RENDICONTI DEL SEMINARIO MATEMATICO

134 D. Papini - F. ZanolinWe study now the auxiliary fixed point problem(7) x = ψ 2 (t, x), x ∈ Xwhere we take, for a moment, t ∈ [−a, a] as a parameter.Observe that, by definition, ψ 2 (t, x) ∈ B[0, R] for every (t, x) and therefore,for any r > R, it follows thatx − ψ 2 (t, x) ̸= 0,∀ t ∈ [−a, a], ∀ x ∈ ∂ B(0, r).Thus the Leray−Schauder topological degreed 0 := deg(I − ψ 2 (t,·), B(0, r), 0)is well defined and is constant with respect to t ∈ [−a, a]. Using the compact homotopyh λ (x) defined by(λ, x) ↦→ x − λψ 2 (t, x), with λ ∈ [0, 1] and x ∈ B[0, r]we find that h λ (x) ̸= 0 for every λ ∈ [0, 1] and x ∈ ∂ B(0, r) and therefore d 0 =deg(I, B(0, r), 0) = 1. Hence, the Leray−Schauder Théorème Fondamental [42] impliesthat the solution set := {(t, x) ∈ [−a, a] × B(0, r) : x = ψ 2 (t, x)}is nonempty and contains a continuum (compact and connected set) S such thatp 1 (S) = [−a, a],where we have denoted by p 1 : R× X → R, p 1 (t, x) = t, the projection of the productspace onto its first factor (see also [44] for more information about this fundamentalresult). Since the projection of S onto the t-axis covers the interval [−a, a], we obtainBy the definition of P R it is clear also thatS ∩ B l ̸= ∅, S ∩ B r ̸= ∅.(8) p 2 (S) ⊆ B[0, R] ⊆ B(0, r),where we have denoted by p 2 : R × X → X, p 2 (t, x) = x, the projection of theproduct space onto its second factor.Let now ε ∈]0, r − R[ be a fixed number and consider a covering of S by a finitenumber of open balls of the form ]t i − ε, t i + ε[ ×B(x i ,ε), with (t i , x i ) ∈ S. Withoutloss of generality, we can suppose that −a ≤ t 1 < t 2 ...t i−1 < t i ...t N ≤ a, where Nis the number of the balls required for the covering. The setU ε :=N⋃]t i − ε, t i + ε[ ×B(x i ,ε) ⊆] − a − ε, a + ε[ ×B(0, r),i=1