f-Contractive multivalued maps and coincidence points

Abstract

In this paper, we prove a result on the existence of an f-orbit for generalized f-contractive multivalued maps. Then, we establish main results on the existence of coincidence points and common fixed points for generalized f-contractive maps not involving the extended Hausdorff metric and the continuity condition. Our results either generalize or improve a number of metric fixed point results.

MSC: 47H10, 47H09, 54H25.

1 Introduction

Let $(X,d)$ be a metric space. Let $2^X$, $Cl(X)$ and $CB(X)$ denote the collection of nonempty subsets of X, nonempty closed subsets of X, and nonempty closed bounded subsets of X, respectively. Let H be the Hausdorff metric with respect to d, that is,

$$H(A,B)=max\{\underset{x\in A}{sup}d(x,B),\underset{y\in B}{sup}d(y,A)\}$$

for every $A,B\in CB(X)$, where $d(x,B)={inf}_{y\in B}d(x,y)$.

Let $f:X\to X$ be a single-valued map, and let $T:X\to 2^X$ be a multivalued map. A point $x\in X$ is called a fixed point of T if $x\in T(x)$, and the set of fixed points of T is denoted by $Fix(T)$. A point $x\in X$ is called a coincidence point of f and T if $f(x)\in T(x)$. We denote by $C(f\cap T)$ the set of coincidence points of f and T.

We say a sequence $\{x_n\}$ in X is an f-orbit of T at $x_0\in X$ if $f{x}_n\in T{x}_{n-1}$ for all $n\ge 1$. We say that f and T weakly commute if $fTx\subset Tfx$ for all $x\in X$. Clearly, commuting maps f and T weakly commute.

A multivalued map $T:X\to CB(X)$ is called

(i) contraction [1] if for a fixed constant $\lambda \in (0,1)$ and for each $x,y\in X$,

$$H(T(x),T(y))\le \lambda d(x,y).$$

(ii) f-contraction [2] if for a fixed constant $\lambda \in (0,1)$ and for each $x,y\in X$,

$$H(T(x),T(y))\le \lambda d(f(x),f(y)).$$

Using the concept of Hausdorff metric, Nadler [1] established the following fixed point result for multivalued contraction maps, which in turn is a generalization of the well-known Banach contraction principle.

Theorem 1.1 [1]

Let $(X,d)$ be a complete metric space, and let $T:X\to CB(X)$ be a contraction map. Then $Fix(T)\ne \mathrm{\varnothing }$.

This result has been generalized in many directions. Kaneko [2] extended the corresponding results of Jungck [3], Nadler [1] and others as follows.

Theorem 1.2 [2]

Let $(X,d)$ be a complete metric space, and let $T:X\to CB(X)$ be a multivalued f-contraction map which commutes with a continuous map f. Then $C(f\cap T)\ne \mathrm{\varnothing }$.

This result has been generalized in different directions. For example, see [4–10].

On the other hand, Kada et al. [11] introduced the concept of w-distance on a metric space as follows:

Let $(X,d)$ be a metric space. A function $\omega :X\times X\to [0,\mathrm{\infty })$ is called a w-distance on X if it satisfies the following for each $x,y,z\in X$:

($w_1$) $\omega (x,z)\le \omega (x,y)+\omega (y,z)$;

($w_2$) a map $\omega (x,\cdot ):X\to [0,\mathrm{\infty })$ is lower semicontinuous;

($w_3$) for any $ϵ>0$, there exists $\delta >0$ such that $\omega (z,x)\le \delta$ and $\omega (z,y)\le \delta$ imply $d(x,y)\le ϵ$.

Note that, in general, for $x,y\in X$, $\omega (x,y)\ne \omega (y,x)$ and not either of the implications $\omega (x,y)=0\iff x=y$ necessarily hold. We say the w-distance ω on X is a $w_0$-distance if $x=y$ implies $\omega (x,y)=0$. Clearly, the metric d is a w-distance on X. Let $(Y,\parallel \cdot \parallel )$ be a normed space. Then the functions ${\omega }_1,{\omega }_2:Y\times Y\to [0,\mathrm{\infty })$ defined by ${\omega }_1(x,y)=\parallel y\parallel$ and ${\omega }_2(x,y)=\parallel x\parallel +\parallel y\parallel$ for all $x,y\in Y$ are w-distances [11]. Many other examples and properties of the w-distance can be found in [11, 12].

The following useful lemma concerning a w-distance is given in [11].

