Generalized iterative methods and nonlinear functional equations.

Abstract

Perturbations of nonlinear operators are also investigated. If F'(x)(B(0; 1)) (R-HOOK) B(0; c((VBAR)(VBAR)x(VBAR)(VBAR))) and if F is perturbed by a nonlinear operator G satisfying a boundedness condition, then F + G is an open mapping from X onto Y. The case where both F and G are Gateaux differentiable operators satisfying various coercive conditions again yields surjectivity results for the sum F + G. These proofs rely on the existence of contractor inequalities derived from the hypotheses. Finally, if G is a compact operator and I - F is compact, then F + G is surjective; the proof uses methods of algebraic topology.

Let X and Y be Banach spaces, P be a Gateaux differentiable mapping from X to Y and c : {0, (INFIN)) (--->) (0, (INFIN)) be a continuous nonincreasing function for which (INT)('(INFIN)) c(u)du = (INFIN). If P'(x)(B(0; 1)) contains B(0; c((VBAR)(VBAR)x(VBAR)(VBAR))) for each x (epsilon) X, then P is an open mapping of X onto Y. If the differentiability assumption on P is removed and instead P is both open and locally expansive, then P(X) = Y. If A is a continuous mapping from X to X satisfying for each x (epsilon) X, (GREATERTHEQ) c(max{(VBAR)(VBAR)x(VBAR)(VBAR), (VBAR)(VBAR)y(VBAR)(VBAR)}) (VBAR)(VBAR)x - y(VBAR)(VBAR)('2) for some j (epsilon) J(x -y), then A is a homeomorphism of X onto X. The main technique used in establishing these results is a new fixed point theorem which includes Ekland's Theorem as a special case.