Fixed Point Theory for Admissible Type Maps with Applications

Volume 2009, Article ID 439176,22pages doi:10.1155/2009/439176

Research Article

Fixed Point Theory for Admissible Type Maps with Applications

Ravi P. Agarwal

and Donal O’Regan

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

2Department of Mathematics, National University of Ireland, Galway, Ireland

Correspondence should be addressed to Ravi P. Agarwal,agarwal@fit.edu Received 8 December 2008; Accepted 18 June 2009

Recommended by Marlene Frigon

We present new Leray-Schauder alternatives, Krasnoselskii and Lefschetz fixed point theory for multivalued maps between Fr´echet spaces. As an application we show that our results are directly applicable to establish the existence of integral equations over infinite intervals.

Copyrightq2009 R. P. Agarwal and D. O’Regan. 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.

1. Introduction

In this paper, assuming a natural sequentially compact condition we establish new fixed point theorems for Urysohn type maps between Fr´echet spaces. InSection 2we present new Leray- Schauder alternatives, Krasnoselskii and Lefschetz fixed point theory for admissible type maps. The proofs rely on fixed point theory in Banach spaces and viewing a Fr´echet space as the projective limit of a sequence of Banach spaces. Our theory is partly motivated by a variety of authors in the literaturesee1–6and the references therein.

Existence inSection 2is based on a Leray-Schauder alternative for Kakutani mapssee 4,5,7for the history of this resultwhich we state here for the convenience of the reader.

Theorem 1.1. LetBbe a Banach space,Uan open subset ofB,and 0∈U. SupposeT :U → CKB is an upper semicontinuous compact (or countably condensing) map (hereCKBdenotes the family of nonempty convex compact subsets ofB). Then either

A1T has a fixed point inUor

A2there existsu∈∂U(the boundary ofUinB) andλ∈0,1withu∈λTu.

Existence inSection 2will also be based on the topological transversality theoremsee 5,7for the history of this resultwhich we now state here for the convenience of the reader.

LetBbe a Banach space andUan open subset ofB.

Definition 1.2. We letF ∈ KU, Bdenote the set of all upper semicontinuous compact or countably condensingmapsF:U → CKE.

Definition 1.3. We letF∈K_∂UU, BifF∈KU, Bwithx /∈Fxforx∈∂U.

Definition 1.4. A mapF∈K_∂UU, Bis essential inK_∂UU, Bif for everyG∈K_∂UU, Bwith G|∂U F|∂Uthere existsx∈Uwithx∈G x. OtherwiseF is inessential inK∂UU, B i.e., there exists a fixed point freeG∈K_∂UU, BwithG|∂UF|∂U.

Definition 1.5. F, G ∈ K∂UU, B are homotopic inK∂UU, B, written F ∼ G inK∂UU, B, if there exists an upper semicontinuous compactor countably condensingmapN : U× 0,1 → CKBsuch thatN_tu Nu, t : U → CKB belongs toK_∂UU, Bfor each t∈0,1andN₀FwithN₁G.

Theorem 1.6. LetBandUbe as above and letF ∈ K∂UU, B. Then the following conditions are equivalent:

iFis inessential inK_∂UU, B;

iithere exists a mapG∈K_∂UU, Bwithx /∈Gxforx∈UandF ∼GinK_∂UU, B.

Theorem 1.6immediately yields the topological transversality theorem for Kakutani maps.

Theorem 1.7. LetBand Ube as above. Suppose thatF andG are two maps inK∂UU, Bwith F∼GinK_∂UU, B. ThenFis essential inK_∂UU, Bif and only ifGis essential inK_∂UU, B.

Also existence in Section 2 will be based on the following result of Petryshyn 8, Theorem 3.

Theorem 1.8. LetEbe a Banach space and letC⊆Ebe a closed cone. LetUandVbe bounded open subsets inEsuch that 0 ∈ U ⊆ U ⊆ V and let F : W → CKCbe an upper semicontinuous, k-set contractive (countably) map; here 0≤k <1,W V∩CandWdenotes the closure ofWinC.

Assume that

1y ≥ x ∀y∈Fxandx∈∂Ωandy ≤ x ∀y∈Fxandx∈∂W(hereΩ U∩C and∂W denotes the boundary ofWinC) or

2y ≤ x ∀y∈F xandx∈∂Ωandy ≥ x ∀y∈Fxandx∈∂W.

ThenFhas a fixed point inW\Ω.

Also inSection 2we consider a class of maps which contain the Kakutani maps.

Suppose that X and Y are Hausdorfftopological spaces. Given a class X of maps, XX, Ydenotes the set of mapsF :X → 2^Y nonempty subsets ofYbelonging toX, and Xcthe set of finite compositions of maps inX. A classUof maps is defined by the following properties:

iUcontains the classCof single-valued continuous functions;

iieachF∈ Ucis upper semicontinuous and compact valued;

iiifor any polytopeP,F ∈ UcP, Phas a fixed point, where the intermediate spaces of composites are suitably chosen for eachU.

