Open Access

Fixed Point Results for Multivalued Maps in Metric Spaces with Generalized Inwardness Conditions

Fixed Point Theory and Applications20102010:183217

DOI: 10.1155/2010/183217

Received: 19 November 2009

Accepted: 12 February 2010

Published: 21 February 2010

Abstract

We establish fixed point theorems for multivalued mappings defined on a closed subset of a complete metric space. We generalize Lim's result on weakly inward contractions in a Banach space. We also generalize recent results of Azé and Corvellec, Maciejewski, and Uderzo for contractions and directional contractions. Finally, we present local fixed point theorems and continuation principles for generalized inward contractions.

1. Introduction and Preliminaries

In the following denotes a complete metric space. The open ball centered in of radius is denoted . For , two nonempty, closed subsets of , the generalized Hausdorff metric is defined by

(1.1)

Definition 1.1.

Let ; we say that the multivalued map is a contraction if has nonempty, closed values, and there exists such that
(1.2)

The constant is called the constant of contraction.

The well known Nadler fixed point Theorem [1] says that a multivalued contraction on to itself has a fixed point. However, to insure the existence of a fixed point to a multivalued contraction defined on a closed subset of , extra assumptions are needed.

In 2000, Lim [2] obtained the following fixed point theorem for weakly inward multivalued contractions in Banach spaces using the transfinite induction.

Theorem 1.2.

Let be a nonempty closed subset of a Banach space and a multivalued contraction with closed values. Assume that is weakly inward, that is,
(1.3)

Then has a fixed point.

Observe that in the definition of weakly inward maps, linear intervals play a crucial role. Indeed, for some and if and only if

(1.4)

Moreover, .

From this observation, generalizations of this result to complete metric spaces were recently obtained with simpler proofs by Azé and Corvellec [3], and by Maciejewski [4]. They generalized the inwardness condition using the metric left-open segment

(1.5)

which should be nonempty for every and "close enough" of . They also obtained results for directional -contractions in the sense of Song [5]. In 2005, Uderzo [6] established a local fixed point theorem for directional -contractions.

In this paper, we generalize their results. More precisely, we first generalize the inwardness conditions used in [24]. In particular, for with , one can have . Also, we slightly generalize the notion of -directional contractions.

Finally, we present local fixed point theorems and continuation principles generalizing results of Maciejewski [4] and Uderzo [6].

Here is the well known Caristi Theorem [7] which will play a crucial role in the following.

Theorem 1.3 (Caristi [7]).

Let and a map lower semicontinuous and bounded from below such that
(1.6)

Then has a fixed point.

This result, which is equivalent to the Ekeland variational Principle [8, 9], can also be deduced from the Bishop-Phelps theorem. The following formulation appeared in [10] (see also [11]) while the original formulation appeared in a different form in [12] (see also [13]).

Theorem 1.4 (Bishop and Phelps).

Let be lower semicontinuous and bounded from below, and . Then for any , there exists such that

(i) ;

(ii) for every .

The interested reader can find a multivalued version of Caristi's fixed point theorem in an article of Mizoguchi and Takahashi [14].

2. Generalizations of Inward Contractions

In this section, we obtain fixed point results for contractions defined on a closed subset of a metric space satisfying a generalized inwardness condition.

Theorem 2.1.

Let be a closed subset of , and let be a multivalued contraction with constant . Assume that there exits such that for every ,
(2.1)

Then has a fixed point.

Proof.

Assume that has no fixed point. Choose such that . Consider on the metric
(2.2)

Since is a contraction with closed values, is a complete metric space.

Let . By assumption, there exists such that

(2.3)
Since and
(2.4)
there exists such that
(2.5)
Therefore,
(2.6)

Defining and , respectively, by

(2.7)

we deduce from Caristi's theorem (Theorem 1.3) that has a fixed point which is a contradiction since . So, has a fixed point.

As a corollary, we obtain Maciejewski's result [4] which generalizes Lim's fixed point theorem for weakly inward multivalued contractions in Banach spaces [2].

Corollary 2.2 (Maciejewski [4]).

Let be a closed subset of , and let be a multivalued contraction such that for every ,
(2.8)

Then has a fixed point.

Proof.

Let be a constant of contraction of . Fix . One can choose . If , there exists such that
(2.9)
Thus, there exists such that . So
(2.10)

Thus satisfies (2.1).

