Second-order Neutral Delay Differential Equations Research Articles

A high order method for oscillatory delay differential equations

This paper addresses the oscillatory properties of a class of second-order neutral delay differential equations (NDDEs) and introduces a numerical scheme based on a fourth-order predictor-corrector multistep method for solving NDDEs of this type. An example in the family of oscillatory equations has been integrated by using the proposed numerical scheme. Results confirm the theoretical assumptions.

Nonoscillation and Oscillation Criteria for a Class of Second-Order Nonlinear Neutral Delay Differential Equations with Positive and Negative Coefficients

In this paper, we investigate some nonoscillatory and oscillatory solutions for a class of second-order nonlinear neutral delay differential equations with positive and negative coefficients. By means of the method of contraction mapping principle and some integral inequality techniques, we extend the recent results provided in the literature.

New Sufficient Conditions for Oscillation of Second-Order Neutral Delay Differential Equations

In this work, new sufficient conditions for the oscillation of all solutions of the second-order neutral delay differential equations with the non-canonical operator are established. Using a generalized Riccati substitution, we obtained criteria that complement and extend some previous results in the literature.

New Criteria for Sharp Oscillation of Second-Order Neutral Delay Differential Equations

In this paper, new oscillation criteria for second-order half-linear neutral delay differential equations are established, using a recently developed method of iteratively improved monotonicity properties of a nonoscillatory solution. Our approach allows removing several disadvantages which were commonly associated with the method based on a priori bound for the nonoscillatory solution, and deriving new results which are optimal in a nonneutral case. It is shown that the newly obtained results significantly improve a large number of existing ones.

On the Oscillation of second order nonlinear neutral delay differential equations

AbstractA class of second-order neutral delay differential equations is considered. New oscillation criteria are established to complement and improve some known results in the literature. Two examples are given to support our results.

New Oscillation Theorems for Second-Order Differential Equations with Canonical and Non-Canonical Operator via Riccati Transformation

In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form (a(ξ)((v(ξ)+b(ξ)v(ϑ(ξ)))′))′+c(ξ)G1(v(κ(ξ)))+d(ξ)G2(v(ς(ξ)))=0 under canonical and non-canonical operators, that is, ∫ξ0∞dξa(ξ)=∞ and ∫ξ0∞dξa(ξ)<∞. We use the Riccati transformation to prove our main results. Furthermore, some examples are provided to show the effectiveness and feasibility of the main results.

On the qualitative behavior of the solutions to second-order neutral delay differential equations

Differential equations of second order appear in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation. First, we have taken a single delay and later the results are generalized for multiple delays. Some examples are given and open problems are presented.

On the Asymptotic Behavior of a Class of Second-Order Non-Linear Neutral Differential Equations with Multiple Delays

In this work, we present some new sufficient conditions for the oscillation of a class of second-order neutral delay differential equation. Our oscillation results, complement, simplify and improve recent results on oscillation theory of this type of non-linear neutral differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments

Abstract In this paper, we describe a method to solve the problem of finding periodic solutions for second-order neutral delay-differential equations with piecewise constant arguments of the form x″(t) + px″(t − 1) = qx([t]) + f(t), where [⋅] denotes the greatest integer function, p and q are nonzero real or complex constants, and f(t) is complex valued periodic function. The method reduces the problem to a system of algebraic equations. We give explicit formula for the solutions of the equation. We also give counter examples to some previous findings concerning uniqueness of solution.

An improved approach for studying oscillation of second-order neutral delay differential equations

The paper is devoted to the study of oscillation of solutions to a class of second-order half-linear neutral differential equations with delayed arguments. New oscillation criteria are established, and they essentially improve the well-known results reported in the literature, including those for non-neutral differential equations. The adopted approach refines the classical Riccati transformation technique by taking into account such part of the overall impact of the delay that has been neglected in the earlier results. The effectiveness of the obtained criteria is illustrated via examples.

