A LERAY-SCHAUDER ALTERNATIVE FOR WEAKLY-STRONGLY SEQUENTIALLY CONTINUOUS WEAKLY COMPACT MAPS

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WEAKLY-STRONGLY SEQUENTIALLY CONTINUOUS WEAKLY COMPACT MAPS

RAVI P. AGARWAL, DONAL O’REGAN, AND XINZHI LIU Received 23 June 2004 and in revised form 17 November 2004

A new applicable Leray-Schauder alternative is presented for weakly-strongly sequentially continuous maps. This result is then used to establish a general existence principle for operator equations.

1. Introduction

This paper presents new fixed point results for weakly sequentially upper semicontinuous maps defined on locally convex Hausdorﬀtopological spaces which are angelic when furnished with the weak topology. Moreover, we establish an applicable Leray-Schauder alternative (Theorem 2.12) for a certain subclass of these maps. Our alternative combines the advantages of the strong topology (i.e., the sets are open in the strong topology) with the advantages of the weak topology (i.e., the maps are weakly-strongly sequentially continuous and weakly compact). InSection 3, we illustrate how easilyTheorem 2.12can be applied in practice.

Finally, we recall the following definition from the literature [9].

Definition 1.1. A Hausdorﬀtopological spaceXis said to be angelic if for every relatively countably compact setC⊆X, the following hold:

(i)Cis relatively compact,

(ii) for eachx∈C, there exists a sequence{xn}n≥1⊆Csuch thatxn→x.

Remark 1.2. All metrizable locally convex spaces equipped with the weak topology are angelic (see the Eberlein-ˇSmulian theorem).

2. Fixed point theory

We begin with some fixed point results which will be needed to obtain our applicable nonlinear alternative of Leray-Schauder type (seeTheorem 2.12).

Theorem2.1. LetEbe a locally convex linear Hausdorﬀtopological space which is angelic when furnished with the weak topology, and letCbe a weakly compact, convex subset ofE.

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Then any weakly sequentially upper semicontinuous mapF:C→K(C)has a fixed point (hereK(C)denotes the family of nonempty, convex, weakly compact subsets ofC).

Remark 2.2. RecallF:C→K(C) is weakly sequentially upper semicontinuous if for any weakly closed setAofC,F⁻¹(A) is sequentially closed for the weak topology onC.

Notice that the proof ofTheorem 2.1is immediate from Himmelberg’s fixed point theorem [10] and the next result.

Theorem2.3. LetEbe a locally convex linear Hausdorfftopological space which is angelic when furnished with the weak topology, and letDbe a weakly compact subset ofE. IfF: D→2Ê(here2Êdenote the family of nonempty subsets ofE) is a weakly sequentially upper semicontinuous map, thenF:D→K(E)is a weakly upper semicontinuous map.

Proof. LetAbe a weakly closed subset of E. We first show thatF⁻¹(A) is sequentially closed inD(with respect to the strong topology). (Recall that a subsetMis sequentially closed inE(with respect to the strong topology) if wheneverx_n∈Mforn∈N= {1, 2,. . .} andx_n→x(strong topology), thenx∈M.)

Let yn∈F⁻¹(A) and yn→y(strong topology). Then yny(i.e., yn→yin (E,w)).

Now sinceF:D→2^E is weakly sequentially upper semicontinuous (i.e.,F⁻¹(A) is sequentially closed in (E,w)), we have y∈F⁻¹(A). Consequently ifAis a weakly closed subset ofD, thenF⁻¹(A) is sequentially closed inE(of course also weakly sequentially closed).

Now sinceD is weakly compact, we have thatF⁻¹(A)^w is weakly compact. Let x∈ F⁻¹(A)^w. Now sinceEis angelic when furnished with the weak topology, there exists a sequencexn∈F⁻¹(A) withxnx. Also sinceF⁻¹(A) is weakly sequentially closed, we havex∈F⁻¹(A). ThusF⁻¹(A)^w=F⁻¹(A), soF⁻¹(A) is weakly closed. ThusF:D→2^Eis

a weakly upper semicontinuous map.

Our next result replaces the weak compactness of the spaceCwith a weak compactness assumption on the operatorF. We present a number of results (see also [2,5,6,11,12]).

Theorem2.4. LetEbe a locally convex linear Hausdorﬀtopological space which is angelic when furnished with the weak topology, and suppose the Krein-ˇSmulian property holds, and letCbe a closed, convex subset ofE. Then any weakly compact, weakly sequentially upper semicontinuous mapF:C→K(C)has a fixed point.

