RENDICONTI DEL SEMINARIO MATEMATICO

138 D. Papini - F. Zanolinholds. Then there exists ˜z = (˜t, ˜x) ∈ W ⊆ D ∩ R, with φ(˜z) = ˜z.Proof. Given the compact operator φ : D ∩ R → R × X we defineψ(t, x) := φ(ρ(t, x)) = φ(t,ρ 2 (t, x)),for (t, x) ∈ D ′ := ρ −1 (D ∩ R).Clearly, ψ is compact on D ′ = D ′ ∩ B[a, R]. We also defineW ′ := ρ −1 (W) ⊆ B[a, R] =: B.Consider now a continuous parameterized curve θ = (θ 1 ,θ 2 ) : [0, 1] → B[a, R] suchthat θ(0) ∈ B l (that is, θ 1 (0) = −a) and θ(1) ∈ B r (that is, θ 1 (1) = a). Then for thecurve ϑ : [0, 1] ∋ s ↦→ ρ(θ(s)), it holds thatϑ(s) ∈ R, ∀ s ∈ [0, 1] and ϑ(0) ∈ R l , ϑ(1) ∈ R r .By assumption (H) referred to R, there exists a restriction of ϑ to an interval [s 0 , s 1 ] ⊆[0, 1] such that ϑ(s) ∈ W for every s ∈ [s 0 , s 1 ] and, moreover,as well asφ(ϑ(s)) ∈ R, ∀ s ∈ [s 0 , s 1 ],φ(ϑ(s 0 )) ∈ R l and φ(ϑ(s 1 )) ∈ R r , or φ(ϑ(s 1 )) ∈ R l and φ(ϑ(s 0 )) ∈ R r .Just to fix one of the two possible cases for the rest of the proof, suppose that the firstpossibility occurs (the treatment of the other case is exactly the same, modulo minorchanges in the role of s 0 and s 1 ). Then, by the definition of ϑ, W ′ and ψ, we can alsowrite thatθ(s) ∈ W ′ , ∀ s ∈ [s 0 , s 1 ],andψ(θ(s)) ∈ R ⊆ B, ∀ s ∈ [s 0 , s 1 ]ψ(θ(s 0 )) ∈ R l ⊆ B l , ψ(θ(s 1 )) ∈ R l ⊆ B r .We have thus proved that assumption (H) of Theorem 6 is satisfied with respect tothe operator ψ and the cylinder B[a, R] and therefore Theorem 6 guarantees that thereexists for ψ a fixed point ˜z = (˜t, ˜x) ∈ W ′ ⊆ D ′ , with ψ(˜z) = ˜z. As a last step, wejust recall that the range of ψ coincides with the range of φ and that ρ (as a retraction)is the identity on R. This implies that ˜z = (˜t, ˜x) ∈ W ⊆ D ∩ R with φ(˜z) = ˜z.THEOREM 8. Let C ̸= ∅ be a closed convex subset of the normed space X. LetC := [−a, a] × C and defineC l := {(−a, x) : x ∈ C},C r := {(a, x) : x ∈ C}.Assume thatφ is compact on D ∩ Cand there is a closed subset W ⊆ D ∩ C such that the assumption