Transversality and cone maps

Abstract

Let X be a real Banach space and let K be a cone in X. We will be concerned in this paper with characteristic value problems for nonlinear operators which map K (or a subset thereof) into K. In general we can allow the operator to depend on the characteristic value parameter so that we examine an equation of the type x = 2 F(2, x), where F is a continuous map from [0, ~ ) x K into K. We will always assume that F takes bounded sets into precompact sets. For the main theorem we further assume that F(2, 0 )=0 for 2 > 0 and that F has an " x " derivative at x = 0, in the direction of the cone. The derivative is assumed independent of 2 and is denoted by DF(O). Since DF(O) is a compact, linear map of K into itself the results of KREIN 8~; RUTMAN [1] show that if DF(O) has positive spectral radius r, then there is a vector veK{0} such that v = 2 x DF(O)v, where 2 x = r k Theorem2.3 is an extension of this result. To describe Theorem2.3 we let 5 e' denote the set of solutions of x = 2 F(2, x) having 2 > 0 and xeK-{0}, and let 6* be the closure of 5 p' in [0, ~ ) x K. If we suppose that DF(O) has positive spectral radius 2~ -1, then Theorem 2.3 asserts that 5 e contains a continuum (i.e. a closed, connected set) of solutions C~SP such that C is unbounded and (,t 1, 0) is in C. In Theorem 2.33 we assume in addition that F has a derivative at " x = Go" with a corresponding lowest characteristic value 2~. In this case there exist continua in 6e emanating from each of the points (A1,0) and (2~ ~ oo). If R > max(X 1, 2~~ and R is not a characteristic value of DF(O) or "DF(oo)", then either each continuum meets the set {R} • (K -{ 0 } ) or the two continua join to form a single continuum in 5 e containing (21,0) and "(2~ ~ ~ ) " . The question of " large" solutions for nonlinear eigenvalue problems was discussed by KRASNOSELSKn ([5], p. 207) and, recently, in the papers [6]-[11], though under different hypotheses from those used here. TOLAND [7] and RAmNOWn'Z [8] observe that "inversion" of vectors in the unit sphere of the Banach space enables one to treat solutions at ov using known results for solutions near x = 0 (cf. [12], [2]). In Theorem2.33 we also use a "change of variable" to treat 0 and ~ at the same time. An essential ingredient in the proofs of Theorems 2.3 and 2.33 is the transversality density theorem [3] together with elementary facts from differential