Nonlinear elliptic boundary value problems via cone analysis

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IN THE study of fixed point equation Ax = x, Leray-Schauder degree theory has proved to be very effective if A is a compact or condensing operator acting in some Banach space, while the fixed point index theory has been very useful for operators leaving a retract invariant. For this information, see [4, 6, 11, 14, 15, 17, 191. In the past, whenever a cone was considered, people were mainly interested in positive fixed points. In this paper, however, we are going to use a combination of Leray-Schauder degree and the index theory to yield some new fixed points. Applications are given to nonlinear elliptic boundary value problems. Throughout this paper, X is a real Banach space, P C X a cone, i.e. a closed and convex subset such that AP c P for all ,! L 0, and P fl (-P) = (01. We always assume that P is total (i.e. P P = X) and A: X --) X is compact (i.e. continuous and mapping bounded sets into precompact sets). We use A’(O) and A’(a) to denote the Frechet derivative of A at x = 0 and 05, respectively. Let us notice that if A also maps P into P with A0 = 0, r@‘(O)) > 1 (where r(B) represents the spectral radius of an operator B) and A = 1 is not an eigenvalue of A’(O), then, for any small open !C? c X with 0 E 0, we have deg(Z A, CJ,O) = kl while, i(A,c2nP,P) = 0.