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**

^{1}**and Donal O’Regan**

^{2}*1**Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA*

*2**Department 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*

A1*T* *has a fixed point inUor*

A2*there existsu*∈*∂U(the boundary ofUinB) andλ*∈0,1*withu*∈*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.

Let*B*be a Banach space and*U*an open subset of*B.*

*Definition 1.2. We letF* ∈ *KU, B*denote the set of all upper semicontinuous compact or
countably condensingmaps*F*:*U* → *CKE.*

*Definition 1.3. We letF*∈*K** _{∂U}*U, Bif

*F*∈

*KU, B*with

*x /*∈

*Fx*for

*x*∈

*∂U.*

*Definition 1.4. A mapF*∈*K** _{∂U}*U, Bis essential in

*K*

*U, Bif for every*

_{∂U}*G*∈

*K*

*U, Bwith*

_{∂U}*G|*

*∂U*

*F|*

*∂U*there exists

*x*∈

*U*with

*x*∈

*G*x. Otherwise

*F*is inessential in

*K*

*∂U*U, B i.e., there exists a fixed point free

*G*∈

*K*

*U, Bwith*

_{∂U}*G|*

*∂U*

*F|*

*∂U*.

*Definition 1.5.* *F, G* ∈ *K**∂U*U, B are homotopic in*K**∂U*U, B, written *F* ∼ *G* in*K**∂U*U, B,
if there exists an upper semicontinuous compactor countably condensingmap*N* : *U*×
0,1 → *CKB*such that*N** _{t}*u

*Nu, t*:

*U*→

*CKB*belongs to

*K*

*U, Bfor each*

_{∂U}*t*∈0,1and

*N*

_{0}

*F*with

*N*

_{1}

*G.*

* Theorem 1.6. LetBandUbe as above and letF* ∈

*K*

*∂U*U, B. Then the following conditions are

*equivalent:*

i*Fis inessential inK** _{∂U}*U, B;

ii*there exists a mapG*∈*K** _{∂U}*U, B

*withx /*∈

*Gxforx*∈

*UandF*∼

*GinK*

*U, B.*

_{∂U}Theorem 1.6immediately yields the topological transversality theorem for Kakutani maps.

**Theorem 1.7. Let**Band*Ube as above. Suppose thatF* *andG* *are two maps inK**∂U*U, B*with*
*F*∼*GinK** _{∂U}*U, B. Then

*Fis essential inK*

*U, B*

_{∂U}*if and only ifGis essential inK*

*U, B.*

_{∂U}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*∈*∂Ωand*y ≤ x ∀y∈*Fxandx*∈*∂W(here*Ω *U*∩*C*
*and∂W* *denotes the boundary ofWinC) or*

2y ≤ x ∀y∈*F xandx*∈*∂Ωand*y ≥ 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 maps*F* :*X* → 2* ^{Y}* nonempty subsets of

*Y*belonging toX, and X

*c*the set of finite compositions of maps inX. A classUof maps is defined by the following properties:

iUcontains the classCof single-valued continuous functions;

iieach*F*∈ U*c*is upper semicontinuous and compact valued;

iiifor any polytope*P,F* ∈ U*c*P, Phas a fixed point, where the intermediate spaces
of composites are suitably chosen for eachU.

*Definition 1.9.* *F* ∈ U^{κ}* _{c}*X, Yif for any compact subset

*K*of

*X, there is aG*∈ U

*c*K, Ywith

*Gx*⊆

*Fx*for each

*x*∈

*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}* _{c}*U, E

*is an upper semicontinuous countably condensing map withx /*∈

*λFxforx*∈

*∂Uand*

*λ*∈0,1. Then

*Fhas 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 map*p*:Γ → *X*is
called a Vietoris mapwritten*p*:Γ⇒*X*if the following two conditions are satisfied:

ifor each*x*∈*X, the setp*^{−1}xis acyclic,

ii*p*is a proper map, that is, for every compact*A*⊆*X*one has that*p*^{−1}Ais compact.

Let*DX, Y* be the set of all pairs*X* ⇐* ^{p}* Γ →

^{q}*Y*where

*p*is a Vietoris map and

*q*is 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

^{}

*, q*

^{}if there are maps

*f*: Γ → Γ

^{}and

*g*:Γ

^{}→ Γsuch that

*q*

^{}◦

*fq,p*

^{}◦

*fp,q*◦

*gq*

^{}

*,*and

*p*◦

*gp*

^{}. The equivalence class of a diagramp, q∈

*DX, Y*with respect to∼is denoted by

*φ*

*X*⇐* ^{p}* Γ−→

^{q}*Y*

:*X*−→*Y* 1.1

or*φ* p, qand is called a morphism from*X* to*Y*. We let*MX, Y*be the set of all such
morphisms. For any*φ* ∈*MX, Y*a set*φx qp*^{−1}xwhere*φ* p, qis called an image
of*x*under a morphism*φ.*

Consider vector spaces over a field*K. Let* *E*be a vector space and*f* : *E* → *E* an
endomorphism. Now let*Nf *{x∈*E*:*f*^{n}x 0 for some *n}*where*f*^{n}is the*nth iterate*
of *f, and letE* *E*\*Nf. Sincef*Nf ⊆ *Nf*one has the induced endomorphism
*f*: *E* → *E. We call* *f* admissible if dim*E <* ∞; for such*f* we define the generalized trace
Trfof*f*by putting Trf tr*f* where tr stands for the ordinary trace.