Definition 1.9. F ∈ U^κ_cX, Yif for any compact subsetKof X, there is aG ∈ UcK, Ywith Gx⊆Fxfor eachx∈K.

The classU^κ_c is due to Park9and his papers include many examples in this class.

Examples ofU^κ_c maps are the Kakutani maps, the acyclic maps, the approximable maps, and the maps admissible in the sense of Gorniewicz.

Existence in Section 2is based on a Leray-Schauder alternative10which we state here for the convenience of the reader.

Theorem 1.10. Let E be a Banach space, Uan open convex subset of E,and 0 ∈ U. Suppose F ∈ U^κ_cU, Eis an upper semicontinuous countably condensing map withx /∈λFxforx∈∂Uand λ∈0,1. ThenFhas a fixed point inU.

Also existence inSection 2will be based on some Lefschetz type fixed point theory. Let X, Y,and Γbe Hausdorfftopological spaces. A continuous single-valued mapp:Γ → Xis called a Vietoris mapwrittenp:Γ⇒Xif the following two conditions are satisfied:

ifor eachx∈X, the setp⁻¹xis acyclic,

iipis a proper map, that is, for every compactA⊆Xone has thatp⁻¹Ais compact.

LetDX, Y be the set of all pairsX ⇐^p Γ →^q Y wherepis a Vietoris map and qis continuous. We will denote every such diagram byp, q. Given two diagramsp, qand p, q, where X ⇐^p Γ →^q Y, we write p, q ∼ p, qif there are maps f : Γ → Γ and g:Γ → Γsuch thatq◦fq,p◦fp,q◦gq,andp◦gp. The equivalence class of a diagramp, q∈DX, Ywith respect to∼is denoted by

φ

X⇐^p Γ−→^q Y

:X−→Y 1.1

orφ p, qand is called a morphism fromX toY. We letMX, Ybe the set of all such morphisms. For anyφ ∈MX, Ya setφx qp⁻¹xwhereφ p, qis called an image ofxunder a morphismφ.

Consider vector spaces over a fieldK. Let Ebe a vector space andf : E → E an endomorphism. Now letNf {x∈E:fⁿx 0 for some n}wherefⁿis thenth iterate of f, and letE E\Nf. SincefNf ⊆ Nfone has the induced endomorphism f: E → E. We call f admissible if dimE < ∞; for suchf we define the generalized trace Trfoffby putting Trf trf where tr stands for the ordinary trace.

Letf {fq} : E → Ebe an endomorphism of degree zero of a graded vector space E {Eq}. We callf a Leray endomorphism ifiallf_qare admissible andiialmost allE_q are trivial. For suchfwe define the generalized Lefschetz numberΛfby

Λ f

q

−1^qTr fq

. 1.2

LetHbe the ˘Cech homology functor with compact carriers and coefficients in the field of rational numbers K from the category of Hausdorfftopological spaces and continuous maps to the category of graded vector spaces and linear maps of degree zero. Thus HX {HqX} is a graded vector space, with H_qXbeing the q-dimensional ˘Cech homology group with compact carriers of X. For a continuous map f :X → X, Hf is the induced linear map f{fq} where f_q :H_qX → H_qX.

The ˘Cech homology functor can be extended to a category of morphismssee11, page 364and also note that the homology functor H extends over this category, that is, for a morphism

φ

X⇐^p Γ−→^q Y

:X −→Y 1.3

we define the induced map

H φ

φ:HX−→HY 1.4

by putting φq◦p⁻¹ .

Let φ : X → Y be a multivalued mapnote for eachx ∈ X we assumeφx is a nonempty subset ofY. A pair p, qof single valued continuous maps of the form X ←^p Γ →^q Yis called a selected pair ofφwrittenp, q⊂φif the following two conditions hold:

ipis a Vietoris map,

iiqp⁻¹x⊂φxfor anyx∈X.

Definition 1.11. An upper semicontinuous compact mapφ :X → Y is said to be admissible and we writeφ∈AdX, Yprovided that there exists a selected pairp, qofφ.

Definition 1.12. An upper semicontinuous mapφ : X → Y is said to be admissible in the sense of Gorniewiczand we writeφ ∈ADX, Yprovided that there exists a selected pair p, qofφ.

Definition 1.13. A mapφ ∈ AdX, Xis said to be a Lefschetz map if for each selected pair p, q ⊂ φ the linear mapq p⁻¹ : HX → HX the existence of p⁻¹ follows from the Vietoris theoremis a Leray endomorphism.

Ifφ:X → Xis a Lefschetz map, we define the Lefschetz setΛφ orΛXφby Λ

φ

Λ

qp⁻¹ : p, q

⊂φ

. 1.5

Definition 1.14. A Hausdorfftopological spaceXis said to be a Lefschetz space provided that everyφ∈AdX, Xis a Lefschetz map andΛφ/{0}that impliesφhas a fixed point.

