## Large-period forced oscillations to higher-order pendulum-type equations.(English)Zbl 0878.34023

This paper deals with equation $x^{(n)}+ \sum^{n-1}_{j=1} a_jx^{(n-j)}+ h(x)=p(t),\quad (n>3),\tag{1}$ where $$a_j$$ are positive damping constants, $$h(x)$$ is a continuous function and $$p(t)$$ is continuous and periodic or bounded function.
The main results are given in lemma, and in theorems 1 and 2.
Lemma: Let the roots of the polynomials $\lambda^{n-p}+\sum^{n-p}_{j=1} a_j\lambda^{n-j-p}$ be negative for all $$p=1,\dots,n-1$$. Then the derivatives $$x^{(k)}(t)$$, $$k=1,\dots,n-1$$, of all solutions $$x(t)$$ of equation $x^{(n)}+\sum^{n-1}_{j=1} a_jx^{(n-j)}= g(t,x),$ are uniformly ultimately bounded and $\limsup_{t\to\infty}|x^{(k)}(t)|\leq D_k:= {k\over a_{n-j}} C_r\quad(k=1,\dots,n-1).\tag{2}$ Theorem 1: Let the assumptions of the lemma be satisfied. If there exist a positive constant $$R$$ and a point $$\overline x$$ such that $(3)\qquad h(\overline x+R)<0,\;h(\overline x-R)>0,\qquad (4)\qquad R\geq \Delta(R),$ then equation (1) admits a $$T$$-periodic solution, provided $$p(t)\equiv p(t+T)$$ and $$\int^T_0 p(t)dt=0$$.
Theorem 2. Let the assumptions of the lemma be satisfied. If there exist a positive constant $$R$$ and a point $$\overline x$$ such that (3) and $$R>\Delta(R)$$ hold, then equation (1) admits a bounded solution on a positive ray, provided $$p(t)$$ and $$\int^t_{t_0} p(s)ds$$ are bounded on the interval $$[t_0,\infty)$$, where $$t_0$$ may be very big.
The proofs of the lemma and theorems 1 and 2 are discussed in detail. Many interesting results are given in remarks and examples.
Reviewer: G.Arazov (Baku)

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems