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Some New Weakly Contractive Type Multimaps and Fixed Point Results in Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 412898 (2010)
Abstract
Some new weakly contractive type multimaps in the setting of metric spaces are introduced, and we prove some results on the existence of fixed points for such maps under certain conditions. Our results extend and improve several known results including the corresponding recent fixed point results of Pathak and Shahzad (2009), Latif and Abdou (2009), Latif and Albar (2008), Cirić (2008), Feng and Liu (2006), and Klim and Wardowski (2007).
1. Introduction
Let be a metric space. Let denote a collection of nonempty subsets of , a collection of nonempty closed subsets of and a collection of nonempty closed bounded subsets of Let be the Hausdorff metric with respect to , that is,
for every where
An element is called a fixed point of a multivalued map (multimap) if . We denote
A sequence in is called an orbit of at if for all . A map is called lower semicontinuous if for any sequence such that we have .
Using the concept of Hausdorff metric, Nadler [1] established the following multivalued version of the Banach contraction principle.
Theorem 1.1.
Let be a complete metric space and let be a map such that for a fixed constant and for each
Then
This result has been generalized in many directions. For instance, Mizoguchi and Takahashi [2] have obtained the following general form of the Nadler's theorem.
Theorem 1.2.
Let be a complete metric space and let . Assume that there exists a function such that for every
and for all
Then
Many authors have been using the Hausdorff metric to obtain fixed point results for multivalued maps. But, in fact, for most cases the existence part of the results can be proved without using the concept of Hausdorff metric. Recently, Feng and Liu [3] extended Nadler's fixed point theorem without using the concept of the Hausdorff metric. They proved the following result.
Theorem 1.3.
Let be a complete metric space and let be a map such that for any fixed constants and for each there is satisfying the following conditions:
Then provided that a realvalued function on , is lower semicontinuous.
Recently, Klim and Wardowski [4] generalized Theorem 1.3 as follows.
Theorem 1.4.
Let be a complete metric space and let . Assume that the following conditions hold:
(I)if there exist a number and a function such that for each ,
(II)for any there is satisfying
Then provided that a realvalued function on , is lower semicontinuous.
The above results have been generalized in many directions; see for instance [5–9] and references therein.
In [10], Kada et al. introduced the concept of distance on a metric space as follows.
A function is called  on if it satisfies the following for each :
a map is lower semicontinuous; that is, if there is a sequence in with , then ;
for any there exists such that and imply
Note that, in general for , and neither of the implications necessarily hold. Clearly, the metric is a distance on . Let be a normed space. Then the functions defined by and for all are distances [10]. Many other examples and properties of the distance can be found in [10, 11].
The following lemmas concerning distance are crucial for the proofs of our results.
Lemma 1.5 (see [10]).
Let and be sequences in and let and be sequences in converging to Then, for the distance on the following conditions hold for every :
(a)if and for any then in particular, if and then ;
(b)if and for any then converges to ;
(c)if for any with then is a Cauchy sequence;
(d)if for any then is a Cauchy sequence.
Lemma 1.6 (see [12]).
Let be a closed subset of and be a distance on Suppose that there exists such that . Then where
Using the concept of distance, the authors of this paper most recently extended and generalized Theorem 1.4 and [8, Theorem ] as follows.
Theorem 1.7 (see [13]).
Let be a complete metric space with a distance . Let be a multivalued map satisfying that for any constant and for each there is such that
where and is a function from to with for every Suppose that a realvalued function on defined by is lower semicontinuous. Then there exists such that Further, if then .
Let Let satisfy that
(i) and for each
(ii) is nondecreasing on
(iii) is subadditive; that is,
We define
Remark 1.8.

(a)
It follows from (ii) property of that for each
(1.10)

(b)
If and is continuous at , then due to the following two facts must be continuous at each point of . First, every subadditive and continuous function at 0 such that is right upper and left lower semicontinuous [14]. Second, each nondecreasing function is left upper and right lower semicontinuous.

