Open Access

Krasnosel'skii-Type Fixed-Set Results

Fixed Point Theory and Applications20102010:394139

https://doi.org/10.1155/2010/394139

Received: 8 February 2010

Accepted: 23 August 2010

Published: 26 August 2010

Abstract

Some new Krasnosel'skii-type fixed-set theorems are proved for the sum , where is a multimap and is a self-map. The common domain of and is not convex. A positive answer to Ok's question (2009) is provided. Applications to the theory of self-similarity are also given.

1. Introduction

The Krasnosel'skii fixed-point theorem [1] is a well-known principle that generalizes the Schauder fixed-point theorem and the Banach contraction principle as follows.

Krasnosel'skii Fixed-Point Theorem

Let be a nonempty closed convex subset of a Banach space , , and . Suppose that

(a) is compact and continuous;

(b) is a -contraction;

(c) for every .

Then there exists such that .

This theorem has been extensively used in differential and functional differential equations and was motivated by the observation that the inversion of a perturbed differential operator may yield the sum of a continuous compact map and a contraction map. Note that the conclusion of the theorem does not need to hold if the convexity of is relaxed even if is the zero operator. Ok [2] noticed that the Krasnosel'skii fixed-point theorem can be reformulated by relaxing or removing the convexity hypothesis of and by allowing the fixed-point to be a fixed-set. For variants or extensions of Krasnosel'skii-type fixed-point results, see [39], and for other interesting results see [1013].

In this paper, we prove several new Krasnosel'skii-type fixed-set theorems for the sum , where is a multimap and is a self-map. The common domain of and is not convex. Our results extend, generalize, or improve several fixed-point and fixed-set results including that given by Ok [2]. A positive answer to Ok's question [2] is provided. Applications to the theory of self-similarity are also given.

2. Preliminaries

Let be a nonempty subset of a metric space , a normed space, the boundary of , the interior of , the closure of , the set all nonempty subsets of , the set of nonempty bounded subsets of , the family of nonempty closed subsets of , the family of nonempty compact subsets of , the set of real numbers, and . A map is called the Kuratoswki measure of noncompactness on if
(2.1)

for every , where denotes the diameter of . Let and . We write . We say that (a) is a fixed point of if , and the set of fixed points of will be denoted by ; (b) is nonexpansive if for all ; (c) is -contraction if for all and some ; (d) is -condensing if it is continuous and, for every with , and ; (e) is -set-contractive if it is continuous and, for every , , and ; (f) is compact if is a compact subset of .

Definition 2.1.

Let , and let be either "a nondecreasing map satisfying for every '' or "an upper semicontinuous map satisfying for every .'' One says that is a -contraction if for all .

Remark 2.2.

A mapping is said to be a -contraction in the sense of Garcia-Falset [6] if there exists a function satisfying either " is continuous and for " or "there exists with and nondecreasing such that " for which the inequality holds for all , . Our definition for -contraction is different in some sense from that of Garcia-Falset.

Lemma 2.3 (see [2]).

Let be a nonempty closed subset of a normed space . If is compact and continuous, then there exists a minimal such that .

Theorem 2.4 (see [14]).

Let be a nonempty bounded closed convex subset of a Banach space . Suppose that is an -condensing map. Then has a fixed point in .

Theorem 2.5 (see [1517]).

Let be a complete metric space. If is a -contraction, then has a unique fixed point in .

Theorem 2.6 (see [14]).

Let be a closed subset of a Banach space such that is bounded, open, and containing the origin. Suppose that is an -condensing map satisfying for all and . Then has a fixed point in .

Theorem 2.7 (see [14]).

Let be a closed subset of a Banach space such that is bounded, open, and containing the origin. Suppose that is a 1-set-contractive map satisfying for all and . If is closed, then has a fixed point in .

3. Fixed-Set Results

We now reformulate the Krasnosel'skii fixed-point theorem by allowing the fixed-point to be a fixed-set and removing the convexity hypothesis of . Under suitable conditions, we look for a nonempty compact subset of such that
(3.1)
or
(3.2)

Theorem 3.1.

Let be a nonempty closed subset of a Banach space , , and . Suppose that

(a) is compact and continuous;

(b) is -condensing and is a bounded subset of ;

(c) .

Then there exists such that .

Proof.

Fix . Let denote the set of closed subsets of for which and . Note that is nonempty since . Take . As is closed, , and , we have . Let . Notice that is a bounded subset of containing . So is a closed subset of , , and
(3.3)
This shows that and . Since is a bounded subset of and is compact, we have
(3.4)
As is -condensing, it follows that . Thus is a compact subset of . As the Vietoris topology and the Hausdorff metric topology coincide on [18, page 17 and page 41], is compact and hence closed. Define by . It follows that
(3.5)
for every . Since is continuous and is compact-valued and continuous, both and are compact subsets of and hence . Moreover, the maps and are continuous, so is continuous. By Lemma 2.3, there exists such that since is compact and hence closed. Let . As , we have
(3.6)

However is a compact subset of [18, page 16], so .

Corollary 3.2 ([2, Theorem  2.4]).

Let be a nonempty closed subset of a Banach space , , and . Suppose that

(a) is compact and continuous;

(b) is compact and continuous;

(c) .

Then there exists such that .

In the following corollary, we assume that whenever is upper semicontinuous.

Corollary 3.3.

Let be a nonempty closed subset of a Banach space , , and . Suppose that

(a) is compact and continuous;

(b) is a -contraction and is bounded;

(c) .

Then there exists such that .

Remark 3.4.

