Externalities, Nonconvexities, and Fixed Points

as follows: First, let γ be such that ε∗ Xγ≤ min{ 1 1, , δ 0 },forsomeγ =1, 2,...,k EE0 k HE0and choose k γ so thatk 1γ< ε∗ Xγ. Second, for each ν >k γ ,chooseE ν ∈ B hw ∗ ×w ∗ ( 1 ν ,E0 ) ∩ D eE and H ν ∈ B hw ∗ ×w ∗ ( 1 ν ,E0 ) ∩ Sand for each ν and k ≥ k ν := max{ν,k FEν },chooseF vk ∈ B hw ∗ ×w ∗ ( 1 ν ,Eν ) ∩ D.−→ E 0 , H νh w ∗ ×w ∗E 0 as ν →∞, and that for each ν, F vk −→h w ∗ ×w ∗By compactness, WLOG we can assume that E νF vk v−→h w ∗ ×w ∗Thus, we haveh w ∗ ×w ∗(F vk ,H ν ) ≤ h w ∗ ×w ∗(F vk ,E ν )+h w ∗ ×w ∗(Ev ,H ν ) → 0 (*)as ν →∞and k →∞,k ≥ k v ,whereh w ∗ ×w ∗(Ev ,H ν ) ≤ h w ∗ ×w ∗(Ev ,E 0 )+h w ∗ ×w ∗(E0 ,H ν ) → 0. (**)−→ E 0 ,andh w ∗ ×w ∗E ν as k →∞, k ≥ k v .Now observe that for all v, F vk ∈ B hw ∗ ×w ∗ (k 1,Eν ) ∩ D together with the h w ∗ ×w ∗-w ∗ -upper semicontinuity of minimal USCO n(·) on S imply that {z vk } = n(F vk ) →w ∗z v0 ∈ n(E ν ) as k → ∞, k ≥ k v . Moreover, E ν ∈ B hw ∗ ×w ∗ (ν 1 ,E0 ) ∩ DE e for allv together with the h w ∗ ×w ∗-d w∗-upper semicontinuity of minimal USCO n(·) implythat {x ν } = n(E ν ) → x0 ∈ n(E 0 ) as kw ∗ v →∞. Also by compactness, WLOG we canassume that N(H ν ) → N 0 ⊆ N(E 0 ) implying that for ν sufficiently large,hw ∗N(H ν ) ⊂ B w ∗(ε 3 ,Ls{N(H ν )}).But now it follows from (88) and the continuity of the excess functions, e w ∗(N(·),N(·))on S × S that there exists a δ 4 > 0 and positive integers, k ν 3 and ν 3 , such that forν ≥ ν 3 ,andfork ≥ k ν 3,h w ∗ ×w ∗(F vk ,H ν ) < δ 4 n implying that e w ∗(N(F vk ),N(H ν )) < ε 4 ,so thatdist w ∗(z vk ,N(H ν )) < ε 4 or z vk ∈ B w ∗(ε 4 ,N(H ν )).Choosing ε 4 so that B w ∗(ε 4 ,N(H ν )) ⊆ B w ∗(ε 3 ,Ls{N(H ν )}), wehave⎫⎪⎬⎪⎭(88)z vk ∈ B w ∗(ε 3 ,Ls{N(H ν )}). (89)But we can also choose ν ≥ ν 3 ,andk ≥ k ν 3large enough so thatρ w ∗(z vk ,x 0 ) ≤ ρ w ∗(z vk ,x ν )+ρ w ∗(x ν ,x 0 ) < ε 2 ,where x 0 ∈ nE e (E0 ) ∩ B w ∗(ε 0 ,m(E 0 )). Thuswehaveforν ≥ ν 3 ,andk ≥ k ν 3z vk ∈ B w ∗(ε 2 ,n eE (E 0 ) ∩ B w ∗(ε 0 ,m(E 0 ))). (90)Together, (89) and (90) contradict (87).53