Remark 2.5. The Krein-ˇSmulian property states that the closed convex hull of a weakly compact set is weakly compact.

Remark 2.6. IfEis a Banach space, then we know [7, page 434] that the Krein-ˇSmulian property holds. For other examples, see [8, page 553] and [9, page 82].

Proof. There exists a weakly compact subset A of C with F(C)⊆A⊆C. The Krein- ˇSmulian property guarantees that co(A) is weakly compact. Notice also thatF: co(A)→ K(co(A)), soTheorem 2.1guarantees that there existsx∈co(A) withx∈F(x).

Theorem2.7. LetEbe a locally convex linear Hausdorﬀtopological space which is angelic when furnished with the weak topology, and letCbe a closed convex subset ofEwithx0∈C.

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SupposeF:C→K(C)is a weakly sequentially upper semicontinuous map with the following property holding:

A⊆C, A=cox0

∪F(A) impliesAis weakly compact. (2.1) ThenFhas a fixed point.

Proof. ConsiderᏲthe family of all closed convex subsetsΩofCwithx0∈ΩandF(x)⊆ Ωfor allx∈Ω. Note thatᏲ= ∅sinceC∈Ᏺ. LetC0= ∩Ω∈ᏲΩ. The argument in [11]

guarantees that

C0=cox0

∪FC0

. (2.2)

Now (2.1) guarantees thatC0 is weakly compact and notice that (2.2) impliesF(C0)⊆ C0. AlsoF:C0→K(C0) is weakly sequentially upper semicontinuous so Theorem 2.1 guarantees the existence of anx0∈C0withx0∈Fx0. Theorem2.8. LetEbe a locally convex linear Hausdorﬀtopological space which is angelic when furnished with the weak topology, and letCbe a closed convex subset ofEwithx0∈C.

SupposeF:C→K(C)is a weakly sequentially upper semicontinuous map with the following properties holding:

A⊆C, A=cox0

∪F(A) impliesA^wis weakly compact, (2.3) F⁻¹A^wis weakly closed for any weakly compact subsetAofC. (2.4) ThenFhas a fixed point.

Proof. Let

D0= x0

, Dn=cox0

∪FDn−1

forn∈ {1, 2,. . .},

D= ∪^∞_n₌0Dn. (2.5)

The argument in [2, page 918] guarantees that D=cox0

∪F(D), (2.6)

so (2.3) implies thatD^wis weakly compact. Consider the mapF:D^w→K(D^w) given by

F(x)=F(x)∩D^w. (2.7)

We need of course to check thatF(x)= ∅for eachx∈D^w. Notice that (2.6) implies that F(D)⊆D⊆D^w so D⊆F⁻¹(D^w). Also F⁻¹(D^w) is a weakly closed from (2.4) so D^w⊆F⁻¹(D^w), that is,F(x)= ∅for eachx∈D^w.

Also notice thatF:D^w→K(D^w) is weakly sequentially upper semicontinuous (note that (F)⁻¹(A)=F⁻¹(A)∩D^w for any subsetAofD^w).Theorem 2.1implies that there

existsx∈D^wwithx∈F(x)⊆F(x).

Theorem2.9. LetEbe a locally convex linear Hausdorﬀtopological space which is angelic when furnished with the weak topology and suppose that the Krein-ˇSmulian property holds,

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and letCbe a closed convex subset ofEwithx0∈C. SupposeF:C→K(C)is a weakly se- quentially upper semicontinuous map with (2.4) satisfied and also assume that the following properties hold:

A⊆C, A=cox0

∪F(A) withA^w=Q^wand

Q⊆Acountable, impliesA^wis weakly compact (2.8) and

for any relatively weakly compact subsetAofE,

there exists a countable setB⊆AwithB^w=A^w. (2.9) ThenFhas a fixed point.

Proof. LetDnandDbe as inTheorem 2.8and notice that (2.6) holds. We claimDnis relatively weakly compact for eachn∈ {0, 1, 2,. . .}. The casen=0 is immediate. SupposeD_k is relatively weakly compact for somek∈ {0, 1,. . .}. ThenTheorem 2.3guarantees thatF: D^w_k →K(E) is weakly upper semicontinuous so [4] guarantees thatF(D^w_k) is weakly compact. Now since the Krein-ˇSmulian property holds, thenD_k+1is relatively weakly compact. ThusDnis relatively weakly compact for eachn∈ {0, 1, 2,. . .}. Now (2.9) implies that there existsCn;Cncountable withCn⊆DnandC^w_n =D_n^w. LetC= ∪^∞n=0Cn. The argument in [2, page 922] guarantees thatC^w=D^w. This (together) with (2.8) and (2.6) implies thatD^wis weakly compact. LetF:D^w→K(D^w) be given byF(x)=F(x)∩D^w. Notice also thatF:D^w→K(D^w) is weakly sequentially upper semicontinuous soTheorem 2.1 implies that there existsx∈D^wwithx∈F(x)⊆F(x).

