Local expansions and accretive mappings

Let X and Y be complete metric spaces with Y metrically convex, let D ⊂ X be open, fix u 0 ∈ X , and let d ( u ) = d ( u 0 , u ) for all u ∈ D . Let f : X → 2 Y be a closed mapping which maps open subsets of D onto open sets in Y , and suppose f is locally expansive on D in the sense that there exists a continuous nonincreasing function c : R + → R + with ∫ + ∞ c ( s ) ds = + ∞ such that each point x ∈ D has a neighborhood N for which dist ( f ( u ) , f ( v ) ) ≥ c ( max { d ( u ) , d ( v ) } ) d ( u , v ) for all u , v ∈ N . Then, given y ∈ Y , it is shown that y ∈ f ( D ) iff there exists x 0 ∈ D such that for x ∈ X \ D , dist ( y , f ( x 0 ) ) ≤ dist ( u , f ( x ) ) . This result is then applied to the study of existence of zeros of (set-valued) locally strongly accretive and ϕ -accretive mappings in Banach spaces