Nonlinear difference equations with asymptotically stable solutions

We establish conditions of asymptotic stability for all solutions of the equation X _n+1=F(X _n ), n≥0, in the Banach space E in the case where r(F′(x))<1 ∀ x ∈ E, r′(x) is the spectral radius of F′(x). An example of an equation with an unstable solution is given. Ukrainian Academy of the Water Industry, Rovno. Translated from Ukrainskii Matematicheskii zhurnal, Vol. 49, No. 7, pp. 970–980. July, 1997.     

Asymptotic equivalence of differential equations and asymptotically almost periodic solutions

In this paper we use Rab’s lemma [M. Ráb, Über lineare perturbationen eines systems von linearen differentialgleichungen, Czechoslovak Math. J. 83 (1958) 222–229; M. Ráb, Note sur les formules asymptotiques pour les solutions d’un systéme d’équations différentielles linéaires, Czechoslovak Math. J. 91 (1966) 127–129] to obtain new sufficient conditions for the asymptotic equivalence of linear and quasilinear systems of ordinary differential equations. Yakubovich’s result [V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1966; V.A. Yakubovich, On the asymptotic behavior of systems of differential equations, Mat. Sb. 28 (1951) 217–240] on the asymptotic equivalence of a linear and a quasilinear system is developed. On the basis of the equivalence, the existence of asymptotically almost periodic solutions of the systems is investigated. The definitions of biasymptotic equivalence for the equations and biasymptotically almost periodic solutions are introduced. Theorems on the sufficient conditions for the systems to be biasymptotically equivalent and for the existence of biasymptotically almost periodic solutions are obtained. Appropriate examples are constructed.     

Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations

Using the final value theorem of Laplace transform, it is firstly shown that nonhomogeneous fractional Cauchy problem does not have nonzero periodic solution. Secondly, two basic existence and uniqueness results for asymptotically periodic solution of semilinear fractional Cauchy problem in an asymptotically periodic functions space. Furthermore, existence and uniqueness results are extended to a closed, nonempty and convex set which is a subset of a Fréchet space. Some examples are given to illustrate the results.     

On the existence of asymptotically stable solutions of certain integral equations

In this paper two existence results of asymptotically stable solutions of certain integral equations are presented.     

Pseudo asymptotically periodic solutions for Volterra integro‐differential equations

In this paper, we propose a new class of functions called pseudo ‐asymptotically ω‐periodic function in the Stepanov sense and explore its properties in Banach spaces including composition results. Furthermore, the existence and uniqueness of the pseudo ‐asymptotically ω‐periodic mild solutions to Volterra integro‐differential equations is investigated. Applications to integral equations arising in the study of heat conduction in materials with memory are shown. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence results of periodic solutions for non-autonomous differential delay equations with asymptotically linear properties

In this paper, we study a class of non-autonomous differential delay equations which can be changed to Hamiltonian systems. By estimating Maslov-type index of the related Hamiltonian systems at infinity and at origin, we establish the existence of periodic solutions of the differential delay equations.     

Linear integral equations with asymptotically almost periodic solutions

Conditions are given that ensure that bounded solutions of $\text{∫}{}_{\mathrm{?\xi }}{}^{\mathrm{\xi }}\text{x(t ? ξ)dA(ξ) = ?(t)}$ are asymptotically almost periodic. The result strengthens and extends a recent theorem of Levin and Shea. Generalizations to systems of integral equations as well as to integrodifferential systems are included.     

Existence of asymptotically almost periodic solutions of third-order neutral differential equations with piecewise constant argument

通过构造差分方程的渐近概周期序列解,研究了三阶中立型逐段常变量微分方程渐近概周期解的存在性.     

Nonlinear fourth-order differential equations with solutions in the form of transcendents

We present two hierarchies of ordinary differential equations and give the relations between these hierarchies. We find rational and special solutions of one hierarchy. Solutions of two differential equations are shown to be essentially transcendental functions with respect to the integration constants. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 72–87, January, 1999.     

Krylov-Bogolyubov averaging of asymptotically autonomous differential equations

We apply the Krylov and Bogolyubov asymptotic integration procedure to asymptotically autonomous systems. First, we consider linear systems with quasi-periodic coefficient matrix multiplied by a scalar factor vanishing at infinity. Next, we study the asymptotically autonomous Van-der-Pol oscillator.

     

Multiple periodic solutions for asymptotically linear Duffing equations with resonance

We investigate multiple periodic solutions of asymptotically linear Duffing equation with resonance using index theory and Morse theory and obtain a new result.     

Existence of almost periodic solutions for strong stable impulsive differential equations

Conditions for strong stability and the existence of almostperiodic solutions of systems of impulsive differential equationswith impulsive effect at fixed moments are obtained. The investigationsare carried out by means of piecewise continuous functions whichare analogues of Lyapunov functions.     

Abstract differential equations with almost-periodic solutions

In this paper we present some quite simple results concerning almost-periodic solutions of abstract differential equations. We start with a general proposition about linear equations, then we establish some new facts about bounded or relatively compact solutions which become almost-periodic; finally, we study “separation from 0 properties” of non-trivial almost-periodic solutions for equations with bounded or unbounded operator coefficients.