Abstract
Some new sufficient conditions are established for the oscillation of fourth order neutral differential equations with continuously distributed delay of the form r t N x ‴ t α ′ + ∫ a b q t , ϑ x β δ t , ϑ d ϑ = 0 , where t ≥ t 0 and N x t : = x t + p t x φ t . An example is provided to show the importance of these results.
Highlights
The theory of differential equations is an adequate mathematical apparatus for the simulation of processes and phenomena observed in biotechnology, neural networks, physics etc, see [1]
One area of active research in recent times is to study the sufficient criterion for oscillation of delay differential equations, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]
We establish the asymptotic behavior of fourth-order neutral differential equation of the form α 0 Z b r (t) Nx000 (t)
Summary
The theory of differential equations is an adequate mathematical apparatus for the simulation of processes and phenomena observed in biotechnology, neural networks, physics etc, see [1]. We discuss some important papers: Chatzarakis et al [9] proved the equation (1) where α = β, is oscillatory, if. Moaaz et al in [19] extended the Riccati transformation to obtain new oscillatory criteria for (1) as condition ". Where n is even, they proved it oscillatory by using the Riccati transformation if either lim inf. We apply the previous results to the equation ( x (t) + px ( φt))(n) + bx (δt) = 0, t ≥ 1, where n = 4, p = 7/8, φ = 1/e, δ = 1/e2 and b = q0 /υ4 , we find:. By using the Riccati transformations, we establish a new oscillation criterion for a class of fourth-order neutral differential equations (1).
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