Krasnosel’skii-Type Fixed-Set Results

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Volume 2010, Article ID 394139,9pages doi:10.1155/2010/394139

Research Article

Krasnosel’skii-Type Fixed-Set Results

M. A. Al-Thagafi and Naseer Shahzad

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Naseer Shahzad,[email protected] Received 8 February 2010; Revised 16 August 2010; Accepted 23 August 2010 Academic Editor: W. A. Kirk

Copyrightq2010 M. A. Al-Thagafi and N. Shahzad. 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.

Some new Krasnosel’skii-type fixed-set theorems are proved for the sumST, whereS is a multimap andTis a self-map. The common domain ofSandT is not convex. A positive answer to Ok’s question2009is 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

LetMbe a nonempty closed convex subset of a Banach spaceE,S:M → E, andT :M → E.

Suppose that

aSis compact and continuous;

bT is ak-contraction;

cSxTy∈Mfor everyx, y∈M.

Then there existsx^∗∈Msuch thatSx^∗Tx^∗x^∗.

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 Mis relaxed even ifT is the zero operator. Ok2noticed that the Krasnosel’skii fixed-point theorem can be reformulated by relaxing or removing the convexity hypothesis ofM and by allowing

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the fixed-point to be a fixed-set. For variants or extensions of Krasnosel’skii-type fixed-point results, see3–9, and for other interesting results see10–13.

In this paper, we prove several new Krasnosel’skii-type fixed-set theorems for the sumST, whereS is a multimap and T is a self-map. The common domain ofS and T is not convex. Our results extend, generalize, or improve several fixed-point and fixed-set results including that given by Ok2. A positive answer to Ok’s question2is provided.

Applications to the theory of self-similarity are also given.

2. Preliminaries

LetMbe a nonempty subset of a metric spaceX : X, d,E : E, · a normed space,

∂Mthe boundary ofM, intMthe interior ofM, clMthe closure ofM, 2^X \ {∅}the set all nonempty subsets ofX,BXthe set of nonempty bounded subsets ofX,CDXthe family of nonempty closed subsets ofX,KXthe family of nonempty compact subsets ofX,Rthe set of real numbers, andR : 0,∞.A mapαK :BM → Ris called the Kuratoswki measure of noncompactness onMif

αKA:inf

>0 :A⊆ⁿ

i1

Ai and diamAi≤

, 2.1

for everyA ∈ BM, where diamAi denotes the diameter ofAi. LetT : M → X andS : M → 2^X\ {∅}. We writeSM: ∪{Sx : x∈ M}. We say thatax∈ Mis a fixed point ofT ifxTx, and the set of fixed points ofT will be denoted byFT;bT is nonexpansive if dTx, Ty ≤ dx, y for all x, y ∈ M;c T is k-contraction ifdTx, Ty ≤ kdx, y for allx, y ∈ Mand some k ∈ 0,1;d T isαK-condensing if it is continuous and, for every A∈ BMwithαKA>0,TA∈ BXandαKTA< αKA;eT is 1-set-contractive if it is continuous and, for everyA∈ BM,TA∈ BX, andαKTA≤αKA;fSis compact if clSMis a compact subset ofX.

Definition 2.1. Let T : M → X, and letϕ : R → R be either “a nondecreasing map satisfying limn→ ∞ϕⁿt 0 for every t > 0” or “an upper semicontinuous map satisfying ϕt < tfor everyt >0.” One says thatT is aϕ-contraction ifdTx, Ty≤ ϕdx, yfor all x, y∈M.

Remark 2.2. A mappingT :M → Xis said to be aϕ-contraction in the sense of Garcia-Falset 6if there exists a functionϕ:R → R satisfying either “ϕis continuous andϕt< tfor t >0” or “there existsψ :R → Rwithψ0 0 and nondecreasing such that 0< ψr ≤ r−ϕr” for which the inequalitydTx, Ty≤ϕdx, yholds for allx, y∈M.Our definition forϕ-contraction is diﬀerent in some sense from that of Garcia-Falset.

Lemma 2.3see2. LetMbe a nonempty closed subset of a normed spaceE. IfT :M → 2^M\{∅}

is compact and continuous, then there exists a minimalA∈ KMsuch thatAclTA.