In applications, it is diﬃcult and sometimes impossible to construct a setCso thatF takesCback intoC. As a result, it makes sense to discuss mapF:C→K(E). We present three Leray-Schauder alternatives. Our first result is for weakly sequentially upper semicontinuous maps, whereas our second and third results are for completely continuous maps (to be defined later).

Theorem2.10. LetEbe a locally convex linear Hausdorﬀtopological space which is angelic when furnished with the weak topology and suppose the Krein-ˇSmulian property holds, and letCbe a closed convex subset ofE,U a weakly open subset ofC,0∈U, andU^w weakly compact (hereU^wdenotes the weak closure ofUinC). SupposeF:U^w→K(C)is a weakly sequentially upper semicontinuous map which satisfies the following property:

x /∈λFx for everyx∈∂U,λ∈(0, 1); (2.10) here∂Udenotes the weak boundary ofUinC. ThenFhas a fixed point inU^w.

Proof. SupposeF does not have a fixed point in∂U(otherwise we are finished), sox /∈ λFxfor everyx∈∂Uandλ∈[0, 1]. Consider

A=

x∈U^w:x∈tF(x) for somet∈[0, 1]. (2.11)

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NowA= ∅since 0∈UandTheorem 2.3guarantees thatF:U^w→K(C) is weakly upper semicontinuous. ThusAis weakly closed, and in fact weakly compact sinceU^wis weakly compact.

AlsoA∩∂U = ∅so there exists (since (E,w), the space Eendowed with the weak topology, is completely regular) a weakly continuous mapµ:U^w→[0, 1] withµ(∂U)=0 andµ(A)=1. Let

J(x)=





µ(x)F(x), x∈U^w,

{0}, x∈C\U^w. (2.12)

Clearly, J:C→K(C) is a weakly compact, weakly sequentially upper semicontinuous map.Theorem 2.4guarantees that there existsx∈Cwithx∈J(x). Notice thatx∈U since 0∈U. As a resultx∈µ(x)F(x), sox∈A. Thusµ(x)=1 and sox∈F(x).

Remark 2.11. Notice that the assumption thatU^wis weakly compact can be removed in Theorem 2.10ifF:U^w→K(C) is weakly upper semicontinuous.

In applications, it is extremely diﬃcult to construct the weakly open setUinTheorem 2.10. This motivated us to construct a Furi-Pera-type theorem in [3]. In this paper, we present a new approach to maps which arise naturally in applications. Of course we would like also to remove the weak compactness of the domain space inTheorem 2.10and re- place it with the map being weakly compact. Our next theorem establishes such a result for a certain subclass of weakly sequential maps. The theorem combines the advantages of the strong topology (the sets are open in the strong topology) with the advantages of the weak topology (the maps are weakly-strongly sequentially continuous and weakly compact). As a result, we get a new applicable (seeSection 3) fixed point theorem. We present the result for single-valued maps.

Theorem2.12. LetEbe a locally convex linear Hausdorﬀtopological space which is angelic when furnished with the weak topology. LetCbe a closed convex subset ofE,U a convex subset ofC, andU an open (strong topology) subset ofEwith0∈U. SupposeF:U→C is a weakly-strongly sequentially continuous map (i.e.,F:U→Cis completely continuous, i.e., ifx_n,x∈U withx_nx, thenFx_n→Fx, i.e., for any closed setAofC, we have that F⁻¹(A)is weakly sequentially closed); here U denotes the closure ofU inC. In addition, suppose eitherUis weakly compact orF:U→Cis weakly compact with the Krein-ˇSmulian property holding. Also assume that

x=λFx forx∈∂_CU,λ∈(0, 1); (2.13) here∂_CUdenotes the boundary (strong topology) ofUinC. ThenFhas a fixed point inU.

Remark 2.13. Note that int_CU=U(interior in the strong topology) sinceUis open inC so as a result,∂CU=∂EU; here∂EUdenotes the boundary ofUinE.

