The Topological Degree and Galerkin Approximations for Noncompact Operators in Banach Spaces by Felix E. Browder and W. v. Petryshyn

Let X and F be real Banach spaces, G a bounded open subset of X, cl(G) its closure in X, bdry(G) its boundary in X. We consider mappings T, (nonlinear, in general), of cl(G) into Y which are A-proper, in the sense defined below, with respect to a given approximation scheme of generalized Galerkin type. We define a generalized concept of topological degree for such mappings with respect to the given approximation scheme, and show that this degree (which may be multivalued) has the basic properties of the classical Leray-Schauder degree (where the latter is defined on the narrower class of maps of X into X of the form 7 + C , with I the identity and C compact). For a wide class of A -proper mappings T of the form T—H+C, with H an A -proper homeomorphism of a suitable type and C compact, we show that the degree is single-valued and coincides with another generalized degree studied in Browder [9] and BrowderNussbaum [ l l ] , and in particular is independent of the approximation scheme involved. In particular, this holds if H is strongly accretive from I t o J (cf. Browder [4], [5], [ó], [8]), including as a very special case all maps H of the form H^I—U, with U a strict contraction. DEFINITION 1. Let X and Y be Banach spaces. By an (oriented) approximation scheme for mappings from X to F, we mean: an increasing sequence {Xn} of oriented finite dimensional subspaces of X, an increasing sequence { Yn} of oriented finite dimensional subspaces of F, and a sequence of linear projection maps {Qn} with Qn mapping Y on Yn such that d im(X n )=d im(F n ) for all n, (JnXn is dense in X, and Qny-*y as n—> oo for all y in Y. DEFINITION 2. Let G be a bounded open subset of X, Ta mapping of cl(G) into F. Then T is said to be A-proper with respect to a given approximation scheme in the sense of Definition 1 if for any sequence \tij} of positive integers with nj—> and a corresponding sequence [xnj\ in cl(G) with each xnj in Xnj such that QniTxnj converges strongly in Y to an element y, there exists an infinite subsequence {n^)} such that XnHh) converges strongly to x in X as k—><*> and T(x) =y. The concept of A -proper mapping is a slight variant of the condition (H) of Petryshyn [18], and both are modifications of the definition of P-compact mapping in Petryshyn [ l5] , [16], and [17]. A sira-