Borsuk’s antipodal fixed points theorems for compact or condensing set-valued maps

Definition 1 (see [1, 2], and also [8, 9] for metric spaces).

1. Given V,W∈𝒩, a function s:X→Y is said to be a (V,W)-approximative selection of F, if for any x∈X, s(x)∈(F[x+V)∩X]+W)∩Y or, equivalently,

   Gr⁡s⊂[Gr⁡F+(V×W)]∩(X×Y).

2. A function F:X→Y is said to be approachable if for any (V,W)∈𝒩×𝒩, there exists a continuous (V,W)-approximative selection for F. We denote by 𝒜⁢(X,Y) the class of such set-valued maps. We write 𝒜⁢(X) if X=Y.

Example 1.

Let F:X→Y be a set-valued map. Then F is approachable in the following cases:

1. F is u.s.c. with nonempty convex values and X,Y are metric spaces, see [4].

2. F is u.s.c. with nonempty convex values, X is a paracompact space and Y is convex, see [7].

3. F is u.s.c. with nonempty contractible compact values, X is dominated by a finite polyhedron and Y is an ANR, see [1].

Lemma 1.

Let U be an open bounded symmetric balanced subset of Rn+1. Then any antipodal single-valued and continuous function f:∂⁡U→Rn has a zero value, that is, a point x¯∈∂⁡U such that f⁢(x¯)=0.

Definition 2.

Let E be a Hausdorff locally convex topological space with a fundamental basis 𝒩 of convex symmetric neighborhoods of the origin, X a symmetric nonempty subset of E and Y a nonempty subset of E.

1. A set-valued map F:X→Y is said to be antipodally approachable if for all V,W∈𝒩, there exists a continuous (V,W)- approximative selection for F which is antipodal on ∂⁡X. We denote by 𝒜a⁢a⁢(X,Y) the class of such set-valued maps.

2. A set-valued map F:X→Y is said to be antipodally approximable if its restriction F|K to any compact subset K of X is antipodally approachable on ∂⁡K.

Theorem 1.

Let U be an open bounded symmetric balanced subset of Rn+1 and let F:U¯→Rn+1 be a set-valued map that is u.s.c. with nonempty closed values. If F∈Aa⁢a⁢(X,Y), then F has a zero value in ∂⁡U, i.e., a point x¯∈∂⁡U such that 0∈F⁢(x¯), and a fixed point on U¯, i.e., a point x^∈U¯ such that x^∈F⁢(x^).

Proof.

Let 0 denote the zero map from U¯ into {0}. If F does not have a zero, then d⁢(Gr⁡(0),Gr⁡(F))>0. Let ϵ=13⁢d⁢(Gr⁡(0),Gr⁡(F)). Then

By hypothesis, the set-valued map F is approachable on U¯ by functions which are antipodal on ∂⁡U. Then, for V=W=Bℝn+1⁢(O,ϵ), there exists a continuous (V,W) approximative selection fV,W such that

Consequently, Gr⁡(fV,W)∩Gr⁡(0)=∅ and hence fV,W does not have a zero value.

Define j:U¯→ℝn+2 by j⁢(x)=x+d⁢(x,∂⁡U)⁢e→n+2. Then j is one-to-one continuous and j⁢(x)=x on ∂⁡U. Let V={x+u⁢e→n+2:x∈U,|u|<d⁢(x,∂⁡U)}. Then V is open, bounded, symmetric, and balanced in ℝn+2 and ∂⁡V=J⁢(U¯)∪{-J⁢(U¯)}.

Define h:∂⁡V→ℝn+1 by

Then h is a continuous antipodal single-valued function with no zero values. This contradicts Lemma 1. Thus, F has a zero value.

Furthermore, by the above conclusion, the set-valued map G⁢(x)=F⁢(x)-x admits a zero value, i.e., there exists x¯∈U¯ such that 0∈G⁢(x¯), hence x¯∈F⁢(x¯), and this proves the theorem. ∎

Theorem 2.

Let U be an open bounded symmetric balanced subset in a finite dimensional vector space Y. Let also F:U¯→Y be a set-valued map that is u.s.c. with nonempty closed values. If F∈Aa⁢a⁢(X,Y), then F has a zero value and a fixed point.

Proof.

Let ℬ={x1,x2,…,xn} be a basis of Y, which allows us to consider Φ, the usual linear homeomorphism between ℝn and Y. Let V=Φ-1⁢(U). Then it is easy to show that it is bounded in the usual sense. Moreover, consider the composite function G=Φ-1∘F∘Φ. Then it is routine to check that (V,G) satisfies the assumptions of Theorem 1, which leads to the conclusion. ∎

Theorem 3.

