DIRECTIONS FOR MULTIMAPS
RAVI P. AGARWAL AND DONAL O’REGAN Received 11 December 2001
We establish Birkhoff-Kellogg type theorems on invariant directions for a gen- eral class of maps. Our results, in particular, apply to Kakutani, acyclic, O’Neill, approximable, admissible, andᐁκcmaps.
1. Introduction
This paper presents Birkhoff-Kellogg type theorems on invariant directions for a large class of maps. A number of results which will enable to deduce results for upper semicontinuous maps which are either (a) Kakutani, (b) acyclic, (c) O’Neill, or (d) admissible (strongly) in the sense of Gorniewicz are given.
The results in this paper, when the map is compact, complement and extend the previously known results in [8,14,16]. Also using the results in [7], we are able to presentinvariant directionresults for countably condensing maps.
For the remainder of this section, we present some definitions and some known facts. LetX andY be subsets of Hausdorfftopological vector spacesE1
andE2, respectively. We will look at mapsF:X→K(Y), hereK(Y) denotes the family of nonempty compact subsets ofY. We sayF:X→K(Y) isKakutaniif F is upper semicontinuous with convex values. A nonempty topological space is said to be acyclic, if all its reduced ˘Cech homology groups over the rationals are trivial. NowF:X→K(Y) isacyclicifFis upper semicontinuous with acyclic values. The mapF:X→K(Y) is said to be anO’Neillmap ifFis continuous and if the values ofFconsist of one orm-acyclic components (heremis fixed).
Given two open neighborhoodsU and V of the origins inE1 and E2, re- spectively, a (U,V)-approximate continuous selection [6] ofF:X→K(Y) is a continuous functions:X→Y satisfying
s(x)∈
F(x+U)∩X+V∩Y, ∀x∈X. (1.1)
Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:7 (2003) 435–448 2000 Mathematics Subject Classification: 47H10 URL:http://dx.doi.org/10.1155/S1085337503205030
We sayF:X→K(Y) isapproximableif its restrictionF|K, to any compact sub- setK of X, admits a (U,V)-approximate continuous selection for every open neighborhoodUandVof the origins inE1andE2, respectively.
For our next definition, letXandY be metric spaces. A continuous single- valued map p:Y→X is called a Vietoris map if the following two conditions are satisfied:
(i) for eachx∈X, the setp−1(x) is acyclic;
(ii) pis a proper map, that is, for every compactA⊆X,p−1(A) is compact.
Definition 1.1. A multifunction φ:X→K(Y) is admissible (strongly) in the sense of Gorniewicz ifφ:X→K(Y) is upper semicontinuous, and if there exist a metric spaceZand two continuous mapsp:Z→Xandq:Z→Ysuch that
(i) pis a Vietoris map;
(ii) φ(x)=q(p−1(x)) for any x∈X.
Remark 1.2. It should be noted [10, page 179] thatφupper semicontinuous is redundant inDefinition 1.1.
SupposeXandY are Hausdorfftopological spaces. Given a classᐄof maps, ᐄ(X,Y) denotes the set of mapsF:X→2Y(nonempty subsets ofY) belonging toᐄ, andᐄcthe set of finite compositions of maps inᐄ. A classᐁof maps is defined by the following properties:
(i)ᐁcontains the classᏯof single-valued continuous functions;
(ii) eachF∈ᐁcis upper semicontinuous and compact valued;
(iii) for any polytopeP,F∈ᐁc(P,P) has a fixed-point where the intermedi- ate spaces of composites are suitably chosen for eachᐁ.
Definition 1.3. The mapF∈ᐁκc(X,Y) if for any compact subsetKofX, there is aG∈ᐁc(K,Y) withG(x)⊆F(x) for eachx∈K.
Examples ofᐁκc maps are the Kakutani, the acyclic, the O’Neill maps, and the maps admissible in the sense of Gorniewicz.
