NONTRIVIAL SOLUTIONS OF INCLUSIONS INVOLVING PERTURBED MAXIMAL MONOTONE OPERATORS

Let X be a real reflexive Banach space and X* its dual space. Let L : X superset of D(L) -> X* be a densely defined linear maximal monotone operator, and T : X superset of D(T) > 2(x*), 0 is an element of D(T) and 0 is an element of T(0), be strongly quasibounded maximal monotone and positively homogeneous of degree 1. Also, let C : X superset of D(C) -> X* be bounded, demicontinuous and of type (S+) w.r.t. to D(L). The existence of nonzero solutions of Lx broken vertical bar Tx broken vertical bar Cx is an element of 0 is established in the set G1 \ G2, where G2 subset of C G(1) with (G) over bar2 subset of G(1), G(1), G(2) are open sets in X, 0 epsilon G2, and G1 is bounded. In the special case when L = 0, a mapping G : (G) over bar1 > X* of class (P) introduced by Hu and Papageorgiou is also incorporated and the existence of nonzero solutions of Tx+Cx+Gx epsilon 0, where T is only maximal monotone and positively homogeneous of degree alpha E (0, 1], is obtained. Applications to elliptic partial differential equations involving p-Laplacian with p epsilon (1,2] and time -dependent parabolic partial differential equations on cylindrical domains are presented.