By A.M. Fink

ISBN-10: 3540067299

ISBN-13: 9783540067290

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We will denote these components by Kj and write deg(f, U, y) = deg(f, U, Kj ) if y ∈ Kj . 19 (Product formula) Let U ⊆ Rn be a bounded and open set and denote by Gj the connected components of Rn \f (∂U ). If g ◦ f ∈ Dy (U, Rn ), then deg(g ◦ f, U, y) = deg(f, U, Gj ) deg(g, Gj , y), j where only finitely many terms in the sum are nonzero. 52) 30 Chapter 2. The Brouwer mapping degree Proof. Since f (U ) is is compact, we can find an r > 0 such that f (U ) ⊆ Br (0). Moreover, since g −1 (y) is closed, g −1 (y) ∩ Br (0) is compact and hence can be covered by finitely many components {Gj }m j=1 .

Proof. Pick ρ > M and observe deg(1l + F, Bρ (0), 0) = deg(1l, Bρ (0), 0) = 1 using the compact homotopy H(t, x) = tF (x). Here 0 ∈ H(t, ∂Bρ (0)) due to the a priori bound. ✷ Now we can extend the Brouwer fixed-point theorem to infinite dimensional spaces as well. 7 (Schauder fixed point) Let K be a closed, convex, and bounded subset of a Banach space X. If F ∈ C(K, K), then F has at least one fixed point. The result remains valid if K is only homeomorphic to a closed, convex, and bounded subset.

Hence we have an a priimplies |x(t)| ≤ exp(M ori bound which implies existence by the Leary–Schauder principle. Since ε was arbitrary we are done. ✷ 42 Chapter 3. 1 Introduction and motivation In this chapter we turn to partial differential equations. In fact, we will only consider one example, namely the stationary Navier–Stokes equation. Our goal is to use the Leray–Schauder principle to prove an existence and uniqueness result for solutions. Let U (= ∅) be an open, bounded, and connected subset of R3 .

### Almost Periodic Differential Equations by A.M. Fink

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