For a subsetK of a topological spaceX, we denote by CovX(K) the directed set of all coverings ofKby open sets ofX(usually we write Cov(K)=CovX(K)).
Given two mapsF,G:X→2Yandα∈Cov(Y),FandGare said to beα-close, if for anyx∈X, there existsUx∈α,y∈F(x)∩Ux, andw∈G(x)∩Ux.
By a space, we mean a Hausdorfftopological space. In what follows,Qde- notes a class of topological spaces. A spaceY is anextension spaceforQ(writ- tenY∈ES(Q)) if for any pair (X,K) inQwithK⊆Xclosed, any continuous function f0:K→Y extends to a continuous functionf :X→Y.
A spaceYis anapproximate extension spaceforQ(and we writeY∈AES(Q)) if for anyα∈Cov(Y) and any pair (X,K) inQwithK⊆Xclosed and any con- tinuous function f0:K→Y, there exists a continuous function f :X→Y such that f|Kisα-close to f0.
Definition 1.4. LetVbe a subset of a Hausdorfftopological vector spaceE. Then we sayV isSchauder admissibleif for every compact subsetK ofV and every coveringα∈CovV(K), there exists a continuous function (called the Schauder projection)πα:K→V such that
(i)παandi:K→V areα-close;
(ii)πα(K) is contained in a subsetC⊆V withC∈AES (compact).
IfV∈AES (compact), thenVis trivially Schauder admissible. IfVis an open convex subset of a Hausdorfflocally convex topological spaceE, then it is well known thatVis Schauder admissible.
The following fixed-point result was established in [5].
Theorem1.5. LetV be a Schauder admissible subset of a Hausdorfftopological vector spaceEandF∈ᐁκc(V,V)a compact map. ThenFhas a fixed point.
A nonempty subsetXof a Hausdorfftopological vector spaceEis said to be admissibleif for every compact subsetKofXand every neighborhoodV of 0, there exists a continuous maph:K→X withx−h(x)∈V for allx∈K and h(K) is contained in a finite-dimensional subspace ofE. The nonempty subset Xis said to beq-admissibleif any nonempty compact, convex subsetΩofXis admissible.
In [12], we proved the following fixed-point result.
Theorem1.6. LetΩbe aq-admissible, closed, convex subset of a Hausdorfftopo- logical vector spaceEwithx0∈Ω. SupposeF∈ᐁκc(Ω,Ω)with the following prop- erty holding:
A⊆Ω, A=cox0
∪F(A),impliesAis compact. (1.2) ThenFhas a fixed point inΩ.
Let (E,d) be a pseudometric space. ForS⊆E, letB(S,)= {x∈E:d(x,S)≤ },>0, whered(x,S)=infy∈Yd(x, y). The measure of noncompactness of the setM⊆Eis defined byα(M)=infQ(M) where
Q(M)=
>0 :M⊆B(A,) for some finite subsetAofE. (1.3) LetEbe a locally convex Hausdorfftopological vector space and letPbe a defin- ing system of seminorms onE. SupposeF:S→2E, hereS⊆E. The mapFis said to be a countablyP-concentrative mapping ifF(S) is bounded, and for p∈P, for each countably bounded subsetX ofS, we haveαp(F(X))≤αp(X), and for p∈P, for each countably bounded non-p-precompact subsetX ofS(i.e.,Xis not precompact in the pseudonormed space (E, p)), we haveαp(F(X))< αp(X), hereαp(·) denotes the measure of noncompactness in the pseudonormed space (E, p).
Finally for completeness, we also give the definition of countablyk-set con- tractive maps. LetX be a metric space and PB(X) the bounded subsets ofX.
The Kuratowskii measure of noncompactness is the mapα:PB(X)→[0,∞) de- fined by
α(A)=inf>0 :A⊆ ∪ni=1Xi,diamXi
≤
, (1.4)
hereA∈PB(X). LetSbe a nonempty subset ofXandH:S→2X. The mapH is called countablyk-set contractive (k≥0) ifH(S) is bounded andα(H(Ω))≤ kα(Ω) for all countably bounded setsΩofS.
