# Krasnosel'skii-Type Fixed-Set Results

- MA Al-Thagafi
^{1}and - Naseer Shahzad
^{1}Email author

**2010**:394139

**DOI: **10.1155/2010/394139

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

**Received: **8 February 2010

**Accepted: **23 August 2010

**Published: **26 August 2010

## Abstract

## 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;

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

*Kuratoswki measure of noncompactness*on if

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 .

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

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 ;

Proof.

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;

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;

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;

(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.

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 .

Theorem 3.6.

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

(a) is compact and continuous;

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

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

Then

(i)there exists a minimal such that and ;

(ii)there exists a maximal 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 ;

Then

(i)there exists a minimal such that and ;

(ii)there exists a maximal such that .

Proof.

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 ;

Then

(i)there exists a minimal such that and ;

Proof.

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).

*iterated 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

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.

(a) is a compact and continuous multimap;

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.

## 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

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