RENDICONTI DEL SEMINARIO MATEMATICO

132 D. Papini - F. Zanolininherited by Z). When no confusion may occur, we also set clA and intA for cl Z A andint Z A, respectively.For a metric space (X, d), we denote by B(x 0 , R) := {x ∈ X : d(x, x 0 ) < r}the open ball of center x 0 ∈ X and radius r > 0 and by B[x 0 , R] := {x ∈ X :d(x, x 0 ) ≤ r} the corresponding closed ball. Given a map ψ : X ⊇ D ψ → Y, withX, Y metric spaces and a given subset D of the domain D ψ of ψ, we say that ψ iscompact on D if it is continuous on D and ψ(D) is relatively compact in Y, that is,cl(ψ(D)) is compact.Let Z be a topological space, let θ 1 : [a 1 , b 1 ] → Z and θ 2 : [a 2 , b 2 ] → Zbe two continuous mappings (parameterized curves). We write θ 1 ∼ θ 2 if there is ahomeomorphism h of [a 1 , b 1 ] onto [a 2 , b 2 ] (a change of variable in the parameter) suchthat θ 2 (h(t)) = θ 1 (t), ∀ t ∈ [a 1 , b 1 ]. It is easy to check that ∼ is in fact an equivalencerelation and that θ 1 ([a 1 , b 1 ]) = θ 2 ([a 2 , b 2 ]) whenever θ 1 ∼ θ 2 . By a path γ in Z wemean (formally) the equivalence class γ = [θ] of a continuous parameterized curveθ : [a, b] → Z. In this case, with small abuse in the notation, we write γ ⊆ Z. Sincethe image set θ([a, b]) is the same for each θ : [a, b] → Z with γ = [θ], the set¯γ := {θ([a, b]) : θ ∈ γ}is well defined. Given a set A ⊆ Z and a path γ ⊆ Z, we write γ ∩ A ̸= ∅ to meanthat ¯γ ∩ A ̸= ∅, that is, for every parameterized curve θ representing γ we have thatθ(t) ∈ A for some t in the interval-domain of θ. Given a path σ ⊆ Z, we say thatγ ⊆ Z is a sub-path of σ and write γ ⊆ σ if there is θ : [a, b] → Z with [θ] = σsuch that the restriction θ| [c,d] , for some [c, d] ⊆ [a, b], represents γ. According tothese positions, given the paths γ,σ ⊆ Z and a set W ⊂ Z, the condition γ ⊆ σ ∩ W,means that γ is a sub-path of σ with values in W. If Z, Y are topological spaces andφ : Z ⊇ D φ → Y is a continuous map, then for any path γ ⊆ D φ and θ : [a, b] → D φsuch that [θ] = γ, we have that φ ◦ θ : [a, b] → Y is a continuous map. It is easy tocheck that φ ◦ θ 1 ∼ φ ◦ θ 2 when θ 1 ∼ θ 2 and therefore φ(γ) := [φ ◦ θ] is well defined.At last we recall a known definition. Let Z be a topological space. We say thatZ is arcwise connected if, given any two points P, Q ∈ Z with P ̸= Q, there is acontinuous map θ : [a, b] → Z such that θ(a) = P and θ(b) = Q. In such a situation,we’ll also write P, Q ∈ γ, where γ = [θ]. In the case of a Hausdorff topologicalspace Z, the image set θ([a, b]) turns out to be a locally connected metric continuum(a Peano space according to [29]). Then, the above definition of arcwise connectednessis equivalent to the fact that, given any two points P, Q ∈ Z with P ̸= Q, there existsan arc (that is the homeomorphic image of a compact interval) contained in Z andhaving P and Q as extreme points (see, e.g., [18, p.29], [29, pp.115–131] or [71]).2. A fixed point theorem in normed spaces and its variants2.1. Main resultsLet (X,‖ · ‖) be a normed space and suppose thatφ = (φ 1 ,φ 2 ) : R × X ⊇ D φ → R × X