Abstract
Without using the concept of Hausdorff metric, we prove some results on the existence of fixed points for generalized contractive multivalued maps with respect to u-distance. Consequently, several known fixed point results are either generalized or improved.
MSC:47H10, 54H25.
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1 Introduction
Using the concept of Hausdorff metric, Nadler [11] introduced the notion of multivalued contraction maps and proved a multivalued version of the well-known Banach contraction principle, which states that each closed bounded valued contraction map on a complete metric space has a fixed point. Since then various fixed point results concerning multivalued contractions have appeared.
Without using the concept of Hausdorff metric, most recently Feng and Liu [4] extended Nadler’s fixed point result, while Klim and Wardowski [7] generalized their corresponding fixed point result in [4].
In [6] Kada et al. introduced the notion of w-distance and improved several classical results including Caristi’s fixed point theorem. Suzuki and Takahashi [12] introduced single-valued and multivalued weakly contractive maps with respect to w-distance and proved fixed point results for such maps. Consequently, they generalized the Banach contraction principle and Nadler’s fixed point result. Generalizing the concept of w-distance, Suzuki [13] introduced the notion of τ-distance on a metric space and improved several classical results including the corresponding results of Suzuki and Takahashi [12]. In literature, several other kinds of distances and various versions of known results have appeared. For example, see [1–3, 10] and references therein. Most recently, Ume [14] generalized the notion of τ-distance by introducing the concept of u-distance.
In this paper, first we prove our key lemma for multivalued general contractive maps with respect to u-distance and then prove some results on the existence of fixed points for such multivalued maps. Consequently, several known fixed point results get either improved or generalized including the corresponding results of Feng and Liu [4], Klim and Wardowski [7], Latif and Albar [9], Suzuki and Takahashi [12], Latif and Abdou [8], and Nadler [11].
Let be a metric space. We denote the collection of nonempty subsets of X, nonempty closed subsets of X and nonempty closed bounded subsets of X by , and respectively. Let H be the Hausdorff metric with respect to d, that is,
for every , where .
A point is called a fixed point of if . We denote .
A sequence in X is called an orbit of T at if for all . A map is called T-orbitally lower semicontinuous if, for any orbit of T and , imply that .
Most recently, Ume [14] generalized the notion of τ-distance by introducing u-distance as follows.
A function is called a u-distance on X if there exists a function such that the following hold for :
;
and for each , and for every , there exists such that , , and imply
imply
imply
or
imply
imply
or
imply
Remark 1.1 [14]
-
(a)
Suppose that θ from into is a mapping satisfying . Then there exists a mapping η from into such that η is nondecreasing in its third and fourth variable respectively, satisfying , where stand for substituting η for θ in respectively.
-
(b)
In the light of (a), we may assume that θ is nondecreasing in its third and fourth variables, respectively, for a function θ from into satisfying .
-
(c)
Each τ-distance p on a metric space is also a u-distance on X.
We present some examples of u-distances which are not τ-distances. For details, see [14].
Example 1.2 Let with the usual metric. Define by . Then p is a u-distance on X but not a τ-distance on X.
Example 1.3 Let X be a normed space with norm . Then a function defined by for every is a u-distance on X but not a τ-distance.
It follows from the above examples and Remark 1.1(c) that u-distance is a proper extension of τ-distance. Other useful examples on u-distance are also given in [14].
Let be a metric space and let p be a u-distance on X. A sequence in X is called p-Cauchy [14] if there exists a function θ from into satisfying and a sequence of X such that
or
The following lemmas concerning u-distance are crucial for our results.
Lemma 1.4 [14]
Let be a metric space and let p be a u-distance on X. If is a p-Cauchy sequence in X, then is a Cauchy sequence.
Lemma 1.5 [5]
Let be a metric space and let p be a u-distance on X. If is a p-Cauchy sequence and is a sequence satisfying
then is also a p-Cauchy sequence and .
Lemma 1.6 [14]
Let be a metric space and let p be a u-distance on X. Suppose that a sequence of X satisfies
or
Then is a p-Cauchy sequence.
We say that a multivalued map is generalized p-contractive if there exist a u-distance p on X and a constant such that, for any , there is satisfying
where and k is a function from to with for each .