Also we present Krasnoselskii compression and expansion theorems inSection 2in the Fr´echet space setting. Let E E,| · | be a normed linear space and C⊆E a closed cone.

For r >0 let BC0, r {x∈C: |x| ≤r} and it is well known that BC0, r B0, R∩C where B0, r {x∈E: |x| ≤r}. Our next result,Theorem 1.8, was established in12and Theorem 1.10can be found in13.

Theorem 1.15. LetE E,| · |be a normed linear space,C⊆Ea closed cone,r,Rconstants, and 0< r < R. Suppose thatF∈ U^κ_cB0, R∩C, Cis compact with

⎧⎨

⎩

y≥ |x| ∀y∈Fx, x∈∂BC0, r,

y≤ |x| ∀y∈Fx, x∈∂BC0, R. 1.6

ThenFhas a fixed point in BCr,R{x∈C:r≤ x ≤R}.

Theorem 1.16. LetE E,| · |be a normed linear space,C⊆Ea closed cone,r,Rconstants, and 0< r < R. Suppose thatF∈ADC, Cis completely continuous with

⎧⎨

⎩

y≤ |x| ∀y∈Fx, x∈∂BC0, r,

y≥ |x| ∀y∈Fx, x∈∂BC0, R. 1.7

ThenFhas a fixed point in BC_r,R.

Now letI be a directed set with order≤and let {Eα}_α∈Ibe a family of locally convex spaces. For eachα∈I, β∈Ifor which α≤βlet πα,β:Eβ → Eα be a continuous map. Then the set

x xα∈

α∈I

Eα:xαπα,β

xβ

∀α, β∈I, α≤β

1.8 is a closed subset of

α∈IE_α and is called the projective limit of {Eα}_α∈I and is denoted by lim_←E_α or lim←{Eα, π_α,β} or the generalized intersection14, page 439

α∈IE_α.

2. Fixed Point Theory in Fr ´echet Spaces

Let E E,{| · |_n}_n∈N be a Fr´echet space with the topology generated by a family of seminorms {| · |_n : n ∈ N}; here N {1,2, . . .}. We assume that the family of seminorms satisfies

|x|₁≤ |x|₂≤ |x|₃ ≤ · · · for everyx∈E. 2.1

A subset X of E is bounded if for every n∈N there exists rn >0 such that|x|n ≤ rn

for all x ∈X. For r >0 and x ∈E we denote Bx, r {y ∈E : |x−y|n ≤ r ∀n ∈N}.

To E we associate a sequence of Banach spaces {En,| · |_n} described as follows. For every n∈Nwe consider the equivalence relation∼ndefined by

x∼ny iffx−y

n0. 2.2

We denote by Eⁿ E/∼n,| · |_n the quotient space, and by En,| · |_n the completion of Eⁿ with respect to| · |_nthe norm on Eⁿ induced by| · |_nand its extension toEnis still denoted by

| · |_n. This construction defines a continuous map μ_n :E → En. Now since2.1is satisfied the seminorm| · |_n induces a seminorm on Em for every m ≥ n again this seminorm is denoted by| · |_n. Also2.2defines an equivalence relation on Em from which we obtain a continuous map μ_n,m : Em → En since Em/∼n can be regarded as a subset of En. Now μ_n,mμ_m,k μ_n,k if n≤ m ≤ k and μ_n μ_n,mμ_m if n ≤ m. We now assume the following condition holds:

⎧⎨

⎩

for eachn∈N, there exists a Banach spaceEn,|·|_n and an isomorphism

between normed spaces

jn:En−→En. 2.3 Remark 2.1. iFor convenience the norm on E_n is denoted by| · |_n.

iiIn our applications EnEⁿ for each n∈N.

iiiNote if x∈En orEⁿthen x∈E. However if x∈En then x is not necessaily in E and in fact E_n is easier to use in applicationseven though E_n is isomorphic to En. For example if EC0,∞, then Eⁿ consists of the class of functions in E which coincide on the interval0, nand EnC0, n.

Finally we assume

⎧⎨

⎩

E1⊇E2 ⊇ · · ·and for eachn∈N, j_nμ_n,n1j_n1⁻¹ x

n≤ |x|_n1 ∀x∈E_n1. 2.4

here we use the notation from14, i.e., decreasing in the generalized senseLet lim_←En

or _∞

1En where _∞

1 is the generalized intersection 14denote the projective limit of {En}_n∈N note π_n,m j_nμ_n,mj_m⁻¹ : E_m → E_n for m ≥ n and note lim_←E_n ∼ E, so for convenience we write Elim_←En.

