Second-order Delay Equations Research Articles

Oscillation Criteria of Fourth-Order Differential Equations with Delay Terms

The aim of this paper is to derive oscillation criteria of the following fourth-order differential equation with delay term r x z ′ ′ ′ x γ ′ + ∑ i = 1 n q i x f z η i x = 0 , under the assumption ∫ x 0 ∞ r − 1 / γ s d s = ∞ . The results are based on comparison with the oscillatory behaviour of second-order delay equations and the generalised Riccati transformation. Not only do the provided theorems provide an entirely new technique but also they vastly improve on a number of previously published conclusions. We give three examples to illustrate our findings.

Different techniques for studying oscillatory behavior of solution of differential equations

The aim of this work is to study oscillatory behavior of solutions for a fourth-order neutral nonlinear differential equation (b(x)(wm−1(x))γ)′+ ∑i=1jqi(x)f(w(gi(x)))=0, x≥x0. The results obtained are based on the Riccati transformation, integral averaging technique and the theory of comparison with second-order delay equations. The obtained results complements and generalize the earlier ones. Some examples are illustrated to show the applicability of the obtained results.

Oscillation Theorems for Advanced Differential Equations with p-Laplacian Like Operators

The main objective of this paper is to establish new oscillation results of solutions to a class of even-order advanced differential equations with a p-Laplacian like operator. The key idea of our approach is to use the Riccati transformation and the theory of comparison with first and second-order delay equations. Some examples are provided to illustrate the main results.

New Oscillation Criteria for Advanced Differential Equations of Fourth Order

The main objective of this paper is to establish new oscillation results of solutions to a class of fourth-order advanced differential equations with delayed arguments. The key idea of our approach is to use the Riccati transformation and the theory of comparison with first and second-order delay equations. Four examples are provided to illustrate the main results.

New Comparison Theorems for the Even-Order Neutral Delay Differential Equation

The aim of this study was to examine the asymptotic properties and oscillation of the even-order neutral differential equations. The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order delay equations. Our results improve and complement some well-known results. We obtain Hille and Nehari type oscillation criteria to ensure the oscillation of the solutions of the equation. One example is provided to illustrate these results.

Geometric integration of the electromagnetic two-body problem

The equations of motion of the two-body problem of Dirac’s electrodynamics of point charges consist of a delay equation for the proton and a delay equation for the electron. These equations involve the third derivative of the charges’s position and have runaway solutions, which make forward numerical integration troublesome. Dirac’s equations of motion are algebraic-delay equations, involving a degenerate linear form of the past accelerations. A Fredholm alternative yields a system of second-order delay equations of motion plus a constraint on the initial segment of trajectory. Here we use the Fredholm constraint as a geometric tool to derive covariant second-order equations of motion in position for backward time integration. We also extend the backward integration scheme to include a generalized version of Dirac’s theory that includes two delays.

Sturm comparison theorems for second-order delay equations

We prove two Sturm theorems for linear delay equations of the form ( p( t) u′( t))′ + ∑ q i ( t) u( τ i ( t)) + ∝ τ( a) t k( s, t) u( s) ds = 0. The method uses a Riccati-type equation and the theory of differential inequalities. An application of the first theorem to a nonlinear boundary value problem gives an extension of Moroney's uniqueness result for second-order nonlinear equations without delay. We then give a method for comparing the oscillation of a linear second-order delay equation with a corresponding linear equation, extending a result of Mahfoud. We also sharpen a result of Wong.