Let X be a closed bounded symmetric balanced set in a Hausdorff locally convex topological vector space E and let F:X→E be an u.s.c. set-valued map with nonempty closed values. Suppose that F∈Aa⁢a⁢(X,E) and that F is compact, i.e., F⁢(X)¯ is compact. Then F admits a fixed point.

Proof.

We will construct the fixed point as a limit of what is called an approximated fixed point. Since F⁢(X)¯ is compact, then for each V∈𝒩, there exists a finite subset CV of F⁢(X) such that (y+V)∩CV≠∅ for each y∈F⁢(X). Let HCV be the vector space spanned by CV. In what follows in this proof, we will refer to the topology of HCV.

Consider the set-valued map FV:X∩HCV→HCV defined by

Then FV is u.s.c. with nonempty compact values. Note that X∩HCV (resp. ∂HSV⁡(X∩HCV) the boundary of X∩HCV with respect to the topology of X∩HCV) is a compact subset of X (resp. ∂⁡X). Since F is antipodally approachable on ∂⁡X, FV is approachable on X∩HCV by a selection which is antipodal on ∂HCV⁡(X∩HCV). Note that since 0∈X∩HCV, either 0∈intHCV⁢(X∩HCV) or 0∈∂HCV⁡(X∩HCV). In the first case, we can apply Theorem 2 with U=intHCV⁢(X∩HCV) to obtain a fixed point xV of FV. In the second case, FV is antipodally approachable, so there exists an antipodal approximative selection sV of FV such that sV⁢(x)∈FV⁢(x+V)+V, and in particular, sV⁢(0)=0∈FV⁢(0+V)+V.

Hence, in both cases, we have the existence for each V∈𝒩, of xV∈X such that xV∈FV⁢(xV+V)+V. Since V is symmetric, there exists (v,w)∈V2 such that yV∈F⁢(zV), where yV=xV+v and zV=xV+w. A standard argument based on the compactness of F⁢(X)¯, the upper semi-continuity of F and the closeness of its values (see, for example, [2]) ends the proof. ∎

Definition 3 (see [10] and [6]).

1. If C is a lattice with a minimal element, denoted by 0, a map Φ:2E→C is called a measure of non-compactness provided that the following conditions hold for any A,B∈2E:

   1. Φ⁢(conv¯⁢A)=Φ⁢(A),

   2. Φ⁢(A)=0 if and only if A is precompact,

   3. Φ⁢(A∪B)=max⁡{Φ⁢(A),Φ⁢(B)}.

2. F:X→Y is said to be Φ-condensing provided that if A⊂X with Φ⁢(F⁢(A))≥Φ⁢(A), then A is relatively compact.

Lemma 2.

Assume that X is a nonempty subset of E and that F:X→E is a Φ-condensing set-valued map. Then there exists a nonempty symmetric compact and convex subset K of E such that F⁢(K∩X)⊂K.

Proof.

Let x0∈X be fixed. Let us consider the family 𝔉 of all closed, convex and symmetric subsets C of E such that x0∈C and F⁢(C∩X)⊂C. Clearly, 𝔉≠∅ since C0=conv¯⁢(F⁢(X)∪{x0})∈𝔉. Note that C0 is symmetric since F is antipodal preserving. Now let K=⋂C∈𝔉. Then K is convex and closed and x0∈K. If x∈X∩K, then F⁢(x)⊂C for all C∈𝔉, so that F⁢(K∩X)⊂K, and thus K∈𝔉. It remains to prove that K is compact. If K is not compact, then Φ⁢(F⁢(K))≱Φ⁢(K), since F is Φ condensing. Let K′=conv¯⁢(F⁢(X∩K)∪{x0}). Then K′⊂K, and therefore K=K′ and Φ(K)=Φ(K′)=Φ(F(K∩X))≤Φ(F(K), which contradicts Φ⁢(F⁢(K))≱Φ⁢(K).

Furthermore, K can be chosen so that it is symmetric, otherwise, let K0=K∩(-K) and the conclusion remains valid for K0. ∎

Theorem 4.

Let X be a closed symmetric balanced set in a Hausdorff locally convex topological vector space E and let F:X→E be a set-valued map. Suppose that the following conditions are satisfied:

1. F is u.s.c. with nonempty closed values.

2. F is Φ -condensing.

3. F is antipodally approximable.

Then F admits a fixed point.

Proof.

Suppose that F does not have a fixed point, i.e., for each x∈X, x∉F⁢(x). By Lemma 2, there exists a nonempty symmetric, convex and compact subset K of E such that F⁢(K∩X)⊂K. Since K∩X is compact and F is antipodally approximable, F|K∩X∈𝒜a⁢a⁢(K∩X,E). Furthermore, F⁢(K∩X)¯ is compact. By Theorem 3, F|K∩X admits a fixed point. ∎