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Common fixed point of the generalized MizoguchiTakahashi’s type contractions
Fixed Point Theory and Applications volume 2014, Article number: 195 (2014)
Abstract
In this work, we introduce a new class of contractive multivalued mappings, called generalized MizoguchiTakahashi’s contractions, which is an extension of many known results in the literature. Finally, a partial answer to the conjecture which was introduced by Rouhani and Moradi is given.
MSC:47H10, 54C60.
1 Introduction
In 1922, Banach established the most famous fundamental fixed point theorem (the socalled the Banach contraction principle [1]) which has played an important role in various fields of applied mathematical analysis. It is known that the Banach contraction principle has been extended in many various directions by several authors (see [2–7]). An interesting direction of research is the extension of the Banach contraction principle of multivalued maps, known as Nadler’s fixed point theorem [8], MizoguchiTakahashi’s fixed point theorem [9]; see M Berinde and V Berinde [3], Ćirić [4], Reich [10], Daffer and Kaneko [5], Rhoades [11], Rouhani and Moradi [12], AminiHarandi [1, 7], Moradi and Khojasteh [13], and Du [6] and references therein.
Let us recall some basic notations, definitions, and wellknown results needed in this paper. Throughout this paper, we denote by N and R, the sets of positive integers and real numbers, respectively. Let (X,d) be a metric space. For each x\in X and A\subseteq X, let d(x,A)={inf}_{y\in A}d(x,y). Denote by \mathcal{N}(X) the class of all nonempty subsets of X, by \mathcal{C}(X) the family of all nonempty closed subsets of X, and by \mathcal{CB}(X) the family of all nonempty, closed, and bounded subsets of X. A function \mathcal{H}:\mathcal{CB}(X)\times \mathcal{CB}(X)\to [0,\mathrm{\infty}) defined by
is said to be the Hausdorff metric on \mathcal{CB}(X) induced by d on X where D(x,A)=inf\{d(x,y):y\in A\} for each A\in \mathcal{CB}(X). A point v in X is a fixed point of a map T if v=Tv (when T:X\to X is a singlevalued map) or v\in Tv (when T:X\to \mathcal{N}(X) is a multivalued map). The set of fixed points of T is denoted by \mathcal{F}(T) and the set of common fixed points of two multivalued mappings T, S is denoted by \mathcal{F}(T,S).
Definition 1 [[7], Du]
A function \phi :[0,\mathrm{\infty})\to [0,1) is said to be an \mathcal{MT}function (or ℛfunction) if {lim\hspace{0.17em}sup}_{s\to {t}^{+}}\phi (s)<1 for all t\in [0,\mathrm{\infty}).
It is evident that if \phi :[0,\mathrm{\infty})\to [0,1) is a nondecreasing function or a nonincreasing function, then φ is an \mathcal{MT}function. So the set of \mathcal{MT}functions is a rich class.
In 1989, Mizoguchi and Takahashi [9] proved a famous generalization of Nadler’s fixed point theorem which gives a partial answer to Reich’s problem [14].
Theorem 1 [[9], Mizoguchi and Takahashi]
Let (X,d) be a complete metric space, \phi :[0,\mathrm{\infty})\to [0,1) be a \mathcal{MT}function and T:X\to \mathcal{CB}(X) be a multivalued map. Assume that
for all x,y\in X. Then \mathcal{F}(T)\ne \mathrm{\varnothing}.
A mapping T:X\to X is said to be a weak contraction if there exists 0\le \alpha <1 such that
for all x,y\in X, where
Two multivalued mappings T,S:X\to \mathcal{CB}(X) are called generalized weak contractions if there exists 0\le \alpha <1 such that
where
Also two mappings T,S:X\u27f6\mathcal{CB}(X) are called generalized φweak contractive if there exists a map \phi :[0,+\mathrm{\infty})\u27f6[0,+\mathrm{\infty}) with \phi (0)=0 and \phi (t)>0 for all t>0 such that
for all x,y\in X.
The concepts of weak and φweak contractive mappings were defined by Daffer and Kaneko [5] in 1995.
In 2010 Rouhani and Moradi [12] introduced an extension of Daffer and Kaneko’s result for two multivalued weak contraction mappings of a complete metric space X into \mathcal{CB}(X) without assuming x\mapsto d(x,Tx) to be l.s.c.
Theorem 2 Let (X,d) be a complete metric space. Suppose that T,S:X\to \mathcal{CB}(X) are contraction mappings in the sense that, for some 0\le \alpha <1,
for all x,y\in X (i.e., weak contractions). Then there exists a point {x}_{0}\in X such that {x}_{0}\in T{x}_{0} and {x}_{0}\in S{x}_{0}.
The paper is organized as follows. In Section 2, we first introduce the Generalized MizoguchiTakahashi’s Contraction (GMT for short) as an extension of Mizoguchi and Takahashi’s type and of work by Daffer and Kaneko. Section 3 is dedicated to the study of some new fixed point theorems, which generalize and improve MizoguchiTakahashi’s fixed point theorem, Nadler’s fixed point theorem, and some wellknown results. Furthermore, we give a partial answer to the conjecture introduced by Rouhani and Moradi (see [[12], Theorem 4.1 and Section 5]). Consequently, some of our results in this paper are original in the literature, and we obtain many results in the literature as special cases.
2 Main result
In this section, we first explain the concept of GMT contraction (see also [9, 15]).
Definition 2 A function \vartheta :\mathsf{R}\times \mathsf{R}\to \mathsf{R} is called a GMT function if the following conditions hold:
(ϑ 1) 0<\vartheta (t,s)<1 for all s,t>0;
(ϑ 2) for any bounded sequence \{{t}_{n}\}\subset (0,+\mathrm{\infty}) and any nonincreasing sequence \{{s}_{n}\}\subset (0,+\mathrm{\infty}), we have
We denote the set of all GMT functions by \stackrel{\u02c6}{\mathsf{GMT}(\mathsf{R})}.
Here, we give simple examples of manageable functions.
Example 1 [[15], Example B]
Let \phi :[0,\mathrm{\infty})\to [0,1) be an \mathcal{MT}function, then \vartheta (t,s)=\phi (s) is a GMTfunction.
Example 2 Let g(x)=\frac{ln(x+10)}{x+9} for all x>9. Define
It is clear that 0<\vartheta (t,s)<1. Also for any bounded sequence \{{t}_{n}\}\subset (0,+\mathrm{\infty}) and any nonincreasing sequence \{{s}_{n}\}\subset (0,+\mathrm{\infty}), if 1<{t}_{n}<{s}_{n}, then {lim}_{n\to \mathrm{\infty}}{s}_{n}={inf}_{n\in \mathsf{N}}{s}_{n}=a for some a\in [0,+\mathrm{\infty}). We have
Otherwise, \vartheta (t,s)=g(s) and since g is continuous, we get
which means that (ϑ 2) holds. Hence \vartheta \in \stackrel{\u02c6}{\mathsf{GMT}(\mathsf{R})}.
Definition 3 Let (X,d) be a metric space. The mapping T:X\to \mathcal{CB}(X) is called a WGMTcontraction if there exists \vartheta \in \stackrel{\u02c6}{\mathsf{GMT}(\mathsf{R})} such that
for each x,y\in X.
Theorem 3 Let (X,d) be a complete metric space and let T,S:X\to \mathcal{CB}(X) and suppose there exists \vartheta \in \stackrel{\u02c6}{\mathsf{GMT}(\mathsf{R})} such that
for each x,y\in X. Then T, S have a common fixed point.
