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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 3 x in D(T) n G is considered, where T . X D D(T) -* 2X is a possibly discontinuous m-dissipative operator and C: G -X is completely continuous. It is assumed that X is uniformly convex, D(T) n G $& 0 and (T + C)(D(T) n OG) C G. 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 IF {= 7Z+ -+ R-; /3(r) -O 0 as r -* oo}. The effect of a weak boundary condition of the type (u + Cx, x) > -/3(11xfl)flxl12 on the range of operators T + C is studied for m-accretive and maximal monotone operators T. Here, ,B E F, x C D(T) with sufficiently large norm and u C Tx. Various new eigenvalue results are given involving the solvability of Tx + ACx 3 0 with respect to (A, x) E (0, oo) 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.

New results in the perturbation theory of maximal monotone and $M$-accretive operators in Banach spaces