Lemma 1.1 [11]

Let $(X,d)$ be a metric space, and let ω be a w-distance on X. Let $\{x_n\}$ and $\{y_n\}$ be sequences in X, and let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be sequences in $[0,\mathrm{\infty })$ converging to zero. Then, for the w-distance ω on X, the following hold for every $x,y,z\in X$:

(a) if $\omega (x_n,y)\le {\alpha }_n$ and $\omega (x_n,z)\le {\beta }_n$ for any $n\in \mathbb{N}$, then $y=z$; in particular, if $\omega (x,y)=0$ and $\omega (x,z)=0$, then $y=z$;

(b) if $\omega (x_n,y_n)\le {\alpha }_n$ and $\omega (x_n,z)\le {\beta }_n$ for any $n\in \mathbb{N}$, then $\{y_n\}$ converges to z;

(c) if $\omega (x_n,x_m)\le {\alpha }_n$ for any $n,m\in \mathbb{N}$ with $m>n$, then $\{x_n\}$ is a Cauchy sequence;

(d) if $\omega (y,x_n)\le {\alpha }_n$ for any $n\in \mathbb{N}$, then $\{x_n\}$ is a Cauchy sequence.

For $x\in X$ and $A\in 2^X$, we denote, $\omega (x,A)={inf}_{y\in A}\omega (x,y)$. Now, let $T:X\to Cl(X)$ be a multivalued map, and let $f:X\to X$ be a single-valued map. We say

(iii) T is w-contractive [12] if there exist a w-distance ω on X and $\lambda \in (0,1)$ such that for any $x,y\in X$ and $u\in T(x)$, there is $v\in T(y)$ with

$$\omega (u,v)\le \lambda \omega (x,y).$$

(iv) T is generalized f-contractive if there exist a $w_0$-distance ω on X and $\lambda \in (0,1)$ such that for any $x,y\in X$, $u\in T(x)$, there is $v\in T(y)$ with

$$\omega (u,v)\le \lambda M_f(x,y),$$

where

$$\begin{array}{rcl}{M}_f(x,y)& =& max\{\omega (f(x),f(y)),\omega (f(x),T(x)),\omega (f(y),T(y)),\\ \frac{1}{2}[\omega (f(x),T(y))+\omega (f(y),T(x))]\}.\end{array}$$

Using the concept of w-distance, Suzuki and Takahashi [12] improved Nadler’s fixed point result as follows.

Theorem 1.3 Let $(X,d)$ be a complete metric space. Then for each w-contractive map $T:X\to Cl(X)$, the set $Fix(T)\ne \mathrm{\varnothing }$.

This result has been generalized by many authors, for example, see [13–16]. In this paper, first we establish a lemma with respect to a w-distance, which is an improved version of the lemma given in [3], and then we prove a key lemma on the existence of an f-orbit for generalized f-contractive maps. Finally, we present our main results on the existence of coincidence points and common fixed points for generalized f-contractive maps not involving the extended Hausdorff metric. As a consequence, we obtain a fixed point result. Our results either generalize or improve a number of known results.

2 Results

Using the concept of w-distance, first we improve a corresponding result of Jungck [3] as follows.

Lemma 2.1 Let $(X,d)$ be a complete metric space with a w-distance ω. If there exist a sequence $\{x_n\}$ in X and a constant λ, $0<\lambda <1$, such that for all $n\in \mathbb{N}$,

$$\omega (x_n,x_{n+1})\le \lambda \omega (x_{n-1},x_n),$$

then the sequence $\{x_n\}$ converges in X.

Proof It is enough to show that $\{x_n\}$ is a Cauchy sequence in X. Note that for each $n\in \mathbb{N}$, we have

$$\begin{array}{rcl}\omega (x_n,x_{n+1})& \le & \lambda \omega (x_{n-1},x_n)\\ \le & {\lambda }^2\omega (x_{n-2},x_{n-1})\\ ⋮\\ \le & {\lambda }^n\omega (x_0,x_1).\end{array}$$

Thus

$$\omega (x_n,x_{n+1})\le {\lambda }^n\omega (x_0,x_1).$$

Consequently, for $m\ge n$, we get

$$\begin{array}{rcl}\omega (x_n,x_m)& \le & \omega (x_n,x_{n+1})+\omega (x_{n+1},x_{n+2})+\cdots +\omega (x_{m-1},x_m)\\ \le & {\lambda }^n\omega (x_0,x_1)+{\lambda }^{n+1}\omega (x_0,x_1)+\cdots +{\lambda }^{m-1}\omega (x_0,x_1),\end{array}$$

and thus

$$\omega (x_n,x_m)\le \frac{{\lambda }^n}{1-\lambda }\omega (x_0,x_1).$$

Since $0<\lambda <1$, we have ${\lambda }^n\to 0$ as $n\to \mathrm{\infty }$. And thus by Lemma 1.1, $\{x_n\}$ is a Cauchy sequence in X. Since X is complete, the sequence $\{x_n\}$ converges to a point in X. □

The following lemma is crucial for our main results.