Theorem 2.4see14. LetMbe a nonempty bounded closed convex subset of a Banach spaceE. Suppose thatT :M → Mis anαK-condensing map. ThenThas a fixed point inM.

Theorem 2.5see15–17. LetX be a complete metric space. IfT : X → X is aϕ-contraction, thenT has a unique fixed point inX.

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Theorem 2.6see14. LetMbe a closed subset of a Banach spaceEsuch that intMis bounded, open, and containing the origin. Suppose that T : M → E is an αK-condensing map satisfying Tx /μxfor allx∈∂Mandμ >1.ThenThas a fixed point inM.

Theorem 2.7see14. LetMbe a closed subset of a Banach spaceEsuch that intMis bounded, open, and containing the origin. Suppose that T : M → Eis a 1-set-contractive map satisfying Tx /μxfor allx∈∂Mandμ >1. IfI−TMis closed, thenThas a fixed point inM.

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 ofM. Under suitable conditions, we look for a nonempty compact subsetAofMsuch that

SA TA A 3.1

or

I−TA SA. 3.2 Theorem 3.1. LetM be a nonempty closed subset of a Banach spaceE,S : M → CDE, and T :M → E.Suppose that

bT isαK-condensing andTMis a bounded subset ofE;

cSM TM⊆M.

Then there existsA∈ KMsuch thatSA TA A.

Proof. Fixy∈SM TM.LetAdenote the set of closed subsetsCofMfor whichy∈C andSC TC ⊆ C.Note thatA is nonempty sinceM ∈ A.TakeC0 : ∩_C∈AC. As C0 is closed,y∈C0, andSC0 TC0⊆C0, we haveC0 ∈ A. LetL: clSC0 TC0∪ {y}.

Notice that clSM TMis a bounded subset ofMcontainingL.SoLis a closed subset ofC0,y∈L, and

SL TL⊆SC0 TC0⊆L. 3.3

This shows thatLC0∈ AandKL⊆ KM.SinceLis a bounded subset ofMand clSL is compact, we have

αKL αK

cl

SL TL∪ y αKSL TL

≤αKSL αKTL

αKclSL αKTL 0αKTL.

3.4

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AsT isαK-condensing, it follows thatαKL 0.ThusLis a compact subset ofM. As the Vietoris topology and the Hausdorﬀmetric topology coincide onKL 18, page 17 and page 41,KLis compact and hence closed. DefineF :KL → 2^MbyFA :SA TA.It follows that

FA SA TA⊆SL TL⊆L 3.5

for everyA ∈ KL.SinceT is continuous and Sis compact-valued and continuous, both SAandTAare compact subsets ofEand henceF :KL → KL.Moreover, the maps A → SAandA → TAare continuous, soF is continuous. ByLemma 2.3, there exists C ∈ KKLsuch thatC clFC FCsinceFCis compact and hence closed. Let A:∪C∈CC.AsCFC, we have

A

C∈C

FC F

C∈C

C

FA SA TA. 3.6

HoweverAis a compact subset ofL18, page 16, soA∈ KM.

Corollary 3.2see2, Theorem 2.4. LetM be a nonempty closed subset of a Banach spaceE, S:M → CDE, andT :M → E.Suppose that

bT is compact and continuous;

cSM TM⊆M.

In the following corollary, we assume that lim inf_{t→ ∞}t−ϕt>0 wheneverϕis upper semicontinuous.

Corollary 3.3. LetMbe a nonempty closed subset of a Banach space E,S : M → CDE, and T :M → E.Suppose that

bT is aϕ-contraction andTMis bounded;

cSM TM⊆M.

Remark 3.4. The following statements are equivalent19:

iT is aϕ-contraction, whereϕis nondecreasing, right continuous such thatϕt< t for allt >0 and limt→ ∞t−ϕt>0;

iiTis aϕ-contraction, whereϕis upper semicontinuous such thatϕt< tfor allt >0 and lim inf_{t→ ∞}t−ϕt>0.

Note thatCorollary 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 3.3.

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Theorem 3.5. LetMbe a nonempty closed subset of a normed space E,S : M → CDE, and T :M → E.Suppose that

bclSM⊆I−TM;

c I−T⁻¹is a continuous single-valued map onSM.