Let*f* {f*q*} : *E* → *E*be an endomorphism of degree zero of a graded vector space
*E* {E*q*}. We call*f* a Leray endomorphism ifiall*f** _{q}*are admissible andiialmost all

*E*

*are trivial. For such*

_{q}*f*we define the generalized Lefschetz numberΛfby

Λ
*f*

*q*

−1* ^{q}*Tr

*f*

*q*

*.* 1.2

Let*H*be 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 *
{H*q*X} is a graded vector space, with *H** _{q}*Xbeing 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*

*{f*

_{}*q*} where

*f*

*:*

_{q}*H*

*X →*

_{q}*H*

*X.*

_{q}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 *φ**q*◦*p*^{−1}* _{}* .

Let *φ* : *X* → *Y* be a multivalued mapnote for each*x* ∈ *X* we assume*φx* is a
nonempty subset of*Y*. A pair p, qof single valued continuous maps of the form *X* ←* ^{p}*
Γ →

^{q}*Y*is called a selected pair of

*φ*writtenp, q⊂

*φ*if the following two conditions hold:

i*p*is a Vietoris map,

ii*qp*^{−1}x⊂*φx*for any*x*∈*X.*

*Definition 1.11. An upper semicontinuous compact mapφ* :*X* → *Y* is said to be admissible
and we write*φ*∈*AdX, Y*provided that there exists a selected pairp, qof*φ.*

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

*Definition 1.13. A mapφ* ∈ *AdX, X*is said to be a Lefschetz map if for each selected pair
p, q ⊂ *φ* the linear map*q*_{}*p*^{−1}* _{}* :

*HX*→

*HX the existence of*

*p*

^{−1}

*follows from the Vietoris theoremis a Leray endomorphism.*

_{}If*φ*:*X* → *X*is a Lefschetz map, we define the Lefschetz set**Λφ orΛ***X*φby
**Λ**

*φ*

Λ

*q*_{}*p*^{−1}* _{}* :

*p, q*

⊂*φ*

*.* 1.5

*Definition 1.14. A Hausdorﬀ*topological space*X*is said to be a Lefschetz space provided that
every*φ*∈*AdX, X*is a Lefschetz map and**Λφ***/*{0}that implies*φ*has 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}*B0, R∩*

_{c}*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* *BC**r,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 let*I* be a directed set with order≤and let {E*α*}* _{α∈I}*be a family of locally convex
spaces. For each

*α*∈

*I, β*∈

*I*for which

*α*≤

*β*let

*π*

*α,β*:

*E*

*β*→

*E*

*α*be a continuous map. Then the set

*x* *x**α*∈

*α∈I*

*E**α*:*x**α**π**α,β*

*x**β*

∀α, β∈*I, α*≤*β*

1.8 is a closed subset of

*α∈I**E** _{α}* and is called the projective limit of {E

*α*}

*and is denoted by lim*

_{α∈I}_{←}

*E*

*or lim←{E*

_{α}*α*

*, π*

*} or the generalized intersection14, page 439*

_{α,β}*α∈I**E** _{α}*.

**2. Fixed Point Theory in Fr ´echet Spaces**

Let *E* E,{| · |* _{n}*}

*be a Fr´echet space with the topology generated by a family of seminorms {| · |*

_{n∈N}*:*

_{n}*n*∈

*N}; here*

*N*{1,2, . . .}. We assume that the family of seminorms satisfies

|x|_{1}≤ |*x|*_{2}≤ |*x|*_{3} ≤ · · · for every*x*∈*E.* 2.1

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

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 {E*n**,*| · |* _{n}*} described as follows. For every

*n*∈

*N*we consider the equivalence relation∼

*n*defined by

*x∼**n**y* iﬀ*x*−*y*

*n*0. 2.2

We denote by **E*** ^{n}* E/∼

*n*

*,*| · |

*the quotient space, and by E*

_{n}*n*

*,*| · |

*the completion of*

_{n}**E**

*with respect to| · |*

^{n}*the norm on*

_{n}**E**

*induced by| · |*

^{n}*and its extension to*

_{n}**E**

*n*is still denoted by

| · |* _{n}*. This construction defines a continuous map

*μ*

*:*

_{n}*E*→

**E**

*n*. Now since2.1is satisfied the seminorm| · |

*induces a seminorm on*

_{n}**E**

*m*for every

*m*≥

*n*again this seminorm is denoted by| · |

*. Also2.2defines an equivalence relation on*

_{n}**E**

*m*from which we obtain a continuous map

*μ*

*:*

_{n,m}**E**

*m*→

**E**

*n*since

**E**

*m*

*/∼*

*n*can be regarded as a subset of

**E**

*n*. Now

*μ*

_{n,m}*μ*

_{m,k}*μ*

*if*

_{n,k}*n*≤

*m*≤

*k*and

*μ*

_{n}*μ*

_{n,m}*μ*

*if*

_{m}*n*≤

*m. We now assume the following*condition holds:

⎧⎨

⎩

for each*n*∈*N,* there exists a Banach spaceE*n**,*|·|* _{n}*
and an isomorphism

between normed spaces

*j**n*:**E***n*−→*E**n**.* 2.3
*Remark 2.1.* iFor convenience the norm on *E** _{n}* is denoted by| · |