Lemma 2.2 Let $(X,d)$ be a complete metric space, and let $T:X\to Cl(X)$ be a generalized f-contractive map such that $T(X)\subset f(X)$. Then there exists an f-orbit $\{x_n\}$ of T at $x_0\in X$ such that $\{f(x_n)\}$ converges in X.

Proof Let $x_0\in X$ and choose $y_0\in T(x_0)$. Since $T(x_0)\subset f(X)$, then there exists $x_1\in X$ such that $f(x_1)=y_0\in T(x_0)$, and thus, by the definition of T, there exists $y_1\in T(x_1)$ such that

$$\omega (f(x_1),y_1)\le \lambda M_f(x_0,x_1),$$

where $0<\lambda <1$. Since $T(x_1)\subset f(X)$, there exists $x_2\in X$ such that $f(x_2)=y_1\in T(x_1)$. Thus

$$\omega (f(x_1),f(x_2))\le \lambda M_f(x_0,x_1).$$

Similarly, using the definition of T and the fact that $T(X)\subset f(X)$, there exists $x_3\in X$ such that $f(x_3)\in T(x_2)$ and

$$\omega (f(x_2),f(x_3))\le \lambda M_f(x_1,x_2).$$

Continuing this process, we get a sequence $\{x_n\}$ in X such that for all n, $f(x_{n+1})\in T(x_n)$ and

$$\omega (f(x_n),f(x_{n+1}))\le \lambda M_f(x_{n-1},x_n),$$

that is,

$$\begin{array}{rcl}\omega (f(x_n),f(x_{n+1}))& \le & \lambda max\{\omega (f(x_{n-1}),f(x_n)),\omega (f(x_{n-1}),T(x_{n-1})),\omega (f(x_n),T(x_n)),\\ \frac{1}{2}[\omega (f(x_{n-1}),T(x_n))+\omega (f(x_n),T(x_{n-1}))]\}.\end{array}$$

Note that

$$\begin{array}{rcl}\omega (f(x_n),f(x_{n+1}))& \le & \lambda max\{\omega (f(x_{n-1}),f(x_n)),\omega (f(x_{n-1}),f(x_n)),\omega (f(x_n),f(x_{n+1})),\\ \frac{1}{2}[\omega (f(x_{n-1}),f(x_{n+1}))+\omega (f(x_n),f(x_n))]\}\\ =& \lambda max\{\omega (f(x_{n-1}),f(x_n)),\omega (f(x_n),f(x_{n+1})),\frac{1}{2}\left[\omega (f(x_{n-1}),f(x_{n+1}))\right]\},\end{array}$$

and we get

$$\omega (f(x_n),f(x_{n+1}))\le \lambda max\{\omega (f(x_{n-1}),f(x_n)),\frac{1}{2}\left[\omega (f(x_{n-1}),f(x_{n+1}))\right]\}.$$

Also, note that

$$\begin{array}{rcl}\omega (f(x_n),f(x_{n+1}))& \le & \lambda max\{\omega (f(x_{n-1}),f(x_n)),\frac{1}{2}[\omega (f(x_{n-1}),f(x_n))+\omega (f(x_n),f(x_{n+1}))]\}\\ \le & \lambda max\left\{[\omega (f(x_{n-1}),f(x_n)),\omega (f(x_n),f(x_{n+1}))]\right\}.\end{array}$$

Thus, for each $n\in \mathbb{N}$, we get

$$\omega (f(x_n),f(x_{n+1}))\le \lambda \omega (f(x_{n-1}),f(x_n)).$$

(1)

Since the sequence $\{f(x_n)\}$ is in the complete metric space X satisfying the inequality (1), it follows from Lemma 2.1 that $\{f(x_n)\}$ converges in X. □

Remark 2.1 Since for each $n\in \mathbb{N}$ we have

$$\omega (f(x_n),f(x_{n+1}))\le \lambda \omega (f(x_{n-1}),f(x_n)),$$

following the proof of Lemma 2.1, we obtain the following two useful inequalities.

$$\omega (f(x_n),f(x_{n+1}))\le {\lambda }^n\omega (f(x_0),f(x_1))$$

(2)

and for $m\ge n$

$$\omega (f(x_n),f(x_m))\le \frac{{\lambda }^n}{1-\lambda }\omega (f(x_0),f(x_1)).$$

(3)

Without using the extended Hausdorff metric and continuity conditions, we prove a coincidence result which improves many known results including Theorem 1.2 due to [2], Theorem 3.3 in [17] and Theorem 2 in [7].