For each X ⊆E and each n∈N we set Xn jnμnX, and we let Xn, intXn and

∂X_n denote, respectively, the closure, the interior, and the boundary of X_n with respect to

| · |_nin En. Also the pseudointerior of X is defined by

pseudo−intX

x∈X:j_nμ_nx∈X_n\∂X_n for everyn∈N

. 2.5

The setXis pseudoopen ifX pseudo−intX. Forr > 0 andx∈En we denoteBnx, r {y∈E_n:|x−y|n≤r}.

We now show how easily one can extend fixed point theory in Banach spaces to applicable fixed point theory in Fr´echet spaces. In this case the mapFn will be related to F by the closure property2.11.

Theorem 2.2. LetEand En be as described above, X a subset of E and F : Y → 2^E where intX_n ⊆Y_n for each n∈N. Also for each n∈N assume that there exists F_n : intX_n → 2^En and suppose the following conditions are satisfied:

0∈pseudo−intX, 2.6

⎧⎨

⎩

for eachn∈ {2,3, . . .}ify∈intX_nsolvesy∈F_nyinE_n thenj_kμ_k,nj_n⁻¹

y

∈intX_k fork∈ {1, . . . , n−1}, 2.7

⎧⎨

⎩

for eachn∈N, F_n: intX_n −→CKEn

is an upper semicontinuous countably condensing map, 2.8

for eachn∈N, y /∈λFny inEn ∀λ∈0,1, y∈∂intXn, 2.9

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⎩

for any sequence y_n

n∈N with y_n∈intX_n and y_n∈F_ny_n in E_n forn∈N and for everyk∈N there exists a subsequence Nk⊆ {k1, k2, . . .}, Nk⊆N_k−1 for

k∈ {1,2, . . .}, N0N, and a zk∈intXk with jkμk,nj_n⁻¹

yn

−→zk in Ek as n−→ ∞inNk,

2.10

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⎩

if there exists aw∈Y and a sequence yn

n∈N

with y_n ∈intX_n and y_n ∈ F_ny_n in E_n such that for everyk∈N there exists a subsequence S⊆ {k1, k2, . . .} ofN withj_kμ_k,nj_n⁻¹

y_n

−→w in E_k as n−→ ∞in S, thenw∈Fw inE.

2.11

ThenFhas a fixed point inE.

Remark 2.3. Notice that to check2.10we need to show that for each k ∈ Nthe sequence {jkμ_k,nj_n⁻¹yn}_n∈Nk−1⊆intX_k is sequentially compact.

Proof. From Theorem 1.1for each n ∈ N there exists yn ∈ intXn with yn ∈ Fnyn we applyTheorem 1.1with U intX_n and note j_nμ_n0 ∈ X_n\∂X_n intX_n. Let us look at {yn}_n∈N. Notice y1 ∈ intX1 and j1μ1,kj_k⁻¹yk ∈ int X1 for k ∈ {2,3, . . .} from2.7.

Now 2.10 with k 1 guarantees that there exists a subsequence N1 ⊆ {2,3, . . .} and a z1 ∈ int X1 with j1μ1,nj_n⁻¹yn → z1 in E1 as n → ∞ in N1. Look at {yn}_n∈N1. Now j₂μ_2,nj_n⁻¹yn ∈ int X₂ for k ∈ N₁. Now 2.10 withk 2 guarantees that there exists a subsequence N2 ⊆ {3,4, . . .} of N1 and a z2 ∈int X2 with j2μ2,nj_n⁻¹yn → z2 in E2

as n → ∞ in N₂. Note from2.4and the uniqueness of limits that j₁μ_1,2j₂⁻¹z₂ z₁ inE₁ since N₂⊆N₁ note j₁μ_1,nj_n⁻¹yn j₁μ_1,2j₂⁻¹j₂μ_2,nj_n⁻¹yn for n∈N₂. Proceed inductively to obtain subsequences of integers

N1⊇N2⊇ · · ·, Nk⊆ {k1, k2, . . .}, 2.12

andz_k∈int X_kwithj_kμ_k,nj_n⁻¹yn → z_kinE_kasn → ∞inN_k.Notej_kμ_k,k1j_k1⁻¹ z_k1 z_kin Ekfork∈ {1,2, . . .}.

Fix k∈N. Note

z_kj_kμ_k,k1j_k1⁻¹ z_k1j_kμ_k,k1j_k1⁻¹ j_k1μ_k1,k2j_k2⁻¹ z_k2 jkμ_k,k2j_k2⁻¹ z_k2· · ·jkμk,mj_m⁻¹zmπk,mzm

2.13

for every m≥k. We can do this for eachk ∈N. As a resulty zk∈lim_←E_nEand also notey∈Y sincezk∈int Xk⊆Ykfor eachk∈N. Also since yn ∈Fn yn inEnfor n∈Nk

andj_kμ_k,nj_n⁻¹yn → z_kyinE_kasn → ∞inN_kone has from2.11thaty∈FyinE.