*.*

_{n}iiIn our applications **E***n***E*** ^{n}* for each

*n*∈

*N.*

iiiNote if *x*∈**E***n* or**E*** ^{n}*then

*x*∈

*E. However if*

*x*∈

*E*

*n*then

*x*is not necessaily in

*E*and in fact

*E*

*is easier to use in applicationseven though*

_{n}*E*

*is isomorphic to*

_{n}**E**

*n*. For example if

*EC0,*∞, then

**E**

*consists of the class of functions in*

^{n}*E*which coincide on the interval0, nand

*E*

*n*

*C0, n.*

Finally we assume

⎧⎨

⎩

*E*1⊇*E*2 ⊇ · · ·and for each*n*∈*N,*
*j*_{n}*μ*_{n,n1}*j*_{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_{←}*E**n*

or _{∞}

1*E**n* where _{∞}

1 is the generalized intersection 14denote the projective limit of
{E*n*}* _{n∈N}* note

*π*

_{n,m}*j*

_{n}*μ*

_{n,m}*j*

_{m}^{−1}:

*E*

*→*

_{m}*E*

*for*

_{n}*m*≥

*n*and note lim

_{←}

*E*

*∼*

_{n}*E, so for*convenience we write

*E*lim

_{←}

*E*

*n*.

For each *X* ⊆*E* and each *n*∈*N* we set *X**n* *j**n**μ**n*X, and we let *X**n*, int*X**n* and

*∂X** _{n}* denote, respectively, the closure, the interior, and the boundary of

*X*

*with respect to*

_{n}| · |* _{n}*in

*E*

*n*. Also the pseudointerior of

*X*is defined by

pseudo−intX

*x*∈*X*:*j*_{n}*μ** _{n}*x∈

*X*

*\*

_{n}*∂X*

*for every*

_{n}*n*∈

*N*

*.* 2.5

The set*X*is pseudoopen if*X* pseudo−intX. For*r >* 0 and*x*∈*E**n* we denote*B**n*x, 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 map*F**n* will be related to
*F* by the closure property2.11.

**Theorem 2.2. Let**Eand*E**n* *be as described above,* *X* *a subset of* *E* *and* *F* : *Y* → 2^{E}*where*
int*X** _{n}* ⊆

*Y*

_{n}*for each*

*n*∈

*N. Also for each*

*n*∈

*N*

*assume that there exists*

*F*

*: int*

_{n}*X*

*→ 2*

_{n}

^{E}

^{n}*and suppose the following conditions are satisfied:*

0∈pseudo−intX, 2.6

⎧⎨

⎩

*for eachn*∈ {2,3, . . .}*ify*∈int*X*_{n}*solvesy*∈*F*_{n}*yinE*_{n}*thenj*_{k}*μ*_{k,n}*j*_{n}^{−1}

*y*

∈int*X*_{k}*fork*∈ {1, . . . , n−1}, 2.7

⎧⎨

⎩

*for eachn*∈*N, F** _{n}*: int

*X*

*−→*

_{n}*CKE*

*n*

*is an upper semicontinuous countably condensing map,* 2.8

*for eachn*∈*N, y /*∈*λF**n**y* *inE**n* ∀λ∈0,1, y∈*∂*int*X**n**,* 2.9

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

*for any sequence*
*y*_{n}

*n∈N* *with* *y** _{n}*∈int

*X*

_{n}*and*

*y*

*∈*

_{n}*F*

_{n}*y*

_{n}*in*

*E*

_{n}*forn*∈

*N*

*and*

*for everyk*∈

*N*

*there exists a subsequence*

*N*

*k*⊆ {

*k*1, k2, . . .}, N

*k*⊆

*N*

_{k−1}*for*

*k*∈ {1,2, . . .}, N0*N,* *and* *a z**k*∈int*X**k* *with*
*j**k**μ**k,n**j*_{n}^{−1}

*y**n*

−→*z**k* *in* *E**k* *as* *n*−→ ∞*inN**k**,*

2.10

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

*if there exists aw*∈*Y* *and a sequence*
*y**n*

*n∈N*

*with* *y** _{n}* ∈int

*X*

_{n}*and*

*y*

*∈*

_{n}*F*

_{n}*y*

_{n}*in*

*E*

_{n}*such that*

*for everyk*∈

*N*

*there exists a subsequence*

*S*⊆ {

*k*1, k2, . . .}

*ofN*

*withj*

_{k}*μ*

_{k,n}*j*

_{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 check*2.10we need to show that for each *k* ∈ *N*the sequence
{j*k**μ*_{k,n}*j*_{n}^{−1}y*n*}_{n∈N}* _{k−1}*⊆int

*X*

*is sequentially compact.*

_{k}*Proof. From* Theorem 1.1for each *n* ∈ *N* there exists *y**n* ∈ int*X**n* with *y**n* ∈ *F**n**y**n* we
applyTheorem 1.1with *U* int*X** _{n}* and note

*j*

_{n}*μ*

*0 ∈*

_{n}*X*

*\*

_{n}*∂X*

*int*

_{n}*X*

*. Let us look at {y*

_{n}*n*}

*. Notice*

_{n∈N}*y*1 ∈ int

*X*1 and

*j*1

*μ*1,k

*j*

_{k}^{−1}y

*k*∈ int

*X*1 for

*k*∈ {2,3, . . .} from2.7.

