f-Contractive multivalued maps and coincidence points
© Kutbi; licensee Springer 2013
Received: 2 October 2012
Accepted: 13 March 2013
Published: 2 April 2013
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.
for every , where .
Let be a single-valued map, and let be a multivalued map. A point is called a fixed point of T if , and the set of fixed points of T is denoted by . A point is called a coincidence point of f and T if . We denote by the set of coincidence points of f and T.
We say a sequence in X is an f-orbit of T at if for all . We say that f and T weakly commute if for all . Clearly, commuting maps f and T weakly commute.
A multivalued map is called
Using the concept of Hausdorff metric, Nadler  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 
Let be a complete metric space, and let be a contraction map. Then .
Theorem 1.2 
Let be a complete metric space, and let be a multivalued f-contraction map which commutes with a continuous map f. Then .
On the other hand, Kada et al.  introduced the concept of w-distance on a metric space as follows:
Let be a metric space. A function is called a w-distance on X if it satisfies the following for each :
() a map is lower semicontinuous;
() for any , there exists such that and imply .
Note that, in general, for , and not either of the implications necessarily hold. We say the w-distance ω on X is a -distance if implies . Clearly, the metric d is a w-distance on X. Let be a normed space. Then the functions defined by and for all are w-distances . 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 .
Lemma 1.1 
Let be a metric space, and let ω be a w-distance on X. Let and be sequences in X, and let and be sequences in converging to zero. Then, for the w-distance ω on X, the following hold for every :
(a) if and for any , then ; in particular, if and , then ;
(b) if and for any , then converges to z;
(c) if for any with , then is a Cauchy sequence;
(d) if for any , then is a Cauchy sequence.
For and , we denote, . Now, let be a multivalued map, and let be a single-valued map. We say
Using the concept of w-distance, Suzuki and Takahashi  improved Nadler’s fixed point result as follows.
Theorem 1.3 Let be a complete metric space. Then for each w-contractive map , the set .
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 , 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.
Using the concept of w-distance, first we improve a corresponding result of Jungck  as follows.
then the sequence converges in X.
Since , we have as . And thus by Lemma 1.1, is a Cauchy sequence in X. Since X is complete, the sequence converges to a point in X. □
The following lemma is crucial for our main results.
Lemma 2.2 Let be a complete metric space, and let be a generalized f-contractive map such that . Then there exists an f-orbit of T at such that converges in X.
Since the sequence is in the complete metric space X satisfying the inequality (1), it follows from Lemma 2.1 that converges in X. □
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 , Theorem 3.3 in  and Theorem 2 in .
which is impossible, and thus , that is, y is a coincidence point of f and T. □
Corollary 2.1 Let be a complete metric space, let ω be a w-distance on X, and let be a multivalued map satisfying the following:
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 , which implies , then f and T have a common fixed point.
that is, is a fixed point of T. Also note that is a fixed point of f and thus is a common fixed point of T and f. □
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.
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