Proof Let {x}_{0}\in X be arbitrary and {x}_{1}\in S{x}_{0}. Since T{x}_{1}\ne \mathrm{\varnothing} choose {x}_{2}\in T{x}_{1}. If {x}_{1}={x}_{2} we have nothing to prove, because
Thus {M}_{T,S}({x}_{1},{x}_{2})=D({x}_{2},S{x}_{2}). If D({x}_{2},S{x}_{2})=0, then {x}_{2}\in S{x}_{2} and {x}_{2}\in T{x}_{1}=T{x}_{2} and so {x}_{2} is the common fixed point of T, S. Hence without loss of generality, we can assume that {M}_{T,S}({x}_{1},{x}_{2})>0. Also if \mathcal{H}(T{x}_{1},S{x}_{2})=0, then T{x}_{1}=S{x}_{2} and since {x}_{1}={x}_{2} we have {x}_{2}\in T{x}_{1}=T{x}_{2} and {x}_{2}\in T{x}_{1}=S{x}_{2} and again {x}_{2} is the common fixed point of T, S. Thus we can assume that \mathcal{H}(T{x}_{1},S{x}_{2})>0. Therefore, by any choice of {x}_{1}, {x}_{2} we can assume that

{M}_{T,S}({x}_{1},{x}_{2})>0,

\mathcal{H}(T{x}_{1},S{x}_{2})>0.
Taking
(note that {\u03f5}_{1}>0), there exists {x}_{3}\in S{x}_{2} such that {x}_{3}\ne {x}_{2} such that
By the above argument, we can assume that

{M}_{T,S}({x}_{2},{x}_{3})>0,

\mathcal{H}(T{x}_{2},S{x}_{3})>0.
Choose {x}_{4}\in T{x}_{3}. Taking
(note that {\u03f5}_{2}>0), there exists {x}_{5}\in S{x}_{4} such that {x}_{5}\ne {x}_{4} and
By induction, if {x}_{2k1},{x}_{2k},{x}_{2k+1}\in X is known to satisfy {x}_{2k}\in T{x}_{2k1}, {x}_{2k+1}\in S{x}_{2k}, \mathcal{H}(T{x}_{2k2},S{x}_{2k1})>0, {M}_{T,S}({x}_{2k2},{x}_{2k1})>0, and
then, by taking
one can obtain {x}_{2k+3}\in S{x}_{2k+2} with {x}_{2k+3}\ne {x}_{2k+2} such that
Hence by induction, we can establish a sequence \{{x}_{2n+1}\} in X satisfying, for each n\in \mathsf{N},

{x}_{2n+1}\in S{x}_{2n},

\mathcal{H}(T{x}_{2n2},S{x}_{2n1})>0,

{M}_{T,S}({x}_{2n2},{x}_{2n1})>0,
and
By (7), we have
By combining (11) and (12), we get
By repeating the above argument (replacing S by T) one can easily verified that
Note that
Also
It means that d({x}_{2n1},{x}_{2n})={M}_{T,S}({x}_{2n1},{x}_{2n}) and {M}_{T,S}({x}_{2n+1},{x}_{2n})=d({x}_{2n+1},{x}_{2n}). Hence for each n\in \mathsf{N} we have
which means that the sequence {\{d({x}_{n1},{x}_{n})\}}_{n\in \mathsf{N}} is strictly decreasing in (0,+\mathrm{\infty}). So
By (12) and (15), we have
which means that {\{\mathcal{H}(T{x}_{n},S{x}_{n1})\}}_{n\in \mathsf{N}} is a bounded sequence. By (ϑ 2), we have
Now, we claim \gamma =0. Suppose \gamma >0. Then, by (18) and taking the limsup in both sides of (13), we get
a contradiction. Hence we prove
To complete the proof it suffices to show that {\{{x}_{n}\}}_{n\in \mathsf{N}} is a Cauchy sequence in X.