2. Hausdorfflocally convex topological vector spaces
In this section, we present a variety of Birkhoff-Kellogg type theorems on in- variant directions. Throughout,Ewill be a Hausdorfflocally convex topological vector space,Cwill be a closed convex subset ofE,U⊆Cwill be convex,Uwill be an open subset ofE, and 0∈U. Notice intCU=UsinceUis open inC. Also we wish to consider mapsF:U→K(C) which are upper semicontinuous and either (a) approximable, (b) admissible (strongly) in the sense of Gorniewicz, or more generally (c)ᐁκc, hereUdenotes the closure ofUinCandK(C) denotes the family of nonempty compact subsets ofC.
To take care of all the above maps (and even more general types), we intro- duce the following definition.
Definition 2.1. The mapF∈LS(U,C) ifF:U→K(C) is upper semicontinuous and satisfies condition (D). We assume condition (D) is
for any mapF∈LSU,Cand any continuous single-valued
mapr:E→U, rFsatisfies condition (D). (2.1) Certainly if condition (D) means (a), (b), or (c) above, then (2.1) holds (see [2,6,10,15]).
Throughout this section, we will assume the mapF:U→K(C) satisfies one of the following conditions:
(H1)Fis compact;
(H2) ifD⊆UandD⊆co({0} ∪F(D)), thenDis compact; or
(H3)F is countablyP-concentrative and Eis Fr´echet (here P is a defining system of seminorms).
Fixi∈ {1,2,3}.
Definition 2.2. We sayF∈LSi(U,C) ifF∈LS(U,C) satisfies (Hi).
Remark 2.3. Throughout this section, it is possible to replaceFupper semicon- tinuous inDefinition 2.1withFclosed and taking compact sets into relatively compact sets.
The following result was established in [4].
Theorem2.4. Fixi∈ {1,2,3}and letEbe a Hausdorfflocally convex topological vector space,Ca closed convex subset ofE,U⊆Cconvex,Uan open subset ofE, 0∈U, and assume (2.1) holds. SupposeF∈LSi(U,C)and assume the following condition holds:
any mapΦ∈LSiU,Uhas a fixed point. (2.2) Then either
(i)Fhas a fixed point inU; or
(ii)there existx∈∂Uandλ∈(0,1)withx∈λFx;
here∂Udenotes the boundary ofUinC.
Example 2.5. Suppose condition (D) inDefinition 2.1meansF:U→K(C) be- longs to ᐁcκ(U,C). Now sinceᐁκc is closed under compositions, then (2.1) is true. Ifi=1, we know from [15] that (2.2) holds. Ifi=2, we know from [13]
that (2.2) is satisfied. Ifi=3, we know from [11] that (2.2) holds. As a result, Theorem 2.4contains most of the Leray-Schauder alternatives (see [4,14,16,17]
and the references therein).
For our next result, assume condition (D) is such that
for any mapF∈LSU,Cand anyλ∈R,λFsatisfies condition (D). (2.3) Certainly if condition (D) means (a) or (b) above, then (2.3) is satisfied.
Now fromTheorem 2.4, we obtain the following Birkhoff-Kellogg type theo- rem. Some of the ideas here were borrowed from the literature (see [14] and the references therein).
Theorem2.6. LetEbe a Hausdorfflocally convex topological vector space,Ca closed convex subset ofE,U⊆Cconvex,Uan open subset ofE,0∈U, and assume (2.1), (2.2) (withi=1), and (2.3) hold. SupposeF∈LS1(U,C)and assume the following condition holds:
∃µ∈R, withµFU∩U= ∅. (2.4) Then there existλ∈(0,1) andx∈∂U with(λ−1µ−1)x∈Fx (i.e.,F|∂U has an eigenvalue); hereµ=0is chosen as in (2.4).
