RENDICONTI DEL SEMINARIO MATEMATICO

140 D. Papini - F. ZanolinREMARK 3. Variants of Theorem 8 and Theorem 9 can be obtained, as a consequenceof Theorem 7, using condition (H ′ ) instead of condition (H). For instance,Theorem 9 could be accompanied by the following.THEOREM 10. Under the same positions of Theorem 9, suppose that there is aclosed subset W ⊆ D ∩ K such that φ is continuous on W and the assumption(H ′ ) for every continuum σ ⊆ K with σ ∩ K l ̸= ∅ and σ ∩ K r ̸= ∅, there is acontinuum Ŵ ⊆ σ ∩ W with φ(Ŵ) ⊆ K and φ(Ŵ) ∩ K l ̸= ∅, φ(Ŵ) ∩ K r ̸= ∅,holds. Then there exists ˜z = (˜t, ˜x) ∈ W ⊆ D ∩ K, with φ(˜z) = ˜z.3. Extensions, remarks and consequences3.1. The “stretching along the paths” fixed point propertyLet Z be a topological space. According to a well known definition, Z has the fixedpoint property (FPP) if every continuous map of Z into itself has at least a fixed point.The FPP is invariant by homeomorphisms and it is preserved under continuous retractions.Thus, by the Brouwer fixed point theorem, we know that any topological spacewhich is homeomorphic to (a retract of) a closed ball of a finite dimensional normedspace has the FPP. It is the aim of this section to show that something similar (evenif not exactly the same) holds with respect to the assumption of “stretching along thepaths” (H) and the corresponding fixed point result in Theorem 6.We consider now the following situation.DEFINITION 1. Assume that Z is a topological space and Z − l, Z − r are twononempty disjoint subsets of Z. We setand defineZ − := Z − l∪ Z − r˜Z := (Z, Z − ).We call ˜Z a two-sided oriented space or simply an oriented space. In view of ourapplications below which concern the case of arcwise connected spaces and where thefamily of paths connecting Z − lto Z − r is involved, we call ˜Z a path-oriented space whenZ is arcwise connected.We say that ˜Z has the fixed point property for maps stretching along the paths (inthe sequel referred as FPP-γ ) if Z is arcwise connected and, for every pair (D,ψ),satisfying the following conditions:(i 1 ) D ⊆ Z;(i 2 ) ψ : D → Z is continuous;(i 3 ) there is a closed set W ⊆ D such that, for every path σ ⊆ Z with σ ∩ Z − l̸= ∅and σ ∩ Z − r ̸= ∅, there is a sub-path γ ⊆ σ ∩ W with ψ(γ) ∩ Z − l̸= ∅ andφ(γ) ∩ Z − r ̸= ∅;