A global existence theorem for a nonautonomous differential equation in a Banach space

Abstract:

Suppose that X is a real or complex Banach space and that A is a continuous function from $[0,\infty ) \times X$ into X. Suppose also that there is a continuous real valued function $\rho$ defined on $[0,\infty )$ such that $A(t, \cdot ) - \rho (t)I$ is dissipative for each t in $[0,\infty )$. In this note we show that, for each z in X, there is a unique differentiable function u from $[0,\infty )$ into X such that $u(0) = z$ and $u’(t) = A(t,u(t))$ for all t in $[0,\infty )$. This is an improvement of previous results on this problem which require additional conditions on A.