BIRKHOFF-KELLOGG THEOREMS ON INVARIANT DIRECTIONS FOR MULTIMAPS

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 present invariant direction results for countably condensing maps. For the remainder of this section, we present some definitions and some known facts. Let X and Y be subsets of Hausdorff topological vector spaces E1 and E2, respectively. We will look at maps F : X → K(Y), here K(Y) denotes the family of nonempty compact subsets of Y . We say F : X → K(Y) is Kakutani if F is upper semicontinuous with convex values. A nonempty topological space is said to be acyclic, if all its reduced C̆ech homology groups over the rationals are trivial. Now F : X → K(Y) is acyclic if F is upper semicontinuous with acyclic values. The map F : X → K(Y) is said to be an O’Neill map if F is continuous and if the values of F consist of one or m-acyclic components (here m is fixed). Given two open neighborhoods U and V of the origins in E1 and E2, respectively, a (U,V)-approximate continuous selection [6] of F : X → K(Y) is a continuous function s : X → Y satisfying


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 present invariant direction results for countably condensing maps.
For the remainder of this section, we present some definitions and some known facts.Let X and Y be subsets of Hausdorff topological vector spaces E 1 and E 2 , respectively.We will look at maps F : X → K(Y ), here K(Y ) denotes the family of nonempty compact subsets of Y .We say F : X → K(Y ) is Kakutani if 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.Now F : X → K(Y ) is acyclic if F is upper semicontinuous with acyclic values.The map F : X → K(Y ) is said to be an O'Neill map if F is continuous and if the values of F consist of one or m-acyclic components (here m is fixed).
Given two open neighborhoods U and V of the origins in E 1 and E 2 , respectively, a (U, V )-approximate continuous selection [6] of F : (1.1) We say F : X → K(Y ) is approximable if its restriction F| K , to any compact subset K of X, admits a (U, V )-approximate continuous selection for every open neighborhood U and V of the origins in E 1 and E 2 , respectively.For our next definition, let X and Y be metric spaces.A continuous singlevalued map p : Y → X is called a Vietoris map if the following two conditions are satisfied: (i) for each x ∈ X, the set p −1 (x) is acyclic; (ii) p is a proper map, that is, for every compact A ⊆ X, p −1 (A) is compact.
is upper semicontinuous, and if there exist a metric space Z and two continuous maps p : Z → X and q : Z → Y such that (i) p is a Vietoris map; (ii) φ(x) = q(p −1 (x)) for any x ∈ X.
Suppose X and Y are Hausdorff topological spaces.Given a class ᐄ of maps, ᐄ(X,Y ) denotes the set of maps F : X → 2 Y (nonempty subsets of Y ) belonging to ᐄ, and ᐄ c the set of finite compositions of maps in ᐄ.A class ᐁ of maps is defined by the following properties: (i) ᐁ contains the class Ꮿ of single-valued continuous functions; (ii) each F ∈ ᐁ c is upper semicontinuous and compact valued; (iii) for any polytope P, F ∈ ᐁ c (P, P) has a fixed-point where the intermediate spaces of composites are suitably chosen for each ᐁ.
Examples of ᐁ κ c maps are the Kakutani, the acyclic, the O'Neill maps, and the maps admissible in the sense of Gorniewicz.
For a subset K of a topological space X, we denote by Cov X (K) the directed set of all coverings of K by open sets of X (usually we write Cov(K) = Cov X (K)).Given two maps F, G : X → 2 Y and α ∈ Cov(Y ), F and G are said to be α-close, if for any x ∈ X, there exists By a space, we mean a Hausdorff topological space.In what follows, Q denotes a class of topological spaces.A space Y is an extension space for Q (written A space Y is an approximate extension space for Q (and we write Y ∈ AES(Q)) if for any α ∈ Cov(Y ) and any pair (X,K) in Q with K ⊆ X closed and any continuous function f 0 : K → Y , there exists a continuous function f :

