On periodic solutions of \(n\)th order ordinary differential equations. (English) Zbl 0953.34028

Kiguradze, Ivan

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 40, No. 1-8, 309-321 (2000).

In this paper, the author studies the existence and uniqueness of a periodic solution to the following equations \[ u^{(n)} = \sum_{k=1}^np_k(t)u^{(k-1)} + q(t),\tag{1} \] and \[ u^{(n)} = f(t,u,\cdots,u^{(n-1)}),\tag{2} \] where the functions \(p_k(t)\), \(k=1,\cdots,n,\) and \(f\) are \(\omega\)-periodic in \(t\). Under the condition \[ \int_0^\omega p_1(t)\neq 0 \] and other conditions, he obtains the result that the linear equation (1) has only one \(\omega\)-periodic solution. For equation (2), he also establishes a similar result. The proof is based on the topological degree and properties of the \(\omega\)-periodic solutions to the following differential inequalities \[ u^{(n)}\text{sgn}(\pm u(t))\geq g_0(t)|u(t)|^\lambda - g_1(t, |u^\prime(t)|,\cdots, |u^{(n-1)}(t)|), \]
\[ |u^{(n)}(t)|\leq g(t, |u(t)|,\cdots, |u^{(n-1)}(t)|). \]

Reviewer: Bin Liu (Beijing)

MSC:

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

Keywords:

periodic solutions; sublinear equations

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