Now 2.10 with *k* 1 guarantees that there exists a subsequence *N*1 ⊆ {2,3, . . .} and a
*z*1 ∈ int *X*1 with *j*1*μ*1,n*j*_{n}^{−1}y*n* → *z*1 in *E*1 as *n* → ∞ in *N*1. Look at {y*n*}_{n∈N}_{1}. Now
*j*_{2}*μ*_{2,n}*j*_{n}^{−1}y*n* ∈ int *X*_{2} for *k* ∈ *N*_{1}. Now 2.10 with*k* 2 guarantees that there exists a
subsequence *N*2 ⊆ {3,4, . . .} of *N*1 and a *z*2 ∈int *X*2 with *j*2*μ*2,n*j*_{n}^{−1}y*n* → *z*2 in *E*2

as *n* → ∞ in *N*_{2}. Note from2.4and the uniqueness of limits that *j*_{1}*μ*_{1,2}*j*_{2}^{−1}*z*_{2} *z*_{1} in*E*_{1}
since *N*_{2}⊆*N*_{1} note *j*_{1}*μ*_{1,n}*j*_{n}^{−1}y*n* *j*_{1}*μ*_{1,2}*j*_{2}^{−1}*j*_{2}*μ*_{2,n}*j*_{n}^{−1}y*n* for *n*∈*N*_{2}. Proceed inductively
to obtain subsequences of integers

*N*1⊇*N*2⊇ · · ·*,* *N**k*⊆ {*k*1, k2, . . .}, 2.12

and*z** _{k}*∈int

*X*

*with*

_{k}*j*

_{k}*μ*

_{k,n}*j*

_{n}^{−1}y

*n*→

*z*

*in*

_{k}*E*

*as*

_{k}*n*→ ∞in

*N*

_{k}*.*Note

*j*

_{k}*μ*

_{k,k1}*j*

_{k1}^{−1}

*z*

_{k1}*z*

*in*

_{k}*E*

*k*for

*k*∈ {1,2, . . .}.

Fix *k*∈*N. Note*

*z*_{k}*j*_{k}*μ*_{k,k1}*j*_{k1}^{−1} *z*_{k1}*j*_{k}*μ*_{k,k1}*j*_{k1}^{−1} *j*_{k1}*μ*_{k1,k2}*j*_{k2}^{−1} *z*_{k2}*j**k**μ*_{k,k2}*j*_{k2}^{−1} *z** _{k2}*· · ·

*j*

*k*

*μ*

*k,m*

*j*

_{m}^{−1}

*z*

*m*

*π*

*k,m*

*z*

*m*

2.13

for every *m*≥*k. We can do this for eachk* ∈*N. As a resulty* z*k*∈lim_{←}*E*_{n}*E*and also
note*y*∈*Y* since*z**k*∈int *X**k*⊆*Y**k*for each*k*∈*N. Also since* *y**n* ∈*F**n* *y**n* in*E**n*for *n*∈*N**k*

and*j*_{k}*μ*_{k,n}*j*_{n}^{−1}y*n* → *z*_{k}*y*in*E** _{k}*as

*n*→ ∞in

*N*

*one has from2.11that*

_{k}*y*∈

*Fy*in

*E.*

*Remark 2.4. From the proof we see that condition*2.7can 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*

*above with int*

_{n}*X*

*a subset of the closure of*

_{n}*Y*

*n*in

*E*

*n*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*

*∈ int*

_{k}*X*

*,*

_{k}*y*z

*k*∈ lim

_{←}

*E*

_{n}*E*and

*π*

*k,m*y

*m*→

*z*

*k*in

*E*

*k*as

*m*→ ∞ and we can conclude that

*y*∈

*Y*

*Y*note that

*q*∈

*Y*if and only if for every

*k*∈

*N*there exists x

*k,m*∈

*Y*,

*x*

_{k,m}*π*

*x*

_{k,n}*n,m*for

*n*≥

*k*with

*x*

*→*

_{k,m}*j*

_{k}*μ*

*q in*

_{k}*E*

*as*

_{k}*m*→ ∞.

*Remark 2.6. Suppose in*Theorem 2.2we replace2.10with

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

for any sequence
*y*_{n}

*n∈N* with *y** _{n}*∈int

*X*

*and*

_{n}*y*

*∈*

_{n}*F*

_{n}*y*

*in*

_{n}*E*

*for*

_{n}*n*∈

*N*and

for every*k*∈*N*there exists a subsequence
*N** _{k}*⊆ {

*k*1, k2, . . .}, N

*k*⊆

*N*

*for*

_{k−1}*k*∈ {1,2, . . .}, N0*N,* and a*z**k*∈int *X**k* with
*j**k**μ**k,n**j*_{n}^{−1}

*y**n*

−→*z**k* in*E**k* as*n*−→ ∞in*N**k**.*

2.14

In addition we assume *F* : *Y* → 2* ^{E}* with int

*X*

*n*⊆

*Y*

*n*for each

*n*∈

*N*is replaced by

*F*:

*X*→ 2

*and suppose2.11is true with*

^{E}*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 *z**k* ∈int *X**k* and
*y*∈*X.*

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

⎧⎨

⎩

for each*n*∈ {2,3, . . .}if *y*∈int *X**n* solves*y*∈*F**n**y*in*E**n*

then*j*_{k}*μ*_{k,n}*j*_{n}^{−1}
*y*

∈int *X** _{k}* for

*k*∈ {1, . . . , n−1}, 2.15

and the result above is again true.