R. P. Agarwal and D. O'Regan 437
Definition 1.4.Let V be a subset of a Hausdorff topological vector space E. Then we say V is Schauder admissible if for every compact subset K of V and every covering α ∈ Cov V (K), there exists a continuous function (called the Schauder projection) π α : K → V such that (i) π α and i : V is an open convex subset of a Hausdorff locally convex topological space E, then it is well known that V is Schauder admissible.
The following fixed-point result was established in [5].
Theorem 1.5.Let V be a Schauder admissible subset of a Hausdorff topological vector space E and F ∈ ᐁ κ c (V, V ) a compact map.Then F has a fixed point.A nonempty subset X of a Hausdorff topological vector space E is said to be admissible if for every compact subset K of X and every neighborhood V of 0, there exists a continuous map h : K → X with x − h(x) ∈ V for all x ∈ K and h(K) is contained in a finite-dimensional subspace of E. The nonempty subset X is said to be q-admissible if any nonempty compact, convex subset Ω of X is admissible.
In [12], we proved the following fixed-point result.
Theorem 1.6.Let Ω be a q-admissible, closed, convex subset of a Hausdorff topological vector space E with x 0 ∈ Ω. Suppose F ∈ ᐁ κ c (Ω,Ω) with the following property holding: Then F has a fixed point in Ω.
Let (E,d) be a pseudometric space.For S ⊆ E, let B(S, ) = {x ∈ E : d(x,S) ≤ }, > 0, where d(x,S) = inf y∈Y d(x, y).The measure of noncompactness of the set where Let E be a locally convex Hausdorff topological vector space and let P be a defining system of seminorms on E. Suppose F : S → 2 E , here S ⊆ E. The map F is said to be a countably P-concentrative mapping if F(S) is bounded, and for p ∈ P, for each countably bounded subset X of S, we have α p (F(X)) ≤ α p (X), and for p ∈ P, for each countably bounded non-p-precompact subset X of S (i.e., X is not precompact in the pseudonormed space (E, p)), we have α p (F(X)) < α p (X), here α p (•) denotes the measure of noncompactness in the pseudonormed space (E, p).
Finally for completeness, we also give the definition of countably k-set contractive maps.Let X be a metric space and P B (X) the bounded subsets of X.
The Kuratowskii measure of noncompactness is the map α here A ∈ P B (X).Let S be a nonempty subset of X and H : for all countably bounded sets Ω of S.

