A simple proof of a semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities. (English) Zbl 0713.34045

Mawhin, J.

Arch. Math., Brno 25, No. 4, 235-238 (1989).

The author proves that a generalized version of a semi-Fredholm principle for the existence of periodic solutions for forced systems with homogeneous nonlinearities obtained by A. C. Lazer and P. J. McKenna [Proc. Am. Math. Soc. 106, 119-125 (1989; Zbl 0683.34027)] can be proved by a simple homotopy argument, which answers a question raised by those authors. The result of this paper is the following: “If U and V are in \(C^ 1(R^ n,R)\), positive homogeneous of degree two, semidefinite and such that the system \(u''(t)+U'(u'(t))+V'(u(t))=0,\) has no T-periodic solution other than 0, then for each \(p\in L^ 1(0,T;R^ n)\) the problem \[ u''(t)+U'(u'(t))+V'(u(t))=p(t),\quad u(0)-u(T)=u'(0)- u'(T)=0 \] has at least one solution.”

Reviewer: Chungyou He

MSC:

34C25

Periodic solutions to ordinary differential equations

Keywords:

semi-Fredholm principle; forced systems with homogeneous nonlinearities

Citations:

Zbl 0683.34027