Periodic solutions for higher order differential equations with deviating argument

By using the coincidence degree theory of Mawhin, we study the existence of periodic solutions for higher order differential equations with deviating argument . Some new results on the existence of periodic solutions of the equations are obtained. Meanwhile, an example is given to illustrate our results.     

Eventually positive solutions of odd order neutral differential equations

The odd order neutral differential equations are considered. New necessary and sufficient conditions, and comparison theorems of existence of eventually positive solutions are obtained. Nonexistence criteria of eventually positive solutions are also established.     

Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument

The aim of this paper is to establish comparison principles on property A$A$, between a nonlinear differential equation of the third order with deviating argument (with delay, advanced or mixed argument) and the corresponding linear equation without deviating argument. On the basis of these comparison principles the sufficient conditions for delay, advanced and mixed equations to have property A$A$ are presented. The results obtained are compared with existing ones in the framework of the papers.     

Oscillation of second-order nonlinear noncanonical differential equations with deviating argument

The purpose of this paper is to study the second-order nonlinear noncanonical differential equation ${\left(r\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }+p\left(t\right)f\left(y\left(\tau \left(t\right)\right)\right)=0$under the condition ${\int }_{}^{\infty }\frac{1}{r\left(t\right)}\mathrm{d}t<\infty$. Contrary to most existing results, oscillation of the studied equation is attained via only one condition. We consider both delay and advanced differential equations. A particular example of Euler type equation is provided in order to illustrate the significance of our main results.     

Periodic solutions of a nonlinear second-order differential equation with deviating argument

With the help of the coincidence degree continuation theorem, the existence of periodic solutions of a nonlinear second-order differential equation with deviating argument

x^″(t)+f₁(x(t))x^′(t)+f₂(x(t))(x^′(t)⁾²+g(x(t−τ(t)))=0,     

On the oscillation of solutions of first order differential equations with deviating arguments

1.IntroductionConsiderthefirstorderdifferentialequationwithdeviatingargUmelltTheoscillationofEq.(1)wasstudiedextensivelyinthelastthreedecades.See,forexamDleif--101andthereferencescitedtherein.In1972Ladas.LakshlnhanthamanddeceivedApril1,1997.Re~AugUBt...     

On the solvability of nonlinear partial differential equations of a high order with deviating argument in lowest terms

We prove the solvability of a high order partial differential equation with discrete deviation of the argument in lowest terms. The proof is carried out by the method of separation of variables. We also study the boundary-value problemfor a particular case which corresponds to a fourth-order nonlinear equation.     

On proper oscillatory and vanishing-at-infinity solutions of differential equations with a deviating argument

Sufficient conditions are found for the existence of multiparametric families of proper oscillatory and vanishing-at-infinity solutions of the differential equation $$u^{(n)} (t) = g\left( {t, u(\tau _0 (t)), \ldots ,u^{(m - 1)} (\tau _{m - 1} (t))} \right)$$ , wheren≥4,m is the integer part of π/2,g:R ₊×R ^m →R is a function satisfying the local Carathéodory conditions, and τ _i :R ₊→R(i=0,...,m?1) are measurable functions such that τ _i (t) →+∞ fort→+∞(i=0,...,m?1).     

Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument

By means of Mawhin's continuation theorem, we study some second order differential equations with a deviating argument:

x^″(t)=f(t,x(t),x(t−τ(t)),x^′(t))+e(t).     

Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions

This paper is concerned with the existence of solutions for systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions. By establishing new comparison results and applying the monotone iterative technique, we obtain the sufficient conditions for the existence of extremal system of solutions.