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.     

Asymptotic behavior of solutions of difference equations in Banach spaces

In this paper we consider the first order difference equation

and give necessary and sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given earlier by Popenda and Schmeidel. As an application we give necessary and sufficient conditions for the second order difference equation

to have asymptotically constant solutions.

     

Regularity of solutions of evolution integrodifferential equations with deviating argument

In this paper we prove the existence of mild and classical solutions of delay integrodifferential equation and delay integrodifferential evolution equations with nonlocal condition in Banach spaces. The regularity solutions of integrodifferential evolution equations interconnected with viscoelastic material is derived to guarantee the stabilization. The results are established by using the resolvent operators and the fixed point principles. Finally, an example is given to show the potential of the proposed techniques.     

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 first order nonlinear differential equations with a deviating argument

The following first order nonlinear differential equation with a deviating argument $ x'(t) + p(t)[x(\tau (t))]^\alpha = 0 $ is considered, where α > 0, α ≠ 1, p ∈ C[t ₀; ∞), p(t) > 0 for t ≧ t ₀, τ ∈ C[t ₀; ∞), lim _t→∞ τ(t) = ∞, τ(t) < t for t ≧ t ₀. Every eventually positive solution x(t) satisfying lim _t→∞ x(t) ≧ 0. The structure of solutions x(t) satisfying lim _t→∞ x(t) > 0 is well known. In this paper we study the existence, nonexistence and asymptotic behavior of eventually positive solutions x(t) satisfying lim _t→∞ x(t) = 0.     

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).     

Asymptotic equivalence of nonlinear evolution equations in Banach spaces

We show how the approach of Yosida Approximation of the derivative serves to obtain new results for evolution systems. We give criteria for the asymptotic equivalence of two different evolution systems, i.e., $$\lim_{t \to \infty} \|U_A(t, s)x - U_B(t, s)x\| =0,$$ where the evolution systems are generated by two different families of nonlinear and multivalued time-dependent operators A(t), and B(t).     

Banach空间上泛函微分方程的渐近性

§ 1　IntroductionIn the last few years,invariant sets and attractors of (functional) differentialequations have been extensively discussed and various interesting results on the invariantsets and attractors,and estimates on the basin of attraction have been reported(see,forinstance,[1 ,2 ,4,6,1 0 ,1 3 ,1 5,1 8] ) .However,not much hasbeen developed in the directionof giving criteria on the existence of invariant sets and attractors for the functionaldifferential equations even though there ar…     

Asymptotic periodicity for some evolution equations in Banach spaces

This work deals with the existence and uniqueness of pseudo-almost periodic and asymptotically ω-periodic mild solutions to some evolution equations in Banach spaces.     

On solutions of a neutral differential equation with deviating argument

We prove a theorem on the existence and asymptotic behaviour of solutions of a differential equation with a deviating argument of neutral type. The considered equation contains both delayed and advanced arguments. The method used in the proof of our main result depends on conjunction of the classical Schauder fixed point theorem with the technique of measures of noncompactness.