From the proof of Theorem 2.1, one sees that one can weaken the assumption that is a contraction, and hence one can generalize a result due to Azé and Corvellec [3].

Theorem 2.3.

Let be a closed subset of and let be a multivalued map with nonempty values and closed graph. Assume that there are constants and such that for every ,
(2.11)

Then has a fixed point.

Corollary 2.4 (Azé and Corvellec [3]).

Let be a closed subset of , and let be a multivalued map with nonempty values and closed graph. Assume that there exists and such that and for every and every there exist and such that
(2.12)

Then has a fixed point.

Proof.

Choose . It is easy to see that satisfies (2.11) for every .

Remark 2.5.

Observe that in Theorems 2.1 and 2.3, one can have for some ,
(2.13)
So
(2.14)

Therefore, (2.8) and (2.12) are not satisfied.

Example 2.6.

Let , , defined by . is a contraction with constant . Take and . Observe that , and . So, (2.8) and (2.12) are not satisfied. On the other hand, choose . Let with . For all , there exists such that . So, taking , one has
(2.15)

Hence satisfies all the assumptions of Theorem 2.1 and in particular condition (2.1).

In the previous results, is a contraction or has to satisfy a type of contractive condition in some direction, namely,

(2.16)

A careful look at their proofs permits to realize that a wider class of maps can be considered. Indeed, it is easy to see that the previous results are corollaries of the following theorem which is a direct consequence of Theorem 1.3.

Theorem 2.7.

Let be a closed subset of and let be a multivalued map with nonempty values and closed graph. Assume that there exists an equivalent metric on such that for every and every ,
(2.17)

Then has a fixed point.

Corollary 2.8.

Let be a closed subset of and let be a multivalued map with nonempty values and closed graph. Assume that there exists such that for every ,
(2.18)

Then has a fixed point.

Corollary 2.9.

Let be a closed subset of , and let be a continuous map. Assume that there exists such that for every , there exists such that
(2.19)

Then has a fixed point.

Example 2.10.

Let be defined by . Obviously, is expansive and satisfies the assumptions of the previous corollary. It does not satisfies (2.1) and (2.11).

3. Intersection Conditions

Observe that even though Theorem 2.7 generalizes Theorems 2.1 and 2.3, Condition (2.17) is quite restrictive in the multivalued context since every has to satify a suitable condition. Here is a fixed point result where at least one element of has to be in a suitable set.

Theorem 3.1.

Let be a closed subset of , and let be a multivalued contraction with constant . Assume that there exits such that for every ,
(3.1)

Then has a fixed point.

Proof.

Assume that has no fixed point. Let . By assumption, there exist and such that
(3.2)
Therefore,
(3.3)
Defining and , respectively, by
(3.4)

we deduce from Caristi's Theorem (Theorem 1.3) that has a fixed point which is a contradiction since . So, has a fixed point.

Example 3.2.

Let , , defined by . Observe that (2.1) is not satisfied. Indeed, for and , we have and for every .

Choose . Let with , then one has

(3.5)
Choose if , and otherwise. So, taking , one has
(3.6)

Thus satisfies all assumptions of Theorem 3.1, and in particular it satisfies (3.1) but does not satisfy (2.1).

Corollary 3.3.

Let be a closed subset of , and let be a multivalued contraction with constant . Assume that there exits such that for every ,
(3.7)

Then has a fixed point.

The previous theorem generalizes a result of Downing and Kirk [15].

Corollary 3.4 (Downing and Kirk [15]).

Let be a closed subset of a Banach space and a multivalued contraction such that for every
(3.8)

Then has a fixed point.

Proof.

Let be a constant of contraction of . Fix . For such that , there exists such that and there exist sequences in and in such that . Choose big enough such that . So and
(3.9)

So, (3.7) is satisfied and the conclusion follows from Corollary 3.3.

Example 3.5.

Let , and
(3.10)
Observe that is a contraction with constant . For ,
(3.11)
Observe that for every , . So (3.7) and hence (3.8) are not satisfied. Now, fix . For , choose . Observe that if . However, for every , choosing , one has
(3.12)

Thus, Condition (3.1) is satisfied.

Observe that if is a single-valued contraction satisfying (2.1) then for every such that ,

(3.13)

An analogous condition in the multivalued context leads to the following result.

