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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 $$ L_0x(t)=x(t), \quad L_kx(t)=p_k(t)(L_{k-1}x(t))',\quad k=1,2,\dots,n, $$ $p_i:[t_0,\infty)\to(0,\infty)$, $i=1,2,\cdots,n-1$, $p_n\equiv 1$, $q,e:[t_0,\infty)\to \bbfR=(-\infty,\infty)$ and $F:\bbfR\to \bbfR$ are continuous, with $xF(x)>0$ for $x\ne 0$. 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 {\it 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)].

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory