Periodic solutions of systems of functional differential equations with linear deviations of argument and their properties

We establish sufficient conditions for the existence of periodic solutions of systems of linear and nonlinear functional differential equations with linear deviations of the argument and investigate their properties.     

New solvability conditions for a nonlocal boundary-value problem for nonlinear functional differential equations

New efficient sufficient conditions are obtained for the solvability and the unique solvability of a nonlocal boundary-value problem for nonlinear functional differential equations. Published in Neliniini Kolyvannya, Vol. 11, No. 3, pp. 365–387, July–September, 2008.     

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.     

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 the asymptotic integration of a singularly perturbed system of linear differential equations with deviating argument

We prove the existence and infinite differentiability of a solution of a singularly perturbed linear system of differential equations with deviating arguments and a turning point.     

Geometric calculus for second-order differential equations and generalizations of the inverse problem of Lagrangian mechanics

We review the main features of the geometric calculus which has been introduced over the past 15 years in the study of second-order ordinary differential equations and then explain how a recently introduced generalization of the inverse problem of Lagrangian mechanics can be very concisely dealt with by this calculus in an intrinsic way. This paper is an account of the lecture with the same title presented at the ICDVC-2010 Conference in Hangzhou, May 12–14 (2010).     

On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operators

We establish new efficient conditions sufficient for the unique solvability of the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operators. