Let X be a real Banach space and G1, G2 two nonempty, open and bounded subsets of X such that 0 ∈ G2 and G2 ⊂ G1. The problem (∗) T x + Cx = 0 is considered, where T : X ⊃ D(T) → X is an accretive or monotone operator with 0 ∈ D(T) and T(0) = 0, while C : X ⊃ D(C) → X can be, e.g., one of the following types: (a) compact; (b) continuous and bounded with the resolvents of T compact; (c) demicontinuous, bounded and of type (S+) with T positively homogeneous of degree one; (d) quasi-bounded and satisfies a generalized (S+)-condition w.r.t. the operator T, while T is positively homogeneous of degree one. Solutions are sought for the problem (∗) lying in the set D(T + C) ∩ (G1 \ G2). Nontrivial solutions of (∗) exist even when C(0) = 0. The degree theories of Leray and Schauder, Browder, and Skrypnik as well as the degree theory by Kartsatos and Skrypnik for densely defined operators T, C are used. The last three degree theories do not assume any compactness conditions on the operator C. The excision and additivity properties of these degree theories are employed, and the main results are significant extensions or generalizations of previous results by Krasnoselskii, Guo, Ding and Kartsatos involving the relaxation of compactness conditions and/or conditions on the boundedness of the operator T. Moreover, a new degree theory developed by Kartsatos and Skrypnik has been used to prove a similar result for operators of type T + C, where T : X ⊃ D(T) → 2 X∗ is a multi-valued maximal monotone operator, with 0 ∈ D(T) and 0 ∈ T(0), and C : X ⊃ D(C) → X∗ is a densely defined quasi-bounded and finitely continuous operator of type (S˜+). The problem of existence of nonzero solutions for T x + Cx + Gx 3 0 is also considered. Here, T is maximal monotone, C is bounded demicontinuous of type (S+), and G is of class (P). Eigenvalue and invariance of domain results have also been established for the sum L + T + C : G ∩ D(L) → 2 X∗ , where G ⊂ X is open and bounded, L : X ⊃ D(L) → X∗ densely defined linear maximal monotone, T : X → 2X∗ bounded maximal monotone, and C : G → X∗ bounded demicontinuous of type (S+) w. r. t. D(L).