For each n\in \mathsf{N}, let
Then {\rho}_{n}\in (0,1) for all n\in \mathsf{N}. By (13), we obtain
From (18), we have {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\rho}_{n}<1, so there exist c\in [0,1) and {n}_{0}\in \mathsf{N}, such that
For any n\ge {n}_{0}, since {\rho}_{n}\in (0,1) for all n\in \mathsf{N} and c\in [0,1), taking into account (20) and (21), we conclude that
Put {\alpha}_{n}=\frac{{c}^{n{n}_{0}+1}}{1c}d({x}_{0},{x}_{1}), n\in \mathsf{N}. For m,n\in \mathsf{N} with m>n\ge {n}_{0}, we have from the last inequality
Since c\in [0,1), {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and hence
So \{{x}_{n}\} is a Cauchy sequence in X and converges to {x}^{\ast}\in X. If
and \mathrm{\u266f}(S)=\mathrm{\infty} where \mathrm{\u266f}(S) is the cardinal number of S, then {x}_{n+1}\in T{x}_{n}=T{x}^{\ast} and we have {x}^{\ast}\in T{x}^{\ast}. If \mathrm{\u266f}(S)<\mathrm{\infty} one deduces that there exists \ell \in \mathsf{N} such that {x}_{n}\ne x for all n\in \mathsf{N} with n\ge \ell. It means that \mathcal{H}(T{x}_{n},S{x}^{\ast})>0 and d({x}_{n},{x}^{\ast})>0 for each n\ge \ell. Hence \mathcal{H}(T{x}_{n},S{x}^{\ast})>0 and {M}_{T,S}({x}_{n},{x}^{\ast})>0 for each n\ge \ell and so
Suppose that D(T{x}^{\ast},{x}^{\ast})>0, then
By (23) one can choose \nu \in \mathsf{N} such that {M}_{T,S}({x}_{n},{x}^{\ast})=D(T{x}^{\ast},{x}^{\ast}) for each n\ge \nu. Now by taking \kappa =max\{\nu ,\ell \}, for each n\ge \kappa we have
For each n\in \mathsf{N}, let
Then {\rho}_{n}\in (0,1) for all n\in \mathsf{N}. By (24), we obtain
Since \mathcal{H}(S{x}_{n},T{x}^{\ast})\le D(T{x}^{\ast},{x}^{\ast}) we have \{\mathcal{H}(S{x}_{n},T{x}^{\ast})\} is bounded and \{D(T{x}^{\ast},{x}^{\ast})\} is a fixed sequence. From ({\eta}_{2}), we have {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\lambda}_{n}<1, so there exist r\in [0,1) and {n}_{0}\in \mathsf{N}, such that
Taking the limit in both sides of (25) we have
Thus r\ge 1 and this is a contradiction. So we have D({x}^{\ast},T{x}^{\ast})=0 and then {x}^{\ast}\in T{x}^{\ast}. The same argument can be applied for S and one can easily verify that {x}^{\ast}\in S{x}^{\ast} and so T, S have a common fixed point {x}^{\ast}\in X. □
In the following, an example is given covering our result.
Example 3 Suppose that X=[0,1]\cup \{4\} and let T,S:X\to \mathcal{CB}(X) be defined as follows:
Suppose that
and \vartheta (t,s)=1\frac{\phi (s)}{s} for all t,s>0. For any bounded sequence \{{t}_{n}\}\subset (0,+\mathrm{\infty}) and any nonincreasing sequence \{{s}_{n}\}\subset (0,+\mathrm{\infty}), we have
First suppose 0\le x\le 1 and 0\le y\le 1, then
In the other case, suppose x=4 and 0\le y\le 1, then
Other cases are easily verified as the above arguments. Henceforth, T is a WGMTcontraction and enjoys all conditions of Theorem 3. Also, T, S have a common fixed point \{0\}.
3 Consequences
Here we deduce some of the known and unknown results by Theorem 3.
Corollary 1 Let (X,d) be a complete metric space and T,S:X\to \mathcal{CB}(X) be two multivalued mappings such that
where \alpha \in [0,1). Then T, S have at least a common fixed point in X.
