PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 77(91) (2005), 79–85

A KRASNOSELSKIĬ CONE COMPRESSION RESULT FOR MULTIMAPS IN THE S-KKM CLASS

Donal O’Regan, Naseer Shahzad, and Ravi P. Agarwal

Communicated by Stevan Pilipović

Abstract. The aim of this article is obtain a Krasnoselskiĭ cone compression theorem for multimaps in the class S-KKM.

1. Introduction

This article discusses various Krasnoselskiĭ cone compression theorems for compact as well as k − Φ-contractive multimaps in the S-KKM class. The class of S-KKM maps was introduced and studied by Chang et al. [5] and further investigated by Chang et al. [4] and Shahzad [12]. The Krasnoselskiĭ cone compression theorem is well known for U_{c}^{k} maps [9] and other classes [1, 10]. We mention that S-KKM class contains the U_{c}^{k} maps. The ideas presented in this paper follow closely those in [9].

2. Preliminaries

Let E be a Hausdorff locally convex space. For a nonempty set Y ⊆ E, 2^{Y} denotes the family of nonempty subsets of Y. If L is a lattice with a minimal element 0, a mapping Φ : 2^{E} → L is called a generalized measure of noncompactness provided that the following conditions hold:

(a) Φ(A) = 0 if and only if Ā is compact.

(b) Φ(\overline{co}(A)) = Φ(A); here \overline{co}(A) denotes the closed convex hull of A.

(c) Φ(A ∪ B) = max{Φ(A), Φ(B)}.

It follows that if A ⊆ B, then Φ(A) ⩽ Φ(B). Let C be a nonempty subset of a Banach space X. The Kuratowski measure of noncompactness is the map α : 2^{X} → R_{+} defined by

α(A) = inf{ϵ > 0 : A can be covered by a finite number

of sets each of diameter less than ϵ}

2000 Mathematics Subject Classification: Primary 47H10.

for A ∈ 2^{X}. The Hausdorff measure of noncompactness is the map χ : 2^{X} → R_{+} defined by

χ(A) = inf{ϵ > 0 : A can be covered by a finite number

of balls with radius less than ϵ}

for A ∈ 2^{X}. Examples of the generalized measure of noncompactness are the Kuratowski measure and the Hausdorff measure of noncompactness (see [11]).

Let C be a nonempty subset of a Hausdorff locally convex space E and F : C → 2^{E}. Then F is called Φ-condensing provided that Φ(A) = 0 for any A ⊆ C with Φ(F (A)) ⩾ Φ(A). It is clear that a compact mapping is Φ-condensing and also every mapping defined on a compact set is necessarily Φ-condensing. Suppose that L is a lattice with a minimal element 0 and that for each l ∈ L and λ ∈ R, with λ > 0, there is defined an element λl ∈ L. A mapping F : C → 2^{E} is called a k-Φ-contractive map (k ∈ R with k > 0) provided that Φ(F (A)) ⩽ kΦ(A) for each A ⊆ C and F (C) is bounded. Obviously, if C is complete, F is k-Φ-contractive, with 0 < k < 1, and Φ = α or χ, then F is Φ-condensing.

Let X and Y be subsets of Hausdorff topological vector spaces E_{1} and E_{2} respectively. Let F : X → K(Y); here K(Y) denotes the family of nonempty compact subsets of Y. We say F 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 is acyclic if F is upper semicontinuous with acyclic values. The map F 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 of F : X → K(Y) is a continuous function s : X → Y satisfying

s(x) ∈ (F [(x + U) ∩ X] + V) ∩ Y, for every x ∈ X.

We say F is approximable if it is a closed map and 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 single valued map p : Y → X is called a Vietoris map if the following two conditions hold:

(i) for each x ∈ X, the set p^{−1}(x) is acyclic (ii) p is a proper map i.e., for every compact A ⊆ X we have that p^{−1}(A) is

compact.