Proof. Letµbe the Minkowski functional onUand letr:E→Ube given by r(x)= x

max1,µ(x) forx∈E. (2.14)

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Note thatr:E→U is continuous. Also sinceF:U→Cis weakly-strongly sequentially continuous, we have immediately thatrF:U→Uis weakly sequentially continuous. No- tice also thatrF:U→Uis a weakly compact map ifF:U→Cis weakly compact; note thatF(U)^wis weakly compact so the weak compactness ofrFfollows from

rFU^w ⊆co{0} ∪FU^w (2.15) and the Krein-ˇSmulian property.

We applyTheorem 2.1ifUis weakly compact andTheorem 2.4ifF:U→Cis weakly compact. Thus there existsx∈Uwithx=rF(x). Thusx=r(y) with y=F(x) andx∈ U=U∪∂U (note that intCU=U sinceU is also open in C). Now either y∈U or y /∈U. Ify∈U, thenr(y)=ysox=y=F(x), and we are finished. Ify /∈U, thenr(y)= y/µ(y) withµ(y)>1. Then x=λy (i.e.,x=λF(x)) with 0< λ=1/µ(y)<1; note that x∈∂_CUsinceµ(x)=µ(λy)=1 (note that∂_CU=∂_EUsince intCU=U). This of course

contradicts (2.13).

Remark 2.14. The argument above breaks down in the multivalued case (i.e., whenF: U→K(C)) sincerF:U→2^U but the values may not be convex. We will consider the multivalued case at a later stage using a diﬀerent argument.

Theorem2.15. LetEbe a locally convex linear Hausdorﬀtopological space which is angelic when furnished with the weak topology. LetCbe a closed convex subset ofE,U a convex subset ofC, andU an open (strong topology) subset ofEwith0∈U. SupposeF:U→C is a weakly-strongly sequentially continuous map and assume that (2.13) and the following condition hold:

D⊆U, D⊆co{0} ∪F(D) impliesD^wis weakly compact (2.16) ThenFhas a fixed point inU.

Proof. Letµandrbe as inTheorem 2.12and note thatrF:U→Uis a weakly sequentially continuous map.

LetA⊆UwithA=co({0} ∪rF(A)). Now sincerF(A)⊆co({0} ∪F(A)), we have A⊆co{0} ∪co{0} ∪F(A)=co{0} ∪F(A), (2.17) so (2.16) guarantees thatA^w(=A) is weakly compact.Theorem 2.7guarantees that there existsx∈Uwithx=rF(x). Essentially, the same reasoning as inTheorem 2.12completes

the proof.

3. Application

In this section, we show how easilyTheorem 2.12can be applied in practice. We remark here that when one uses the standard Leray-Schauder (strong topology) alternative [1] in the literature, most of the work involves checking that the map is compact. This work is removed if one usesTheorem 2.12(seeTheorem 3.1).

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Consider the Dirichlet boundary value problem

y+f(t,y,y)=0 a.e. on [0, 1],

y(0)=y(1)=0, (3.1)

where f : [0, 1]×R²→Ris anL^p-Carath´eodory function withp >1. By this we mean (i)t→ f(t,u,v) is measurable for all (u,v)∈R²,

(ii) (u,v)→ f(t,u,v) is continuous for a.e.t∈[0, 1],

(iii) for anyr >0, there existshr∈L^p[0, 1] with|f(t,u,v)| ≤hr(t) for a.e.t∈[0, 1]

and all|u| ≤rand|v| ≤r.

By a solution to (3.1) we mean a functiony∈W^2,p[0, 1] (i.e.,y∈AC[0, 1] andy∈ L^p[0, 1]), which satisfies the diﬀerential equation a.e. andy(0)=y(1)=0.

Define the operators

H1,H2:L^p[0, 1]−→C[0, 1]⊆L^p[0, 1] (3.2) by

H1u(t)= ₁

0G(t,s)u(s)ds, H2u(t)= ₁

0Gt(t,s)u(s)ds, (3.3) where

G(t,s)=





(t−1)s, 0≤s≤t≤1,

(s−1)t, 0≤t≤s≤1. (3.4)

It is easy to see that solving (3.1) is equivalent to finding a solutionu∈L^p[0, 1] to

u= −ft,H1(u),H2(u). (3.5)

Note that ifu is a solution of (3.5), then y(t)=₁

0G(t,s)u(s)dsis a solution of (3.1), whereas ifwis a solution of (3.1), thenv=wis a solution of (3.5).

