Let X be a real reflexive Banach space and X * its dual space. Let L : X ⊃ D(L) → X * be a densely defined linear maximal monotone operator, and T : X ⊃ D(T) → 2 X * , 0 ∈ D(T) and 0 ∈ T (0), be strongly quasibounded maximal monotone and positively homogeneous of degree 1. Also, let C : X ⊃ D(C) → X * be bounded, demicontinuous and of type (S +) w.r.t. to D(L). The existence of nonzero solutions of Lx + T x + Cx 0 is established in the set G 1 \ G 2 , where G 2 ⊂ G 1 with G 2 ⊂ G 1 , G 1 , G 2

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... re open sets in X, 0 ∈ G 2 , and G 1 is bounded. In the special case when L = 0, a mapping G : G 1 → X * of class (P) introduced by Hu and Papageorgiou is also incorporated and the existence of nonzero solutions of T x+Cx+Gx 0, where T is only maximal monotone and positively homogeneous of degree α ∈ (0, 1], is obtained. Applications to elliptic partial differential equations involving p-Laplacian with p ∈ (1, 2] and time-dependent parabolic partial differential equations on cylindrical domains are presented.