Hausdorff locally convex topological vector spaces
In this section, we present a variety of Birkhoff-Kellogg type theorems on invariant directions.Throughout, E will be a Hausdorff locally convex topological vector space, C will be a closed convex subset of E, U ⊆ C will be convex, U will be an open subset of E, and 0 Also we wish to consider maps F : U → K(C) which are upper semicontinuous and either (a) approximable, (b) admissible (strongly) in the sense of Gorniewicz, or more generally (c) ᐁ κ c , here U denotes the closure of U in C and K(C) denotes the family of nonempty compact subsets of C.
Throughout this section, we will assume the map F : U → K(C) satisfies one of the following conditions:  [4,14,16,17] and the references therein).
For our next result, assume condition (D) is such that for any map F ∈ LS U,C and any λ ∈ R,λF satisfies condition (D).(2.3) Certainly if condition (D) means (a) or (b) above, then (2.3) is satisfied.Now from Theorem 2.4, we obtain the following Birkhoff-Kellogg type theorem.Some of the ideas here were borrowed from the literature (see [14] and the references therein).
In Theorem 2.6, if condition (D) means that the map F : U → K(C) belongs to ᐁ κ c (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 subclass Ꮽ(U,C) of ᐁ κ c (U,C) (e.g., Ꮽ could 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 map F : Then there exists λ ∈ (0,1) and x ∈ ∂U with (λ −1 µ −1 )x ∈ Fx.
Proof.Let µ = 0 be chosen as in (i), and notice that µF ∈ LS(U,C) from (i).We claim (2.6)  Remark 2.12.It is also possible to use Theorem 2.9 to obtain an analogue of Theorem 2.10 for the subclass Ꮽ.We leave the details to the reader.
Proof.We know [7] that there exists a continuous retraction r : B → S. Let G=Fr and notice that G ∈ LS(B,E) from (2.9).Now we claim that there exists µ > 0 with µF(S) ∩ B = ∅.(2.11)If this is true, then and so Theorem 2.6 (applied to G with U = B and C = E) guarantees that there exist λ ∈ (0,1) and The proof is finished.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 y n ∈ F(S) and w n ∈ B with y n = (1/n)w n ).
Remark 2.14.In Theorem 2.13, we can replace B by any open set U of E with 0 ∈ U (here E is any Hausdorff locally 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 with µF(∂U) ∩ U = ∅.Remark 2.15.In Theorem 2.13, F ∈ LS 1 (B,E) could be replaced by F ∈ LS 1
Example 2.17.In Theorem 2.13, if condition (D) means that the map F : B → K(E) belongs to ᐁ κ c (U,C), then we know that (2.1), (2.2), and (2.9) are satisfied.It is possible to use Theorem 2.9 to obtain an analogue of Theorem 2.13 for the subclass Ꮽ of ᐁ κ c .We leave the details to the reader.In [7], the authors show that if E is an infinite-dimensional normed linear space, then there exists a Lipschitzian retraction r : B → S with Lipschitz constant k 0 (E), here B and S are as in Theorem 2.13.In fact there exists a k 0 with k 0 (E) ≤ k 0 for any space E (as described above).We refer the reader to [9, Chapter 21] for a discussion of upper and lower bounds for k 0 (E), note in particular that k 0 (E) ≥ 3.For our next theorem, we let r : B −→ S be a Lipschitzian retraction with Lipschitz constant k 0 (E). (2.13) , and assume that (2.1), (2.2) (with i = 3), Theorem 2.10(i), (2.9), and (2.13) hold; here B = {x ∈ E : x < 1} and S = {x ∈ E : x = 1}.In addition, suppose the following two conditions are satisfied: Then F has an invariant direction.
Proof.Let G = Fr where r is as in (2.13).Notice that G ∈ LS(B,E) and it is easy to check that G is countably kk 0 (E)-set contractive.Thus, G ∈ LS 3 (B,E).Now apply Theorem 2.10 to G. Remark 2.19.In Theorem 2.18, F ∈ LS 1 (B,E) could be replaced by F ∈ LS 1

(S, E).
Remark 2.20.Theorem 2.18 is the first invariant direction result, to our knowledge, for countably contractive maps.
Remark 2.21.We note that the results in this section improve those in [8,14,16].

Hausdorff topological vector spaces
Throughout this section, E will be a Hausdorff topological vector space, C a closed convex subset of E, U an open subset of C, and 0 ∈ U.This section also presents Birkhoff-Kellogg type theorems, and in some cases the results in Section 2 will be improved.Fix i ∈ {1, 2,3}.
Definition 3.2.We say that F ∈ GA i (U,C) if F ∈ GA(U,C) satisfies (Hi), here (Hi) is as in Section 2.
Definition 3.3.We say that ) with G| ∂U = F| ∂U , there exists x ∈ U with x ∈ G(x).Remark 3.5.Throughout this section, it is possible to replace F upper semicontinuous in Definition 3.1 with F closed and taking compact sets into relatively compact sets.
The following result was established in [4].
Theorem 3.6.Fix i ∈ {1, 2,3} and let E be a Hausdorff topological vector space, C a closed convex subset of E, U an open subset of C, 0 ∈ U, and assume (3.1)  (3) Suppose condition (C) in Definition 3.1 means that F : U → K(C) is admissible in the sense of Gorniewicz, E is a Fréchet space (P a defining system of seminorms), U is convex, and C = E. Now [10] guarantees that (3.1) is true.Now we show that (3.2) is satisfied if i = 1,2, or 3 (in fact if i = 1, it is enough (see Theorem 1.5) for E to be a metrizable locally convex topological vector space).
To see (3.2), let θ ∈ GA i ∂U (U,E) with θ| ∂U = {0}.We must show that there exists x ∈ U with x ∈ θ(x).Let µ be the Minkowski functional on U and let r : E → U be given by Consider G = rθ.We know [10] that G is admissible in the sense of Gorniewicz, and as a result {0}) for any subset A of E, we have Thus, D is compact since θ ∈ GA 2 (U,E).Now [12, Theorem 2.1] and [13, Theorem 2.2] (or alternatively Theorem 1.5, Theorem 1.6 if i = 1 or 2) guarantee that there exists x ∈ U with x ∈ G(x) = rθ(x).Thus, x = r(y) for some y ∈ θx, here x ∈ U = U ∪ ∂U (note C = E here).Suppose x ∈ ∂U.Then µ(x) = 1 and so , since r(y) = y max 1,µ(y) .
Proof.Apply Theorem 3.6 to µF (see the proof of Theorem 2.10).
upper semicontinuous and satisfies condition (D).We assume condition (D) is for any map F ∈ LS U,C and any continuous single-valued map r : E → U, rF satisfies condition (D).

