NEW RESULTS IN THE PERTURBATION THEORY OF MAXIMAL MONOTONE AND M-ACCRETIVE OPERATORS IN BANACH SPACES

摘要

Let X be a real Banach space and G a bounded, open and convex subset of X. The solvability of the fixed point problem (*) Tx + Cx contains x in D(T) boolean AND (G) over bar is considered, where T : X superset of D(T) --> 2(X) is a possibly discontinuous m-dissipative operator and C : (G) over bar --> X is completely continuous. It is assumed that X is uniformly convex, D(T) boolean AND G not equal 0 and (T + C)(D(T) boolean AND partial derivative G) subset of (G) over bar. A result of Browder, concerning single-valued operators T that are either uniformly continuous or continuous with X* uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let Gamma = {beta : R(+) --> R(+) ; beta(r) --> 0 as r --> infinity}. The effect of a weak boundary condition of the type (u + Cx,x) greater than or equal to -beta(x)x(2) on the range of operators T + C is studied for m-accretive and maximal monotone operators T. Here, beta E is an element of Gamma, x is an element of D(T) with sufficiently large norm and u is an element of Tx. Various new eigenvalue results are given involving the solvability of Tx + lambda Cx contains 0 with respect to (lambda,x) is an element of (0,infinity) x D(T). Several results do not require the continuity of the operator C. Four open problems are also given, the solution of which would improve upon certain results of the paper. [References: 41]