New criteria for the oscillation and asymptotic behavior of second-order neutral differential equations with several delays

In this paper, necessary and sufficient conditions for asymptotic behavior are established of the solutions to second-order neutral delay differential equations of the form d dt ( r(t) ( d dt [x(t) − p(t)x(τ (t))])γ ) + ∑m i=1 qi(t)fi ( x(σi(t))) = 0 for t ≥ t0. We consider two cases when fi(u)/uβ is nonincreasing for γ > β , and nondecreasing for β > γ , where β and γ are quotients of two positive odd integers. Our main tool is Lebesgue’s dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem.

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