Definition 2.1. A multifunction ϕ : X → K(Y) is admissible (strongly) in the sense of Gorniewicz, if ϕ : X → K(Y) is upper semicontinuous, and if there exists a metric space Z and two continuous maps p : Z → X and q : Z → Y such that

(i) p is a Vietoris map, and

(ii) ϕ(x) = q(p^{−1}(x)) for any x ∈ X.

Remark 2.1. It should be noted [8, p. 179] that ϕ upper semicontinuous is superfluous in Definition 2.1.

Suppose X and Y are Hausdorff topological spaces. Given a class X of maps, X (X, Y) denotes the set of maps F : X → 2^{Y} belonging to X, and X_{c} the set of finite compositions of maps in X. A class U of maps is defined by the following properties:

(i) U contains the class C of single valued continuous functions; (ii) each F ∈ U_{c} is upper semicontinuous and compact valued; and (iii) for any polytope P, F ∈ U_{c}(P, P) has a fixed point, where the intermediate

spaces of composites are suitably chosen for each U.

Definition 2.2. F ∈ U_{c}^{κ}(X, Y) if for any compact subset K of X, there is a G ∈ U_{c}(K, Y) with G(x) ⊆ F (x) for each x ∈ K.

Examples of U_{c}^{κ} maps are the Kakutani maps, the acyclic maps, the O’Neill maps, and the maps admissible in the sense of Gorniewicz.

Definition 2.3. Let X be a convex subset of a Hausdorff topological vector space and Y a topological space. If S, T : X → 2^{Y} are two set-valued maps such that T (co(A)) ⊆ S(A) for each finite subset A of X, then we say that S is a generalized KKM map w.r.t. T. The map T : X → 2^{Y} is said to have the KKM property if for any generalized KKM w.r.t. T map S, the family {S̄(x) : x ∈ X} has the finite intersection property. We let KKM(X, Y) = {T : X → 2^{Y} : T has the KKM property}.

Remark 2.2. If X is a convex space, then U_{c}^{κ}(X, Y) ⊂ KKM(X, Y) (see [5]).

Definition 2.4. Let X be a nonempty set, Y a nonempty convex subset of a Hausdorff topological vector space and Z a topological space. If S : X → 2^{Y}, T : Y → 2^{Z}, F : X → 2^{Z} are three set-valued maps such that T (co(S(A))) ⊆ F (A) for each nonempty finite subset A of X, then F is called a generalized S-KKM map w.r.t. T. If the map T : X → 2^{Z} is such that for any generalized S-KKM w.r.t. T map F, the family {F̄(x) : x ∈ X} has the finite intersection property, then T is said to have the S-KKM property. The class S-KKM(X, Y, Z) = {T : Y → 2^{Z} : T has the S-KKM property}.

Remark 2.3. If X = Y and S is the identity mapping 1_{X}, then S-KKM(X, Y, Z) = KKM(X, Z). Also KKM(Y, Z) is a proper subset of S-KKM(X, Y, Z) for any S : X → 2^{Y} and so S-KKM(X, Y, Z) is a very large class of maps which includes other important classes of multimaps (see [4, 5] for examples).

Remark 2.4. Let X be a convex space, Y a convex subset of a Hausdorff locally convex space, and Z a normal space. Suppose s : Y → Y is surjective, F ∈ s-KKM(Y, Y, Z) is closed, and f ∈ C(X, Y). Then F ∘f ∈ 1_{X} −KKM(X, X, Z) (see [5]).

The following result [4] will be needed in the sequel.

Lemma 2.1. Let C be a nonempty, closed, convex subset of a Hausdorff locally convex space E Suppose s : C → C is surjective and F ∈ s-KKM(C, C, C) is a closed Φ-condensing map. Then F has a fixed point in C.

3. Main Results

Let C be a cone in a normed space E = (E, ∥ · ∥). For ρ > 0 let

B_{ρ} = {x ∈ C : ∥x∥ < ρ}, B̄_{ρ} = {x ∈ C : ∥x∥ ⩽ ρ},
S_{ρ} = {x ∈ C : ∥x∥ = ρ}, EB_{ρ} = {x ∈ C : ∥x∥ ⩾ ρ}.

