Periodic derivative of solutions to nonlinear differential equations. (English) Zbl 0727.34029

The differential equation of the form \((1)\quad x^{(n)}=f(t,x,x',...,x^{(n-1)})\) \((n>1)\), \(x(\theta)-x(0)-\omega =x^{(i)}(\theta)-x^{(i)}(0)=0,\quad i=1,...,n-1,\) where \(f(t+\theta,x+\omega,x',...,x^{(n-1)})\equiv f(t,x,x',...,x^{(n- 1)})\in C({\mathbb{R}}^{n+1})\) satisfies locally a Lipschitz condition with respect to \(x,x',...,x^{(n-1)}\) and \(\theta\), \(\omega\) are positive reals are considered. The question of existence of a type of solution of equation (1), the so-called D-periodic (derivo-periodic) solutions, is discussed in the article. Up to now sufficient conditions for the existence of such solution have been found in an autonomous case in a nonautonomous case with a small parameter. The purpose of the article is to get sufficient conditions for the existence of D-periodic solutions for (D-)periodically forced equations not necessarily involving a small parameter. The topological degree theory is used to prove the basic theorem.


34C25 Periodic solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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