Externalities, Nonconvexities, and Fixed Points

(2) For all z ∈ Z, F ∗ (z) is the unique, irreducible continuum about η(z) andF ∗ (z) ⊆ N (z).(3) For all z ∈ Z, F ∗ (z) is minimally essential for N (z) in C w ∗ f(X) -i.e.,F ∗ (z) ∈ M N (z) (C w ∗ f(X)).Proof. (1) Because Z is a locally connected, compact metric space, the connectednessof F ∗ (z) follows Theorem 3 in Hola (1987). Thus, we have F ∗ (z) ∈ C w ∗ f(X) for allz ∈ Z. Because the hyperspace of subcontinua, C w ∗ f(X), is a convex metric space(see Theorems 2 and 3 and Corollary 4 above), the fact that F ∗ (z) is a dendrite (i.e.,a locally connected, hereditarily unicoherent subcontinua) for all z is an immediateconsequence of Theorem 5 above. Moreover, by Whyburn (1942) (also see Charatonikand Charatonik, Theorem 1.3, 1998), all subcontinua of F ∗ (z) are dendrites. Thefact that all subcontinua of F ∗ (z) are also dendrites implies, by Theorem 2.5 of Duda(1962), that F ∗ (z) is strongly convex for all z.(2) We have by expression (59) that η(z) ⊆ F ∗ (z). We will show thatF ∗ (z) =κ(η(z)) := ∩ {M ∈ C w ∗ f(X) :η(z) ⊆ M} for all z ∈ Z,where κ(·) is the Goodykoontz (1977) mapping, continuous with respect to the Hausdorffmetric h w ∗ on P w ∗ f(X) because X is a dendrite (see expression 12), and wherefor each z ∈ Z, κ(η(z)) ∈ C w ∗ f(X) is the unique subcontinuum irreducible aboutη(z). Because F ∗ (z) ∈ C w ∗ f(X) and η(z) ⊆ F ∗ (z) for all z ∈ Z, the equalityF ∗ (z) =κ(η(z)) canfailonlyifforsomez 0 , κ(η(z 0 )) is a proper subset of F ∗ (z 0 ).Letx 0 ∈ F ∗ (z 0 )\κ(η(z 0 )). Because (z 0 ,x 0 ) ∈ F ∗ there exists a sequence {(z n ,f kn (z n ))} nsuch that (z n ,f kn (z n )) → (z 0 ,x 0 ). By compactness of X we can assume WLOG that{f ∗ (z n )} n converges to some l 0 . Therefore, (z 0 ,l 0 ) ∈ Gr ρZ ×w ∗f∗ = Grη. Because(z 0 ,x 0 ) /∈ Gr ρZ ×w ∗f ∗ = Grη,there exists open balls, B δ 0(z 0 ) and B ε 0(x 0 ),ofsufficiently small radii, δ 0 > 0 andε 0 > 0, such that£Bδ 0(z 0 ) × B ε 0(x 0 ) ¤ ∩ Gr ρZ ×w ∗f∗ = ∅.or equivalently,£Bδ 0(z 0 ) × B ε 0(x 0 ) ¤ ∩ Grη = ∅.Therefore, for some η 0 > 0, we have for all (z,x) ∈ B δ 0(z 0 ) × B ε 0(x 0 )Given that (z n ,f k n(z n ))inf (z,x)∈GrρZ ×w ∗f ∗ [ρ Z (z,z)+ρ w ∗(x, x)] > η 0 .→ρZ ×w ∗(z0 ,x 0 ), for some integer N 0 sufficiently large, wehave for all n ≥ N 0 , (z n ,f k n(z n )) ∈ B δ 0(z 0 ) × B ε 0(x 0 ), implying that for all n ≥ N 0and for all (z, x) ∈ Gr ρZ ×w ∗f ∗[ρ Z (z n ,z)+ρ w ∗(f k n(z n ),x)] > η 0 .35