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

Authors: David Lowell Lovelady and Robert H. Martin
Journal: Proc. Amer. Math. Soc. 35 (1972), 445-449
MSC: Primary 34G05
MathSciNet review: 0303035
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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.

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Keywords: Nonautonomous differential equations, dissipative operators, global existence theorems, an application of nonlinear semigroups
Article copyright: © Copyright 1972 American Mathematical Society

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