RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 139(H) for every path σ ⊆ C with σ ∩ C l ̸= ∅ and σ ∩ C r ̸= ∅, there is a sub-pathγ ⊆ σ ∩ W with φ(γ) ⊆ C and φ(γ) ∩ C l ̸= ∅, φ(γ) ∩ C r ̸= ∅,holds. Then there exists ˜z = (˜t, ˜x) ∈ W ⊆ D ∩ C, with φ(˜z) = ˜z.Proof. By assumption, the operator φ = (φ 1 ,φ 2 ) is compact on D∩C, hence, φ(D∩C)is bounded in R × X. In particular, there is R > 0 such that,||φ 2 (t, x)|| < R, ∀ z = (t, x) ∈ D ∩ C.As a consequence of the Dugundji Extension Theorem, the closed convex set C ′ :=C ∩ B[0, R] is a retract of B[0, R] (actually, it is a retract of the whole space X,but for us it is more convenient to restrict the retraction to B[0, R]). We denote by̺ : B[0, R] → C ′ such a continuous retraction.If we define now C ′ := [−a, a] × C ′ andC ′ l := {(−a, x) : x ∈ C′ }, C ′ r := {(a, x) : x ∈ C′ },as well as D ′ := D ∩ C ′ and W ′ := W ∩ C ′ , we find that(H) for every path σ ⊆ C ′ with σ ∩ Cl ′ ̸= ∅ and σ ∩ C r ′ ̸= ∅, there is a sub-pathγ ⊆ σ ∩ W ′ with φ(γ) ⊆ C ′ and φ(γ) ∩ Cl ′ ̸= ∅, φ(γ) ∩ C r ′ ̸= ∅,holds. At last, we define the continuous retractionρ = (ρ 1 ,ρ 2 ) : B[a, R] := [−a, a] × B[0, R] → B[a, R],ρ 1 (t, x) = t, ρ 2 (t, x) = ̺(x)and easily check that now the set C ′ plays here the same role as the set R in Lemma 1.Then, according to Lemma 1, there exists ˜z = (˜t, ˜x) ∈ W ′ ⊆ W, with φ(˜z) = ˜z. Theproof is complete.THEOREM 9. Let K ̸= ∅ be a compact convex subset of the normed space X.Let K := [−a, a] × K and defineK l := {(−a, x) : x ∈ K},K r := {(a, x) : x ∈ K}.Suppose that there is a closed subset W ⊆ D ∩ K such that φ is continuous on W andthe assumption(H) for every path σ ⊆ K with σ ∩ K l ̸= ∅ and σ ∩ K r ̸= ∅, there is a sub-pathγ ⊆ σ ∩ W with φ(γ) ⊆ K and φ(γ) ∩ K l ̸= ∅, φ(γ) ∩ K r ̸= ∅,holds. Then there exists ˜z = (˜t, ˜x) ∈ W ⊆ D ∩ K, with φ(˜z) = ˜z.Proof. This result is an immediate consequence of Theorem 8, with the position C :=K .