Asymptotics of solutions of nonautonomous second-order ordinary differential equations close to linear equations

We find asymptotic representations for certain classes of solutions of nonautonomous second-order differential equations that are close, in a certain sense, to linear equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 230–241, April–June, 2008.     

Asymptotic representations of solutions of nonautonomous second-order differential equations with nonlinearities close to power-type nonlinearities

Asymptotic representations are found for solutions of nonautonomous second-order differential equations with nonlinearities close to power-type nonlinearities. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 3–15, January–March, 2009.     

Regularity and asymptotic behavior of solutions of nonautonomous differential equations

The existence of a nonautonomous approximate inertial manifold is shown for problems of the formu + Au + N(t,u)=0, in whichA is a self-adjoint operator with compact resolvent in a Hilbert spaceH. The operatorN(t, u) = G(u) + F(t, u) is nonlinear withG a monotone gradient that is locally Lipschitz fromD(A ^1/2) intoH, andF:⁺×HH a Lipschitz perturbation that is Hölder continuous int. Weak solutions are shown to be uniformly locally Hölder continuous intoD(A) with equicontinuity in families of solutions with ¦u(0)¦ r.A priori estimates of ¦Au(t)¦ are also verified and used in a skew-product flow to show there is a global attractor whose component elements form a equicontinuous family of solutions.     

The boundedness and asymptotic behavior of solutions of differential system of second-order with variable coefficients

In this paper, the differential system of second-order withvariable coefficients is studied. and some criteria of theboundedness and asymptotic behavior for solutions are given.Consider a system of differential equationsdx_1/dt=p_(11)(t)x_1 p_(12)(t)x_2dx_2/dt=p_(21)(t)x_1 p_(22)(t)x_2Now we studg the boundedness and asymptotic behavior of its so-lutions. In the case of Pij(t)being periodic functions. it wasinvestigated by Burdina; in the case of Pij(t) being arbitraryfunctions. it has not been investigated yet. Besides. the me-thod used by Burdina is only oppropriate for the former but notfor the latter case. In this paper we shall give a method whichis appropriate for both cases.     

ON THE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF CERTAIN FIFTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

The sufficient conditions are given for all solutions of certain non- autonomous differential equation to be uniformly bounded and convergence to zero as t →∞ ?. The result given includes and improves that result obtained by Abou-El-Ela & Sadek .     

On asymptotic properties of solutions of functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation x′(t) = ax(t) + bx(t − r) + cx′(t − r) + px(qt) + hx′(qt) + f ₁(x(t), x(t − r), x′(t − r), x(qt), x′(qt)) in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 1, pp. 144–160, January–March, 2007.     

On the asymptotic properties of solutions of linear functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation {fx153-01} in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 147–150, April–June, 2008.     

On asymptotic properties of solutions of linear differential functional equations with linearly transformed argument

We establish new properties of C ¹[−1, +∞)-solutions of the linear functional differential equation ẋ(t) = ax(t) + bx(qt) + hx(t−1) + cẋ(qt) + rẋ(t−1) in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 170–177, April–June, 2006.     

ON PERIODIC SOLUTIONS OF SEVERAL CLASSES 0F RICCATI'S EQAUTION AND SECOND一ORDER DIFFERENTIAL EQUATIONS

In this paper, the problem on periodic solutions of several classes of Riccati's equation with periodic coefficients is discussed, and the conditions, under which several classes of secondorder equations with periodic coefficients have periodic solutions, are given.     

On the asymptotic behavior of solutions of linear second-order boundary-value problems on a semi-infinite strip

This paper treats the asymptotic behavior of solutions of a linear secondorder elliptic partial differential equation defined on a two-dimensional semiinfinite strip. The equation has divergence form and variable coefficients. Such equations arise in the theory of steady-state heat conduction for inhomogeneous anisotropic materials, as well as in the theory of anti-plane shear deformations for a linearized inhomogeneous anisotropic elastic solid. Solutions of such equations that vanish on the long sides of the strip are shown to satisfy a theorem of Phragmén-Lindelöf type, providing estimates for the rate of growth or decay which are optimal for the case of constant coefficients. The results are illustrated by several examples. The estimates obtained in this paper can be used to assess the influence of inhomogeneity and anisotropy on the decay of end effects arising in connection with Saint-Venant's principle.     

On the asymptotic solutions for a class of second order differential equations with slowly varying coefficients

In this paper we study the asymptotic expansions of the solutions for a class of secondorder ordinary differential equations with slowly varying coefficients.The defect of theknown works on these problems is noted,and the results in[1—4]are improved andextended by means of the modified method of multiple scales.     

On local asymptotic expansions of solutions of singularly perturbed systems of differential equations with degeneration at a point

We propose an algorithm for the construction of asymptotic solutions of singularly perturbed systems of differential equations in the case where the characteristic equation has simple roots. In contrast to previous investigations, in which the matrix multiplying the derivatives becomes degenerate on the entire interval, we study the case where degeneration occurs at a single point.     

The exponential asymptotic solution of differential equation

In this paper, the exponential asymptotic solution (E.A.S.) of differential equation is discussed. Firstly, E.A.S. of the second-order differential equation is studied and the orthogonal conditions of the uniformly valid E.A.S. are found out. Next, E.A.S. in matched asymptotic method is discussed. Finally, some examples are given.     

Asymptotic representations of solutions of one class of second-order ordinary differential equations

We establish asymptotic representations for some classes of solutions of nonautonomous second-order differential equations close, in a certain sense, to linear equations.     

On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations

We consider nonlinear wave and Klein-Gordon equations with general nonlinear terms, localized in space. Conditions are found which provide asymptotic stability of stationary solutions in local energy norms. These conditions are formulated in terms of spectral properties of the Schrödinger operator corresponding to the linearized problem. They are natural extensions to partial differential equations of the known Lyapunov condition. For the nonlinear wave equation in three-dimensional space we find asymptotic expansions, as t, of the solutions which are close enough to a stationary asymptotically stable solution.     

On the construction of asymptotic solutions of two-point boundary-value problems for degenerate singularly perturbed systems of differential equations

We consider the problem of the existence of a solution of a two-point boundary-value problem for degenerate singularly perturbed linear systems of differential equations. We obtain asymptotic formulas for this solution.     

On global solutions of functional differential equations

We construct a system of linear differential equations all solutions of which are global solutions of a system of functional differential equations. We substantiate the existence of this system and examine some properties of its solutions.