A global existence theorem for a nonautonomous differential equation in a Banach space
1972; American Mathematical Society; Volume: 35; Issue: 2 Linguagem: Inglês
10.1090/s0002-9939-1972-0303035-5
ISSN1088-6826
AutoresDavid Lowell Lovelady, Robert H. Martin,
Tópico(s)Nonlinear Differential Equations Analysis
ResumoSuppose that X is a real or complex Banach space and that A is a continuous function from [ 0 , ∞ ) × X [0,\infty ) \times X into X . Suppose also that there is a continuous real valued function ρ \rho defined on [ 0 , ∞ ) [0,\infty ) such that A ( t , ⋅ ) − ρ ( t ) I A(t, \cdot ) - \rho (t)I is dissipative for each t in [ 0 , ∞ ) [0,\infty ) . In this note we show that, for each z in X , there is a unique differentiable function u from [ 0 , ∞ ) [0,\infty ) into X such that u ( 0 ) = z u(0) = z and u ′ ( t ) = A ( t , u ( t ) ) u’(t) = A(t,u(t)) for all t in [ 0 , ∞ ) [0,\infty ) . This is an improvement of previous results on this problem which require additional conditions on A .
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