Nonlinear Neutral Equations Research Articles

Non-Linear Neutral Differential Equations with Damping: Oscillation of Solutions

The oscillation of non-linear neutral equations contributes to many applications, such as torsional oscillations, which have been observed during earthquakes. These oscillations are generally caused by the asymmetry of the structures. The objective of this work is to establish new oscillation criteria for a class of nonlinear even-order differential equations with damping. We employ different approach based on using Riccati technique to reduce the main equation into a second order equation and then comparing with a second order equation whose oscillatory behavior is known. The new conditions complement several results in the literature. Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

A multi-domain Legendre spectral collocation method for nonlinear neutral equations with piecewise continuous argument

ABSTRACTIn this paper, for the neutral equations with piecewise continuous argument, we construct a spectral collocation method by combining the shifted Legendre–Gauss–Radau interpolation and a multi-domain division. Based on the non-classical Lipschitz condition, the convergence results of the method are derived. The results show that the method can arrive at high accuracy under the suitable conditions. Several numerical examples further illustrate the obtained theoretical results and the computational effectiveness of the method.

Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

Abstract In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation d dt x ( t ) = − a ( t ) h ( x ( t ) ) + d dt Q ( t , x ( t − τ ( t ) ) ) + G ( t , x ( t ) , x ( t − τ ( t ) ) ) . $${{\rm{d}} \over {{\rm{dt}}}}{\rm{x}}({\rm{t}}) = - {\rm{a}}\;({\rm{t}})\;{\rm{h}}\;({\rm{x}}\;({\rm{t}})) + {{\rm{d}} \over {{\rm{dt}}}}{\rm{Q}}\;({\rm{t}},\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))) + {\rm{G}}\;({\rm{t}},\;{\rm{x}}({\rm{t}}),\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))).$$ We invert this equation to construct a sum of a completely continuous map and a large contraction which is suitable for applying the modificatition of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G.

On implicit abstract neutral nonlinear differential equations

In this paper we continue our developments in Hernandez and O'Regan (J Funct Anal 261:3457---3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernandez and O'Regan (J Funct Anal 261:3457---3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial "nonlinear" neutral differential equations. Some applications involving partial neutral differential equations are presented.

Dynamics of Third-Order Nonlinear Neutral Equations

The aim of this paper is to study oscillatory and asymptotic properties of the third-order nonlinear neutral equation with continuously distributed delays of the form(r(t)([xt+∫abpt,μxτt,μdμ]′′)α)′+∫cdqt,ξfαxgt,ξdξ=0. Applying suitable generalized Riccati transformation and integral averaging technique, we present new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of this equation. Obtained results essentially improve and complement earlier ones.

Existence of positive solutions of higher-order nonlinear neutral equations

In this work, we consider the existence of positive solutions of higher-order nonlinear neutral differential equations. In the special case, our results include some well-known results. In order to obtain new sufficient conditions for the existence of a positive solution, we use Schauder’s fixed point theorem.

Existence of positive periodic solutions in neutral nonlinear equations with functional delay

Existence of positive periodic solutions in neutral nonlinear equations with functional delay

The existence of nonoscillatory solutions of higher order nonlinear neutral equations

In this work, we consider the existence of nonoscillatory solutions of variable coefficient higher order nonlinear neutral differential equations. Our results include as special cases some well-known results for linear and nonlinear equations of first, second and higher order. We use the Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.

Oscillatory Solutions of Neutral Equations with Polynomial Nonlinearities

Existence uniqueness of an oscillatory solution for nonlinear neutral equations by fixed point method is proved.

Forced oscillation of nth-order nonlinear differential equations

Some new oscillation criteria for forced nonlinear equations of the form x (n) (t) − q(t)|x(t)| λ sgn x(t) = f(t), λ >1 , are established. We impose no restriction on the forcing term f(t) as those assumed in [1,2]. We further employ the technique presented here to establish new oscillation criteria for forced nonlinear neutral equations of the form (x(t) + cx[t − τ ]) (n) + a(t)x(t) + b(t)x[t − τ] = f(t) + q(t)|x(t)| λ sgn x(t) + p(t)|x[t − τ]| σ sgn x[t − τ] , where λ > 1 and σ > 1.

Positive Solutions of a Nonlinear Neutral Equation with Positive and Negative Coefficients

In this paper, a class of nonlinear neutral equations with positive and negative coefficients is considered. Several conditions for the existence and nonexistence of eventually positive solutions are obtained. These conditions extend and improve some of the existing criteria in the literature.

Asymptotic behavior of nonoscillatory solutions of nth-order neutral nonlinear equations with oscillating coefficients

Asymptotic behavior of nonoscillatory solutions of nth-order neutral nonlinear equations with oscillating coefficients

On Oscillation of First Order Nonlinear Neutral Equations

On Oscillation of First Order Nonlinear Neutral Equations

Existence of nonoscillatory solutions of first order nonlinear neutral equations

AbstractConsider the nonlinear neutral equationwhere pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).