Interval oscillation criteria for higher-order forced nonlinear differential equations. (English) Zbl 1075.34031

Consider the forced nonlinear differential equation \[ L_nx(t)+\delta q(t)F(x(t))=e(t), \quad \delta=\pm 1, \tag{1} \] where \[ L_0x(t)=x(t), \quad L_kx(t)=p_k(t)(L_{k-1}x(t))',\quad k=1,2,\dots,n, \] \(p_i:[t_0,\infty)\to(0,\infty)\), \(i=1,2,\cdots,n-1\), \(p_n\equiv 1\), \(q,e:[t_0,\infty)\to \mathbb{R}=(-\infty,\infty)\) and \(F:\mathbb{R}\to \mathbb{R}\) are continuous, with \(xF(x)>0\) for \(x\neq 0\). By using Young’s inequality and using a class of particular functions, some new criteria are established for the oscillation of higher-order forced nonlinear differential equations of the form (1) that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of \([t_0,\infty)\), rather than on the whole half-line. In particular, the coefficient \(q(t)\) is not restricted as the conditions are assumed in papers by A. G. Kartsatos [Stab. Dyn. Syst., Theor. Appl., Proc. Conf. Miss. State Univ. 1975, 17–72 (1977; Zbl 0361.34031) and Proc. Am. Math. Soc. 33, 377–383 (1972; Zbl 0234.34040)].


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations