EXISTENCE OF MONOTONE SOLUTIONS FOR NONLINEAR PERTURBED DIFFERENTIAL INCLUSIONS (monotone solution/admissible set valued map)

Let Z be a separable Banach space whose dual is uni-formly convex, A: D(A) e X ^ 2 ^ an m-dissipative operator generating a compact semigroup 5(i):Z)(A)->D(A), i>0, /) a locally closed set in D{A) and F: D ^ 2 ^ a nonempty, closed convex and bounded valued mapping which is strongly-weakly upper-semicon-tinuous and locally bounded on D. Let «: ̂ » be a preorder on D, relation characterized by the set-valued map F: D-> 2^, defined by P(§)- {r \ ^ D \ t, < r\} whose graph is closed in Z) X Z). Whe denote by w(-, 0, ^, y-i- p) the unique mild solution of uXf) G Au{t) + y + /7, satisfying w(0, 0, < ̂ , j + /?) = < ̂ and we prove: Theorem. Under the general assumptions above a sufficient condition in order that for each ^ e D there exists at least one mild solution u of uXt) € Au{t) + F{u{t)) satisfying w(0) = t, and u(s) •< u(t) for each s < t is the «bounded w-hypermonotonicity condition » below: (MwíHMC) There exists a locally bounded function M: D —> R^. satisfying: for each ^ e D there exists ye F(^) such that for each 5> 0 each weak neighborhood VofO there exist te (0, S \ and p e V with \\p\ \ < M{^) and u{t, 0, ( ̂ , 3; + /7) e P(^). 1