Fixed Point Results for Generalized Contractive Multimaps in Metric Spaces

Abstract

The concept of generalized contractive multimaps in the setting of metric spaces is introduced, and the existence of fixed points for such maps is guaranteed under certain conditions. Consequently, our results either generalize or improve a number of fixed point results including the corresponding recent fixed point results of Ciric (2008), Latif-Albar (2008), Klim-Wardowski (2007), and Feng-Liu (2006). Examples are also given.

1. Introduction and Preliminaries

Let be a metric space, a collection of nonempty subsets of , a collection of nonempty closed bounded subsets of , a collection of nonempty closed subsets of a collection of nonempty compact subsets of and the Hausdorff metric induced by Then for any

(1.1)

where

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 it implies that .

Using the concept of Hausdorff metric, Nadler [1] established the following fixed point result for multivalued contraction maps, known as Nadler's contraction principle which in turn is a generalization of the well-known Banach contraction principle.

Theorem 1.1 (see [1]).

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

Using the concept of the Hausdorff metric, many authors have generalized Nadler's contraction principle in many directions. 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 [2] extended Nadler's fixed point theorem without using the concept of Hausdorff metric. They proved the following result.

Theorem 1.2.

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:

(1.2)

Then provided a real-valued function on , is lower semicontinuous.

Recently, Klim and Wardowski [3] generalized Theorem 1.2 and proved the following two results.

Theorem 1.3.

Let be a complete metric space and let . Assume that the following conditions hold:

(i)there exist a number and a function such that for each

(1.3)

(ii)for any there is satisfying

(1.4)

Then provided a real-valued function on , is lower semicontinuous.

Theorem 1.4.

Let be a complete metric space and let . Assume that the following conditions hold:

(i)there exists a function such that for each

(1.5)

(ii)for any there is satisfying

(1.6)

Then provided a real-valued function on , is lower semicontinuous.

Note that Theorem 1.3 generalizes Nadler's contraction principle and Theorem 1.2. Most recently, Ciric [4] obtained some interesting fixed point results which extend and generalize the cited results. Namely, [4, Theorem  5] generalizes [5, Theorem  5], [4, Theorem  6] generalizes [4, Theorems  1.2,  1.3], and [3, theorem  7] generalizes Theorem 1.4.

In [6], 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 a sequence in with , then ;

()for any there exists such that and imply

Note that, in general for , and not either 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 [6]. Many other examples and properties of the -distance can be found in [6, 7].

The following lemma is crucial for the proofs of our results.

Lemma 1.5 (see [8]).

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

Most recently, the authors of this paper generalized Latif and Albar [9, Theorem  1.3] as follows.

Theorem 1.6 (see [10]).

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

(1.7)

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 .

The aim of this paper is to present some more general results on the existence of fixed points for multivalued maps satisfying certain conditions. Our results unify and generalize the corresponding results of Mizoguchi and Takahashi [5], Klim and Wardowski [3], Latif and Abdou [10], Ciric [4], Feng and Liu [2], Latif and Albar [9] and several others.

2. The Results

First we prove a theorem which is a generalization of Ciric [4, Theorem  5] and due to Klim and Wardowski [3, Theorem  1.4].

Theorem 2.1.

Let be a complete metric space with a -distance Let be a multivalued map. Assume that the following conditions hold:

(i)there exists a function such that for each

(2.1)

(ii)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 there exists such that

(2.3)

From (2.3) we get

(2.4)

Define a function by

(2.5)

Using the facts that for each and we have

(2.6)

(2.7)

From (2.4) and (2.5), we have

(2.8)

Similarly, for , there exists such that

(2.9)

Thus

(2.10)

Continuing this process we can get an orbit of in satisfying the following:

(2.11)

(2.12)

for each integer . Since for each and from (2.12), we have for all

(2.13)

Thus the sequence of nonnegative real numbers is decreasing and bounded below, thus convergent. Therefore, there is some such that

(2.14)

From (2.11), as for all we get

(2.15)

Thus, we conclude that the sequence of nonnegative reals is bounded. Therefore, there is some such that

(2.16)

Note that for each so we have Now we will show that Suppose that Then we get

(2.17)

Now consider Suppose to the contrary, that Then and so from (2.14) and (2.16) there is a positive integer such that

(2.18)

(2.19)

Then from (2.19), (2.11) and (2.18), we get

(2.20)

Thus for all

(2.21)

that is,

(2.22)

and we get

(2.23)

Thus for all

(2.24)

Thus, from (2.12) and (2.24), we get

(2.25)

where Clearly as From (2.18) and (2.25), we have for any

(2.26)

Since and there is a positive integer such that Now, since for each by (2.26) we have

(2.27)

a contradiction. Hence, our assumption is wrong. Thus Now we will show that Since then from (2.16) we can read as

(2.28)

so, there exists a subsequence of such that

(2.29)

Now from (2.7) we have

(2.30)

and from (2.12), we have

(2.31)

Taking the limit as and using (2.14), we get

(2.32)

If we suppose that then from last inequality, we have

(2.33)

which contradicts with (2.30). Thus Then from (2.14) and (2.15), we have

(2.34)

and thus

(2.35)

Now, let

(2.36)

Then by (2.7), Let be such that Then there is some such that

(2.37)

Thus it follows from (2.12),

(2.38)

By induction we get

(2.39)

Now, using (2.15) and (2.39), we have

(2.40)

Now, we show that is a Cauchy sequence, for all we get

(2.41)

Hence we conclude, as that is Cauchy sequence. Due to the completeness of , there exists some such that . Since is lower semicontinuous and from (2.34), we have

(2.42)

and thus, Since and is closed, it follows from Lemma 1.5 that

We also have the following interesting result by replacing the hypothesis (iii) of Theorem 2.1 with another natural condition.

