On the Theory of Accretive Operators Involving Differential Equations in Banach Spaces
In this paper we consider the initial value problem given by
d/dt[u(t)] + A(t)u(t) = 0 with u(0) = a
for t ∈ [0, T] where we assume the following restrictions: (A(t)) is a family of operators with domains and ranges in a Banach space X with X* uniformly convex; the domain of A(t) is independent of t; A(t) is "m-accretive" for each t; and the existence of a constant C such that
||A(t)x − A(s)x|| ≤ C|t − s|(1 + ||x|| + ||A(s)x||)
for all x ∈ D and s, t ∈ [0, T].
In Theorem 1 we show a solution to the initial value problem exists almost everywhere in [0, T]. in Theorem 2 we show the uniqueness of the solution found in Theorem 1. In Theorem 3 we place the extra restriction on X that it be uniformly convex itself. Under these conditions there exists a solution to the initial value problem everywhere in [0, T] except, possibly, at a countable number of points. Finally we will discuss the relationship between "m-accretive" operators and contraction semigroups.