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Existence for nonoscillatory solutions of second-order nonlinear differential equations. (English) Zbl 1111.34049

Summary: The existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation

[r(t)(x(t)+P(t)x(t-τ)) ' ] ' + i=1 m Q i (t)f i (x(t-σ i ))=0,tt 0 ,

where m1 is an integer,τ>0, σ i 0, r,P,Q i C([t 0 ,),), f i C(,), i=1,2,,m, is studied. Some new sufficient conditions for the existence of a nonoscillatory solution are obtained for general P(t) and Q i (t), i=1,2,,m, which means that we allow oscillatory P(t) and Q i (t), i=1,2,,m. In particular, our results improve essentially and extend some known results in the recent references.

34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations