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The study of fixed points for multivalued mappings in a Menger probabilistic metric space endowed with a graph
 Hajer Argoubi^{1},
 Mohamed Jleli^{2} and
 Bessem Samet^{2}Email author
https://doi.org/10.1186/s136630150361y
© Argoubi et al. 2015
 Received: 28 February 2015
 Accepted: 23 June 2015
 Published: 11 July 2015
Abstract
We study the existence of fixed points for multivalued mappings \(f: S \to S\), where \((S,F,T)\) is a complete Menger PMspace with a tnorm of Htype T and S is endowed with a directed graph \(G=(V(G),E(G))\) such that \(V(G)=S\) and \(\Delta= \{ (x,x): x \in S \} \subset E(G)\). The obtained results recover several existing fixed point theorems from the literature. As applications, we obtain a convergence result of successive approximations for certain nonlinear operators defined on a complete metric space. This last result allows us to establish a KeliskyRivlin type result for a class of modified qBernstein operators on the space \(C([0,1])\).
Keywords
 fixed point
 multivalued mapping
 Menger probabilistic metric, directed graph
 modified qBernstein operator
MSC
 47H10
 05C40
 06A06
1 Introduction
In recent years, many results related to metric fixed point theory in partially ordered sets have appeared. The first work in this direction was the 2004 paper of Ran and Reurings [1], where they established a fixed point result, which can be considered as a combination of two fundamental fixed point theorems: the Banach contraction principle and the KnasterTarski fixed point theorem. More precisely, Ran and Reurings considered a class of singlevalued mappings \(f : X \to X\), where \((X,d)\) is a complete metric space endowed with a certain partial order ⪯. The considered mappings are supposed to be continuous, monotone with respect to the partial order ⪯, and satisfying a Banach contraction inequality for every pair \((x,y) \in X \times X\) such that \(x \preceq y\). If for some \(x_{0} \in X\) we have \(x_{0} \preceq fx_{0}\), they proved that the Picard sequence \(\{f^{n} x_{0}\}\) converges to a fixed point of f. By combining this result with the Schauder fixed point theorem, Ran and Reurings obtained some existence and uniqueness results of positive definite solutions to some nonlinear matrix equations. Nieto and RodríguezLópez [2] extended the result of Ran and Reurings to singlevalued mappings that are not necessarily continuous. Under an additional assumption, that is, \(x_{n} \preceq x\) for all n, whenever \(\{x_{n}\}\) is an increasing sequence with respect to the partial order ⪯ and convergent to x, they proved that f has at least one fixed point. For other related results, we refer to [3–6] and references therein.
In [7], Jachymski presented an interesting concept in fixed point theory with some general structures by using the context of metric spaces endowed with a graph. He proved that it is possible to unify a large class of fixed point theorems including the previous cited results by considering singlevalued mappings satisfying a Banach contraction inequality for every pair \((x,y) \in X \times X\) such that \((x,y)\) is an edge of a certain directed graph G. He also presented a new proof of the Kelisky and Rivlin theorem [8] concerning Bernstein operators using a fixed point theorem for linear operators on a Banach space following from a fixed point theorem in a metric space with a graph.
Very recently, Dinevari and Frigon [9] extended some fixed point results of Jachymski [7] to multivalued mappings. They introduced the notions of multivalued Gcontractions and weak Gcontractions for which they established fixed point theorems. They also presented a comparison between fixed point sets obtained from Picard iterations starting from different points. For other related works, we refer to [10, 11] and references therein.
Recently, Kamran et al. [12] extended the results of Jachymski [7] to the setting of Menger probabilistic metric spaces. They introduced the class of probabilistic Gcontraction singlevalued mappings and studied the existence of fixed points for such mappings.
Our aim in this paper is to study the existence of fixed points for nonempty multivalued mappings defined on a complete Menger probabilistic metric space \((S,F,T)\), where T is a tnorm of Htype and S is a set endowed with a directed graph G.
The paper is organized as follows. In Section 2, we recall some basic concepts on Menger probabilistic metric spaces and fix some notations. In Section 3, we introduce the class of multivalued Gcontractions, and we study the existence of fixed points for such mappings. Some interesting consequences are derived from our main result in this section. In particular, we obtain existence results of fixed points for nonempty closed multivalued Gcontractions, a probabilistic version of the fixed point theorem for \((\varepsilon, \lambda)\)uniformly locally contractive multivalued maps due to Nadler, and many other results including also the case of singlevalued mappings. In Section 4, we introduce the class of multivalued weak Gcontractions, for which we study the existence of fixed points. Finally, in Section 5, we present an application to modified qBernstein polynomials. More precisely, we obtain a KeliskyRivlin type result for a class of modified qBernstein operators on the space \(C([0, 1])\).
2 Preliminaries and notations
The introduction of the general concept of statistical metric spaces is due to Karl Menger (1942), who dealt with probabilistic geometry. The new theory of fundamental probabilistic structures was developed later on by many authors. In this section, we start by recalling some basic concepts from Menger probabilistic metric spaces. For more details on such spaces, we refer to [13–16].
 (d1)
F is nondecreasing;
 (d2)
F is left continuous;
 (d3)
\(\inf_{t \in\mathbb{R} } F(t)=0\) and \(\sup_{t \in\mathbb{R}} F(t)=1\).
 (d4)
\(F(0) = 0\),
Definition 2.1
 (t1)
\(T(x,y)=T(y,x)\);
 (t2)
\(T(x, T(y, z))=T(T(x, y), z)\);
 (t3)
\(T(x, y) \leq T(x,z)\) if \(y \leq z\);
 (t4)
\(T(x, 1)=x\).
Definition 2.2
A trivial example of a tnorm of Htype is \(T_{M} = \mathrm{min}\), but there exist tnorms of Htype with \(T \neq T_{M} \) (see, e.g., [13]).
Definition 2.3
 (PM1)
\(F(x, y)=\delta_{0} \Leftrightarrow x=y\);
 (PM2)
\(F(x, y)=F(y, x)\);
 (PM3)
\(F(x, z)(t+s) \geq T(F(x, y)(t), F(y, z)(s))\) for all \(t,s \geq0 \).
Definition 2.4
 (i)
A sequence \(\{x_{n} \} \subset S\) converges to an element \(x \in S\) if for every \(\varepsilon> 0\) and \(\delta\in(0, 1]\), there exists \(N \in\mathbb{N}\) such that \(x_{n} \in N_{x}(\varepsilon ,\delta)\) for every \(n \geq N\).
 (ii)
A sequence \(\{x_{n} \} \subset S\) is a Cauchy sequence if for every \(\varepsilon> 0\) and \(\delta\in(0, 1]\), there exists \(N \in\mathbb{N}\) such that \(F(x_{n}, x_{m})(\varepsilon) > 1  \lambda\), whenever \(n,m \geq N\).
 (iii)
A Menger PMspace is complete if every Cauchy sequence in S converges to a point in S.
 (iv)
A subset A of S is closed if every convergent sequence in A converges to an element of A.
Lemma 2.5
 (i)
\(d_{\lambda}(x,y) < t\) if and only if \(F(x, y)(t) > 1  \lambda\);
 (ii)
\(d_{\lambda}(x, y) = 0\) for all \(\lambda\in(0, 1]\) if and only if \(x = y\);
 (iii)
\(d_{\lambda}(x, y) = d_{\lambda}(y, x)\) for all \(x, y \in S\);
 (iv)if T is of Htype, then for each \(\lambda\in(0, 1]\), there exists \(\mu\in(0, \lambda]\) such that for each \(m \in\mathbb{N}\),$$d_{\lambda}(x_{0}, x_{m}) \leq\sum _{i=1}^{m} d_{\mu}(x_{i1}, x_{i}) \quad \textit{for all } x_{0}, x_{1}, \ldots, x_{m} \in S. $$
Let \((S,F,T)\) be a Menger PMspace. Let G be a directed graph. The set of its vertices and the set of its edges are denoted by \(V (G)\) and \(E(G)\), respectively. We assume that \(S =V (G)\), Δ the diagonal in \(X \times X\) is contained in \(E(G)\), and G has no parallel edges. We identify G with the pair \((V (G),E(G))\).