Remark 2.7. Notice that 0∈Uguarantees thatµ=0 in (2.4).
Proof. Letµ=0 be chosen as in (2.4). Now (2.3) guarantees thatµF∈LS(U,C), and as a resultµF∈LS1(U,C). In addition, (2.4) guarantees thatµFhas no fixed points inU.Theorem 2.4(applied toµF) guarantees that there existsλandx∈
∂Uwithx∈λ(µF)x. As a result, (λ−1µ−1)x∈Fxand the proof is complete.
Example 2.8. InTheorem 2.6, if condition (D) means that the mapF:U→K(C) is either (a) approximable, or (b) admissible in the sense of Gorniewicz, then we know that (2.1), (2.2) (see [3,12,13]), and (2.3) hold.
InTheorem 2.6, if condition (D) means that the mapF:U→K(C) belongs toᐁκc(U,C), then we know that (2.1) and (2.2) hold. Notice that (2.3) may not be true. However, (2.3) (or a slight modification of it, see (2.5)) may work for a subclassᏭ(U,C) ofᐁκc(U,C) (e.g.,Ꮽcould be the Kakutani or acyclic maps or indeed the maps described in the above example). In the proof of our next result, condition (D) means that the mapF:U→K(C) belongs toᐁκc(U,C), so F∈LS(U,C) means thatFis upper semicontinuous andF∈ᐁκc(U,C).
Theorem 2.9. Let E be a Hausdorff locally convex topological vector space, C a closed convex subset ofE,U⊆Cconvex,U an open subset ofE,0∈U,F∈ Ꮽ(U,C)a compact map, and assume (2.4) holds. Suppose the following condition holds:
for any mapF∈ᏭU,C,and anyλ∈R, λF∈ᐁcκU,C. (2.5) Then there existλ∈(0,1)andx∈∂U with(λ−1µ−1)x∈Fx (i.e.,F|∂U has an eigenvalue); hereµ=0is chosen as in (2.4).
Proof. Essentially the same reasoning as inTheorem 2.6establishes the result.
In our next result, we assume (2.3) when|λ| ≤1.
Theorem2.10. Fixi∈ {2,3}and letEbe a Hausdorfflocally convex topological vector space,Ca closed convex subset ofE,U⊆Cconvex,Uan open subset ofE, 0∈U,F∈LSi(U,C), and assume (2.1) and (2.2) hold. In addition, suppose the following conditions are satisfied:
(i)for any mapF∈LS(U,C)and anyλ∈Rwith|λ| ≤1,λFsatisfies condition (D),
(ii)there existsµ∈Rwith|µ| ≤1,µF(U∩U= ∅,
(iii)ifi=2, assume eitherµ >0in (ii) or−F(D)=F(D)for anyD⊆U.
Then there existsλ∈(0,1)andx∈∂Uwith(λ−1µ−1)x∈Fx.
Proof. Letµ=0 be chosen as in (i), and notice thatµF∈LS(U,C) from (i). We claim
µF∈LSiU,C. (2.6)
Ifi=3, then (2.6) is immediate since |µ| ≤1. Next supposei=2 and letD⊆ U withD⊆co({0} ∪µF(D)). Now fromTheorem 2.10(iii), we haveµF(D)⊆ co({0} ∪F(D)), and so
D⊆co{0} ∪co{0} ∪F(D)=coco{0} ∪F(D)=co{0} ∪F(D). (2.7) Now D is compact since F∈LS2(U,C), and so (2.6) holds if i=2. Apply
Theorem 2.4toµF.
Remark 2.11. InTheorem 2.10, condition (iii) can be replaced by the following more general condition:
ifi=2 and ifD⊆UwithD⊆co{0} ∪µF(D),
thenDis compact, hereµis chosen as in (ii) (2.8) (of course with this assumption, we do not need to assume that|µ| ≤1 in (ii) if i=2). For example, ifFisP-concentrative (hereEis Fr´echet), then clearly (2.8) is satisfied (if|µ| ≤1).
Remark 2.12. It is also possible to useTheorem 2.9 to obtain an analogue of Theorem 2.10for the subclassᏭ. We leave the details to the reader.
InTheorem 2.6(resp.,Theorem 2.10), ifµ >0 in (2.4) (resp., (ii)), we say that F|∂Uhas an invariant direction (i.e., has a positive eigenvalue). Some of the ideas here were borrowed from the literature (see [14] and the references therein).
Theorem2.13. LetE=(E, · )be an infinite-dimensional normed linear space, C=E,U=B,F∈LS1(B,E), and assume that (2.1), (2.2) (withi=1), and (2.3) hold; hereB= {x∈E:x<1}. In addition, suppose the following two conditions are satisfied:
for any continuous mapr:B−→S, Frsatisfies condition (D), (2.9)
0∈/ F(S), (2.10)
hereS= {x∈E:x =1}. ThenFhas an invariant direction.
Proof. We know [7] that there exists a continuous retractionr:B→S. LetG=Fr and notice thatG∈LS(B,E) from (2.9). Now we claim that there existsµ >0 with
µF(S)∩B= ∅. (2.11)
If this is true, then
µGB∩B= ∅, (2.12)
and soTheorem 2.6(applied toGwithU=BandC=E) guarantees that there existλ∈(0,1) andx∈∂B=Swithλ−1µ−1x∈Gx=Frx=Fx. The proof is fin- ished. It remains to prove (2.11) but this is immediate since 0∈/ F(S) (i.e., if (2.11) was false, then for each n∈ {1,2,...}, there exist yn∈F(S) andwn∈B
withyn=(1/n)wn).
Remark 2.14. InTheorem 2.13, we can replaceBby any open setU ofEwith 0∈U(hereEis any Hausdorfflocally convex topological vector space) provided that ∂U is a retract of U, and in this case (2.10) is replaced by the following condition:∃µ >0 withµF(∂U)∩U= ∅.
Remark 2.15. InTheorem 2.13,F∈LS1(B,E) could be replaced byF∈LS1(S,E).
Example 2.16. InTheorem 2.13, if condition (D) means that the mapF:B→ K(E) is either (a) approximable, or (b) admissible in the sense of Gorniewicz, then we know that (2.1), (2.2), (2.3), and (2.9) hold.
Example 2.17. InTheorem 2.13, if condition (D) means that the mapF:B→ K(E) belongs toᐁκc(U,C), then we know that (2.1), (2.2), and (2.9) are satisfied.
It is possible to useTheorem 2.9to obtain an analogue ofTheorem 2.13for the subclassᏭofᐁκc. We leave the details to the reader.
In [7], the authors show that ifEis an infinite-dimensional normed linear space, then there exists a Lipschitzian retractionr:B→Swith Lipschitz constant k0(E), hereBandSare as inTheorem 2.13. In fact there exists ak0withk0(E)≤ k0for any spaceE(as described above). We refer the reader to [9, Chapter 21]
for a discussion of upper and lower bounds fork0(E), note in particular that k0(E)≥3. For our next theorem, we let
r:B−→Sbe a Lipschitzian retraction
with Lipschitz constantk0(E). (2.13) Theorem2.18. LetE=(E, · )be an infinite-dimensional normed linear space, C=E,U=B,F∈LS(B,E), and assume that (2.1), (2.2) (withi=3),Theorem 2.10(i), (2.9), and (2.13) hold; hereB= {x∈E:x<1}andS= {x∈E:x = 1}. In addition, suppose the following two conditions are satisfied:
(a)F is countablyk-set contractive with 0≤k <1/k0(E), herek0(E)is as in (2.13);
(b)there existµ >0with0< µ≤1,µF(S)∩B= ∅. ThenFhas an invariant direction.