Remark 2.4. From the proof we see that condition2.7can be removed from the statement of Theorem 2.2. We include it only to explain condition2.10 seeRemark 2.3.

Remark 2.5. Note that we could replace int X_n ⊆ Y_n above with int X_n a subset of the closure of Yn in En if Y is a closed subset of E so in this case we can take Y X if X is a closed subset of E. To see this note z_k ∈ int X_k,y zk ∈ lim_←E_n E and πk,mym → zk in Ek as m → ∞ and we can conclude that y∈Y Y note thatq∈Y if and only if for every k ∈ N there exists xk,m ∈ Y,x_k,m π_k,nxn,m for n ≥ k with x_k,m → j_kμ_kq in E_k as m → ∞.

Remark 2.6. Suppose inTheorem 2.2we replace2.10with

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⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

for any sequence y_n

n∈N with y_n∈int X_n andy_n∈F_ny_n inE_n forn∈Nand

for everyk∈Nthere exists a subsequence N_k⊆ {k1, k2, . . .}, Nk⊆N_k−1 for

k∈ {1,2, . . .}, N0N, and azk∈int Xk with jkμk,nj_n⁻¹

yn

−→zk inEk asn−→ ∞inNk.

2.14

In addition we assume F : Y → 2^E with int Xn ⊆ Yn for each n ∈ N is replaced by F :X → 2^E and suppose2.11is true with w ∈Y replaced by w∈X. Then the result in Theorem 2.2is again true.

The proof follows the reasoning inTheorem 2.2except in this case zk ∈int Xk and y∈X.

Remark 2.7. In fact we could replacein fact we can remove it as mentioned inRemark 2.4 2.7inTheorem 2.2with

⎧⎨

⎩

for eachn∈ {2,3, . . .}if y∈int Xn solvesy∈FnyinEn

thenj_kμ_k,nj_n⁻¹ y

∈int X_k fork∈ {1, . . . , n−1}, 2.15

and the result above is again true.

Remark 2.8. Usually in our applications one has∂Xn ∂intXn soXn int Xn. If X is a pseudoopen subset of E then for each n∈N one hassee15that X_n is a open subset of E_n so int X_nX_n.

Essentially the same reasoning as inTheorem 2.2now usingTheorem 1.7establishes the following result. We will need the following definitions.

Let E and E_n be as described inSection 2. For the definitions below X ⊆E and F: Y → 2^E with int Xn⊆Yn for each n∈Nor int Xn a subset of the closure of Yn in En

if Y is a closed subset of E. In addition assume for each n∈N that F_n: int X_n → 2^En. Definition 2.9. We say F ∈KY, E if for each n∈None hasF_n ∈ Kint X_n, E_n i.e., for each n ∈ N, F_n : int X_n → CKEn is an upper semicontinuous countably condensing map.

Definition 2.10. F ∈ K∂Y, EifF ∈ KY, Eand for eachn ∈ N one hasx /∈Fnxforx ∈

∂ int X_n.

Definition 2.11. F ∈K_∂Y, E is essential in K_∂Y, E if for each n∈ N one has thatF_n ∈ K_∂intXnint X_n, E_nis essential in K_∂intXnint X_n, E_n i.e., for eachn∈N, every mapG_n∈ K∂intXnint Xn, EnwithGn|∂intXn Fn|∂intXnhas a fixed point in int Xn.

Remark 2.12. Note that ifj_nμ_n0 ∈ U_n for each n ∈ N then 0 ∈ K_∂Y, E is essential in K_∂Y, E see7.

Definition 2.13. We assume jnμn0 ∈int Xn for n∈N. F,0 ∈ K∂Y, E are homotopic in K_∂Y, E, written F ∼ G in K_∂Y, E, if for each n ∈ N one has F_n ∼ j_nμ_n0 in K∂intXnint Xn, En.

Theorem 2.14. LetEandE_nbe as described above,Xa subset ofEandF:Y → 2^Ewhere intX_n⊆ Y_nfor eachn∈Nor int X_na subset of the closure ofY_ninE_n(ifY is a closed subset ofE). Also for

eachn∈Nassume that there existsFn: intXn → 2^En and supposeF ∈K∂Y, E,2.6,2.7, and the following condition holds:

F ∼0 inK_∂Y, E. 2.16

Also assume2.10and2.11hold. ThenFhas a fixed point inE.

Proof. Fixn ∈ N. NowRemark 2.12guarantees that the zero mapi.e.,Gx jnμn0is essential inK_∂UnUn, E_nfor eachn ∈ N. NowTheorem 1.7guarantees thatF_nis essential in K_∂UnUn, E_n so in particular there exists y_n∈U_nwith y_n ∈F_ny_n. Essentially the same reasoning as inTheorem 2.2withRemark 2.5establishes the result.

