By using the theory of strongly continuous cosine families of linear operators in Banach space the existence of solutions of a semilinear second order differential initial value problem (1) as well as the existence of solutions of the linear inhomogeneous problem corresponding to (1) are proved. The main result of the paper is contained in Theorem 5.
@article{bwmeta1.element.bwnjournal-article-apmv54z2p155bwm,
author = {Jan Bochenek},
title = {An abstract nonlinear second order differential equation},
journal = {Annales Polonici Mathematici},
volume = {55},
year = {1991},
pages = {155-166},
zbl = {0724.34069},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv54z2p155bwm}
}
Jan Bochenek. An abstract nonlinear second order differential equation. Annales Polonici Mathematici, Tome 55 (1991) pp. 155-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv54z2p155bwm/
[000] [1] T. Kato, Perturbation Theory for Linear Operators, Springer, 1966. | Zbl 0148.12601
[001] [2] R. Rabczuk, Elements of the Theory of Differential Inequalities, PWN, Warszawa 1976 (in Polish). | Zbl 0351.34006
[002] [3] I. Segal, Nonlinear semi-groups, Ann. of Math. 78 (1963), 339-364. | Zbl 0204.16004
[003] [4] C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar. 32 (1978), 75-96. | Zbl 0388.34039
[004] [5] T. Winiarska, Differential Equations with Parameter, Monograph 68, Technical University of Cracow, 1988.
[005] [6] K. Yosida, Functional Analysis, Springer, New York 1980.