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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 n x(t)+δq(t)F(x(t))=e(t),δ=±1,(1)

where

L 0 x(t)=x(t),L k x(t)=p k (t)(L k-1 x(t)) ' ,k=1,2,,n,

p i :[t 0 ,)(0,), i=1,2,,n-1, p n 1, q,e:[t 0 ,)=(-,) and F: are continuous, with xF(x)>0 for x0. 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 ,), 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)].

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