Note that if we take , then the definition of a generalized p-contractive map reduces to the definition of a generalized contractive map due to Klim and Wardowski [7]. In particular, if we take and a constant map , , then the generalized p-contractive map T reduces to the definition of contractive maps due to Feng and Liu [4].
2 The results
First, we prove our key lemma in the setting of metric spaces.
Lemma 2.1 Let be a metric space. Let be a generalized p-contractive map. Then, there exists an orbit of T at such that the sequence of nonnegative real numbers is decreasing to zero and is a Cauchy sequence.
Proof Since for any , is nonempty for any constant . Let be an arbitrary but fixed element of X, there exists such that
Using (16) and (17), we have
Similarly, there is such that
Using (19) and (20), we have
From (16) and (20), it follows that
Continuing this process, we get an orbit of T in X such that ,
Using (23) and (24), we get
and thus for each n
Note that the sequences and are decreasing and bounded, thus convergent. Now, by definition of the function k there exists such that
Thus for any , there exists such that for each
and thus for each , we have
Also, it follows from (25) that for each ,
where . Using (23) and (24), for each , we have
and thus
Now, since , we have , and hence the decreasing sequence converges to 0. Now, we show that is a Cauchy sequence. Note that for each
where . Thus, for any , , we get
and hence
Thus, by Lemma 1.6, is a p-Cauchy sequence, and hence by Lemma 1.4, is a Cauchy sequence. □
Applying Lemma 2.1, we obtain the following main fixed point result.
Theorem 2.2 Let be a complete metric space and let be a generalized p-contractive map. Suppose that a real valued function f on X defined by is T-orbitally lower semicontinuous. Then there exists such that and .
Proof By Lemma 2.1, there exists a Cauchy sequence in X such that the decreasing sequence converges to 0. Due to the completeness of X, there exists some such that . Since f is T-orbitally lower semicontinuous, we have
that is, . Thus there exists a sequence such that . It follows that
Since is a p-Cauchy sequence, it follows from (36) and Lemma 1.5 that is also a p-Cauchy sequence and . Thus, by Lemma 1.4, is a Cauchy sequence in the complete space. Due to the closedness of , there exists such that . Consequently, using we get
and thus . But, since , and , we have . Hence and . □
As a consequence of Theorem 2.2, we obtain the following fixed point result of Klim and Wardowski [7], Theorem 2.1] which contains the fixed point result of Feng and Liu [4], Theorem 3.1] and Nadler’s fixed point theorem.
Corollary 2.3 Let be a complete metric space and let be a multivalued map such that a real-valued function f on X defined by is lower semi-continuous. If there exists such that for any there is satisfying
where ; and k is a function from to with , for every . Then .
Remarks 2.4
-
(a)
Theorem 2.2 also generalizes fixed point theorems of Latif and Abdou [8], Theorem 2.2], Suzuki [13], Theorem 2], Suzuki and Takahashi [12], Theorem 1].
-
(b)
It is worth mentioning that in the proofs of [7], Theorem 2.1], [4], Theorem 3.1], and [8], Theorem 2.2] a full force of the lower semicontinuity of the real-valued function f is not used, but in fact T-orbitally lower semicontinuity of f is enough to obtain the conclusions.
Applying Lemma 2.1, we also obtain the following fixed point result for a multivalued generalized p-contractive map where we use another suitable condition.
Theorem 2.5 Let X be a complete metric space with metric d and let be a generalized p-contractive map. Assume that
for every with . Then .
Proof By Lemma 2.1, there exists an orbit of T which is a Cauchy sequence in a complete metric space X, so there exists some such that . Thus, using and (34), we have
where . Also, we get
Assume that . Then we have
which is impossible, and hence . □
Remarks 2.6 Theorem 2.5 generalizes [8], Theorem 2.4] and [9], Theorem 3.3].
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Acknowledgements
The authors thank the referees for their valuable comments. This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No.9-843-D1432. The authors, therefore, acknowledge with thanks technical and financial support of DSR.
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Bin Dehaish, B.A., Latif, A. Fixed point theorems for generalized contractive type multivalued maps. Fixed Point Theory Appl 2012, 135 (2012). https://doi.org/10.1186/1687-1812-2012-135
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DOI: https://doi.org/10.1186/1687-1812-2012-135