The following statements are equivalent [19]:

(i) is a -contraction, where is nondecreasing, right continuous such that for all and ;

(ii) is a -contraction, where is upper semicontinuous such that for all and .

Note that Corollary 3.3 provides a positive answer to the following question of Ok [2]. We do not know at present if the fixed-set can be taken to be a compact set in the statement of [2, Corollary ].

Theorem 3.5.

Let be a nonempty closed subset of a normed space , , and . Suppose that

(a) is compact and continuous;

(b) ;

(c) is a continuous single-valued map on .

Then

(i)there exists a minimal such that and ;

(ii)there exists a maximal such that .

Proof.

Let . Then, by (b), there exists such that , and, as is a single-valued map on ,
(3.7)

So . Note that is compact-valued and is a compact subset of . The continuity of follows from that of and . Moreover, is a compact subset of , and hence is a compact subset of . By Lemma 2.3, there exists a minimal such that . But, since is continuous and is compact-valued, is compact-valued and maps compact sets to compact sets. Then , is a compact subset of M, so . Thus , and hence .

Let
(3.8)
and . Clearly is nonempty since . Then . Take . It follows that
(3.9)

and hence and . Thus .

Theorem 3.6.

Let be a nonempty closed subset of a normed space , , and . Suppose that

(a) is compact and continuous;

(b) is a -contraction;

(c)if , then ( has a convergent subsequence;

(d) .

Then

(i)there exists a minimal such that and ;

(ii)there exists a maximal such that .

Proof.

Let . By (b), (d), and the closeness of , the map is a -contraction from into . So, by Theorem 2.5, there exists a unique such that . Then , and so . Since the map has a unique fixed-point, its fixed-point set is singleton. So is a single-valued map. To show that is continuous, let be a sequence in such that . Define and . Then , and . We claim that is convergent. First, notice that is bounded; otherwise, has a subsequence such that . As , (c) implies that has a convergent subsequence, a contradiction. Next, as is continuous and one-to-one, it follows from (c) that the sequence converges to . Therefore, is continuous. Now the result follows from Theorem 3.5.

In the following result, we assume that whenever is upper semicontinuous.

Theorem 3.7.

Let be a nonempty compact subset of a Banach space , , and . Suppose that

(a) is continuous;

(b) is a -contraction;

(c) .

Then

(i)there exists a minimal such that and ;

(ii)there exists a maximal such that .

(iii)there exists such that .

Proof.

Parts (i) and (ii) follow from Theorem 3.6. Part (iii) follows from Theorem 3.1.

Theorem 3.8.

Let be a closed subset of a Banach space such that is bounded, open, and containing the origin, , and . Suppose that

(a) is compact and continuous;

(b) is an -condensing map satisfying for all ;

(c) is a continuous single-valued map on ;

(d) .

Then

(i)there exists a minimal such that and ;

(ii)there exists a maximal such that .

(iii)there exists such that .

Proof.

Let . As is -condensing, part (d) and the closeness of imply that the map is an -condensing self-map of . Moreover, this map satisfies for all and ; otherwise, there are and such that . This implies that
(3.10)

which contradicts the second part of (b). It follows from Theorem 2.6 that there exists such that . Then , and so . Now parts (i) and (ii) follow from Theorem 3.5. Part (iii) follows from Theorem 3.1.

Theorem 3.9.

Let be a closed subset of a Banach space such that is bounded, open, and containing the origin, , and . Suppose that

(a) is compact and continuous;

(b) is a -set-contractive map satisfying for all ;

(c) is closed, and is a continuous single-valued map on ;

(d) .

Then

(i)there exists a minimal such that and ;

(ii)there exists such that .

Proof.

Let . As is 1-set-contractive, part (d) and the closeness of imply that the map is a 1-set-contractive self-map of . Moreover, this map satisfies for all and ; otherwise, there are and such that . This implies that
(3.11)

which contradicts the second part of (b). It follows from Theorem 2.7 that there exists such that . Then , and so . Now the result follows from Theorem 3.5.

Definition 3.10 (self-similar sets).

Let be a nonempty closed subset of a Banach space . If are finitely many self-maps of , then the list is called aniterated function system (IFS). This IFS is continuous (resp., contraction, -condensing, etc.) if each is so. A nonempty subset of is said to be self-similar with respect to the IFS if
(3.12)

Remark 3.11.

It is well known that there exists a unique compact self-similar set with respect to any contractive IFS; see [20].

Example 3.12.

Consider an IFS such that

(a) is a compact and continuous multimap;

(b) for each .

Then the existence of a compact self-similar set with respect to the IFS is ensured by letting to be zero in each of the following situations.

(i)Suppose that is an -condensing map such that is bounded. Then Theorem 3.1 ensures the existence of a compact subset of such that
(3.13)
(ii)Suppose that is a -contraction satisfying condition (c) of Theorem 3.6. Then there exists a minimal compact subset of such that
(3.14)
(iii)Suppose that is a closed subset of a Banach space such that is bounded, open, and containing the origin, is an -condensing map satisfying for all , and is a continuous single-valued map on . Then Theorem 3.8 ensures the existence of a minimal compact subset of such that
(3.15)
(iv)Suppose that is a closed subset of a Banach space such that is bounded, open, and containing the origin, is a 1-set-contractive map satisfying for all , is closed, and is a continuous single-valued map on . Then Theorem 3.9 ensures the existence of a minimal compact subset of such that
(3.16)

Declarations

Acknowledgments

The authors thank the referee for his valuable suggestions. This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah under project no. 3-017/429.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University

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Copyright

© M. A. Al-Thagafi and N. Shahzad. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.