Remark 2.15. Notice that2.6and2.17could be replaced by F ∼G in K∂Y, E of course we assume G∈K_∂Y, E and we must specify G_n for n∈N here.

Remark 2.16. Condition2.7can be removed from the statement ofTheorem 2.14.

Remark 2.17. Note thatRemark 2.6holds in this situation also.

As an application ofTheorem 2.2we discuss the integral equation

yt _∞

0

Kt, sf s, ys

ds fort∈0,∞. 2.17

Theorem 2.18. Let 1≤p <∞be a constant and 1< q≤ ∞the conjugate top. Suppose the following conditions are satisfied:

for eacht∈0,∞, the maps−→Kt, sis measurable, 2.18

sup

t∈0,∞

_∞

0

|Kt, s|^qds 1/q

< ∞, 2.19

_∞

0

K t, s

−Kt, s^qds−→0 ast−→t, for eacht∈0,∞, 2.20

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⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

f:0,∞×R → R is aL^p-Carath´eodory function:

by this one means athe mapt−→f

t, y

is measurable for ally∈R;

bthe mapy−→f t, y

is continuous for a.e. t∈0,∞;

cfor eachr >0 there existsμr ∈L^p0,∞such that y≤r impliesf

t, y≤μrtfor a.e. t∈0,∞,

2.21

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

there exists a continuous nondecreasing function ψ:0,∞−→0,∞and aφ∈L^p0,∞with f

s, y≤φsψy∀y∈R and a.e. s∈0,∞,

2.22

∃r >0 withr > K1ψr whereK1 sup

t∈0,∞

_∞

0

φs|Kt, s|ds. 2.23

Then2.17has at least one solution in C0,∞.

Remark 2.19. One could also obtain a multivalued version ofTheorem 2.18by using the ideas in the proof below with the ideas in16.

Proof. Here E C0,∞, E^k consists of the class of functions in E which coincide on the interval0, k, Ek C0, kwith of course πn,m jnμn,mj_m⁻¹ : Em → En defined by π_n,mx x|_0,n. We will applyTheorem 2.2with

X{u∈C0,∞:|u|_n≤r for eachn∈N}, 2.24 here |u|nsup_t∈0,n|ut|. Fix n∈N and note

X_nX_n{u∈C0, n:|u|_n≤r} 2.25

with

int Xn{u∈C0, n:|u|_n< r}. 2.26 Let F_n: int X_n → E_n be given by

F_n yt _n

0

Kt, sf s, ys

ds. 2.27

Also let Y X we will useRemark 2.5and let F :Y → E be given by Fyt

_∞

0

Kt, sf s, ys

ds. 2.28

Clearly2.6and2.7hold, and a standard argument in the literature guarantees that F : int X_n → E_n is continuous and compact so 2.8 holds. To show2.9 fix n ∈ N and suppose that there exists x∈∂ int X_n so|x|nrand λ∈0,1 with xλF_nx. Then for t∈0, none has

|xt| ≤ψ|x|_{n n}

0

|Kt, s|φsds≤ψ|x|_nK1, 2.29

so |x|n≤ψ|x|nK1, that is,r ≤ψrK1. This contradicts2.23, so2.9holds. To show2.10 consider a sequence {yn}_n∈N with y_n ∈C0, n,y_nF_ny_n on0, nand |yn|n< r. Now to show2.10we will show for a fixed k∈N that {jkμk,nj_n⁻¹yn}_n∈S⊆int Xkis sequentially compact for any subsequence S of {k, k1, . . .}. Note for n∈S that jkμk,nj_n⁻¹yn yn|_0,k so {jkμ_k,nj_n⁻¹yn}_n∈S is uniformly bounded since |yn|n≤r for n∈S implies |yn|k≤r for n∈S. Also {jkμk,nj_n⁻¹yn}_n∈Sis equicontinuous on0, ksince for n∈ S andt, x∈0, k note there existshr ∈L^p0,∞ with|fs, yns| ≤hrs for a.e. s∈0, none has

j_kμ_k,nj_n⁻¹ y_nt

−j_kμ_k,nj_n⁻¹ y_nx

≤ _n

0

|Kt, s−Kx, s|f

s, ynsds

≤ _∞

0

hrs^pds

1/p_∞

0

|Kt, s−Kx, s|^qds 1/q

.