Proof It suffices to take \vartheta (t,s)=\alpha and apply Theorem 3. □
Corollary 2 Let (X,d) be a complete metric space and T,S:X\to \mathcal{CB}(X) be two multivalued mappings such that
where \phi :[0,+\mathrm{\infty})\to [0,1) be an \mathcal{MT}function, then T has a fixed point in X.
Proof It suffices to take \vartheta (t,s)=\phi (s) and apply Theorem 3. □
Corollary 3 [[13], Generalized weak contraction]
Let (X,d) be a complete metric space and T,S:X\to \mathcal{CB}(X) be two multivalued mappings such that
where \phi :[0,+\mathrm{\infty})\to [0,1) be a function such that \phi (s)<s and {lim\hspace{0.17em}sup}_{s\to {t}^{+}}\frac{\phi (s)}{s}<1. Then T has a fixed point in X.
Proof It suffices to take \vartheta (t,s)=\frac{\phi (s)}{s} and apply Theorem 3. □
4 A partial answer to a known conjecture
In 2010, Rouhani and Moradi [12] proved the following theorem.
Corollary 4 Let (X,d) be a complete metric space and let T:X\to X and S:X\to \mathcal{CB}(X) be two mappings such that, for all x,y\in X,
(i.e. generalized φweak contractions) where \phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) is l.s.c. with \phi (0)=0, \phi (t)<t and \phi (t)>0 for all t>0. Then there exists a unique point x\in X such that Tx=x\in Sx.
Motivated by the above the authors extended Rhoades’s theorem by assuming φ to be only l.s.c., as well as Zhang and Song’s [16] theorem to the case where one of the mappings is multivalued. They also asserted the following: ‘Future directions to be pursued in the context of this research include the investigation of the case where both mappings in Zhang and Song’s theorem are multivalued.’ By research in the literature such as [1, 7] and specially [16] (see [[16], Problem 3.2]) one deduces the following problem, a conjecture in the literature.
Problem (A) Let (X,d) be a complete metric space and let T,S:X\to \mathcal{CB}(X) be two mappings such that, for all x,y\in X,
(i.e. generalized φweak contractions) where \phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) is l.s.c. with \phi (0)=0, \phi (t)<t and \phi (t)>0, for all t>0. Then do T and S have a common fixed point?
Definition 4 We say that \phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) is a weak l.s.c. function if for each bounded sequence \{{t}_{n}\}\subset (0,\mathrm{\infty}),
The collection of all weak l.s.c. functions is denoted by {\mathcal{W}}_{\mathrm{lsc}}(\mathsf{R}).
In the following theorem a partial solution to Problem (A) is given as an application of Theorem 3.
Corollary 5 Let (X,d) be a complete metric space and let T,S:X\to \mathcal{CB}(X) be two mappings such that, for all x,y\in X,
(i.e. generalized φweak contractions) where \phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}), with \phi (0)=0, \phi (t)<t and \phi \in {\mathcal{W}}_{\mathrm{lsc}}(\mathsf{R}). Then \mathcal{F}ix(T,S)\ne \mathrm{\varnothing}.
Proof Define \vartheta (t,s)=(1\frac{\phi (t)}{t}), for all t,s>0. Since for each bounded sequence \{{t}_{n}\}\subset (0,\mathrm{\infty}), {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}\phi ({t}_{n})>0, thus {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}\frac{\phi ({t}_{n})}{{t}_{n}}>0. Thus,
It means that \vartheta \in \stackrel{\u02c6}{\mathsf{GMT}(\mathsf{R})}. Also
Applying Theorem 3 yields \mathcal{F}ix(T,S)\ne \mathrm{\varnothing}. □
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Javahernia, M., Razani, A. & Khojasteh, F. Common fixed point of the generalized MizoguchiTakahashi’s type contractions. Fixed Point Theory Appl 2014, 195 (2014). https://doi.org/10.1186/168718122014195
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DOI: https://doi.org/10.1186/168718122014195
Keywords
 multivalued mapping
 Hausdorff metric
 MizoguchiTakahashi’s type
 \mathcal{MT}function
 GMTcontraction