Theorem 3.6.

Let be a closed subset of , and let be a multivalued contraction with constant . Assume that there exits such that for every , or
(3.14)

Then has a fixed point.

Proposition 3.7.

Theorems 3.1 and 3.6 are equivalent.

Proof.

It is clear that if (3.1) is satisfied, then (3.14) is also satisfied. Thus, Theorem 3.6 implies Theorem 3.1.

Now, if assumptions of Theorem 3.6 are satisfied with some . Fix such that . Let . If , there exists such that

(3.15)
Choose such that . So
(3.16)

where . Hence assumptions of Theorem 3.1 are satisfied with .

As before, looking at the proof of Theorem 3.1, we see that we can relax the assumption that is a contraction.

Theorem 3.8.

Let be a closed subset of and let be a multivalued map with nonempty, closed values such that the map is lower semicontinuous. Assume that there exist and such that for every ,
(3.17)

Then has a fixed point.

We obtain as corollary a result due to Song [5] which generalizes a fixed point result due to Clarke [16].

Corollary 3.9 (Song [5]).

Let be a closed nonempty subset of , and let be a multivalued with nonempty, closed, bounded values such that

(i) is -upper semicontinuous, that is, for every and every there exists such that for every ;

(ii)there exist , and such that for every with , there exists satisfying
(3.18)

Then has a fixed point.

Uderzo [6] generalized Song's result introducing the notion of directional multi-valued -contraction (this means that satisfies the following condition (ii)). This notion generalizes the notion of directional contractions used by Song [5] (Condition (ii) in Corollary 3.9). We show how Uderzo's result can be obtained from Theorem 3.8.

Corollary 3.10 (Uderzo [6]).

Let be a closed nonempty subset of , and let be a multivalued with nonempty, closed, bounded values such that

(i) is -upper semicontinuous;

(ii)there exist , and such that for every with , there exists satisfying
(3.19)

(iii)there exist and such that ;

(iv) .

Then has a fixed point.

Proof.

It is known that the -upper semicontinuity of implies that is lower semicontinuous. Let . This set is closed and nonempty.

Let be such that . Assumption (ii) implies that there exists such that

(3.20)
So . This inequality with (ii) and (iv) implies that
(3.21)

So .

Denote

(3.22)
Fix . Since , choose such that
(3.23)
So, by (ii),
(3.24)

So, the restriction satisfies the assumptions of Theorem 3.8, and hence has a fixed point.

4. Local Fixed Point Theorems for Generalized Inward Contractions

In this section, we present local versions of fixed point theorems for generalized inward contractions.

Theorem 4.1.

Let be a closed subset of , , , and let be a multivalued map with nonempty values and closed graph. Assume that there exist and an equivalent metric on such that

(i) for every , and ;

(ii) ;

(iii)for every and every ,
(4.1)

Then has a fixed point.

Proof.

Choose and such that . Fix such that
(4.2)
Consider
(4.3)

This space endowed with the metric is a nonempty complete metric space since .

Applying the Bisholp-Phelps Theorem (Theorem 1.4) insures the existence of satisfying

(4.4)

If then is a fixed point of .

If not, by assumption (iii), there exists and such that

(4.5)
This inequality combined with the fact that implies that
(4.6)
From (4.2), (4.6) and Assumption (i) we deduce
(4.7)

So, , and by (iii), . Thus, by (4.6), .

Therefore, (4.5) contracdicts (4.4). So, has a fixed point.

Corollary 4.2.

Let be a closed subset of , , , and let be a multivalued map with nonempty values and closed graph. Assume that there exist such that

(i) ;

(ii)for every and every ,
(4.8)

Then has a fixed point.

As corollaries, we obtain local versions of Theorems 2.1 and 2.3.

Theorem 4.3.

Let be a closed subset of , , , and let be a multivalued map with nonempty values and closed graph. Assume that there are constants and such that

(i) ;

(ii)for all ,
(4.9)

Then has a fixed point.

Proof.

Choose such that . Let , . Consider on the metric
(4.10)

The conclusion follows from Theorem 4.1 if we show that Condition (iii) holds. Let and , and let be given by Assumption (ii). If , choose , and if , choose such that . So,

(4.11)

In the case where is a contraction, the previous result can be stated more simply.