(c)
For any and for each sequence in satisfying we have
For a metric space , we denote In the sequel, we consider if and if
Assuming that the function is continuous and satisfies the conditions (i) and (ii) above, Zhang [15] proved some fixed point results for singlevalued maps which satisfy some contractive type condition involving such function . Recently, using Pathak and Shahzad [9] generalized Theorem 1.4.
In this paper, we prove some results on the existence of fixed points for contractive type multimaps involving the function where and the function is a distance on a metric space Our results either generalize or improve several known fixed point results in the setting of metric spaces, (see Remarks 2.3 and 2.6).
2. The Results
Theorem 2.1.
Let be a complete metric space with a distance Let be a multimap. Assume that the following conditions hold:
(I)there exist a number and a function such that for each
(II)there exists a function such that for any there exists satisfying
(III)the map defined by is lower semicontinuous.
Then there exists such that Further if then
Proof.
Let be any initial point. Then from (II) we can choose such that
From (2.3) and (2.4), we have
Similarly, there exists such that
From (2.6) and (2.7), we obtain
From (2.4) and (2.6), it follows that
Continuing this process, we get an orbit of at satisfying the following:
From (2.10), we get
Note that for each ,
Thus the sequences of nonnegative real numbers and are decreasing. Now, since is nondecreasing, it follows that and are decreasing sequences and are bounded from below, thus convergent. Now, by the definition of the function there exists such that
Thus, for any there exists such that
and thus for all we have
Also, it follows from (2.11) that for all
where Note that for all we have
Thus
where Now, since we have and we get the decreasing sequence converging to . Thus we have
Note that for all
where Now, for any
Clearly, and thus we get that
that is, is Cauchy sequence in . Due to the completeness of , there exists some such that . Due to the fact that the function is lower semicontinuous and (2.19), we have
thus, Since and is closed, it follows from Lemma 1.6 that
If we consider a constant map in Theorem 2.1, then we obtain the following result.
Corollary 2.2.
Let be a complete metric space with a distance . Let be a multimap satisfying that for any constants and for each there is such that
where Suppose that a realvalued function on defined by is lower semicontinuous. Then there exists such that Further, if then .
Remark 2.3.

(a)
Theorem 2.1 extends and generalizes Theorem 1.7. Indeed, if we consider for each in Theorem 2.1, then we can get Theorem 1.7 due to Latif and Abdou [13, Theorem ].

(b)
Theorem 2.1 contains Theorem of Pathak and Shahzad [9] as a special case.

(c)
Corollary 2.2 extends and generalizes Theorem of Latif and Albar [8].
We have also the following fixed point result which generalizes [13, Theorem ].
Theorem 2.4.
Suppose that all the hypotheses of Theorem 2.1 except (III) hold. Assume that
for every with and the function is continuous at . Then
Proof.
Following the proof of Theorem 2.1, there exists a Cauchy sequence and such that and Since is lower semicontinuous, it follows, from the proof of Theorem 2.1 that for all we have
where Since for all and the function is nondecreasing, we have
and thus by using (2.20), we get
Assume that Then, we have
which is impossible and hence
Theorem 2.5.
Let be a complete metric space with a distance Let be a multimap. Assume that the following conditions hold.
(I)there exists a function such that for each
(II)there exists a function such that for any there exists satisfying
(III)the map defined by is lower semicontinuous.
Then there exists such that Further if then
Proof.
Let be any initial point. Following the same method as in the proof of Theorem 2.1, we obtain the existence of a Cauchy sequence such that satisfying
Consequently, there existssuch that . Since is lower semicontinuous, we have
thus, Further by the closedness of and since it follows from Lemma 1.6 that
Remark 2.6.
Theorem 2.5 extends and generalizes fixed point results of Klim and Wardowski [4, Theorem ], Cirić [5, Theorem ], and improves fixed point result of Pathak and Shahzad [9, Theorem ].
Following the same method as in the proof of Theorem 2.4, we can obtain the following fixed point result.
Theorem 2.7.
Suppose that all the hypotheses of Theorem 2.5 except (III) hold. Assume that
for every with and the function is continuous at then
Now we present an example which satisfies all the conditions of the main results, namely, Theorems 2.1 and 2.5 and thus the set of fixed points of is nonempty.
Example 2.8.
Let with the usual metric Define a distance function , by
Let be defined as
Note that Let Define a function by Clearly, Define by
Note that
Clearly, is lower semicontinuous. Note that for each we have
Thus, for , satisfies all the conditions of Theorem 2.1. Now, let then we have Clearly, there exists such that Now
Thus, all the hypotheses of Theorem 2.1 are satisfied and clearly we have Now, if we consider , then all the hypotheses of Theorem 2.5 are also satisfied. Note that in the above example the distance is not a metric .
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Latif, A., Abdou, A.A.N. Some New Weakly Contractive Type Multimaps and Fixed Point Results in Metric Spaces. Fixed Point Theory Appl 2009, 412898 (2010). https://doi.org/10.1155/2009/412898
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DOI: https://doi.org/10.1155/2009/412898
Keywords
 Initial Point
 Fixed Point Theorem
 Lower Semicontinuous
 Cauchy Sequence
 Nondecreasing Function