On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments

This paper provides a method of finding periodical solutions of the second-order neutral delay differential equations with piecewise constant arguments of the form x''(t)+px''(t-1)=qx(2[frac{t+1}{2}])+f(t), where [cdot] denotes the greatest integer function, p and q are nonzero constants, and f is a periodic function of t. This reduces the 2n-periodic solvable problem to a system of n+1 linear equations. Furthermore, by applying the well-known properties of a linear system in the algebra, all existence conditions are described for 2n-periodical solutions that render explicit formula for these solutions.

Oscillation of nonlinear second-order neutral delay differential equations

Oscillation of nonlinear second-order neutral delay differential equations

Oscillation criteria for second-order neutral delay differential equations

Oscillation criteria for second-order neutral delay differential equations

Stability of Human Balance With Reflex Delays Using Galerkin Approximations

Falling is the leading cause of both fatal and nonfatal injury in the elderly, often requiring expensive hospitalization and rehabilitation. We study the stability of human balance during stance using inverted single- and double-pendulum models, accounting for physiological reflex delays in the controller. The governing second-order neutral delay differential equation (NDDE) is transformed into an equivalent partial differential equation (PDE) constrained by a boundary condition and then into a system of ordinary differential equations (ODEs) using the Galerkin method. The stability of the ODE system approximates that of the original NDDE system; convergence is achieved by increasing the number of terms used in the Galerkin approximation. We validate our formulation by deriving analytical expressions for the stability margins of the double-pendulum human stance model. Numerical examples demonstrate that proportional–derivative–acceleration (PDA) feedback generally, but not always, results in larger stability margins than proportional–derivative (PD) feedback in the presence of reflex delays.

Oscillation of forced second-order neutral delay differential equations

The objective of this paper is to study oscillation of a forced second-order neutral differential equation. By using the generalized Riccati substitution and integral technique, a new sufficient condition is obtained which insures that all solutions to the studied equation are oscillatory. An illustrative example is included.

On Eventually Positive Solutions of Quasilinear Second-Order Neutral Differential Equations

We study the second-order neutral delay differential equation[r(t)Φγ(z′(t))]′+q(t)Φβ(x(σ(t)))=0, whereΦα(t)=|t|α-1t,α≥1andz(t)=x(t)+p(t)x(τ(t)). Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral∫∞r-1/γ(t)dt. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case.

Weighted pseudo almost periodic solutions of second-order neutral-delay differential equations with piecewise constant argument

By introducing the method of decomposition of weighted pseudo almost periodic sequence, we present some existence theorems of weighted pseudo almost periodic solutions for second order neutral differential equations with piecewise constant argument of the form d 2 d t 2 ( x ( t ) + p x ( t − 1 ) ) = q x ( 2 [ t + 1 2 ] ) + f ( t ) , where | p | = 1 , [ ⋅ ] denotes the greatest integer function, q is a nonzero constant and f ( t ) is weighted pseudo almost periodic. Our results are new and can be regarded as a complement of some known results even in the special cases of almost periodicity and pseudo almost periodicity.

Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

Some sufficient conditions are established for the oscillation of second-order neutral differential equation , , where . The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of second-order neutral delay differential equations, Rat. Mat. 1 (1985), Grace and Lalli (1987), Ruan (1993), H. J. Li (1996), H. J. Li (1997), Xu and Xia (2008).

On the Oscillation of Second-Order Neutral Delay Differential Equations

Some new oscillation criteria for the second-order neutral delay differential equation , are established, where , , , . These oscillation criteria extend and improve some known results. An example is considered to illustrate the main results.

New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients

We establish the oscillation and asymptotic criteria for the second-order neutral delay differential equations with positive and negative coefficients having the forms[x(t)+∑i∈Rci(t)x(αi(t))]′′+r(t)[x(t)+∑i∈Rci(t)x(αi(t))]′+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=0and[x(t)+∑i∈Rci(t)x(αi(t))]′′+r(t)[x(t)+∑i∈Rci(t)x(αi(t))]′+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=f(t). The obtained new oscillation criteria extend and improve the recent results given in the paperof B. Karpuz et al. (2009).