2.30

The Arzela-Ascoli theorem guarantees that {jkμ_k,nj_n⁻¹yn}_n∈S ⊆ int X_k is sequentially compact. Finally we show 2.11. Suppose there exists w ∈ C0,∞ and a sequence {yn}_n∈N with yn ∈ int Xn and yn Fnyn in C0, n such that for every k ∈ N there exists a subsequence S⊆ {k1, k2, . . .} of N with y_n → w in C0, k as n → ∞ in S. If we show

wt _∞

0

Kt, sfs, wsds fort∈0,∞, 2.31

then 2.11holds. To see 2.31fix t ∈ 0,∞. Consider k ≥ t and n ∈ S as described above. Then y_nF_ny_n for n∈S and so

ynt− _k

0

Kt, sf

s, yns ds

_n

k

Kt, sf

s, yns

ds, 2.32

so

j_kμ_k,nj_n⁻¹ y_nt

− _k

0

Kt, sf

s, j_kμ_k,nj_n⁻¹

y_ns ds ≤

_n

k

|Kt, s|hrsds. 2.33

here2.21 guarantees that there exists h_r ∈ L^p0,∞ with |fs, yns| ≤ h_rs for a.e.

s∈0, nLet n → ∞through S and use the Lebesgue Dominated Convergence theorem to obtain

wt−

_k

0

Kt, sfs, wsds ≤

_∞

k

|Kt, s|hrsds 2.34

since jkμk,nj_n⁻¹yn → winC0, k. Finally let k → ∞ note2.19to obtain

wt− _∞

0

Kt, sfs, wsds0. 2.35

Thus2.11holds. Our result now follows fromTheorem 2.2withRemark 2.5.

Essentially the same reasoning as inTheorem 2.2now usingTheorem 1.8establishes the following result.

Theorem 2.20. LetEandE_nbe as described in the beginning ofSection 2,Ca closed cone inE,U, andV are bounded pseudoopen subsets ofEwith 0 ∈U⊆ U⊆ V, andF :Y → 2^E. Also assume eitherWnVn∩Cn⊆Ynfor eachn∈N(hereWnVn∩Cn) orVn∩Cna subset of the closure of Y_ninE_n(ifY is a closed subset ofE). Also for eachn∈NassumeF_n:W_n → 2^En and suppose that the following conditions hold (hereΩnC_n∩U_n):

⎧⎨

⎩

for eachn∈ {2,3, . . .} ify∈W_n\Ωn solvesy∈F_ny in E_n thenj_kμ_k,nj_n⁻¹

y

∈W_kfork∈ {1, . . . , n−1}, 2.36

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

for eachn∈N, Fn:Wn−→CK

Cn is an upper semicontinuousk-setcountablycontractive map here0≤k <1.

2.37

Also for eachn∈Nassume either

⎧⎨

⎩ y

n≥ |x|_n ∀y∈Fnx, ∀x∈∂Ωn, y

n≤ |x|_n ∀y∈Fnx, ∀x∈∂Wn, 2.38 or

⎧⎨

⎩ y

n≤ |x|_n ∀y∈Fnx, ∀x∈∂Ωn, y

n≥ |x|_n ∀y∈Fnx, ∀x∈∂Wn

2.39

hold. Finally suppose that the following hold:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

for everyk∈N and any subsequenceA⊆ {k, k1, . . .}

if x∈Cn is such thatx∈Wn\Ωn for somen∈A then there exists aγ >0 withjkμk,nj_n⁻¹x

k≥γ,

2.40

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

for any sequence yn

n∈N withyn∈Wn\Ωn

and y_n∈F_ny_n in E_n forn∈N and for everyk∈N there exists a subsequence N_k⊆ {k1, k2, . . .}, Nk⊆N_k−1 for k∈ {1,2, . . .}, N0N, and az_k∈W_k with j_kμ_k,nj_n⁻¹

y_n

−→z_k in E_kas n−→ ∞inN_k,

2.41

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

if there exists aw∈Y and a sequence y_n

n∈N

with yn ∈Wn\Ωn and yn∈Fnyn in En such that for everyk∈N there exists a subsequence

S⊆ {k1, k2, . . .} ofN withjkμk,nj_n⁻¹ yn

−→w in Ek as n−→ ∞in S, thenw∈Fw inE.

2.42

ThenFhas a fixed point in E.

Proof. Fix n∈N. We would like to applyTheorem 1.8. Note that we know from15that Cn

is a cone and U_n and V_n are open and bounded with j_nμ_n0∈U_n⊆U_n⊆V_n.Theorem 1.8 guarantees that there exists y_n ∈W_n\Ωn with y_n ∈F_ny_n in E_n. As inTheorem 2.2there exists a subsequence N1 ⊆ {2,3, . . .} and a z1 ∈ W1 with j1μ1,nj_n⁻¹yn → z1in E1 as n → ∞ in N₁. Also y_n ∈ W_n\Ωn together with2.40yields |j1μ_1,nj_n⁻¹yn|1 ≥ γ for n∈N and so|z1|1 ≥γ. Proceed inductively to obtain subsequences of integers

N1 ⊇N2⊇ · · ·, Nk⊆ {k1, k2, . . .} 2.43

and z_k∈W_k with j_kμ_k,nj_n⁻¹yn → z_k in E_k as n → ∞ in N_k. Note j_kμ_k,k1j_k1⁻¹ z_k1 z_k in E_k for k ∈ {1,2, . . .} and |zk|k ≥ γ. Now essentially the same reasoning as in Theorem 2.2withRemark 2.5guarantees the result.