We have the following properties.
Lemma 2.6
 (i)
\(p_{N}(x,y) \geq P_{N+m}(x,y)\) for every \(m,N \in\mathbb{N}\);
 (ii)
\(p(x,z)= \inf \{p_{k}(x,z): k \in\mathbb{N}, z \in [x]_{G}^{k} \}\).
Proof
3 The study of fixed points for multivalued Gcontractions
In this section, we establish fixed point results for a multivalued contraction with respect to the graph G in a Menger PMspace.
Definition 3.1
Lemma 3.2
Proof
Lemma 3.3
Proof
The following concepts are adaptations of those introduced in [9] to the case of Menger PMspaces.
Definition 3.4
 (i)
Let \(N \in\mathbb{N}\). We say that a sequence \(\{x_{n}\} \subset S\) is a \(G_{N}\)Picard trajectory from \(x_{0}\) if \(x_{n} \in [x_{n1}]_{G}^{N} \cap fx_{n1}\) for all \(n \geq1\). We denote by \(\mathcal{T}_{N}(f, G, x_{0})\) the set of all such \(G_{N}\)Picard trajectories from \(x_{0}\).
 (ii)
We say that a sequence \(\{x_{n}\} \subset S\) is a GPicard trajectory from \(x_{0}\) if \(x_{n} \in[x_{n1}]_{G} \cap fx_{n1}\) for all \(n \geq1\). We denote by \(\mathcal{T}(f,G,x_{0})\) the set of all such GPicard trajectories from \(x_{0}\).
Definition 3.5
 (i)
Let \(N \in\mathbb{N}\). We say that f is \(G_{N}\)Picard continuous from \(x_{0} \in S\) if the limit of any convergent sequence \(\{ x_{n}\} \in\mathcal{T}_{N}(f,G,x_{0})\) is a fixed point of T.
 (ii)
We say that f is GPicard continuous from \(x_{0} \in S\) if the limit of any convergent sequence \(\{x_{n}\} \in\mathcal {T}(f,G,x_{0})\) is a fixed point of T.
Now, we are able to establish our first main result.
Theorem 3.6
 (i)
there exists \(x_{0} \in S\) such that \([x_{0}]_{G}^{N} \cap fx_{0} \neq\emptyset\);
 (ii)
f is \(G_{N}\)Picard continuous from \(x_{0}\).
Proof
Suppose now that \(f: S \to S\) is a multivalued Gcontraction with closed values, and let us consider the assumption (i) of Theorem 3.6 with the following assumption: (ii)′ If \(\{x_{n}\} \subset S\) is a sequence in \(\mathcal {T}_{N}(f,G,x_{0})\) converging to some \(x \in S\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \((x_{n_{k}},x) \in E(G)\) for every \(k \in\mathbb{N}\).
Corollary 3.7
 (i):

there exists \(x_{0} \in S\) such that \([x_{0}]_{G}^{N} \cap fx_{0} \neq\emptyset\);
 (ii)′:

if \(\{x_{n}\} \subset S\) is a sequence in \(\mathcal {T}_{N}(f,G,x_{0})\) converging to some \(x \in S\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \((x_{n_{k}},x) \in E(G)\) for every \(k \in\mathbb{N}\).
Remark 3.8
Condition (ii)′ in Corollary 3.7 can be replaced by: if \(\{x_{n}\} \subset S\) is a sequence in \(\mathcal{T}_{N}(f,G,x_{0})\) converging to some \(x \in S\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \((x_{n_{k}},x) \in E(G)\) for every k large enough.