Theorem 3.1. Let C be a closed convex cone in a normed space E = (E, ∥ · ∥) and let r, R be constants with 0 < r < R. Suppose s : B̄_{R} → B̄_{R} is surjective and F ∈ s-KKM(B̄_{R}, B̄_{R}, C) is a closed and compact map with

(3.1)  F (S_{r}) ⊆ EB_{r} and F (S_{R}) ⊆ B̄_{R}.

Then F has a fixed point in B_{r,R} = {x ∈ C : r ⩽ ∥x∥ ⩽ R}.

Proof. Define g : C → B̄_{R} as follows

g(x) = \begin{cases}r_{0}(x), & if x ∈ B̄_{r} \\ x, & if x ∈ B_{r,R} \\ Rx/∥x∥, & if x ∈ EB_{R}, \end{cases}

where r_{0} : B̄_{r} → S_{r} is a continuous retraction (which exists in our case, indeed if we fix x_{0} ∈ S_{r}, then we may take

r_{0}(x) = \frac{r\{(r − ∥x∥)x_{0} + x\}}{∥(r − ∥x∥)x_{0} + x∥}.

Note (r − ∥x∥)x_{0} + x ≠ 0 since C ∩ (−C) = {0}). Then g is continuous. Since F ∈ s-KKM(B̄_{R}, B̄_{R}, C), by Remark 2.4 G = F ∘g ∈ 1_{C} −KKM(C, C, C). Furthermore, G is closed and compact. Now Lemma 2.1 guarantees that G has a fixed point x ∈ C, i.e., x ∈ G(x). If ∥x∥ < r, then

x ∈ F r_{0}(x) ⊆ F (S_{r}) ⊆ EB_{r}.

This is a contradiction. If ∥x∥ > R, then

x ∈ F (R x/∥x∥) ⊆ F (S_{R}) ⊆ B̄_{R}.

This is a contraction. Hence x ∈ B_{r,R} and x ∈ G(x) = F (x). □

Remark 3.1. The condition in (3.1) that F (S_{R}) ⊆ B̄_{R} may be replaced by

(3.2)  x ∉ λF x for x ∈ S_{R} and λ ∈ (0, 1).

To see this, let x be as in Theorem 3.1. If ∥x∥ > R, then x ∈ F (Rx/∥x∥). This implies that y ∈ λF (y) with y = R x/∥x∥ and λ = R/∥x∥.

Next let E = (E, ∥.∥) be an infinite dimensional normed space. For ρ > 0 let

B_{ρ} = {x ∈ E : ∥x∥ < ρ}, B̄_{ρ} = {x ∈ E : ∥x∥ ⩽ ρ},
S_{ρ} = {x ∈ E : ∥x∥ = ρ}, EB_{ρ} = {x ∈ E : ∥x∥ ⩾ ρ}

Theorem 3.2. Let E = (E, ∥ · ∥) be an infinite dimensional normed space and let r, R be constants with 0 < r < R. Suppose s : B̄_{R} → B̄_{R} is surjective and F ∈ s-KKM(B̄_{R}, B̄_{R}, C) is a closed and compact map with

(3.3)  (S_{r}) ⊆ EB_{r} and F (S_{R}) ⊆ B̄_{R}

Then F has a fixed point in B_{r,R} = {x ∈ E : r ⩽ ∥x∥ ⩽ R}.

Proof. It is known [3] that there exists a continuous retraction r_{0} : B̄_{r} → S_{r}. Essentially the same reasoning as in Theorem 3.1 gives the result. □

We now establish a general version of the above result.

Theorem 3.3. Let E = (E, ∥ · ∥) be an infinite dimensional normed space and let U_{1} and U_{2} be open convex subsets of E with 0 ∈ U_{1} with Ū_{1} ⊂ U_{2}. Suppose s : Ū_{2} → Ū_{2} is surjective and F ∈ s-KKM(Ū_{2}, Ū_{2}, E) is a closed and compact map with

(3.4)  F (∂U_{1}) ⊆ E ∖ U_{1} and F (∂U_{2}) ⊆ Ū_{2}.

Then F has a fixed point in Ū_{2} ∖ U_{1}.

