Third-order Nonlinear Delay Differential Equation Research Articles

New Conditions for Testing the Oscillation of Third-Order Differential Equations with Distributed Arguments

In this paper, we consider a certain class of third-order nonlinear delay differential equations with distributed arguments. By the principle of comparison, we obtain the conditions for the nonexistence of positive decreasing solutions as well as, and by using the Riccati transformation technique, we obtain the conditions for the nonexistence of increasing solutions. Therefore, we get new sufficient criteria that ensure that every solution of the studied equation oscillates. Asymmetry plays an important role in describing the properties of solutions of differential equations. An example is given to illustrate the importance of our results.

Oscillation Results of Third-Order Differential Equations with Symmetrical Distributed Arguments

This paper is concerned with the oscillation and asymptotic behavior of certain third-order nonlinear delay differential equations with distributed deviating arguments. By establishing sufficient conditions for the nonexistence of Kneser solutions and existing oscillation results for the studied equation, we obtain new criteria which ensure that every solution oscillates by using the theory of comparison with first-order delay equations and the technique of Riccati transformation. Some examples are presented to illustrate the importance of main results.

Oscillation and asymptotic behavior of third-order nonlinear delay differential equations with positive and negative terms

We study the oscillation and nonoscillation of third order delay differential equations with positive and negative terms. We establish Kiguradze Lemma, and offer the novel and useful estimate x(τ1(t))x(t)≥τ1(t)t which plays an important role in our main results. First of all, we give Leighton–Wintner type criteria. Then, by using Riccati transformation, we establish new oscillation criteria including Kamenev-type oscillation criteria. Finally, we present a sufficient and necessary condition which guarantees that the nonoscillatory solution x(t) has an upper bound and tends to zero. Examples illustrate the validity and practicability of our results.

Existence of positive periodic solutions for third-order nonlinear delay differential equations with variable coefficients

In this paper, the following third-order nonlinear delay differential equation with periodic coefficients x'''(t) + p(t)x''(t) + q(t)x'(t) + r(t)x(t) = f(t,x(t), x(t - t (t))) + d/dt g (t, x(t - t (t))), is considered. By employing Green's function and Krasnoselskii's fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order nonlinear delay differential equation.

Oscillation criteria for a class of third order damped differential equations

The present study concerns the oscillation of a class of third-order nonlinear delay differential equations with middle term. We offer a new description of oscillation of the third-order equations in terms of oscillation of a related well studied second-order linear differential equation without damping. By using the integral averaging technique, we establish new oscillation results for this equation. Some examples are provided to illustrate the main results.

Oscillation of third-order nonlinear damped delay differential equations

This paper is concerned with the oscillation of certain third-order nonlinear delay differential equations with damping. We give new characterizations of oscillation of the third-order equation in terms of oscillation of a related, well-studied, second-order linear differential equation without damping. We also establish new oscillation results for the third-order equation by using the integral averaging technique due to Philos. Numerous examples are given throughout.

OSCILLATION OF THIRD-ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS

In this paper, we study the oscillatory behavior of a class of third-order nonlinear delay differential equations $$ (a(t) (b(t) y'(t))')' + q(t) y^\gamma(\tau(t)) = 0. $$ Some new oscillation criteria are presented by transforming this equation to the first-order delayed and advanced differential equations. Employing suitable comparison theorems we establish new results on oscillation of the studied equation. Assumptions in our theorems are less restrictive, these criteria improve those in the recent paper [Appl. Math. Comput., 202 (2008), 102-112] and related contributions to the subject. Examples are provided to illustrate new results.

Oscillation theory of third-order nonlinear functional differential equations

In this paper, we are concerned with oscillation of solutions of a certain class of third-order nonlinear delay differential equations of the form $% x^{^{\prime \prime \prime }}(t)+p(t)x^{^{\prime}}(t)+q(t)f(x(\tau (t)))=0$. We establish some new oscillation results that extend and improve some results in the literature in the sense that our results do not require that $% \tau ^{^{\prime }}(t)\geq 0$. Some examples are considered to illustrate the main results and some conjectures and open problems are presented.

Oscillation Criteria of Certain Third-Order Differential Equation with Piecewise Constant Argument

We study the oscillation and asymptotic behavior of third-order nonlinear delay differential equation with piecewise constant argument of the form(r2(t)(r1(t)x'(t))')'+p(t)x'(t)+f(t,x([t]))=0. We establish several sufficient conditions which insure that any solution of this equation oscillates or converges to zero. Some examples are given to illustrate the importance of our results.

On the oscillation of certain class of third-order nonlinear delay differential equations

In this paper we consider the third-order nonlinear delay differential equation (*) $$ ( a(t)\left ( x''(t)\right ) ^{\gamma })' +q(t)x^{\gamma }(\tau (t))=0,\quad t\geq t_0, $$ where $a(t)$, $q(t)$ are positive functions, $\gamma >0$ is a quotient of odd positive integers and the delay function $\tau (t)\leq t$ satisfies $\lim _{t\rightarrow infty }\tau (t)=infty $. We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria. Some examples are considered to illustrate the main results.

Uniform Ultimate Boundedness of Solutions of Third-Order Nonlinear Delay Differential Equations

In this paper, we study the uniform boundedness and uniform ultimate boundedness of solutions of the third-order delay differential equation (*) ::: x +φ(x, ˙ x)¨ x + g( ˙ x(t r(t))) + f (x(t r(t))) = p(t, x, ˙ x, ¨), where 0 r(t) γ, γ is a positive constant, φ(x, ˙ x), g( ˙ x), f (x) and p(t, x, ˙ x, ¨) are contin- uous functions. We obtain some sufficient conditions which ensure that all the solutions of Eq.(*) are uniformly bounded and uniformly ultimately bounded. Our result revise and improve some earlier results. Mathematics Subject Classication 2000: 34K20.

Bound of solutions to third-order nonlinear differential equations with bounded delay

Abstract A theorem is presented that contains some sufficient conditions to ensure bound of solutions to a third-order nonlinear delay differential equation. It is shown that this theorem improves the result of Ponzo [On the stability of certain nonlinear differential equations, IEEE Trans. Automatic Control AC-10 (1965) 470–472] to the bound of solutions for the equation considered.