Corollary 4.4.

Let be a closed subset of , , and . Assume that is a multivalued contraction with constant for which there exits such that

(i) ;

(ii) such that for all .

Then has a fixed point.

We obtain as corollary the following result due to Maciejewski [4].

Corollary 4.5.

Let be a closed subset of , , and . Assume that is a multivalued contraction with constant such that

(i) ;

(ii) for all , where is defined in (2.8).

Then has a fixed point.

Proof.

Choose and be such that . Arguing as in the proof of Corollary 2.2, one sees that Assumption (ii) implies that for every ,
(4.12)

The conclusion follows from Corollary 4.4.

Fixed point results can also be obtained for multivalued maps defined on a ball of and satisfying an intersection condition. Here is a local version of Theorem 3.8.

Theorem 4.6.

Let be a closed subset of , , , and let be a multivalued map with nonempty, closed values such that is lower semicontinuous. Assume that there exist and such that

(i) ;

(ii)for every ,
(4.13)

Then has a fixed point.

Proof.

Choose such that . Consider
(4.14)

The space is a nonempty closed subset of since and is lower semicontinuous.

Applying the Bisholp-Phelps Theorem to insures the existence of such that

(4.15)

If is not a fixed point of , by Assumption (ii), there exist and such that

(4.16)
By Assumption (i) and since ,
(4.17)
So, . From (4.16), we deduce
(4.18)

Hence, .

By (4.16),

(4.19)

this contradicts (4.15) since . So has a fixed point.

5. Continuation Principle for Generalized Inward Contractions

In this section, we obtain continuation principles for families of contractions satisfying a generalized inwardness condition. For and open in , we denote the boundary of relative to . Here is a generalization of Theorem in [17]. The proof is analogous.

Theorem 5.1.

Let be a closed subset of , open in , , , and continuous and increasing. Assume is a multivalued map with nonempty values and closed graph such that

(i) and for all ;

(ii) for all and all ;

(iii)for all , and all ,
(5.1)

Then has a fixed point if and only if has a fixed point.

Proof.

Consider
(5.2)
endowed with the partial order
(5.3)
Assuming that has a fixed point implies that is nonempty. It is easy to show that every totally ordered subset of has a an upper bound. Hence, by Zorn's Lemma, has a maximal element . By (i), . To conclude we need to show that . If not, there is such that , and there exist and such that . Therefore,
(5.4)

Thus, satisfies the assumptions of Theorem 4.3, and hence has a fixed point . This contradicts the maximality of since .

Corollary 5.2.

Let be a uniformly convex Banach space, open, bounded, convex, and closed, convex such that . Assume that is a nonexpansive map such that

(i) and for all ;

(ii)there exists a lower semicontinuous map defined on with and such that for every , or there exists such that .

Then has a fixed point.

Proof.

Observe that for ,
(5.5)

for every since is bounded.

Assumption (ii) implies that for every , there exists , such that for every ,

(5.6)
where . So, is an open cover of . Thus, there exists an increasing sequence in converging to such that is an open cover of . Denote , and . By construction and by Assumption (ii), for every and every , we have that for every such that , there exists such that
(5.7)

The previous theorem applied inductively to insures the existence of such that . The sequence has a weakly converging subsequence still denoted such that . The demi-closedness of (see [18, Theorem ]) implies that has a fixed point.

Similarly to Theorem 5.1, we can prove the following continuation principles using Theorems 4.1 and 4.6, respectively.

Theorem 5.3.

Let be a closed subset of , open in , continuous and increasing, and a multivalued map with nonempty values and closed graph. Assume that there exist and an equivalent metric on such that

(i) and for all ;

(ii) for all and all ;

(iii) for every , and ;

(iv)for every and every ,
(5.8)

Then has a fixed point if and only if has a fixed point.

Theorem 5.4.

Let be a closed subset of , open in , , and and continuous and increasing. Assume is a multivalued with nonempty, closed values such that

(i) for all and all ;

(ii) for all and all ;

(iii) is lower semicontinuous for all ;

(iv)for all , and all ,
(5.9)

Then has a fixed point if and only if has a fixed point.

Declarations

Acknowledgment

This work was partially supported by CRSNG Canada.

Authors’ Affiliations

(1)
Département de Mathématiques et de Statistique, Université de Montréal

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