx_{0})>0\) for some \(x_{0}\in X\);

(iii)
If \(\varphi _{1}(fx)>0\), then there exists \(y\in X\) such that \(\int _{0}^{u} \varPhi _{1}(u,v,y(v))\,dv=\int _{0}^{u} \varPhi _{2}(u,v,x(v))\,dv\) and \(\varphi _{2}(fy)>0\); if \(\varphi _{2}(fx)>0\), then there exists \(y\in X\) such that \(\int _{0}^{u} \varPhi _{1}(u,v,y(v))\,dv=\int _{0}^{u} \varPhi _{2}(u,v,x(v))\,dv\) and \(\varphi _{1}(fy)>0\);

(iv)
There exist \(k>0\) and \(0<\eta <\xi \) with \(\xi \in (0,1)\) such that, for all \(x,y\in X\) and all \(u,v\in [p,q]\), \(1e^{kq}\leq \eta \), and whenever \(x\in X\), \(\int _{0}^{u} \varPhi _{1}(u,v,y(v))\,dv=\int _{0}^{u} \varPhi _{2}(u,v,x(v))\,dv\) with \(\varphi _{1}(fx)>0\) and \(\varphi _{2}(fy)>0\), we have
$$ \bigl\vert \varPhi _{1}\bigl(u,v,y(v)\bigr)\varPhi _{2} \bigl(u,v,y(v)\bigr) \bigr\vert \leq k \bigl\vert fx(v)fy(v) \bigr\vert , $$where \(f, S: X\rightarrow X\) are defined by \(fx(u)=h(u)+\int _{0}^{u} \varPhi _{1}(u,v,x(v))\,dv\) and \(Sx(u)=h(u)+\int _{0}^{u} \varPhi _{2}(u,v,x(v))\,dv\) for all \(u\in [p,q]\), respectively;

(v)
There exists a sequence \(\{x_{n}\}\) in X such that \(\int _{0} ^{u} \varPhi _{2}(u,v,x_{n}(v))\,dv\rightarrow \int _{0}^{u} \varPhi _{2}(u,v,x(v))\,dv\) (\(n \rightarrow \infty \));

(vi)
If \(\int _{0}^{u} \varPhi _{1}(u,v,z(v))\,dv=\int _{0}^{u} \varPhi _{2}(u,v,z(v))\,dv\), then \(\int _{0}^{u} \varPhi _{1}(u,v,fz(v))\,dv=\int _{0}^{u} \varPhi _{1}(u,v,z(v))\,dv\).
Then (5.4) has a solution \(z'\) in X with constraints \(Az'\preceq Bz'\) and \(Cz'\preceq Dz'\).
Proof
It is obvious that \((X,\mathfrak{F},\Delta _{\min })\) is τcomplete and the partial order ⪯ is Fregular. Define \(O,P,Q,R:X\rightarrow X\) as follows:
where \(\vert x \vert \in X\) is the function defined by \(\vert x \vert (u)= \vert x(u) \vert \) for all \(u\in [p,q]\).
By condition (ii), we know that \(Afx_{0}\preceq Bfx_{0}\) for some \(x_{0}\in X\), and by (iii), \((f,S)\) are both ξ\((A,B,C,D,\preceq ,\preceq )\)stable and ξ\((C,D, A,B,\preceq ,\preceq )\)stable for \(\xi \in (0,1)\).
Consider the norm defined by (5.1), where k satisfies condition (iv). It follows from (iv) that for \(x\in X\), \(fy=Sx\) with \(Ofx\preceq Pfx\) and \(Qfy\preceq Rfy\), we obtain
which implies that
for some \(\eta \in (0,\xi )\) and any \(w>0\) by (5.2).
Moreover, it follows from condition (v) that assumption \((A_{1})\) holds. And by (vi), it holds that \(ffz=fz\) for \(z\in C(f,S)\). Therefore, all the hypotheses of Theorem 3.1 and Theorem 3.2 hold, and so there exists a point \(z'\in F(f,S)\) in X with constraints \(Az'\preceq Bz'\) and \(Cz'\preceq Dz'\), which means that the system of integral equations (5.4) has a solution \(z'\) with constraints \(Az'\preceq Bz'\) and \(Cz'\preceq Dz'\). □
Conclusions
In this paper, we have studied a common fixed point problem for a hybrid pair of mappings (a selfmapping and a setvalued mapping) under constraints in the framework of a Menger PMspace, by first investigating a related approximate coincidence point problem under certain constraints, and have derived some new results. It is worth noting that for the existence result of a solution to the common fixed point problem (1.4), the τcontinuity of two mappings O and P (or Q and S) is required rather than posing the τcontinuity on the four selfmappings. We have also constructed some examples and explored an application of the main results. The obtained results in this paper may shed some new light on the study of approximate coincidence point problems and common fixed point problems for a hybrid pair of mappings in the framework of Menger PMspaces.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant Nos. 11701259, 11461045, 11771198, 11661053, China Scholarship Council under Grant No. 201806825038 and the Natural Science Foundation of Jiangxi Province under Grant No. 20181BAB201003. This work was completed while Zhaoqi Wu was visiting MaxPlanckInstitute for Mathematics in the Sciences in Germany.
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MSC
 47H10
 46S50
 47S50
Keywords
 Menger PMspace
 Approximate coincidence point
 Common fixed point
 Partial order
 Setvalued mapping