Open Access

Fixed Point Results in Quasimetric Spaces

Fixed Point Theory and Applications20102011:178306

https://doi.org/10.1155/2011/178306

Received: 21 August 2010

Accepted: 5 October 2010

Published: 14 October 2010

Abstract

In the setting of quasimetric spaces, we prove some new results on the existence of fixed points for contractive type maps with respect to -function. Our results either improve or generalize many known results in the literature.

1. Introduction and Preliminaries

Let be a metric space with metric . We use to denote the collection of all nonempty subsets of for the collection of all nonempty closed subsets of for the collection of all nonempty closed bounded subsets of and for the Hausdorff metric on that is,
(1.1)

where is the distance from the point to the subset

For a multivalued map , we say

is contraction [1] if there exists a constant , such that for all ,
(1.2)
T is weakly contractive [2] if there exist constants , such that for any , there is satisfying
(1.3)

where .

A point is called a fixed point of a multivalued map if . We denote

A sequence in is called an of at if for all integer . A real valued function on is called lower semicontinuous if for any sequence with implies that

Using the Hausdorff metric, Nadler Jr. [1] has established a multivalued version of the well-known Banach contraction principle in the setting of metric spaces as follows.

Theorem 1.1.

Let be a complete metric space, then each contraction map has a fixed point.

Without using the Hausdorff metric, Feng and Liu [2] generalized Nadler's contraction principle as follows.

Theorem 1.2.

Let be a complete metric space and let be a weakly contractive map, then has a fixed point in provided the real valued function on is a lower semicontinuous.

In [3], Kada et al. introduced the concept of -distance in the setting of metric spaces as follows.

A function is called a -distance on if it satisfies the following:

(w1) for all

(w2) is lower semicontinuous in its second variable;

(w3)for any there exists , such that and imply

Note that in general for , and not either of the implications necessarily holds. Clearly, the metric is a -distance on . Many other examples and properties of -distances are given in [3].

In [4], Suzuki and Takahashi improved Nadler contraction principle (Theorem 1.1) as follows.

Theorem 1.3.

Let be a complete metric space and let . If there exist a -distance on and a constant , such that for each and , there is satisfying
(1.4)

then has a fixed point.

Recently, Latif and Albar [5] generalized Theorem 1.2 with respect to -distance (see, Theorem in [5]), and Latif [6] proved a fixed point result with respect to -distance ( see, Theorem in [6]) which contains Theorem 1.3 as a special case.

A nonempty set together with a quasimetric (i.e., not necessarily symmetric) is called a quasimetric space. In the setting of a quasimetric spaces, Al-Homidan et al. [7] introduced the concept of a -function on quasimetric spaces which generalizes the notion of a -distance.

A function is called a -function on if it satisfies the following conditions:

(Q1) for all

(Q2)If is a sequence in such that and for , for some , then

(Q3)for any there exists , such that and imply

Note that every -distance is a -function, but the converse is not true in general [7]. Now, we state some useful properties of -function as given in [7].

Lemma 1.4.

Let be a complete quasimetric space and let be a -function on . Let and be sequences in . Let and be sequences in converging to , then the following hold for any :

(i)if and for all then in particular, if and , then

(ii)if and for all then converges to ;

(iii)if for any with then is a Cauchy sequence;

(iv)if for any then is a Cauchy sequence.

Using the concept -function, Al-Homidan et al. [7] recently studied an equilibrium version of the Ekeland-type variational principle. They also generalized Nadler's fixed point theorem (Theorem 1.1) in the setting of quasimetric spaces as follows.

Theorem 1.5.

Let be a complete quasimetric space and let . If there exist -function on and a constant , such that for each and , there is satisfying
(1.5)

then has a fixed point.

In the sequel, we consider as a quasimetric space with quasimetric .

Considering a multivalued map , we say
  1. (c)
    is weakly -contractive if there exist - function on and constants , , such that for any , there is satisfying
    (1.6)
     
where and
  1. (d)
    is generalized -contractive if there exists a - function on , such that for each and , there is satisfying
    (1.7)
     

where is a function of to , such that for all

Clearly, t he class of weakly - contractive maps contains the class of weakly contractive maps, and the class of generalized -contractive maps contains the classes of generalized -contraction maps [6], -contractive maps [4], and -contractive maps [7].