Remark 2.21. Condition2.36can be removed from the statement ofTheorem 2.20.

Remark 2.22. Note 2.40 is only needed to guarantee that the fixed point y satisfies

|jkμky|k ≥ γ for k ∈ N. If we assume all the conditions inTheorem 2.20 except 2.40 then again F has a fixed point in E but the above property is not guaranteed.

Essentially the same reasoning as inTheorem 2.2just applyTheorem 1.10in this case establishes the following result.

Theorem 2.23. LetEandE_nbe as described above,Xa convex subset ofE,andF :Y → 2^Ewhere intX_n⊆Y_nfor eachn∈Nor intX_n a subset of the closure ofY_ninE_n(ifYis a closed subset ofE).

Also for eachn∈Nassume that there existsFn: intXn → 2^En and suppose that2.6,2.7,2.9, 2.10,2.11and the following condition hold:

⎧⎨

⎩

for eachn∈N, F_n∈ U^κ_c

intX_n, E_n is an

upper semicontinuous countably condensing map. 2.44

ThenFhas a fixed point inE.

Proof. Fix n ∈ N. We would like to apply Theorem 1.10. Note that we know from 15 that int Xn is convex. FromTheorem 1.10 for each n ∈ N there exists yn ∈ int Xn with y_n ∈F_ny_n in E_n. Now essentially the same reasoning as inTheorem 2.2withRemark 2.5 guarantees the result.

Remark 2.24. Note Remarks2.4,2.6, and2.7hold in this situation also.

Now we present some Lefschetz type theorems in Fr´echet spaces. Let E and E_n be as described above.

Definition 2.25. A set A ⊆ E is said to be PRLS if for each n ∈ N, A_n ≡ j_nμ_nA is a Lefschetz space.

Definition 2.26. A set A⊆E is said to be CPRLS if for each n∈N, A_n is a Lefschetz space.

Theorem 2.27. LetEandE_nbe as described above,C ⊆Eis an PRLS, andF :C → 2^E. Also for eachn∈ Nassume that there existsFn :Cn → 2^En and suppose that the following conditions are satisfied:

⎧⎨

⎩

for eachn∈ {2,3, . . .} ify∈C_n solvesy∈F_nyin E_n thenj_kμ_k,nj_n⁻¹

y

∈C_k fork∈ {1, . . . , n−1}, 2.45

for eachn∈N, Fn∈AdCn, Cn, 2.46

for eachn∈N, ΛCnFn/{0}, 2.47

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

for any sequence y_n

n∈N withy_n∈C_n and y_n∈F_ny_n in E_n forn∈N and for every∈N there exists a subsequence N_k⊆ {k1, k2, . . .}, Nk⊆N_k−1 for k∈ {1,2, . . .}, N0N, and azk∈Ck with jkμk,nj_n⁻¹

yn

−→zk in Ekas n−→ ∞inNk,

2.48

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

if there exists aw∈Cand a sequence yn

n∈N

with y_n ∈C_n andy_n∈F_ny_n in E_n such that for everyk∈N there exists a subsequence S⊆ {k1, k2, . . .} ofN withj_kμ_k,nj_n⁻¹

y_n

−→w in E_k as n−→ ∞in S, thenw∈Fw inE.

2.49

ThenFhas a fixed point inE.

Proof. For each n ∈ N there exists y_n ∈ C_n.Now the same reasoning as in Theorem 2.2 guarantees the result.

Remark 2.28. Condition2.45can be removed from the statement ofTheorem 2.27.

Remark 2.29. Suppose inTheorem 2.27, one has

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

for any sequence y_n

n∈N with y_n∈C_n andy_n∈F_ny_n inE_n forn∈N and

for everyk∈N there exists a subsequence Nk⊆ {k1, k2, . . .}, Nk⊆N_k−1 for k∈ {1,2, . . .}, N0 N, and azk∈Ck with jkμk,nj_n⁻¹

yn

−→zk inEkasn−→ ∞inNk

2.50

instead of2.48and F:C → 2^E is replaced by F:Y → 2^E with C⊆Y andCn ⊆Yn for each n∈Nand suppose that2.49is true with w∈C replaced by w∈Y. Then the result inTheorem 2.27is again true.

In fact we could replace C_n ⊆ Y_n above with C_n a subset of the closure of Y_n in E_n if Y is a closed subset of E so in this case we can take Y C if C is a closed subset of E.

In fact in this remark we could replacein fact we can remove it as mentioned in Remark 2.4 2.45with

⎧⎨

⎩

for eachn∈ {2,3, . . .}if y∈C_n solvesy∈F_nyinE_n thenj_kμ_k,nj_n⁻¹

y

∈C_kfork∈ {1, . . . , n−1} 2.51

and the result above is again true.