Theorem 2.1 Suppose that all the hypotheses of Lemma  2.2 hold. Furthermore, if for every $y\in X$ with $f(y)\notin T(y)$

$$inf\{\omega (f(x),y)+\omega (f(x),T(x)):x\in X\}>0.$$

Then $C(f\cap T)\ne \mathrm{\varnothing }$.

Proof By Lemma 2.2, there exists an f-orbit $\{x_n\}$ of T at $x_0\in X$ such that $\{f(x_n)\}$ converges in X. Also note that for each $n\in \mathbb{N}$, we have

$$\omega (f(x_n),f(x_{n+1}))\le \lambda \omega (f(x_{n-1}),f(x_n)),$$

where $0<\lambda <1$. Let $f(x_n)\to y\in X$. Now since $\omega (f(x_n),\cdot )$ is lower semicontinuous, from Remark 2.1 (2), we have

$$\begin{array}{rcl}\omega (f(x_n),y)& \le & \underset{m\to \mathrm{\infty }}{lim}inf\omega (f(x_n),f(x_m))\\ \le & \frac{{\lambda }^n}{1-\lambda }\omega (f(x_0),f(x_1)).\end{array}$$

Since $\lambda <1$, we get $\omega (f(x_n),y)\to 0$ as $n\to \mathrm{\infty }$. Assume that $f(y)\notin T(y)$, then from the hypothesis and Remark 2.1, we get

$$\begin{array}{rcl}0& <& inf\{\omega (f(x),y)+\omega (f(x),T(x)):x\in X\}\\ \le & inf\{\omega (f(x_n),y)+\omega (f(x_n),T(x_n)):n\in \mathbb{N}\}\\ \le & inf\{\omega (f(x_n),y)+\omega (f(x_n),f(x_{n+1})):n\in \mathbb{N}\}\\ \le & inf\{\frac{{\lambda }^n}{1-\lambda }\omega (f(x_o),f(x_1))+{\lambda }^n\omega (f(x_0),f(x_1)):n\in \mathbb{N}\}\\ =& \left\{\frac{2-\lambda }{1-\lambda }\right\}\omega (f(x_0),f(x_1))inf\{{\lambda }^n:n\in \mathbb{N}\}=0,\end{array}$$

which is impossible, and thus $f(y)\in T(y)$, that is, y is a coincidence point of f and T. □

If we take $f=I$ (an identity map on X) in Theorem 2.1, we obtain the following improved version of the corresponding fixed point results in [12, 17, 18].

Corollary 2.1 Let $(X,d)$ be a complete metric space, let ω be a w-distance on X, and let $T:X\to Cl(X)$ be a multivalued map satisfying the following:

(I) for fixed $\lambda \in (0,1)$, for each $x,y\in X$ and $u\in T(x)$, there exists $v\in T(y)$ such that

$$\omega (u,v)\le \lambda M_{\omega }(x,y),$$

where

$$M_{\omega }(x,y)=max\{\omega (x,y),\omega (x,T(x)),\omega (y,T(y)),\frac{1}{2}[\omega (x,T(y))+\omega (y,T(x))]\},$$

(II) $inf\{\omega (x,y)+\omega (x,T(x)):x\in X\}>0$.

Then $Fix(T)\ne \mathrm{\varnothing }$.

Finally, we obtain a common fixed point result.

Theorem 2.2 Suppose that all the hypotheses of Theorem  2.1 hold. Further, if the maps f and T commute weakly and satisfy the condition that $f(x)\ne f^2(x)$, which implies $f(x)\notin T(x)$, then f and T have a common fixed point.

Proof From Theorem 2.1 we have $f(y)\in T(y)$, and thus we get $f(y)=f^2(y)$. Note that

$$f(y)=f(f(y))\in f(T(y))\subseteq T(f(y)),$$

that is, $f(y)$ is a fixed point of T. Also note that $f(y)$ is a fixed point of f and thus $f(y)$ is a common fixed point of T and f. □