RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 137φ(Ŵ) ∩ B r ̸= ∅. This means that there are points Q 1 = (q1 1, q1 2 ), Q 2 = (q1 2, q2 2 ) ∈ Ŵsuch that φ 1 (Q 1 ) = −a and φ 1 (Q 2 ) = a. This implies that p 1 (Q 1 ) − φ 1 (Q 1 ) =q1 1 + a ≥ −a + a = 0 and p 1(Q 2 )−φ 1 (Q 2 ) = q1 2 − a ≤ a − a = 0. By the Bolzano’sTheorem we can conclude that there exists a point ¯z = (¯t, ¯x) ∈ Ŵ ⊆ S ∩ W such that¯t = ψ 1 (¯z) and, by (8), φ 2 (¯z) ∈ B[0, R]. At this point, we can complete our argumentas in the proof of Theorem 6. Indeed, the fact that S is contained in the solution set of (7) implies that¯x = ψ 2 (¯t, ¯x)and therefore ¯z ∈ W is a fixed point of ψ. The fact that φ 2 (¯t, ¯x) ∈ B[0, R], implies (inview of (6) and the remarks at the beginning of the proof of Theorem 6) that ¯z ∈ W isa fixed point of φ.REMARK 2. In our theorems we have confined ourselves to the case of compactmaps. Extensions can be given to more general operators like, locally compact, k-contractive, etc., provided that a decent degree theory is available (see [15, 27, 50] forthe corresponding definitions).2.2. Results related to Theorem 6Like in the case of the Schauder fixed point theorem, we give now some variants ofTheorem 6 (see Theorem 8 and Theorem 9 below). As in Theorem 6 we assume thatX is a normed space andφ : R × X ⊇ D φ → R × Xis a map which is continuous on a set D ⊆ D φ . For the subsequent proofs, we systematicallycheck condition (H) by taking as a representation of the path σ a continuouscurve θ(s) which is parameterized on the interval [0, 1] and look for a suitable restrictionof θ(s) with s ∈ [s 0 , s 1 ] ⊆ [0, 1], as a representation of a sub-path γ ⊆ σ.We start with a preliminary lemma.LEMMA 1. Let ρ = (ρ 1 ,ρ 2 ) : B[a, R] := [−a, a] × B[0, R] → B[a, R] be acontinuous map such that, for each t ∈ [−a, a], ρ 1 (t, x) = t and ρ 2 (t,·) is a retractionof B[0, R] onto its image. DefineandAssume thatR := ∪ t∈[−a,a] {ρ(t, x) : x ∈ B[0, R]} = ρ(B[a, R])R l := ρ(B l ), R r := ρ(B r ).φ is compact on D ∩ Rand there is a closed subset W ⊆ D ∩ R such that the assumption(H) for every path σ ⊆ R with σ ∩ R l ̸= ∅ and σ ∩ R r ̸= ∅, there is a sub-pathγ ⊆ σ ∩ W with φ(γ) ⊆ R and φ(γ) ∩ R l ̸= ∅, φ(γ) ∩ R r ̸= ∅,