Define an operatorF:L^p[0, 1]→L^p[0, 1] by Fu(t)= −ft,H1

u(t),H2

u(t). (3.6)

Consequently, solving (3.1) is equivalent to finding a fixed pointu∈L^p[0, 1] to

u=Fu. (3.7)

Theorem 3.1. Let f : [0, 1]×R²→R be anL^p-Carath´eodory function with p >1 and suppose there is a constantM0, independent ofλ, with

yL^p= 1

0

y(t)^pdt 1/ p

=M0 (3.8)

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for any solutionyto the problem

y+λ f(t,y,y)=0 a.e. on[0, 1],

y(0)=y(1)=0 (3.9)

for anyλ∈(0, 1). Then (3.1) has at least one solution.

Proof. We will applyTheorem 2.12with E=C=L^p[0, 1], U=

u∈L^p[0, 1] :uL^p< M0

. (3.10)

Notice thatU= {u∈L^p[0, 1] :uL^p≤M0}is closed and convex, so weakly closed. More- over,Uis weakly compact (recall that in a reflexive Banach space a subset is weakly compact if and only if it is closed in the weak topology and bounded in the norm topology). Also (3.8) guarantees that (2.13) holds. It remains to show thatF:U→L^p[0, 1] is a weakly-strongly sequentially continuous map. Letyn,y∈UwithynyinL^p[0, 1] (i.e., ₁

0 y_ng dt→₁

0 yg dtfor allg∈L^q[0, 1] with 1/ p+ 1/q=1). We must show thatF y_n→F y inL^p[0, 1]. Notice that

1 0

F y_n(t)−F y(t)^pdt≤ 1

0

f(t,H1

y_n,H2

y_n−ft,H1(y),H2(y)^pdt. (3.11) If we show that

1 0

ft,H1

y_n,H2

y_n−ft,H1(y),H2(y)^pdt−→0 asy_ny, (3.12) then we are finished.

First we show, for eacht∈[0, 1], that yny impliesHi

yn(t)−→Hi

y(t)fori=1, 2. (3.13) We prove (3.13) wheni=1 (the casei=2 is similar). Fixt∈[0, 1]. Then

H1

yn(t)−H1

y(t)= 1

0G(t,s)yn(s)−y(s)ds−→0 (3.14) asy_nysinceG(t,·)∈L^q[0, 1] for fixedt∈[0, 1]. Now (3.13) (together) with the fact that f is anL^p-Carath´eodory function gives

yny=⇒ft,H1 yn

,H2 yn

−→ft,H1(y),H2(y) a.e. on [0, 1]. (3.15) Also foru∈Uandt∈[0, 1], we have

H1

u(t)= 1

0G(t,s)u(s)ds

≤ 1

0|u|^pds 1/ p

sup

t∈[0,1]

1 0

G(t,s)^qds 1/q

≤M0 sup

t∈[0,1]

1 0

G(t,s)^qds 1/q

.

(3.16)

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Thus there exists anr >0 with Hi

u(t)≤r ∀t∈[0, 1],u∈U,i=1, 2. (3.17) Now (3.12) follows immediately from (3.15), (3.17), and the Lebesgue dominated con- vergence theorem.

We may now applyTheorem 2.12to deduce thatFhas a fixed point inU.

The argument inTheorem 3.1establishes the following existence principle for the operator equation

u=Tu, (3.18)

whereT:L^p[0, 1]→L^p[0, 1] withp >1.

Theorem3.2. Suppose there is a constantM0, independent ofλ, with

yL^p=M0 (3.19)

for any solutionyto the problem

y=λT y (3.20)

for anyλ∈(0, 1). In addition, assume thatT:U→L^p[0, 1]is a weakly-strongly sequen- tially continuous map; hereU= {u∈L^p[0, 1] :uL^p≤M0}. Then (3.18) has at least one solution inU.

Remark 3.3. Of course there is an analog ofTheorem 3.2for the operator equation (3.18) where T:E→E withE a reflexive Banach space (e.g., E could be the Sobolev space W^k,p([0, 1],Rⁿ) withk≥0 and 1< p <∞).

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Ravi P. Agarwal: Department of Mathematical Sciences, Florida Institute of Technology, Mel- bourne, FL 32901-6975, USA

E-mail address:[email protected]

Donal O’Regan: Department of Mathematics, National University of Ireland, Galway, Ireland E-mail address:[email protected]

Xinzhi Liu: Department of Applied Mathematics, University of Waterloo, ON, Canada N2L 3G1 E-mail address:[email protected]

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