Periodic solutions of a class of abstract nonlinear equations of the second order. (English) Zbl 0574.34024

Krejčí, Pavel

Čas. Pěst. Mat. 110, 122-129 (1985).

The aim of this paper is to prove the existence of weak periodic solutions of the abstract differential equation (1) \(F(u)=\phi (u)+h\), where \(F(u)\equiv u''+\psi (u')+{\mathcal A}u\), \(u'=du/dt\), \(\psi\) and \(\phi\) are nonlinear mappings of a Hilbert space H into itself with linear growth and \({\mathcal A}\) is a linear elliptic operator from \(V\subset H\) into \(V^*\). The results obtained here are applied to the jumping- nonlinearity problem for ordinary and partial differential equations.

MSC:

34C25	Periodic solutions to ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems

Keywords:

weak periodic solutions; abstract differential equation; linear elliptic operator; jumping-nonlinearity problem