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By Milgram R. (ed.)

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Example text

N2, we thus observe that for n ? N2 there exists xn E Gn such that Tn(xn) = 0. Since T is A-proper, there exists a subsequence x,, and xo E GD such that Xnj -p x0 in X and T(xo) = 0. t. F. In the sequel we will need the following property, which is not hard to prove. (P6) Let T: GD - Y be A-proper and let y is T(aGD) with T(aGD) bounded. Then there is a d > 0 such that, if W: GD -' Y is A proper and 11 Wx- Txjj < d for x e aGD, then Deg(T, GD, y) = Deg(W, GD, y). 2. 3 reduces to the degree Deg(T, G, y) developed in [16a,b].

2. 10 can be found in Schwartz [S], Lloyd [59], and Deimling [24b]. 3. For the reader who is interested in more general approximation schemes within the framework of A-proper mapping theory, where and {Yn] are not subspaces of X and Y (respectively) and and are not projections, see Petryshyn [73m], Chapters 34-36 in [112b], and Ruotsalainen [S]. The study of monotone operations in general reflexive spaces was developed independently by Minty and Browder (see [14g,h; 24c; 58; 73r]). 4. For the study of and references on duality maps, see [15; 73r; 78; 80].

T. a suitable admissible scheme rL depending on L, constructed in what follows. t. rL. This condition is assumed to hold throughout Chapters 2-4, where we extend a number of existence results (in some cases constructively) obtained by other authors for Eq. 1) under conditions on L and/or N which are more restrictive than those used in this chapter. t. FL for each A a (0, 1]. In particular, among other examples, it is shown that the class of semilinear equations studied by a number of authors mentioned in the Introduction are special cases of Eq.

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