Nonlinear Operators
This chapter focuses on important classes of nonlinear operators stating abstract results that offer powerful tools for establishing the existence of solutions to nonlinear equations. Specifically, they are useful in the study of nonlinear elliptic bounda
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Nonlinear Operators
Abstract This chapter focuses on important classes of nonlinear operators stating abstract results that offer powerful tools for establishing the existence of solutions to nonlinear equations. Specifically, they are useful in the study of nonlinear elliptic boundary value problems as demonstrated in the final three chapters of the present book. The first section of the chapter is devoted to compact operators and emphasizes the spectral properties, including the Fredholm alternative theorem. The second section treats nonlinear operators of monotone type, possibly setvalued, among which a prominent place is occupied by maximal monotone, pseudomonotone, generalized pseudomonotone, and (S+ )-operators. The cases of duality maps and p-Laplacian are of high interest in the sequel. The third section contains essential results on Nemytskii operators highlighting their main continuity and differentiability properties. Comments on the material of this chapter and related literature are given in a remarks section.
2.1 Compact Operators In this section we present some results from nonlinear functional analysis that will be useful in the study of boundary value problems that follow. We start with compact maps, which historically are the first nonlinear operators studied in detail (primarily by Leray and Schauder in connection with their degree map). Compact operators, by definition, are close to operators in finite-dimensional spaces. In what follows, we will use the following basic definitions (slightly different versions of these notions exist in the literature). Definition 2.1. Let X and Y be Banach spaces and C a nonempty subset of X. (a) We say that f : C → Y is compact if it is continuous and for every B ⊂ C bounded, f (B) is compact in Y .
D. Motreanu et al., Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, DOI 10.1007/978-1-4614-9323-5__2, © Springer Science+Business Media, LLC 2014
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2 Nonlinear Operators
(b) We say that f : C → Y is completely continuous if for every sequence {xn }n≥1 ⊂ w C such that xn → x ∈ C, we have f (xn ) → f (x) in Y . In general, the two notions introduced above are not comparable. However, if we enrich the structure of X or we restrict the map f , then we can compare them. Proposition 2.2. If X is a reflexive Banach space, Y is a Banach space, C ⊂ X is nonempty, closed, and convex, and f : C → Y is completely continuous, then f is compact. Proof. Evidently, f is continuous. Let B ⊂ C be a bounded set and let {yn }n≥1 ⊂ f (B). Then yn = f (xn ) with {xn }n≥1 ⊂ B. The reflexivity of X and the Eberlein– Šmulian theorem (e.g., Brezis [52, p. 70]) imply that, along a relabeled subsequence, w xn → x ∈ C. Then yn = f (xn ) → f (x) in Y , which proves the compactness of f . For linear operators, compactness implies complete continuity. Proposition 2.3. If X and Y are Banach spaces and L : X → Y is a linear, compact operator, then L is completely continuous. w
Proof. Let xn → x in X. Then we can find r > 0 such that {xn }n
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