Taking \(N = 1\) and \(G = S \times S\) in Corollary 3.7, we obtain the following Nadler fixed point theorem in Menger PMspaces.
Corollary 3.9
 (i)
there exists \((x_{0}, x_{1}) \in S \times S\) such that \(x_{1} \in fx_{0}\) and \(D(x_{0}, x_{1}) < \infty\);
 (ii)there exists \(\kappa\in(0, 1)\) such that$$(x,y) \in S \times S, u \in fx \quad \Longrightarrow\quad \exists v \in fy \mbox{: } F(u, v) (\kappa t) \geq F(x, y) (t), \forall t > 0. $$
Corollary 3.10
 (i)
there exists \(x_{0} \in S\) such that \([x_{0}]_{G} \cap fx_{0} \neq \emptyset\);
 (ii)
f is GPicard continuous from \(x_{0}\).
Proof
From (i), there exists some \(N \in\mathbb{N}\) such that \([x_{0}]_{G}^{N} \cap fx_{0} \neq\emptyset\). Since from (ii) f is GPicard continuous from \(x_{0}\), then it is \(G_{N}\)Picard continuous from \(x_{0}\). Now, the result follows from Theorem 3.6. □
Similarly, from Corollary 3.7, we have the following result.
Corollary 3.11
 (i)
there exists \(x_{0} \in S\) such that \([x_{0}]_{G} \cap fx_{0} \neq \emptyset\);
 (ii)
if \(\{x_{n}\} \subset S\) is a sequence in \(\mathcal {T}(f,G,x_{0})\) converging to some \(x \in S\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \((x_{n_{k}},x) \in E(G)\) for every \(k \in\mathbb{N}\).
From Corollary 3.11, we can obtain a probabilistic version of the fixed point theorem for \((\varepsilon,\lambda)\)uniformly locally contractive multivalued maps due to Nadler [19]. At first, let us introduce some concepts.
Definition 3.12
Definition 3.13
We have the following result.
Corollary 3.14
 (i)
f is \((\varepsilon,\kappa)\)uniformly locally contractive;
 (ii)
there exist \(x_{0} \in S\) and \(\tilde{x} \in fx_{0}\) such that \(\{x_{0},\tilde{x}\}\) is εchainable in S.
Proof
Corollary 3.15
 (i)
if \(\{f^{n}x_{0}\}\) converges to some \(x \in S\), then \(x = fx\);
 (ii)
if \(\{f^{n}x_{0}\}\) converges to some \(x \in S\), then there exists a subsequence \(\{f^{n_{k}}x_{0}\}\) of \(\{f^{n}x_{0}\}\) such that \((f^{n_{k}}x_{0}, x) \in E(G)\) for every \(k \in\mathbb{N}\).
From Corollary 3.15, we can obtain a probabilistic version of Kirk, Srinivasan and Veeramani’s fixed point theorem for cyclic mappings [20].
Corollary 3.16
 (i)
\(f(A_{i}) \subseteq A_{i+1}\) for every \(i = 1,2, \ldots, p\), with \(A_{p+1}=A_{1}\);
 (ii)there exists \(\kappa\in(0, 1)\) such that for all \(i=1,2, \ldots, p\),$$(x,y) \in A_{i} \times A_{i+1} \quad \Longrightarrow \quad F(fx, fy) (\kappa t) \geq F(x,y) (t),\forall t > 0; $$
 (iii)
there exists \(x_{0} \in A_{1}\) such that \(D(x_{0},fx_{0}) < \infty\).
Proof
4 The study of fixed points for multivalued weak Gcontractions
In this section, we study the existence of fixed points for multivalued weak Gcontractions.
Definition 4.1
From the next result, we observe that the class of multivalued weak Gcontractions is larger than the class of multivalued Gcontractions.
Lemma 4.2
Let \((S,F,T)\) be a Menger PMspace and \(f: S \to S\) be a multivalued mapping with nonempty values. If f is a Gcontraction, then f is a weak Gcontraction.
Proof
We have the following fixed point result for the class of weak Gcontractions.
