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

Abstract

Let X be a real Banach space and T : D( T) ⊂ X → 2 X 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 + C x , j x 〉 ≥ 0 , for all u ∈ Tx. Then, we have 0 ∈ ( T + C ) ( D ( T ) ∩ B r ( 0 ) ) , ¯ where B r (0) denotes the open ball of X with centre at zero and radius r > 0. Assume, furthermore, that T : D( T) → 2 X is strongly accretive. Then, 0 ∈ ( T + C)( D( T) ∩ B r (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.