Then

ithere exists a minimalL∈ KMsuch thatI−TL SLandL⊆SL TL;

iithere exists a maximalA∈2^Msuch thatSA TA A.

Proof. Lety∈M.Then, byb, there existsA⊆Msuch thatSy⊆I−TA, and, asI−T⁻¹ is a single-valued map onSM,

I−T⁻¹◦S

y I−T⁻¹ Sy

⊆A⊆M. 3.7

SoI−T⁻¹◦S:M → 2^M\{∅}.Note thatSis compact-valued and clSMis a compact subset ofI−TM.The continuity ofI−T⁻¹◦Sfollows from that ofSandI−T⁻¹. Moreover, I−T⁻¹clSMis a compact subset ofM, and hence clI−T⁻¹◦SMis a compact subset ofM. ByLemma 2.3, there exists a minimalL ∈ KMsuch thatL clI−T⁻¹ ◦SL.

But, sinceI−T⁻¹ is continuous andSis compact-valued,I−T⁻¹◦Sis compact-valued and maps compact sets to compact sets. ThenI−T⁻¹◦SL, is a compact subset of M, so L I−T⁻¹◦SL.ThusI−TL SL, and henceL⊆SL TL.

Let

C:

C∈2^M:C⊆SC TC

3.8

and A : ∪_C∈CC. Clearly Ais nonempty since L ∈ C. ThenA ⊆ SA TA.Take y ∈ SA TA.It follows that

A∪ y

⊆SA TA⊆S A∪

y T

A∪ y

, 3.9

and henceA∪ {y} ∈ Candy∈A.ThusSA TA A.

Theorem 3.6. LetMbe a nonempty closed subset of a normed space E,S : M → CDE, and T :M → E.Suppose that

bT is aϕ-contraction;

cifI−Txn → y, then (xnhas a convergent subsequence;

dSM TM⊆M.

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Then

Proof. Letz ∈ clSM.By b,d, and the closeness of M, the mapx → zTx is aϕ- contraction fromMintoM. So, byTheorem 2.5, there exists a uniquex0 ∈Msuch thatx0 zTx0.Thenzx0−Tx0∈I−TM, and so clSM⊆I−TM.Since the map → zTx has a unique fixed-point, its fixed-point setI−T⁻¹zis singleton. SoI−T⁻¹: clSM → M is a single-valued map. To show thatI−T⁻¹is continuous, letynbe a sequence in clSM such thatyn → y∈I−TM. Definexn: I−T⁻¹ynandx: I−T⁻¹y. ThenI−Txn yn, andI −Tx y.We claim that xn is convergent. First, notice thatxnis bounded;

otherwise,xnhas a subsequencexnksuch thatxnk → ∞. AsI−Txnk → I−Tx,c implies thatxnkhas a convergent subsequence, a contradiction. Next, asI−T is continuous and one-to-one, it follows fromcthat the sequencexnconverges tox. Therefore,I−T⁻¹ is continuous. Now the result follows fromTheorem 3.5.

In the following result, we assume that lim inf_{t→ ∞}t−ϕt>0 wheneverϕis upper semicontinuous.

Theorem 3.7. LetMbe a nonempty compact subset of a Banach spaceE,S : M → CDE, and T :M → E.Suppose that

aSis continuous;

bT is aϕ-contraction;

cSM TM⊆M.

Then

iiithere existsB∈ KMsuch thatSB TB B.

Proof. Partsiandiifollow fromTheorem 3.6. Partiiifollows fromTheorem 3.1.

Theorem 3.8. LetMbe a closed subset of a Banach spaceEsuch that intMis bounded, open, and containing the origin,S:M → CDE, andT:M → E. Suppose that

bT is anαK-condensing map satisfying clSM∩μI−T∂M ∅for allμ >1;

c I−T⁻¹is a continuous single-valued map onSM;

dSM TM⊆M.

Then

iiithere existsB∈ KMsuch thatSB TB B.

