Nonlinear equations involving m-accretive operators

Let X be a Banach space and T an m-accretive operator defined on a subset D(T) of X and taking values in 2^x. For the class of spaces whose bounded closed and convex subsets have the fixed point property for nonexpansive self-mappings, it is shown here that two boundary conditions which imply existence of zeroes for T, appear to be equivalent. This fact is then used to prove that if there exists x₀?D(T) and a bounded open neighborhood U of x₀, such that $\text{¦T(x}{}_0\text{)¦ < r ? ¦T(x)¦}$ for all x??U ∩ D(T), then the open ball B(0; r) is contained in the range of T.