Proof. Define g : E → ∂U_{2} by

g(x) = \begin{cases}r_{1}(x), & if x ∈ Ū_{1} \\ x, & if x ∈ Ū_{2} ∖ U_{1} \\ x/p(x) & if x ∈ E ∖ U_{2}. \end{cases}

where p is the Minkowski functional on Ū_{2} and r_{1} : Ū_{1} → ∂U_{1} is a continuous retraction (which exists [2]). Then g is continuous. Since F ∈ s-KKM(Ū_{2}, Ū_{2}, E), by Remark 2.4 G = F ∘ g ∈ 1_{E} − KKM(E, E, E). Furthermore, G is closed and compact. Now as in Theorem 3.1 G has a fixed point x ∈ E, i.e., x ∈ G(x). If x ∈ U_{1}, then

x ∈ F r_{1}(x) ⊆ F (∂U_{1}) ⊆ E ∖ U_{1}.

This is a contradiction. If x ∈ E ∖ Ū_{2}, then

x ∈ F (x/p(x)) ⊆ F (∂U_{2}) ⊆ Ū_{2}.

This is a contraction. Hence x ∈ Ū_{2} ∖ U_{1} and x ∈ G(x) = F (x). □

It is known [3, 7] that if E is an infinite dimensional normed space, then there exits a Lipschitzian retraction r_{0} : B̄_{r} → S_{r} with Lipschitz constant k_{0} > 1. We are now in a position to improve Theorem 3.2.

Theorem 3.4. Let E = (E, ∥ · ∥) be an infinite dimensional normed space and let r, R be constants with 0 < r < R. Suppose s : B̄_{R} → B̄_{R} is surjective and F ∈ s-KKM(B̄_{R}, B̄_{R}, C) is a closed k − Φ-contractive map, 0 ⩽ k < 1/k_{0}, with

(3.5)  F (S_{r}) ⊆ EB_{r} and F (S_{R}) ⊆ B̄_{R}.

Then F has a fixed point in B_{r,R} = {x ∈ E : r ⩽ ∥x∥ ⩽ R}.

Proof. Let r_{0} : B̄_{r} → S_{r} be the retraction with Lipschitz constant k_{0}. Define g : C → B̄_{R} as follows

g(x) = \begin{cases}r_{0}(x), & if x ∈ B̄_{r} \\ x, & if x ∈ B_{r,R} \\ R x/∥x∥, & if x ∈ EB_{R}. \end{cases}

Then g is continuous. Since F ∈ s-KKM(B̄_{R}, B̄_{R}, C), by Remark 2.4 G = F ∘ g ∈ 1_{E} − KKM(E, E, E). Furthermore, g is k_{0} − Φ-contractive; indeed if Ω is bounded subset of E, then Ω = Ω_{1} ∪ Ω_{2} ∪ Ω_{3}, where Ω_{1} = Ω ∩ B_{r}, Ω_{2} = Ω ∩ B_{r,R}, Ω_{3} = Ω ∩ {x ∈ E : ∥x∥ > R} and

Φ(g(Ω)) ⩽ max{Φ(g(Ω_{1})), Φ(g(Ω_{2})), Φ(g(Ω_{3}))}
⩽ max{k_{0}Φ(Ω_{1}), k_{0}Φ(Ω_{2}), k_{0}Φ(Ω_{3})}
⩽ k_{0}Φ(Ω)

since g(Ω_{3}) ⊆ co(Ω_{3}∪{0})). Consequently, G is kk_{0}−Φ-contractive and also closed. Now Lemma 2.1 guarantees that G has a fixed point x ∈ E, i.e., x ∈ G(x). As before, x ∈ B_{r,R} and x ∈ G(x) = F (x). □

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Department of Mathematics (Received 19 01 2004) National University of Ireland (Revised 03 03 2005) Galway, Ireland

donal.oregan@nuigalway.ie

Department of Mathematics King Abdul Aziz University P.O. Box 80203, Jeddah 21589 Saudi Arabia

nshahzad@kaau.edu.sa

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

agarwal@fit.edu