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Proof. Letz∈clSM.AsTisαK-condensing, partdand the closeness ofMimply that the mapx → zTxis anαK-condensing self-map ofM.Moreover, this map satisfieszTx /μx for allx∈∂Mandμ >1; otherwise, there arex0 ∈∂Mandμ0 >1 such thatzTx0 μ0x0. This implies that

zμ0x0−Tx0

μ0I−T x0 ∈

μ0I−T

∂M 3.10 which contradicts the second part ofb. It follows fromTheorem 2.6that there existsv∈M such thatzTvv.Thenzv−Tv∈I−TM, and so clSM⊆I−TM.Now parts iandiifollow fromTheorem 3.5. Partiiifollows fromTheorem 3.1.

Theorem 3.9. LetMbe a closed subset of a Banach spaceEsuch that intMis bounded, open, and containing the origin,S:M → CDE, andT:M → E.Suppose that

bT is a 1-set-contractive map satisfying clSM∩μI−T∂M ∅for allμ >1;

c I−TMis closed, andI−T⁻¹is a continuous single-valued map onSM;

dSM TM⊆M.

Then

iithere existsA∈2^Msuch thatSA TA A.

Proof. Letz∈clSM.AsTis 1-set-contractive, partdand the closeness ofMimply that the mapx → zTxis a 1-set-contractive self-map ofM. Moreover, this map satisfieszTx /μx for allx∈∂Mandμ >1; otherwise, there arex0 ∈∂Mandμ0 >1 such thatzTx0 μ0x0. This implies that

zμ0x0−Tx0

μ0I−T x0 ∈

μ0I−T

∂M 3.11 which contradicts the second part ofb. It follows fromTheorem 2.7that there existsv∈M such thatzTv v.Thenzv−Tv∈I−TM, and so clSM⊆I−TM.Now the result follows fromTheorem 3.5.

Definition 3.10 self-similar sets. Let M be a nonempty closed subset of a Banach space E. If F1, . . . , Fn are finitely many self-maps of M, then the list M,{F1, . . . , Fn} is called aniterated function system IFS. This IFS is continuousresp., contraction, αK-condensing, etc.if each Fi is so. A nonempty subsetAofMis said to be self-similar with respect to the IFSM,{F1, . . . , Fn}if

F1A∪ · · · ∪FnA A. 3.12 Remark 3.11. It is well known that there exists a unique compact self-similar set with respect to any contractive IFS; see20.

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Example 3.12. Consider an IFSM,{F1, . . . , Fn, Fn1}such that aF1∪ · · · ∪Fnis a compact and continuous multimap;

bFiM Fn1M⊆Mfor eachi1,2, . . . , n.

Then the existence of a compact self-similar set with respect to the IFSM,{F1, . . . , Fn} is ensured by lettingFn1to be zero in each of the following situations.

iSuppose thatF_n1 is anαK-condensing map such thatF_n1Mis bounded. Then Theorem 3.1ensures the existence of a compact subsetAofMsuch that

F1A∪ · · · ∪FnA Fn1A A. 3.13

iiSuppose thatFn1is aϕ-contraction satisfying conditioncofTheorem 3.6. Then there exists a minimal compact subsetLofMsuch that

I−Fn1L F1L∪ · · · ∪FnL. 3.14

iiiSuppose that M is a closed subset of a Banach space E such that intM is bounded, open, and containing the origin,F_n1is anαK-condensing map satisfying clF1M∪ · · · ∪FnM∩μI−F_n1∂M ∅for allμ >1,andI−F_n1⁻¹ is a continuous single-valued map onF1∪ · · · ∪FnM.ThenTheorem 3.8ensures the existence of a minimal compact subsetLofMsuch that

I−F_n1L F1L∪ · · · ∪FnL. 3.15

ivSuppose that M is a closed subset of a Banach space E such that intM is bounded, open, and containing the origin,Fn1is a 1-set-contractive map satisfying clF1M∪ · · · ∪FnM∩μI−F_n1∂M ∅for allμ >1,I−F_n1Mis closed, and I −Fn1⁻¹ is a continuous single-valued map onF1 ∪ · · · ∪FnM.Then Theorem 3.9ensures the existence of a minimal compact subsetLofMsuch that

I−F_n1L F1L∪ · · · ∪FnL. 3.16

Acknowledgments

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

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