*Remark 2.8. Usually in our applications one has∂X**n* *∂*int*X**n* so*X**n* int *X**n*. If *X* is a
pseudoopen subset of *E* then for each *n*∈*N* one hassee15that *X** _{n}* is a open subset
of

*E*

*so int*

_{n}*X*

_{n}*X*

*.*

_{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

*with int*

^{E}*X*

*n*⊆

*Y*

*n*for each

*n*∈

*N*or int

*X*

*n*a subset of the closure of

*Y*

*n*in

*E*

*n*

if *Y* is a closed subset of *E. In addition assume for each* *n*∈*N* that *F** _{n}*: int

*X*

*→ 2*

_{n}

^{E}

^{n}*.*

*Definition 2.9. We say*

*F*∈

*KY, E*if for each

*n*∈

*N*one has

*F*

*∈*

_{n}*Kint*

*X*

_{n}*, E*

*i.e., for each*

_{n}*n*∈

*N,*

*F*

*: int*

_{n}*X*

*→*

_{n}*CKE*

*n*is an upper semicontinuous countably condensing map.

*Definition 2.10.* *F* ∈ *K**∂*Y, Eif*F* ∈ *KY, E*and for each*n* ∈ *N* one has*x /*∈*F**n*xfor*x* ∈

*∂* int *X** _{n}*.

*Definition 2.11.* *F* ∈*K** _{∂}*Y, E is essential in

*K*

*Y, E if for each*

_{∂}*n*∈

*N*one has that

*F*

*∈*

_{n}*K*

_{∂}_{int}

_{X}*int*

_{n}*X*

_{n}*, E*

*is essential in*

_{n}*K*

_{∂}_{int}

_{X}*int*

_{n}*X*

_{n}*, E*

*i.e., for each*

_{n}*n*∈

*N, every mapG*

*∈*

_{n}*K*

*∂*int

*X*

*n*int

*X*

*n*

*, E*

*n*with

*G*

*n*|

*∂*int

*X*

*n*

*F*

*n*|

*∂*int

*X*

*n*has a fixed point in int

*X*

*n*.

*Remark 2.12. Note that ifj*_{n}*μ** _{n}*0 ∈

*U*

*for each*

_{n}*n*∈

*N*then 0 ∈

*K*

*Y, E is essential in*

_{∂}*K*

*Y, E see7.*

_{∂}*Definition 2.13.* We assume *j**n**μ**n*0 ∈int *X**n* 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*

_{n}*μ*

*0 in*

_{n}*K*

*∂*int

*X*

*n*int

*X*

*n*

*, E*

*n*.

**Theorem 2.14. Let**EandE_{n}*be as described above,Xa subset ofEandF*:*Y* → 2^{E}*where intX** _{n}*⊆

*Y*

_{n}*for eachn*∈

*Nor int*

*X*

_{n}*a subset of the closure ofY*

_{n}*inE*

_{n}*(ifY*

*is a closed subset ofE). Also for*

*eachn*∈*Nassume that there existsF**n*: int*X**n* → 2^{E}^{n}*and supposeF* ∈*K**∂*Y, E,2.6,2.7, and
*the following condition holds:*

*F* ∼*0 inK** _{∂}*Y, E. 2.16

*Also assume*2.10*and*2.11*hold. ThenFhas a fixed point inE.*

*Proof. Fixn* ∈ *N. Now*Remark 2.12guarantees that the zero mapi.e.,*Gx * *j**n**μ**n*0is
essential in*K*_{∂U}* _{n}*U

*n*

*, E*

*for each*

_{n}*n*∈

*N. Now*Theorem 1.7guarantees that

*F*

*is essential in*

_{n}*K*

_{∂U}*U*

_{n}*n*

*, E*

*so in particular there exists*

_{n}*y*

*∈*

_{n}*U*

*with*

_{n}*y*

*∈*

_{n}*F*

_{n}*y*

*. Essentially the same reasoning as inTheorem 2.2withRemark 2.5establishes the result.*

_{n}*Remark 2.15. Notice that*2.6and2.17could be replaced by *F* ∼*G* in *K**∂*Y, E of course
we assume *G*∈*K** _{∂}*Y, E and we must specify

*G*

*for*

_{n}*n*∈

*N*here.

*Remark 2.16. Condition*2.7can be removed from the statement ofTheorem 2.14.

*Remark 2.17. Note that*Remark 2.6holds in this situation also.

As an application ofTheorem 2.2we discuss the integral equation

*yt *
_{∞}

0

*Kt, sf*
*s, ys*

*ds* for*t*∈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|^{q}*ds*
1/q

*<* ∞, 2.19

_{∞}

0

*K*
*t*^{}*, s*

−*Kt, s*^{q}*ds*−→*0 ast*−→*t*^{}*,* *for eacht*^{}∈0,∞, 2.20

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

*f*:0,∞×**R** → **R is a**L^{p}*-Carath´eodory function:*

*by this one means*
a*the mapt*−→*f*

*t, y*

*is measurable for ally*∈**R;**

b*the mapy*−→*f*
*t, y*

*is continuous for a.e. t*∈0,∞;

c*for eachr >0 there existsμ**r* ∈*L** ^{p}*0,∞

*such that*

*y*≤

*r*

*impliesf*

*t, y*≤*μ**r*t*for a.e. t*∈0,∞,

2.21

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

*there exists a continuous nondecreasing function*
*ψ*:0,∞−→0,∞*and aφ*∈*L** ^{p}*0,∞

*with*

*f*

*s, y*≤*φsψy*∀*y*∈**R**
*and a.e. s*∈0,∞,

2.22

∃r >*0 withr > K*1*ψr* *whereK*1 sup

*t∈0,∞*

_{∞}

0

*φs|Kt, s|ds.* 2.23

*Then*2.17*has at least one solution in* *C0,*∞.

