Open Access

f-Contractive multivalued maps and coincidence points

Journal of Inequalities and Applications20132013:141

https://doi.org/10.1186/1029-242X-2013-141

Received: 2 October 2012

Accepted: 13 March 2013

Published: 2 April 2013

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.

Keywords

metric spacefixed pointmultivalued contractive mapcoincidence point

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 { sup x A d ( x , B ) , sup y B d ( y , A ) }

for every A , B CB ( X ) , where d ( x , B ) = inf y B d ( x , y ) .

Let f : X X be a single-valued map, and let T : X 2 X be a multivalued map. A point x X is called a fixed point of T if x T ( x ) , and the set of fixed points of T is denoted by Fix ( T ) . A point x X is called a coincidence point of f and T if f ( x ) T ( x ) . We denote by C ( f 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 X if f x n T x n 1 for all n 1 . We say that f and T weakly commute if f T x T f x for all x X . Clearly, commuting maps f and T weakly commute.

A multivalued map T : X CB ( X ) is called

(i) contraction [1] if for a fixed constant λ ( 0 , 1 ) and for each x , y X ,
H ( T ( x ) , T ( y ) ) λ d ( x , y ) .
(ii) f-contraction [2] if for a fixed constant λ ( 0 , 1 ) and for each x , y X ,
H ( T ( x ) , T ( y ) ) λ 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 CB ( X ) be a contraction map. Then Fix ( T ) .

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 CB ( X ) be a multivalued f-contraction map which commutes with a continuous map f. Then C ( f T ) .

This result has been generalized in different directions. For example, see [410].

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 ω : X × X [ 0 , ) is called a w-distance on X if it satisfies the following for each x , y , z X :

( w 1 ) ω ( x , z ) ω ( x , y ) + ω ( y , z ) ;

( w 2 ) a map ω ( x , ) : X [ 0 , ) is lower semicontinuous;

( w 3 ) for any ϵ > 0 , there exists δ > 0 such that ω ( z , x ) δ and ω ( z , y ) δ imply d ( x , y ) ϵ .

Note that, in general, for x , y X , ω ( x , y ) ω ( y , x ) and not either of the implications ω ( x , y ) = 0 x = y necessarily hold. We say the w-distance ω on X is a w 0 -distance if x = y implies ω ( x , y ) = 0 . Clearly, the metric d is a w-distance on X. Let ( Y , ) be a normed space. Then the functions ω 1 , ω 2 : Y × Y [ 0 , ) defined by ω 1 ( x , y ) = y and ω 2 ( x , y ) = x + y for all x , y 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 { α n } and { β n } be sequences in [ 0 , ) converging to zero. Then, for the w-distance ω on X, the following hold for every x , y , z X :

(a) if ω ( x n , y ) α n and ω ( x n , z ) β n for any n N , then y = z ; in particular, if ω ( x , y ) = 0 and ω ( x , z ) = 0 , then y = z ;

(b) if ω ( x n , y n ) α n and ω ( x n , z ) β n for any n N , then { y n } converges to z;

(c) if ω ( x n , x m ) α n for any n , m N with m > n , then { x n } is a Cauchy sequence;

(d) if ω ( y , x n ) α n for any n N , then { x n } is a Cauchy sequence.

For x X and A 2 X , we denote, ω ( x , A ) = inf y A ω ( x , y ) . Now, let T : X Cl ( X ) be a multivalued map, and let f : X X be a single-valued map. We say

(iii) T is w-contractive [12] if there exist a w-distance ω on X and λ ( 0 , 1 ) such that for any x , y X and u T ( x ) , there is v T ( y ) with
ω ( u , v ) λ ω ( x , y ) .
(iv) T is generalized f-contractive if there exist a w 0 -distance ω on X and λ ( 0 , 1 ) such that for any x , y X , u T ( x ) , there is v T ( y ) with
ω ( u , v ) λ M f ( x , y ) ,
where
M f ( x , y ) = max { ω ( f ( x ) , f ( y ) ) , ω ( f ( x ) , T ( x ) ) , ω ( f ( y ) , T ( y ) ) , 1 2 [ ω ( f ( x ) , T ( y ) ) + ω ( f ( y ) , T ( x ) ) ] } .

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 Cl ( X ) , the set Fix ( T ) .

This result has been generalized by many authors, for example, see [1316]. 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 < λ < 1 , such that for all n N ,
ω ( x n , x n + 1 ) λ ω ( 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 N , we have
ω ( x n , x n + 1 ) λ ω ( x n 1 , x n ) λ 2 ω ( x n 2 , x n 1 ) λ n ω ( x 0 , x 1 ) .
Thus
ω ( x n , x n + 1 ) λ n ω ( x 0 , x 1 ) .
Consequently, for m n , we get
ω ( x n , x m ) ω ( x n , x n + 1 ) + ω ( x n + 1 , x n + 2 ) + + ω ( x m 1 , x m ) λ n ω ( x 0 , x 1 ) + λ n + 1 ω ( x 0 , x 1 ) + + λ m 1 ω ( x 0 , x 1 ) ,
and thus
ω ( x n , x m ) λ n 1 λ ω ( x 0 , x 1 ) .

Since 0 < λ < 1 , we have λ n 0 as n . 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 Cl ( X ) be a generalized f-contractive map such that T ( X ) f ( X ) . Then there exists an f-orbit { x n } of T at x 0 X such that { f ( x n ) } converges in X.

