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First we show that F : U → C[0, 1] is continuous. Let yn → y in C[0, 1] with {yn }∞ n=1 ⊆ U . We are required to show that F yn → F y in C[0, 1]. There exists µM ∈ Lq [0, 1] with |yn |0 ≤ M , |y|0 ≤ M and |g(s, yn (s))| ≤ µM (s), |g(s, y(s))| ≤ µM (s) for almost all s ∈ [0, 1] (here n ∈ {1, 2, . }). 8) |F yn (t1 ) − F yn (t2 )| < 3 and |F y(t1 ) − F y(t2 )| < 3 ; here n ∈ {1, 2, . }. 9) |F yn (t) − F y(t)| ≤ sup kt p t∈[0,1] × 1 0 |g(s, yn (s)) − g(s, y(s))| ds q 1 q → 0 as n → ∞. 9), and using the fact that [0, 1] is compact, yields a constant N ≥ 0 such that for all n ≥ N, |F yn (t) − F y(t)| < for all t ∈ [0, 1].

12 Show that the closed, convex hull of a compact set in a Banach space is compact. 14, show that it is enough to assume C is convex, that is, C need not necessarily be closed. 7) may be replaced by D ⊆ C countable and D = co({x0 } ∪ F (D)) imply that D is compact. 5 Nonlinear Alternatives of Leray–Schauder Type To apply the Schauder or M¨ onch fixed point theorem we need to find a closed, convex set that is mapped by the map under investigation back into itself. From an application viewpoint this is extremely difficult to achieve.

36 The Theorems of Brouwer, Schauder and M¨ onch It is easy to see that f is continuous but does not have a fixed point. 6 Let X and Y be normed linear spaces. A map F : X → Y is called compact if F (X) is contained in a compact subset of Y . A compact map F : X → Y is called finite dimensional, if F (X) is contained in a finite dimensional linear subspace of Y . We next extend Brouwer’s fixed point theorem to compact maps in normed linear spaces. This generalisation is due to Schauder. The main idea is to approximate compact maps by maps with finite dimensional ranges.

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