In this paper, we prove some new fixed point results in the setting of quasimetric spaces for weakly -contractive and generalized -contractive multivalued maps. Consequently, our results either improve or generalize many known results including the above stated fixed point results.

2. The Results

First, we prove a fixed point theorem for weakly -contractive maps in the setting of quasimetric spaces.

Theorem 2.1.

Let be a complete quasimetric space and let be a weakly - contractive map. If a real valued function on is lower semicontinuous, then there exists , such that Further, if then is a fixed point of .

Proof.

Let Since is weakly contractive, there is , such that
(2.1)
where Continuing this process, we can get an orbit of at satisfying and
(2.2)
Since and thus we get
(2.3)
If we put , then also we have
(2.4)
Thus, we obtain
(2.5)
and since , hence the sequence which is decreasing, converges to 0. Now, we show that is a Cauchy sequence. Note that
(2.6)
Now, for any integer with , we have
(2.7)
and thus by Lemma 1.4, is a Cauchy sequence. Due to the completeness of , there exists some , such that Now, since is lower semicontinuous, we have
(2.8)

and thus, It follows that there exists a sequence in , such that Now, if then by Lemma 1.4, . Since is closed, we get

Now, we prove the following useful lemma.

Lemma 2.2.

Let be a complete quasimetric space and let be a generalized -contractive map, then there exists an orbit of at , such that the sequence of nonnegative numbers is decreasing to zero and is a Cauchy sequence.

Proof.

Let be an arbitrary but fixed element of and let . Since is generalized as a -contractive, there is , such that
(2.9)
Continuing this process, we get a sequence in , such that and
(2.10)
Thus, for all , we have
(2.11)
Write . Suppose that , then we have
(2.12)
Now, taking limits as on both sides, we get
(2.13)
which is not possible, and hence the sequence of nonnegative numbers which is decreasing, converges to 0. Finally, we show that is a Cauchy sequence. Let . There exists real number such that . Then for sufficiently large , , and thus for sufficiently large , we have Consequently, we obtain , that is,
(2.14)
Now, for any integers ,
(2.15)

and thus by Lemma 1.4, is a Cauchy sequence.

Applying Lemma 2.2, we prove a fixed point result for generalized -contractive maps.

Theorem 2.3.

Let be a complete quasimetric space then each generalized q -contractive map has a fixed point.

Proof.

It follows from Lemma 2.2 that there exists a Cauchy sequence in such that the decreasing sequence converges to 0. Due to the completeness of , there exists some such that Let be arbitrary fixed positive integer then for all positive integers with , we have
(2.16)
Let , then . Now, note that
(2.17)
Since was arbitrary fixed, we have
(2.18)
Note that converges to . Now, since and is a generalized -contractive map, then there is , such that
(2.19)
And for large , we obtain
(2.20)
thus, we get
(2.21)

Thus, it follows from Lemma 1.4 that . Since is closed, we get

Corollary 2.4.

Let be a complete quasimetric space and a -function on . Let be a multivalued map, such that for any and , there is with
(2.22)

where is a monotonic increasing function from to , then has a fixed point.

Finally, we conclude with the following remarks concerning our results related to the known fixed point results.

Remark 2.5.

(1)Theorem 2.1 generalizes Theorem 1.2 according to Feng and Liu [2] and Latif and Albar [5, Theorem ].

(2)Theorem 2.3 generalizes Theorem 1.3 according to Suzuki and Takahashi [4] and Theorem 1.5 according to Al-Homidan et al. [7] and contains Latif's Theorem in [6].

(3)Theorem 2.3 also generalizes Theorem in [8] in several ways.

(4)Corollary 2.4 improves and generalizes Theorem in [9].

Declarations

Acknowledgments

The authors thank the referees for their kind comments. The authors also thank King Abdulaziz University and the Deanship of Scientific Research for the research Grant no. 3-35/429.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University

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Copyright

© A. Latif and S. A. Al-Mezel. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.