Periodic solutions for a differential equation in Banach space

Lightbourne, James H.

doi:10.1090/S0002-9947-1978-0481337-X

Periodic solutions for a differential equation in Banach space
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by James H. Lightbourne

Trans. Amer. Math. Soc. 238 (1978), 285-299

DOI: https://doi.org/10.1090/S0002-9947-1978-0481337-X

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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 \[ \lim \limits _{h \to 0} |x + hA(t,x);\Omega |/h = 0\quad {\text {for}}\;{\text {all}}\;(t,x) \in [0,\infty ) \times \Omega ,\] there exist approximate solutions to the initial value problem \begin{equation}\tag {$IVP$}u’(t) = A(t,u(t)),\quad u(0) = x \in \Omega .\end{equation} 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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