Definition 3 . 1 .
The map F ∈ GA(U,C) if F : U → K(C) is upper semicontinuous and satisfies condition (C), here U denotes the closure of U in C. We assume condition (C) is for any map F ∈ GA U,C and any continuous single-valued map µ : U → [0,1], µF satisfies condition (C).(3.1) Certainly if condition (C) means that the map F : 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.
and E is Fréchet (here P is a defining system of seminorms).Theorem 2.4.Fix i ∈ {1, 2,3} and let E be a Hausdorff locally convex topological vector space, C a closed convex subset of E, U ⊆ C convex, U an open subset of E, 0 ∈ U, and assume (2.1) holds.Suppose F ∈ LS i (U,C) and assume the following there exist x ∈ ∂U and λ ∈ (0,1) with x ∈ λFx; here ∂U denotes the boundary of U in C. Example 2.5.Suppose condition (D) in Definition 2.1 means F : U → K(C) belongs to ᐁ κ c (U,C).Now since ᐁ κ c is closed under compositions, then (2.1) is true.If i = 1, we know from [15] that (2.2) holds.If i = 2, we know from [13] that (2.2) is satisfied.If i = 3, we know from [11] that (2.2) holds.As a result, Theorem 2.4 contains most of the Leray-Schauder alternatives (see [4]S i (U,C) if F ∈ LS(U,C) satisfies (Hi).Remark 2.3.Throughout this section, it is possible to replace F upper semicontinuous in Definition 2.1 with F closed and taking compact sets into relatively compact sets.The following result was established in[4].R. P. Agarwal and D. O'Regan 439 map, and assume(2.4)holds.Suppose the following condition holds:for any map F ∈ Ꮽ U,C , and any λ ∈ R, λF ∈ ᐁ κ c U,C .(2.5)Then there exist λ ∈ (0,1) and x ∈ ∂U with (λ −1 µ −1 )x ∈ Fx (i.e., F| ∂U has an eigenvalue); here µ = 0 is chosen as in(2.4).Proof.Essentially the same reasoning as in Theorem 2.6 establishes the result.In our next result, we assume (2.3) when |λ| ≤ 1.Theorem 2.10.Fix i ∈ {2, 3} and let E be a Hausdorff locally convex topological vector space, C a closed convex subset of E, U ⊆ C convex, U an open subset of E, 0 ∈ U, F ∈ LS i (U,C), and assume (2.1) and (2.2) hold.In addition, suppose the following conditions are satisfied:(i) for any map F ∈ LS(U,C) and any λ ∈ R with |λ| ≤ 1, λF satisfies condition (D), (ii) there exists µ holds.Suppose F ∈ GA i (U,C) and assume the following condition is satisfied: there exist x ∈ ∂U and λ ∈ (0,1) with x ∈ λFx.