Krasnosel'skii-Type Fixed-Set Results

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 [3–9], and for other interesting results see [10–13].

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 [15–17]).

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)