Common fixed point of the generalized Mizoguchi-Takahashi’s type contractions

In 1922, Banach established the most famous fundamental fixed point theorem (the so-called 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 multi-valued maps, known as Nadler’s fixed point theorem [8], Mizoguchi-Takahashi’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], Amini-Harandi [1, 7], Moradi and Khojasteh [13], and Du [6] and references therein.

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 single-valued map) or $v\in Tv$ (when $T:X\to \mathcal{N}(X)$ is a multi-valued map). The set of fixed points of T is denoted by $\mathcal{F}(T)$ and the set of common fixed points of two multi-valued 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 ℛ-function) 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 non-decreasing function or a non-increasing 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]

for all $x,y\in X$. Then $\mathcal{F}(T)\ne \mathrm{\varnothing }$.

for all $x,y\in X$.

The concepts of weak and φ-weak 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 multi-valued 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.

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 Mizoguchi-Takahashi’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 Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, and some well-known 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.

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 $s,t>0$;

We denote the set of all GMT functions by $\widehat{\mathsf{GMT}(\mathsf{R})}$.

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 GMT-function.

which means that (ϑ 2) holds. Hence $\vartheta \in \widehat{\mathsf{GMT}(\mathsf{R})}$.

for each $x,y\in X$.

for each $x,y\in X$. Then T, S have a common fixed point.

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

By the above argument, we can assume that

Hence by induction, we can establish a sequence $\{x_{2n+1}\}$ in X satisfying, for each $n\in \mathsf{N}$,

To complete the proof it suffices to show that ${\{x_n\}}_{n\in \mathsf{N}}$ is a Cauchy sequence in X.

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.

Other cases are easily verified as the above arguments. Henceforth, T is a WGMT-contraction and enjoys all conditions of Theorem 3. Also, T, S have a common fixed point $\{0\}$.

Here we deduce some of the known and unknown results by Theorem 3.

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. □

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]

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. □

In 2010, Rouhani and Moradi [12] proved the following theorem.

(i.e. generalized φ-weak 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 multi-valued. 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 multi-valued.’ 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.

(i.e. generalized φ-weak 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?

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.

(i.e. generalized φ-weak 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 $\mathcal{F}ix(T,S)\ne \mathrm{\varnothing }$.

Applying Theorem 3 yields $\mathcal{F}ix(T,S)\ne \mathrm{\varnothing }$. □