Proof. LetG=Frwhereris as in (2.13). Notice thatG∈LS(B,E) and it is easy to check thatGis countablykk0(E)-set contractive. Thus,G∈LS3(B,E). Now
applyTheorem 2.10toG.
Remark 2.19. InTheorem 2.18,F∈LS1(B,E) could be replaced byF∈LS1(S,E).
Remark 2.20. Theorem 2.18is the first invariant direction result, to our knowl- edge, for countably contractive maps.
Remark 2.21. We note that the results in this section improve those in [8,14,16].
3. Hausdorfftopological vector spaces
Throughout this section, Ewill be a Hausdorfftopological vector space, Ca closed convex subset ofE,Uan open subset ofC, and 0∈U. This section also presents Birkhoff-Kellogg type theorems, and in some cases the results inSection 2will be improved.
Definition 3.1. The mapF∈GA(U,C) ifF:U→K(C) is upper semicontinuous and satisfies condition (C), hereU denotes the closure ofU inC. We assume condition (C) is
for any mapF∈GAU,Cand any continuous single-valued
mapµ:U→[0,1], µFsatisfies condition (C). (3.1) Certainly if condition (C) means that the mapF:U→K(C) is (a) Kakutani, (b) acyclic, (c) O’Neill, (d) approximable, or (e) admissible (strongly) in the sense of Gorniewicz, then (3.1) holds.
Fixi∈ {1,2,3}.
Definition 3.2. We say thatF∈GAi(U,C) ifF∈GA(U,C) satisfies (Hi), here (Hi) is as inSection 2.
Definition 3.3. We say thatF∈GAi∂U(U,C) ifF∈GAi(U,C) withx /∈F(x) for x∈∂U, here∂Udenotes the boundary ofUinC.
Definition 3.4. A mapF∈GAi∂U(U,C) is essential inGAi∂U(U,C) if for every G∈GAi∂U(U,C) withG|∂U=F|∂U, there existsx∈Uwithx∈G(x).
Remark 3.5. Throughout this section, it is possible to replaceFupper semicon- tinuous inDefinition 3.1withFclosed and taking compact sets into relatively compact sets.
The following result was established in [4].
Theorem3.6. Fixi∈ {1,2,3}and letEbe a Hausdorfftopological vector space,C a closed convex subset ofE,Uan open subset ofC,0∈U, and assume (3.1) holds.
SupposeF∈GAi(U,C)and assume the following condition is satisfied:
the zero map is essential inGAi∂UU,C. (3.2) Then either
(i)Fhas a fixed point inU; or
(ii)there existx∈∂Uandλ∈(0,1)withx∈λFx.
Examples. (1) Suppose condition (C) inDefinition 3.1meansF:U→AK(C), hereAK(C) denotes the family of nonempty, acyclic, compact subsets ofC. Then ifi=3 (in particularEis Fr´echet), we know from [3, Theorem 2.2] and [13, Theorem 2.6] that (3.2) (and of course (3.1)) is satisfied.
(2) Suppose condition (C) inDefinition 3.1means thatF:U→K(C) is ap- proximable. Then ifi=3, we know from [3, Theorem 2.2] and [13, Theorem 2.6] that (3.2) (and of course (3.1)) holds.
(3) Suppose condition (C) inDefinition 3.1means thatF:U→K(C) is ad- missible in the sense of Gorniewicz,Eis a Fr´echet space (P a defining system of seminorms),Uis convex, andC=E. Now [10] guarantees that (3.1) is true.
Now we show that (3.2) is satisfied ifi=1,2, or 3 (in fact ifi=1, it is enough (see Theorem 1.5) forEto be a metrizable locally convex topological vector space).
