Some New Weakly Contractive Type Multimaps and Fixed Point Results in Metric Spaces

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,

(1.1)

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

(1.2)

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

(1.3)

and for all

(1.4)

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:

(1.5)

Then provided that a real-valued 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 ,

(1.6)

(II)for any there is satisfying

(1.7)

Then provided that a real-valued 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

(1.8)

where and is a function from to with for every Suppose that a real-valued 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,

(1.9)

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 sub-additive 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 single-valued 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

(2.1)

(II)there exists a function such that for any there exists satisfying

(2.2)

(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

(2.3)

(2.4)

From (2.3) and (2.4), we have

(2.5)

Similarly, there exists such that

(2.6)

(2.7)

From (2.6) and (2.7), we obtain

(2.8)

From (2.4) and (2.6), it follows that

(2.9)

Continuing this process, we get an orbit of at satisfying the following:

(2.10)

From (2.10), we get

(2.11)

Note that for each ,

(2.12)

Thus the sequences of non-negative 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

(2.13)

Thus, for any there exists such that

(2.14)

and thus for all we have

(2.15)

Also, it follows from (2.11) that for all

(2.16)

where Note that for all we have

(2.17)

Thus

(2.18)

where Now, since we have and we get the decreasing sequence converging to . Thus we have

(2.19)

Note that for all

(2.20)

where Now, for any

(2.21)

Clearly, and thus we get that

(2.22)

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

(2.23)

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

(2.24)

where Suppose that a real-valued 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

(2.25)

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

(2.26)

where Since for all and the function is nondecreasing, we have

(2.27)

and thus by using (2.20), we get

(2.28)

Assume that Then, we have

(2.29)

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

(2.30)

(II)there exists a function such that for any there exists satisfying

(2.31)

(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

(2.32)

Consequently, there existssuch that . Since is lower semicontinuous, we have

(2.33)

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

(2.34)

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

(2.35)

Let be defined as

(2.36)

Note that Let Define a function by Clearly, Define by

(2.37)

Note that

(2.38)

Clearly, is lower semicontinuous. Note that for each we have

(2.39)

Thus, for , satisfies all the conditions of Theorem 2.1. Now, let then we have Clearly, there exists such that Now

(2.40)

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 .