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
HTML articles powered by AMS MathViewer

by James H. Lightbourne PDF

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

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).

References

A. Cellina, On the local existence of solutions of ordinary differential equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 293–296 (English, with Russian summary). MR 315237
M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57–94. MR 300166, DOI 10.1007/BF02761448
Tosio Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508–520. MR 226230, DOI 10.2969/jmsj/01940508
M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by A. H. Armstrong; translation edited by J. Burlak. MR 0159197

Translation along trajectories of differential equations

K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751

Differential and integral inequalities

A. Lasota and James A. Yorke, The generic property of existence of solutions of differential equations in Banach space, J. Differential Equations 13 (1973), 1–12. MR 335994, DOI 10.1016/0022-0396(73)90027-2
T. Y. Li, Existence of solutions for ordinary differential equations in Banach spaces, J. Differential Equations 18 (1975), 29–40. MR 369848, DOI 10.1016/0022-0396(75)90079-0
R. H. Martin Jr., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973), 399–414. MR 318991, DOI 10.1090/S0002-9947-1973-0318991-4
Robert H. Martin Jr., Approximation and existence of solutions to ordinary differential equations in Banach spaces, Funkcial. Ekvac. 16 (1973), 195–211. MR 352641
R. H. Martin Jr., Remarks on ordinary differential equations involving dissipative and compact operators, J. London Math. Soc. (2) 10 (1975), 61–65. MR 369849, DOI 10.1112/jlms/s2-10.1.61
Robert H. Martin Jr., Invariant sets for perturbed semigroups of linear operators, Ann. Mat. Pura Appl. (4) 105 (1975), 221–239. MR 390414, DOI 10.1007/BF02414931
B. N. Sadovskiĭ, On a fixed point principle, Funkcional. Anal. i Priložen. 1 (1967), no. 2, 74–76 (Russian). MR 0211302