To see (3.2), letθ∈GAi∂U(U,E) withθ|∂U= {0}. We must show that there exists x∈U withx∈θ(x). Letµ be the Minkowski functional onU and let r:E→Ube given by
r(x)= x
max1,µ(x), forx∈E. (3.3)
ConsiderG=rθ. We know [10] thatGis admissible in the sense of Gorniewicz, and as a resultG∈GA(U,U). Ifi=1, thenGis compact whereas ifi=3, then Gis countableP-concentrative sincer(A)⊆co(A∪ {0}) for any subsetAofE.
Now leti=2 and letD⊆UwithD=co({0} ∪G(D)). Then sincer(A)⊆co(A∪ {0}) for any subsetAofE, we have
D⊆co{0} ∪coθ(D)∪ {0}
=co{0} ∪θ(D). (3.4) Thus,Dis compact sinceθ∈GA2(U,E). Now [12, Theorem 2.1] and [13, Theo- rem 2.2] (or alternativelyTheorem 1.5,Theorem 1.6ifi=1 or 2) guarantee that there existsx∈Uwithx∈G(x)=rθ(x). Thus,x=r(y) for some y∈θx, here x∈U=U∪∂U(noteC=Ehere). Supposex∈∂U. Thenµ(x)=1 and so
1=µ(x)=µr(y)= µ(y)
max1,µ(y), sincer(y)= y
max1,µ(y). (3.5) Thus,µ(y)≥1 and sox=r(y)=y/µ(y). This implies
x∈λθ(x)= {0} sinceθ|∂U= {0}; hereλ= 1
µ(y). (3.6) This is a contradiction since 0∈U. As a resultx∈U. This implies µ(x)<1.
Consequently,
1> µ(x)=µr(y)= µ(y)
max1,µ(y), (3.7)
and soµ(y)<1. Thusr(y)=y, sox=y∈θ(x). As a result, (3.2) holds.
(4) Suppose condition (C) inDefinition 3.1means thatF:U→K(C) is either (a) Kakutani, (b) acyclic, (c) O’Neill, or (d) approximable andC is Schauder admissible. Ifi=1, then we know from [4] that (3.2) and also (3.1) hold.
(5) Suppose condition (C) inDefinition 3.1means thatF:U→K(C) is either (a) Kakutani, (b) acyclic, (c) O’Neill, or (d) approximable andCisq-admissible with the extra condition that
co(K) is compact for any compact subsetKofE. (3.8) Ifi=2, then we know [4,1] (we useTheorem 1.6) that (3.2) and also (3.1) hold.
For our next result, assume condition (C) is
for any mapF∈GAU,Cand anyλ∈R, λFsatisfies condition (C). (3.9) Theorem3.7. Let Ebe a Hausdorff topological vector space,C a closed convex subset ofE,U an open subset ofC,0∈U, and assume (3.1), (3.2) (withi=1), and (3.9) hold. SupposeF∈GA1(U,C)and assume the following condition holds:
∃µ∈R, withµFU∩U= ∅. (3.10) Then there existλ∈(0,1)andx∈∂Uwith(λ−1µ−1)x∈Fx, hereµ=0is chosen as in (3.10).
Proof. ApplyTheorem 3.6toµF(see the proof ofTheorem 2.6).
Example 3.8. In Theorem 3.7, if condition (C) means the mapF:U→K(C) is either (a) Kakutani, (b) acyclic, (c) O’Neill, or (d) approximable, and Cis Schauder admissible, then (3.1), (3.2), and (3.9) hold.
Example 3.9. InTheorem 3.7, if condition (C) means that the mapF:U→K(C) is admissible in the sense of Gorniewicz,Eis a Fr´echet space,Uis convex, and C=E, then (3.1), (3.2), and (3.9) hold.
For our next result, we assume (3.9) when|λ| ≤1.
Theorem3.10. Fixi∈ {2,3}and letEbe a Hausdorfftopological vector space, Ca closed convex subset ofE,Uan open subset ofC,0∈U,F∈GAi(U,C), and assume (3.1) and (3.2) hold. In addition, suppose the following conditions are sat- isfied:
(a)for any mapF∈GA(U,C)and anyλ∈Rwith|λ| ≤1,λFsatisfies condi- tion(C),
(b)there existµ∈Rwith|µ| ≤1,µF(U)∩U= ∅,
(c)ifi=2, assume eitherµ >0in (b) or−F(D)=F(D)for anyD⊆U.
