Periodic solutions to second order differential equations of Liénard type with jumping nonlinearities. (English) Zbl 0625.34046

Fonda, Alessandro; Zanolin, Fabio

Commentat. Math. Univ. Carol. 28, 33-41 (1987).

This paper is devoted to the existence of periodic solutions for differential systems of the form \[ x'=y-F(x)+E_ 1(t),\quad y'=- g(x)+E_ 2(t) \] under the assumption that the limits \(F_ 1=\lim_{x\to -\infty}-F(x)/x,\) \(F_ 2=\lim_{x\to +\infty}F(x)/x,\) \(g_ 1=\lim_{x\to -\infty}g(x)/x,\) \(g_ 2=\lim_{x\to +\infty}g(x)/x\) exist and are finite. The proof is based upon a study of an associated homogeneous system \(x'=y-F_ 1x^--F_ 2x^+\), \(y'=g_ 1x^--g_ 2x^+\) and coincidence degree arguments.

Reviewer: J.Mawhin

MSC:

34C25	Periodic solutions to ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

Liénard equations; jumping nonlinearities; homogeneous system; coincidence degree