The existence of bounded solutions for equations x' = A(t)x + r(x,t) is proved, where the linear part is exponentially dichotomic and the nonlinear term r satisfies some weak conditions.
@article{bwmeta1.element.bwnjournal-article-apmv56z2p103bwm,
author = {B. Przeradzki},
title = {The existence of bounded solutions for differential equations in Hilbert spaces},
journal = {Annales Polonici Mathematici},
volume = {57},
year = {1992},
pages = {103-121},
zbl = {0805.47041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv56z2p103bwm}
}
B. Przeradzki. The existence of bounded solutions for differential equations in Hilbert spaces. Annales Polonici Mathematici, Tome 57 (1992) pp. 103-121. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv56z2p103bwm/
[000] [1] Yu. L. Daletskiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in a Banach Space, Nauka, Moscow 1970 (in Russian).
[001] [2] K. Goebel and W. Rzymowski, An existence theorem for the equations x' = f(x,t) in Banach spaces, Bull. Acad. Polon. Sci. 18 (7) (1970), 367-370. | Zbl 0202.10003
[002] [3] K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), 301-309. | Zbl 56.1124.04
[003] [4] J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Acad. Press, New York and London 1966. | Zbl 0243.34107
[004] [5] B. N. Sadovskiĭ, Ultimately compact and condensing operators, Uspekhi Mat. Nauk 27 (1) (1972), 82-146 (in Russian).
[005] [6] T. Ważewski, Sur la limitation des intégrales des systèmes d'équations différentielles linéaires ordinaires, Studia Math. 10 (1948), 48-59. | Zbl 0036.05703