A Global Existence Theorem for a Nonautonomous Differential Equation in a Banach Space

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