Abstract
By using comparison principles, we analyze the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Due to less restrictive assumptions on the coefficients of the equation and on the deviating argument τ, our criteria improve a number of related results reported in the literature.
Highlights
Higher-order functional differential equations have numerous applications in engineering and natural sciences, see Hale [1]
We are concerned with the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations (a(t) (z′′(t))γ)′ + q(t)xγ(τ (t)) = 0, (1)
Using the result of Philos [18, Theorem 1] once again, we deduce that the associated delay differential equation (23) has a positive solution, which contradicts the main assumption of the theorem. ■
Summary
Higher-order functional differential equations have numerous applications in engineering and natural sciences, see Hale [1]. A second-order differential equation is called oscillatory if all its solutions oscillate This is not the case for third-order equations whose solutions often exhibit different asymptotic behavior. By the result due to Ladas et al [5, Theorem 1], all solutions to the latter equation are oscillatory since the associated characteristic equation λ3 + e−πλ = 0 has no real roots We note that such drastic changes in the asymptotic behavior of solutions are not specific for third-order equations and can be observed for first-order differential equations. Can be both delayed and advanced and that we are concerned in this paper only with the asymptotic behavior of solutions, we tacitly assume that solutions to the equation under study exist and can be continued to infinity. We deal only with eventually positive solutions of (1) since, under our assumption on γ, if x(t) is a solution of Eq (1), so is −x(t)
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