RENDICONTI DEL SEMINARIO MATEMATICO

142 D. Papini - F. ZanolinThe next two lemmas extend to the case of the FPP-γ two corresponding classicalresults about the usual fixed point property. Their proof is quite standard andtherefore it is omitted.LEMMA 2. Let Z, Y be two arcwise connected topological spaces and let h :Z → Y be a homeomorphism. Suppose Zl − and Zr − are nonempty disjoint subsets ofZ and set Yl− = h(Zl − ), Yr− = h(Zr −). Then ˜Z has the FPP-γ if and only if Ỹ has theFPP-γ.Proof. We leave the proof as an exercise.LEMMA 3. Let Z, Y be two arcwise connected topological spaces with Y ⊆ Zand let r : Z → Y be a continuous retraction. Suppose Yl− and Yr− are nonemptydisjoint subsets of Y and set Zl− = r −1 (Yl − ), Zr − = r −1 (Yr − ). Then Ỹ has the FPP-γif ˜Z has the FPP-γ.Proof. The proof follows the same argument (mutatis mutandis) of that of Lemma 1and therefore it is omitted.COROLLARY 3. Let K ̸= ∅ be a compact convex subset of a normed space.Let Z be a compact topological space which is homeomorphic to [−1, 1] × K, via ahomeomorphism h : Z → [−1, 1] × K. DefineThen ˜Z has the FPP-γ.Z − l:= h −1 ({−1} × K), Z − r := h −1 ({1} × K).Lemma 2 and Lemma 3 together with Theorem 6 (or its variants) permit to givesome straightforward examples with some geometrical meaning. For simplicity, weconfine ourselves to subsets of a finite dimensional space E.EXAMPLE 1. Let Z ⊆ E be a compact set which is homeomorphic to the closedunit ball B[0, 1] ⊆ R N , with N ≥ 1. Let P, Q ∈ Z with P ̸= Q be two given points.Let ψ : E ⊇ D ψ → E be a continuous map and suppose that E ⊆ D ψ is a closed setsuch that for every path σ ⊆ Z, with P, Q ∈ σ, there is a sub-path γ ⊆ σ ∩ E withψ(σ) ⊆ Z and P, Q ∈ ψ(γ). Then ψ has at least a fixed point in E ∩ Z.Proof. We discuss only the case N ≥ 2, since for N = 1, the result is obvious. Let usconsider the cylinderC := {(x 1 ,..., x N−1 , x N ) : ||(x 1 ,..., x N−1)|| ≤ 1, |x N | ≤ 1}on which we select as a right and left sides the south and the north bases respectively:C l := {x = (x 1 ,...,, x N ) ∈ C : x N = −1},C r := {x = (x 1 ,...,, x N ) ∈ C : x N = 1}