*Remark 2.19. One could also obtain a multivalued version of*Theorem 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,

*E*

*k*

*C0, k*with of course

*π*

*n,m*

*j*

*n*

*μ*

*n,m*

*j*

_{m}^{−1}:

*E*

*m*→

*E*

*n*defined by

*π*

*x*

_{n,m}*x|*

_{0,n}. We will applyTheorem 2.2with

*X*{u∈*C0,*∞:|u|* _{n}*≤

*r*for each

*n*∈

*N},*2.24 here |u|

*n*sup

*|ut|. Fix*

_{t∈0,n}*n*∈

*N*and note

*X*_{n}*X** _{n}*{u∈

*C0, n*:|u|

*≤*

_{n}*r}*2.25

with

int *X**n*{u∈*C0, n*:|u|_{n}*< r}.* 2.26
Let *F** _{n}*: int

*X*

*→*

_{n}*E*

*be given by*

_{n}*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*

*is continuous and compact so 2.8 holds. To show2.9 fix*

_{n}*n*∈

*N*and suppose that there exists

*x*∈

*∂*int

*X*

*so|x|*

_{n}*n*

*r*and

*λ*∈0,1 with

*xλF*

_{n}*x. Then for*

*t*∈0, none has

|xt| ≤*ψ|x|*_{n}_{n}

0

|Kt, s|φsds≤*ψ|x|** _{n}*K1

*,*2.29

so |x|*n*≤*ψ|x|**n*K1, that is,*r* ≤*ψrK*1. This contradicts2.23, so2.9holds. To show2.10
consider a sequence {y*n*}* _{n∈N}* with

*y*

*∈*

_{n}*C0, n,y*

_{n}*F*

_{n}*y*

*on0, nand |y*

_{n}*n*|

*n*

*< r. Now to*show2.10we will show for a fixed

*k*∈

*N*that {j

*k*

*μ*

*k,n*

*j*

_{n}^{−1}y

*n*}

*⊆int*

_{n∈S}*X*

*k*is sequentially compact for any subsequence

*S*of {k, k1, . . .}. Note for

*n*∈

*S*that

*j*

*k*

*μ*

*k,n*

*j*

_{n}^{−1}y

*n*

*y*

*n*|

_{0,k}so {j

*k*

*μ*

_{k,n}*j*

_{n}^{−1}y

*n*}

*is uniformly bounded since |y*

_{n∈S}*n*|

*n*≤

*r*for

*n*∈

*S*implies |y

*n*|

*k*≤

*r*for

*n*∈

*S. Also*{j

*k*

*μ*

*k,n*

*j*

_{n}^{−1}y

*n*}

*is equicontinuous on0, ksince for*

_{n∈S}*n*∈

*S*and

*t, x*∈0, k note there exists

*h*

*r*∈

*L*

*0,∞ with|fs, y*

^{p}*n*s| ≤

*h*rs for a.e.

*s*∈0, none has

*j*_{k}*μ*_{k,n}*j*_{n}^{−1}
*y** _{n}*t

−*j*_{k}*μ*_{k,n}*j*_{n}^{−1}
*y** _{n}*x

≤
_{n}

0

|Kt, s−*Kx, s|f*

*s, y**n*s*ds*

≤
_{∞}

0

h*r*s^{p}*ds*

1/p_{∞}

0

|Kt, s−*Kx, s|*^{q}*ds*
1/q

*.*

2.30

The Arzela-Ascoli theorem guarantees that {j*k**μ*_{k,n}*j*_{n}^{−1}y*n*}* _{n∈S}* ⊆ int

*X*

*is sequentially compact. Finally we show 2.11. Suppose there exists*

_{k}*w*∈

*C0,*∞ and a sequence {y

*n*}

*with*

_{n∈N}*y*

*n*∈ int

*X*

*n*and

*y*

*n*

*F*

*n*

*y*

*n*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* for*t*∈0,∞, 2.31

then 2.11holds. To see 2.31fix *t* ∈ 0,∞. Consider *k* ≥ *t* and *n* ∈ *S* as described
above. Then *y*_{n}*F*_{n}*y** _{n}* for

*n*∈

*S*and so

*y**n*t−
_{k}

0

*Kt, sf*

*s, y**n*s
*ds*

_{n}

*k*

*Kt, sf*

*s, y**n*s

*ds,* 2.32

so

*j*_{k}*μ*_{k,n}*j*_{n}^{−1}
*y** _{n}*t

−
_{k}

0

*Kt, sf*

*s, j*_{k}*μ*_{k,n}*j*_{n}^{−1}

*y** _{n}*s

*ds*≤

_{n}

*k*

|Kt, s|h*r*sds. 2.33

here2.21 guarantees that there exists *h** _{r}* ∈

*L*

*0,∞ with |fs, y*

^{p}*n*s| ≤

*h*

*s for a.e.*

_{r}*s*∈0, nLet *n* → ∞through *S* and use the Lebesgue Dominated Convergence theorem
to obtain

*wt*−

_{k}

0

*Kt, sfs, wsds*
≤

_{∞}

*k*

|Kt, s|h*r*sds 2.34

since *j**k**μ**k,n**j*_{n}^{−1}y*n* → *w*in*C0, k. Finally let* *k* → ∞ note2.19to obtain

*wt*−
_{∞}

0

*Kt, sfs, wsds*0. 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. Let**EandE_{n}*be 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*
*eitherW**n**V**n*∩*C**n*⊆*Y**n**for eachn*∈*N(hereW**n**V**n*∩*C**n**) orV**n*∩*C**n**a subset of the closure of*
*Y*_{n}*inE*_{n}*(ifY* *is a closed subset ofE). Also for eachn*∈*NassumeF** _{n}*:

*W*

*→ 2*

_{n}

^{E}

^{n}*and suppose that*

*the following conditions hold (here*Ω

*n*

*C*

*∩*

_{n}*U*

_{n}*):*

⎧⎨

⎩

*for eachn*∈ {2,3, . . .} *ify*∈*W** _{n}*\Ω

*n*

*solvesy*∈

*F*

_{n}*y*

*in*

*E*

_{n}*thenj*

_{k}*μ*

_{k,n}*j*

_{n}^{−1}

*y*

∈*W*_{k}*fork*∈ {1, . . . , n−1}, 2.36

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*for eachn*∈*N, F**n*:*W**n*−→*CK*

*C**n* *is an upper*
*semicontinuousk-setcountablycontractive map*
here0≤*k <*1.

2.37

*Also for eachn*∈*Nassume either*

⎧⎨

⎩
*y*

*n*≥ |x|* _{n}* ∀y∈

*F*

*n*

*x,*∀x∈

*∂Ω*

*n*

*,*

*y*

*n*≤ |x|* _{n}* ∀y∈

*F*

*n*

*x,*∀x∈

*∂W*

*n*

*,*2.38

*or*

⎧⎨

⎩
*y*

*n*≤ |*x|** _{n}* ∀y∈

*F*

*n*

*x,*∀x∈

*∂Ω*

*n*

*,*

*y*

*n*≥ |*x|** _{n}* ∀y∈

*F*

*n*

*x,*∀x∈

*∂W*

*n*

2.39

*hold. Finally suppose that the following hold:*

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*for everyk*∈*N* *and any subsequenceA*⊆ {*k, k*1, . . .}

*if* *x*∈*C**n* *is such thatx*∈*W**n*\Ω*n* *for somen*∈*A*
*then there exists aγ >0 withj**k**μ**k,n**j*_{n}^{−1}*x*

*k*≥*γ,*

2.40

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

*for any sequence*
*y**n*

*n∈N* *withy**n*∈*W**n*\Ω*n*

*and* *y** _{n}*∈

*F*

_{n}*y*

_{n}*in*

*E*

_{n}*forn*∈

*N*

*and*

*for everyk*∈

*N*

*there exists a subsequence*

*N*

*⊆ {*

_{k}*k*1, k2, . . .}, N

*k*⊆

*N*

_{k−1}*for*

*k*∈ {1,2, . . .}, N0

*N,*

*and az*

*∈*

_{k}*W*

_{k}*with*

*j*

_{k}*μ*

_{k,n}*j*

_{n}^{−1}

*y*_{n}

−→*z*_{k}*in* *E*_{k}*as* *n*−→ ∞*inN*_{k}*,*

2.41

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

*if there exists aw*∈*Y* *and a sequence*
*y*_{n}

*n∈N*

*with* *y**n* ∈*W**n*\Ω*n* *and* *y**n*∈*F**n**y**n* *in* *E**n* *such that*
*for everyk*∈*N* *there exists a subsequence*

*S*⊆ {k1, k2, . . .} *ofN* *withj**k**μ**k,n**j*_{n}^{−1}
*y**n*

−→*w*
*in* *E**k* *as* *n*−→ ∞*in* *S,* *thenw*∈*Fw* *inE.*

2.42

*ThenFhas a fixed point in* *E.*

*Proof. Fix* *n*∈*N. We would like to apply*Theorem 1.8. Note that we know from15that *C**n*

is a cone and *U** _{n}* and

*V*

*are open and bounded with*

_{n}*j*

_{n}*μ*

*0∈*

_{n}*U*

*⊆*

_{n}*U*

*⊆*

_{n}*V*

*.Theorem 1.8 guarantees that there exists*

_{n}*y*

*∈*

_{n}*W*

*\Ω*

_{n}*n*with

*y*

*∈*

_{n}*F*

_{n}*y*

*in*

_{n}*E*

*. As inTheorem 2.2there exists a subsequence*

_{n}*N*1 ⊆ {2,3, . . .} and a

*z*1 ∈

*W*1 with

*j*1

*μ*1,n

*j*

_{n}^{−1}y

*n*→

*z*1in

*E*1 as

*n*→ ∞ in

*N*

_{1}. Also

*y*

*∈*

_{n}*W*

*\Ω*

_{n}*n*together with2.40yields |j1

*μ*

_{1,n}

*j*

_{n}^{−1}y

*n*|1 ≥

*γ*for

*n*∈

*N*and so|z1|1 ≥

*γ. Proceed inductively to obtain subsequences of integers*

*N*1 ⊇*N*2⊇ · · ·*,* *N**k*⊆ {*k*1, k2, . . .} 2.43

and *z** _{k}*∈

*W*

*with*

_{k}*j*

_{k}*μ*

_{k,n}*j*

_{n}^{−1}y

*n*→

*z*

*in*

_{k}*E*

*as*

_{k}*n*→ ∞ in

*N*

*. Note*

_{k}*j*

_{k}*μ*

_{k,k1}*j*

_{k1}^{−1}

*z*

_{k1}*z*

*in*

_{k}*E*

*for*

_{k}*k*∈ {1,2, . . .} and |z

*k*|

*k*≥

*γ. Now essentially the same reasoning as in*Theorem 2.2withRemark 2.5guarantees the result.

*Remark 2.21. Condition*2.36can 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

|j*k**μ**k*y|*k* ≥ *γ* for *k* ∈ *N. If we assume all the conditions in*Theorem 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. Let**EandE_{n}*be as described above,Xa convex subset ofE,andF* :*Y* → 2^{E}*where*
int*X** _{n}*⊆

*Y*

_{n}*for eachn*∈

*Nor intX*

_{n}*a subset of the closure ofY*

_{n}*inE*

_{n}*(ifYis a closed subset ofE).*

*Also for eachn*∈*Nassume that there existsF**n*: int*X**n* → 2^{E}^{n}*and suppose that*2.6,2.7,2.9,
2.10,2.11*and the following condition hold:*

⎧⎨

⎩

*for eachn*∈*N, F** _{n}*∈ U

^{κ}

_{c}int*X*_{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 *X**n* is convex. FromTheorem 1.10 for each *n* ∈ *N* there exists *y**n* ∈ int *X**n* with
*y** _{n}* ∈

*F*

_{n}*y*

*in*

_{n}*E*

*. Now essentially the same reasoning as inTheorem 2.2withRemark 2.5 guarantees the result.*

_{n}*Remark 2.24. Note Remarks*2.4,2.6, 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*

_{n}*μ*

*A is a Lefschetz space.*

_{n}*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. Let**EandE_{n}*be as described above,C* ⊆*Eis an PRLS, andF* :*C* → 2^{E}*. Also for*
*eachn*∈ *Nassume that there existsF**n* :*C**n* → 2^{E}^{n}*and suppose that the following conditions are*
*satisfied:*

⎧⎨

⎩

*for eachn*∈ {2,3, . . .} *ify*∈*C*_{n}*solvesy*∈*F*_{n}*yin* *E*_{n}*thenj*_{k}*μ*_{k,n}*j*_{n}^{−1}

*y*

∈*C*_{k}*fork*∈ {1, . . . , n−1}, 2.45

*for eachn*∈*N,* *F**n*∈*AdC**n**, C**n*, 2.46

*for eachn*∈*N,* **Λ***C**n*F*n**/*{0}, 2.47

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

*for any sequence*
*y*_{n}

*n∈N* *withy** _{n}*∈

*C*

_{n}*and*

*y*

*∈*

_{n}*F*

_{n}*y*

_{n}*in*

*E*

_{n}*forn*∈

*N*

*and*

*for every*∈

*N*

*there exists a subsequence*

*N*

*⊆ {*

_{k}*k*1, k2, . . .}, N

*k*⊆

*N*

_{k−1}*for*

*k*∈ {1,2, . . .}, N0

*N,*

*and az*

*k*∈

*C*

*k*

*with*

*j*

*k*

*μ*

*k,n*

*j*

_{n}^{−1}

*y**n*

−→*z**k* *in* *E**k**as* *n*−→ ∞*inN**k**,*

2.48

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

*if there exists aw*∈*Cand a sequence*
*y**n*

*n∈N*

*with* *y** _{n}* ∈

*C*

_{n}*andy*

*∈*

_{n}*F*

_{n}*y*

_{n}*in*

*E*

_{n}*such that*

*for everyk*∈

*N*

*there exists a subsequence*

*S*⊆ {

*k*1, k2, . . .}

*ofN*

*withj*

_{k}*μ*

_{k,n}*j*

_{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. Condition*2.45can be removed from the statement ofTheorem 2.27.

*Remark 2.29. Suppose in*Theorem 2.27, one has

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

for any sequence
*y*_{n}

*n∈N* with *y** _{n}*∈

*C*

*and*

_{n}*y*

*∈*

_{n}*F*

_{n}*y*

*in*

_{n}*E*

*for*

_{n}*n*∈

*N*and

for every*k*∈*N* there exists a subsequence
*N**k*⊆ {*k*1, k2, . . .}, N*k*⊆*N** _{k−1}* for

*k*∈ {1,2, . . .}, N0

*N,*and a

*z*

*k*∈

*C*

*k*with

*j*

*k*

*μ*

*k,n*

*j*

_{n}^{−1}

*y**n*

−→*z**k* in*E**k*as*n*−→ ∞in*N**k*

2.50

instead of2.48and *F*:*C* → 2* ^{E}* is replaced by

*F*:

*Y*→ 2

*with*

^{E}*C*⊆

*Y*and

*C*

*n*⊆

*Y*

*n*for each

*n*∈

*N*and 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*

*above with*

_{n}*C*

*a subset of the closure of*

_{n}*Y*

*in*

_{n}*E*

*if*

_{n}*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 each*n*∈ {2,3, . . .}if *y*∈*C** _{n}* solves

*y*∈

*F*

_{n}*y*in

*E*

*then*

_{n}*j*

_{k}*μ*

_{k,n}*j*

_{n}^{−1}

*y*

∈*C** _{k}*for

*k*∈ {1, . . . , n−1} 2.51

and the result above is again true.