Proof Let x 0 X and choose y 0 T ( x 0 ) . Since T ( x 0 ) f ( X ) , then there exists x 1 X such that f ( x 1 ) = y 0 T ( x 0 ) , and thus, by the definition of T, there exists y 1 T ( x 1 ) such that
ω ( f ( x 1 ) , y 1 ) λ M f ( x 0 , x 1 ) ,
where 0 < λ < 1 . Since T ( x 1 ) f ( X ) , there exists x 2 X such that f ( x 2 ) = y 1 T ( x 1 ) . Thus
ω ( f ( x 1 ) , f ( x 2 ) ) λ M f ( x 0 , x 1 ) .
Similarly, using the definition of T and the fact that T ( X ) f ( X ) , there exists x 3 X such that f ( x 3 ) T ( x 2 ) and
ω ( f ( x 2 ) , f ( x 3 ) ) λ 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 ) T ( x n ) and
ω ( f ( x n ) , f ( x n + 1 ) ) λ M f ( x n 1 , x n ) ,
that is,
ω ( f ( x n ) , f ( x n + 1 ) ) λ max { ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n 1 ) , T ( x n 1 ) ) , ω ( f ( x n ) , T ( x n ) ) , 1 2 [ ω ( f ( x n 1 ) , T ( x n ) ) + ω ( f ( x n ) , T ( x n 1 ) ) ] } .
Note that
ω ( f ( x n ) , f ( x n + 1 ) ) λ max { ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n ) , f ( x n + 1 ) ) , 1 2 [ ω ( f ( x n 1 ) , f ( x n + 1 ) ) + ω ( f ( x n ) , f ( x n ) ) ] } = λ max { ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n ) , f ( x n + 1 ) ) , 1 2 [ ω ( f ( x n 1 ) , f ( x n + 1 ) ) ] } ,
and we get
ω ( f ( x n ) , f ( x n + 1 ) ) λ max { ω ( f ( x n 1 ) , f ( x n ) ) , 1 2 [ ω ( f ( x n 1 ) , f ( x n + 1 ) ) ] } .
Also, note that
ω ( f ( x n ) , f ( x n + 1 ) ) λ max { ω ( f ( x n 1 ) , f ( x n ) ) , 1 2 [ ω ( f ( x n 1 ) , f ( x n ) ) + ω ( f ( x n ) , f ( x n + 1 ) ) ] } λ max { [ ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n ) , f ( x n + 1 ) ) ] } .
Thus, for each n N , we get
ω ( f ( x n ) , f ( x n + 1 ) ) λ ω ( 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 N we have
ω ( f ( x n ) , f ( x n + 1 ) ) λ ω ( f ( x n 1 ) , f ( x n ) ) ,
following the proof of Lemma 2.1, we obtain the following two useful inequalities.
ω ( f ( x n ) , f ( x n + 1 ) ) λ n ω ( f ( x 0 ) , f ( x 1 ) )
(2)
and for m n
ω ( f ( x n ) , f ( x m ) ) λ n 1 λ ω ( 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 X with f ( y ) T ( y )
inf { ω ( f ( x ) , y ) + ω ( f ( x ) , T ( x ) ) : x X } > 0 .

Then C ( f T ) .

Proof By Lemma 2.2, there exists an f-orbit { x n } of T at x 0 X such that { f ( x n ) } converges in X. Also note that for each n N , we have
ω ( f ( x n ) , f ( x n + 1 ) ) λ ω ( f ( x n 1 ) , f ( x n ) ) ,
where 0 < λ < 1 . Let f ( x n ) y X . Now since ω ( f ( x n ) , ) is lower semicontinuous, from Remark 2.1 (2), we have
ω ( f ( x n ) , y ) lim m inf ω ( f ( x n ) , f ( x m ) ) λ n 1 λ ω ( f ( x 0 ) , f ( x 1 ) ) .
Since λ < 1 , we get ω ( f ( x n ) , y ) 0 as n . Assume that f ( y ) T ( y ) , then from the hypothesis and Remark 2.1, we get
0 < inf { ω ( f ( x ) , y ) + ω ( f ( x ) , T ( x ) ) : x X } inf { ω ( f ( x n ) , y ) + ω ( f ( x n ) , T ( x n ) ) : n N } inf { ω ( f ( x n ) , y ) + ω ( f ( x n ) , f ( x n + 1 ) ) : n N } inf { λ n 1 λ ω ( f ( x o ) , f ( x 1 ) ) + λ n ω ( f ( x 0 ) , f ( x 1 ) ) : n N } = { 2 λ 1 λ } ω ( f ( x 0 ) , f ( x 1 ) ) inf { λ n : n N } = 0 ,

which is impossible, and thus f ( y ) 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 Cl ( X ) be a multivalued map satisfying the following:

(I) for fixed λ ( 0 , 1 ) , for each x , y X and u T ( x ) , there exists v T ( y ) such that
ω ( u , v ) λ M ω ( x , y ) ,
where
M ω ( x , y ) = max { ω ( x , y ) , ω ( x , T ( x ) ) , ω ( y , T ( y ) ) , 1 2 [ ω ( x , T ( y ) ) + ω ( y , T ( x ) ) ] } ,

(II) inf { ω ( x , y ) + ω ( x , T ( x ) ) : x X } > 0 .

Then Fix ( T ) .

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 ) f 2 ( x ) , which implies f ( x ) T ( x ) , then f and T have a common fixed point.

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

Declarations

Acknowledgements

This paper is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. 491/130/1432. The author therefore acknowledges with thanks DSR for technical and financial support.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University

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