Theorem 2.2.

Suppose that all the hypotheses of Theorem 2.1 except (iii) hold. Assume that

(2.43)

for every with Then

Proof.

Following the proof of Theorem 2.1, there exists a Cauchy sequence with Due to the completeness of , there exists such that Since is lower semicontinuous and it follows for all

(2.44)

Assume that Then, we have

(2.45)

which is impossible and hence

Now, we present an improved version of Ciric [4, Theorem  6] and which also generalizes due to Latif and Abdou [10, Theorem  1.6] and due to Klim and Wardowski [3, Theorem  1.3].

Theorem 2.3.

Let be a complete metric space with a -distance Let , be a multivalued map. Assume that the following condition hold:

(i)there exist functions and with nondecreasing such that

(2.46)

(ii)for any there exists satisfying the following conditions:

(2.47)

(iii)the map defined by is lower semicontinuous.

Then there exists such that Further if then

Proof.

Let be an arbitrary, then there exists such that

(2.48)

From (2.48) we have

(2.49)

Define a function by

(2.50)

Since we have

(2.51)

(2.52)

Thus from (2.49)

(2.53)

Similarly, there exists such that

(2.54)

Then by definition of we get

(2.55)

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

(2.56)

(2.57)

Thus

(2.58)

Since for all we get

(2.59)

Thus the sequence of nonnegative real numbers is decreasing and bounded below, thus convergent. Now, we want to show that the sequence is also decreasing. Suppose to the contrary, that then as is nondecreasing, we have

(2.60)

Now using (2.56), (2.57) and (2.60) with , we get

(2.61)

a contradiction. Thus the sequences is decreasing. Now let

(2.62)

Thus by (2.52), Then for any there exists such that

(2.63)

So, from (2.58), for all we get

(2.64)

Thus by induction we get for all

(2.65)

Since from (2.56) and (2.65), we have

(2.66)

for all Note that Now, we show that is a Cauchy sequence. For all we have

(2.67)

Thus we conclude that is a Cauchy sequence. Now, proceeding the proof of Theorem 2.1, we get some such that and

Following the proof of Theorem 2.2, we can obtain the following result.

Theorem 2.4.

Suppose that all the hypotheses of Theorem 2.3 except (iii) hold. Assume that

(2.68)

for every with Then

Now, we present a result which is a generalization of Theorem 1.4 due to Klim and Wardowski [3] and Ciric [4, Theorem  7].

Theorem 2.5.

Let be a complete metric space with a -distance Let be a multivalued map. Assume that the following conditions hold:

(i)there exists a function such that for each

(2.69)

(ii)for any there exists satisfying

(2.70)

(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

(2.71)

Using the analogous method like in the proof of Lemma  2.1 [10], we obtain the existence of Cauchy sequence such that and satisfying

(2.72)

Consequently, there exists such that . Since is lower semicontinuous, we have

(2.73)

thus, Further by closedness of and since it follows from Lemma 1.5 that

3. Examples

The following example shows that Theorem 2.1 is a genuine generalization of Ciric [4, Theorem  5].

Example 3.1.

Let with the usual metric . Define a function , by

(3.1)

Clearly, is a -distance on and . Let be such that

(3.2)

Define now as follows

(3.3)

Note that

(3.4)

and is lower semicontinuous. Moreover for each we have Take then we have

(3.5)

Further, note that

(3.6)

Hence, for all , satisfies all the conditions of Theorem 2.1. Now, if then we have and

(3.7)

Note that for there is such that

(3.8)

Thus, also satisfies all the conditions of Theorem 2.1 for Hence it follows from Theorem 2.1 that Note that Clearly, does not satisfy the hypotheses of Ciric [4, Theorem  5] because is not the metric .

Finally, we present an example which shows that Theorem 2.5 is a genuine generalization of Theorem 1.4 due to Klim-Wardowski [3].

Example 3.2.

Let with the usual metric . Define a function , by

(3.9)

Then is a -distance on Note that . Now, for any real number define by

(3.10)

and define a constant function by

(3.11)

Note that for all . And for each we have

(3.12)

Thus, is continuous. Now for each there exists satisfying

(3.13)

Therefore, all assumptions of Theorem 2.5 are satisfied and Note that is not compact for all and the -distance is not a metric so do not satisfy the hypotheses of Theorem 1.4.