Theorem 4.3
 (i)
there exists \(x_{0} \in S\) such that \([x_{0}]_{G} \cap fx_{0} \neq \emptyset\);
 (ii)
f is GPicard continuous from \(x_{0}\).
Proof
The next result concerns weak Gcontraction multivalued mappings with nonempty closed values.
Theorem 4.4
 (i)
there exists \(x_{0} \in S\) such that \([x_{0}]_{G} \cap fx_{0} \neq \emptyset\);
 (ii)
for every sequence \(\{x_{n}\} \in\mathcal{T}(f, G, x_{0})\) converging to some \(x \in S\), there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x \in[x_{n_{k}} ]_{G}\) for every \(k \in\mathbb{N}\), and \(p(x_{n_{k}} , x^{*}) \to0\), as \(k \rightarrow\infty\).
Proof
Finally, from Theorem 4.3 and Theorem 4.4, we obtain the following fixed point theorem for singlevalued mappings.
Theorem 4.5
 (i)
if \(\{f^{n} x_{0}\}\) converges to some \(x \in S\), then \(x = fx\);
 (ii)
if \(\{f^{n} x_{0}\}\) converges to some \(x \in S\), then there exists a subsequence \(\{f^{n_{k}}x_{0}\}\) of \(\{f^{n} x_{0}\}\) such that \(p(f^{n_{k}}x_{0}, x) \to0\), as \(k\to\infty\), and \(f^{n_{k}} x_{0} \in[x]_{G}\) for every \(k \in\mathbb{N}\).
5 Applications: KeliskyRivlin type result for modified qBernstein polynomials
As applications, we establish in this section a KeliskyRivlin type result, a certain class of modified qBernstein polynomials.
At first, we have the following result concerning the convergence of successive approximations for a certain family of operators.
Theorem 5.1
 (i)
for every \(x \in X\), the Picard sequence \(\{f^{n} x\}\) converges to a fixed point of f;
 (ii)
for every \(x \in X\), \((x + X_{0}) \cap\operatorname{Fix} f = \{ \lim_{n\to\infty} f^{n} x \} \), where Fixf denotes the set of fixed points of f.
Proof
Remark 5.2
Theorem 5.1 recovers Theorem 4.1 in [7], where X was supposed to be a Banach space and f was supposed to be a linear operator.
In this section, we are interested in establishing Kelisky and Rivlin type results for a class of modified qBernstein polynomials. To formulate our results, we need the following definitions.
Definition 5.3
(Phillips [22])
Note that for \(q = 1\), the polynomials \(B_{n}(1, \varphi)(t)\) are classical Bernstein polynomials.
We introduce the following class of modified qBernstein polynomials.
Definition 5.4
We have the following result.
Theorem 5.5
Proof
Remark 5.6
Note that Theorem 4.1 in [7] cannot be applied in the case of modified qBernstein operators since it requires linear operators defined on a certain Banach space X. Observe that in our case, X is not a linear space.
Remark 5.7
The case of modified 1Bernstein operator was considered recently in [11]. The authors claimed that if \(n \in\mathbb{N}\) for every \(\varphi\in X = C([0, 1])\), the Picard sequence \(\{\mathcal{B}_{n}^{j} (1, \varphi) \}\) converges uniformly to a fixed point of \(\mathcal{B}_{n}(1,\cdot)\) (see Corollary 4 in [11]). For the proof of this claim, the authors used that \(\varphi\mathcal{B}_{n}(1, \varphi) \in X_{0}\) for every \(\varphi\in X\), where \(X_{0}\) is the set of functions \(\phi\in X\) such that \(\phi(0) = \phi(1) = 0\). Unfortunately, the above property is not true. To observe this fact, we have just to consider a function \(\varphi \in X\) such that \(\varphi(0) < 0\) or \(\varphi(1) < 0\). Our Theorem 5.5 for the case \(q = 1\) is a corrected version of Corollary 4 in [11].
Declarations
Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project no. IRG1404.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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