Then there existλ∈(0,1)andx∈∂Uwith(λ−1µ−1)x∈Fx.
Proof. ApplyTheorem 3.6toµF(see the proof ofTheorem 2.10).
Remark 3.11. InTheorem 3.10, (c) can be replaced by the more general condi- tion
ifi=2 and ifD⊆UwithD⊆co{0} ∪µF(D),
thenDis compact, hereµis chosen as in (b) (3.11) (of course with this assumption, we do not need to assume|µ| ≤1 in (b) ifi=2).
Theorem3.12. LetE=(E, · )be an infinite-dimensional normed linear space, C=E,U=B,F∈GA1(B,E), and assume (3.1), (3.2) (withi=1), and (3.9) hold, hereB= {x∈E:x<1}. In addition, suppose the following two conditions are satisfied:
for any continuous mapr:B−→S, Frsatisfies condition (C), (3.12)
0∈/ F(S), (3.13)
hereS= {x∈E:x =1}. ThenFhas an invariant direction.
Proof. Essentially the same reasoning as inTheorem 2.13(except here we use Theorem 3.7instead ofTheorem 2.6) establishes the result.
Remark 3.13. InTheorem 3.12, we can replaceBby any open setU ofEwith 0∈U(hereEis any Hausdorfftopological vector space) provided that∂U is a retract ofU, and in this case (3.13) is replaced by the following condition:∃µ >0 withµF(∂U)∩U= ∅.
Remark 3.14. InTheorem 3.12,F∈GA1(B,E) could be replaced byF∈GA1 (S,E).
Example 3.15. InTheorem 3.12, if condition (C) means that the mapF:B→ K(E) is either (a) Kakutani, (b) acyclic, (c) O’Neill, (d) approximable, or (e) admissible (strongly) in the sense of Gorniewicz, then clearly (3.1), (3.2), (3.9), and (3.12) hold.
We also have the following result whenEis not necessarily infinite dimen- sional.
Theorem3.16. LetE=(E, · )be a normed linear space,C⊆Eis a cone (i.e., closed, convex, invariant under multiplication by nonnegative real numbers and C∩(−C)= {0}),U=BC,F∈GA1(BC,C), and assume (3.1), (3.2) (withi=1), and (3.9) hold, hereBC= {x∈C:x<1}andBC= {x∈C:x ≤1}. In addi- tion, suppose the following two conditions are satisfied:
(a)for any continuous mapr:BC→SC,Frsatisfies condition (C);
(b) 0∈/ F(SC),
hereSC= {x∈C:x =1}. ThenFhas an invariant direction.
Proof. SinceCis a cone, it is well known that there exists a continuous retraction r:BC→SC. LetG=Frand followTheorem 2.13.
Also, as inSection 2, ifEis an infinite-dimensional normed linear space, then there exists a Lipschitzian retractionr:B→Swith Lipschitz constantk0(E), here B= {x∈E:x<1}andS= {x∈E:x =1}.
Theorem3.17. LetE=(E, · )be an infinite-dimensional normed linear space, C=E,U=B,F∈GA(B,E), and assume (2.13), (3.1), (3.2) (withi=3), (a), and (3.12) hold, hereB= {x∈E:x<1}andS= {x∈E:x =1}. In addition, suppose the following two conditions are satisfied:
(a)F is countablyk-set contractive with 0≤k <1/k0(E), herek0(E)is as in (2.13);
(b)there existµ >0with0< µ≤1,µF(S)∩B= ∅. ThenFhas an invariant direction.
Proof. Essentially the same reasoning as inTheorem 2.18establishes the result.
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Ravi P. Agarwal: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, 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]
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