1. Introduction

Throughout this paper, unless otherwise specified, is a metric space with metric . Let , and denote the collection of nonempty subsets of , nonempty closed subsets of , and nonempty closed bounded subsets of , respectively. Let be the Hausdorff metric on , that is,

(1.1)

A multivalued map is called

(i)contraction [1] if for a fixed constant and for each

(1.2)

(ii)generalized contraction [2] if for any

(1.3)

where is a function from to with for every ;

(iii)contractive [3] if there exist constants such that for any there is satisfying

(1.4)

where ;

(iv)generalized contractive [4] if there exist such that for any there is satisfying

(1.5)

where is a function from to with for every

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

A sequence in is called an of at if for all . A map is called lower semicontinuous if for any sequence with imply that .

Using the concept of Hausdorff metric, Nadler Jr. [1] 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 (see [1]).

Let be a complete space and let be a contraction map. 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 (see [2]).

Let be a complete space and let be a generalized contraction map. Then

Another extension of Nadler's result obtained recently by Feng and Liu [3]. Without using the concept of the Hausdorff metric, they proved the following result.

Theorem 1.3 (see [3]).

Let X be a complete space and let be a multivalued contractive map. Suppose that a real-valued function on , , is lower semicontinuous. Then

Most recently, Klim and Wardowski [4] generalized Theorem 1.3 as follows:

Theorem 1.4 (see [4]).

Let X be a complete metric space and let be a multivalued generalized contractive map such that a real-valued function on , is lower semicontinuous. Then

Recently, Kada et al. [5] introduced the concept of -distance on a metric space as follows.

A function is called - on if it satisfies the following for any :

()

() a map is lower semicontinuous;

() for any there exists such that and imply

Using the concept of -distance, they improved Caristi's fixed point theorem, Ekland's variational principle, and Takahashi's existence theorem. In [6], Susuki and Takahashi proved a fixed point theorem for contractive type multivalued maps with respect to -distance. See also [7–12].

Let us give some examples of -distance [5].

  1. (a)

    The metric is a -distance on .

  2. (b)

    Let be normed space with norm Then the functions defined by and for every , are -distance.

The following lemmas concerning -distance are crucial for the proofs of our results.

Lemma 1.5 (see [5]).

Let and be sequences in and let and be sequences in converging to Then, for the w-distance on the following hold for every :

(a)if and for any then in particular, if and then ;

  1. (b)

    if and for any then converges to ;

  2. (c)

    if for any with then is a Cauchy sequence;

  3. (d)

    if for any then is a Cauchy sequence.

Lemma 1.6 (see [9]).

Let be a closed subset of and let be a w-distance on Suppose that there exists such that . Then (where )

We say a multivalued map is generalized -contractive if there exist a -distance on and a constant such that for any there is satisfying

(1.6)

where and is a function from to with for every

Note that if we take then the definition of generalized -contractive map reduces to the definition of generalized contractive map due to Klim and Wardowski [4]. In particular, if we take a constant map then the map is weakly contractive (in short, -contractive) [8], and further if we take then we obtain and is contractive [3].

In this paper, using the concept of -distance, we first establish key lemma and then obtain fixed point results for multivalued generalized -contractive maps not involving the extended Hausdorff metric. Our results either generalize or improve a number of fixed point results including the corresponding results of Feng and Liu [3], Latif and Albar [8], and Klim and Wardowski [4].

2. Results

First, we prove key lemma in the setting of metric spaces.

Lemma 2.1.

Let be a generalized -contractive map. Then, there exists an orbit of in such that the sequence of nonnegative real numbers is decreasing to zero and the sequence is Cauchy.

Proof.

Since for each , is closed, the set is nonempty for any Let be an arbitrary but fixed element of . Since is generalized -contractive, there is such that

(2.1)
(2.2)

Using (2.1) and (2.2), we have

(2.3)

Similarly, there is such that

(2.4)
(2.5)

Using (2.4) and (2.5), we have

(2.6)

From (2.5) and (2.1), it follows that

(2.7)

Continuing this process, we get an orbit of in such that

(2.8)

Using (2.8), we get

(2.9)

and thus for all

(2.10)
(2.11)

Note that the sequences and are decreasing, and thus convergent. Now, by the definition of the function there exists such that

(2.12)

Thus, for any there exists such that

(2.13)

and thus for all we have

(2.14)

Also, it follows from (2.9) that for all ,

(2.15)

where Note that for all we have

(2.16)

and thus

(2.17)

Now, since we have and hence the decreasing sequence converges to . Now, we show that is a Cauchy sequence. Note that for all ,

(2.18)

where Now, for any

(2.19)

and thus by Lemma 1.5, is a Cauchy sequence.

Using Lemma 2.1, we obtain the following fixed point result which is an improved version of Theorem 1.4 and contains Theorem 1.3 as a special case.

Theorem 2.2.

Let be a complete space and let be a generalized -contractive map. Suppose that a real-valued function on defined by is lower semicontinous. Then there exists such that Further, if then .

Proof.

Since is a generalized -contractive map, it follows from Lemma 2.1 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 Since is lower semicontinuous, we have

(2.20)

and thus, Since and is closed, it follows from Lemma 1.6 that

As a consequence, we also obtain the following fixed point result.

Corollary 2.3 (see [8]).

Let be a complete space and let be a -contractive map. If the real-valued function on defined by is lower semicontinous, then there exists such that Further, if then

Applying Lemma 2.1, we also obtain a fixed point result for multivalued generalized -contractive map satisfying another suitable condition.

Theorem 2.4.

Let be a complete space and let be a generalized -contractive map. Assume that

(2.21)

for every with Then

Proof.

By Lemma 2.1, there exists an orbit of , which is a Cauchy sequence in . Due to the completeness of , there exists such that Since is lower semicontinuous and it follows from the proof of Lemma 2.1 that for all

(2.22)

where Also, we get

(2.23)

Assume that Then, we have

(2.24)

which is impossible and hence .

Corollary 2.5 (see [8]).

Let be a complete space and let be -contractive map. Assume that

(2.25)

for every with Then