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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Periodic solutions for a differential equation in Banach space

Author: James H. Lightbourne
Journal: Trans. Amer. Math. Soc. 238 (1978), 285-299
MSC: Primary 34G05
MathSciNet review: 0481337
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Abstract: Suppose X is a Banach space, $ \Omega \subset X$ is closed and convex, and $ A:[0,\infty ) \times \Omega \to X$ is continuous. Then if

$\displaystyle \mathop {\lim }\limits_{h \to 0} \vert x + hA(t,x);\Omega \vert/h = 0\quad {\text{for}}\;{\text{all}}\;(t,x) \in [0,\infty ) \times \Omega ,$

there exist approximate solutions to the initial value problem

$\displaystyle u'(t) = A(t,u(t)),\quad u(0) = x \in \Omega .$ ($ IVP$)

In the case that $ A(t,x) = B(t,x) + C(t,x)$, where B satisfies a dissipative condition and C is compact, we obtain a growth estimate on the measure of noncompactness of trajectories for a class of approximate solutions. This estimate is employed to obtain existence of periodic solutions to (IVP).

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Keywords: Banach space, dissipative, measure of noncompactness, periodic solutions
Article copyright: © Copyright 1978 American Mathematical Society