Large-period forced oscillations to higher-order pendulum-type equations (Q1355052)

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.