Zeros and mapping theorems for perturbations of m-accretive operators in Banach spaces

Let X be a real Banach space and T : D(T) X → 2X be an m-accretive operator. Let C : D(T) X → X be a bounded operator (not necessarily continuous) such that C(T + I)−1 is compact. Suppose that for every x D(T) with x > r, there exists jx Jx such that u+Cx,jx ≥0, for all u Tx. Then, we have where Br(0) denotes the open ball of X with centre at zero and radius r > 0. Assume, furthermore, that T : D(T) → 2X is strongly accretive. Then, 0 (T + C)(D(T) ∩ Br (0)). As applications of the above zero theorem, we derive many new mapping theorems for perturbations of m-accretive operators in Banach spaces. When, T and C are odd operators, we also obtain some new mapping theorems.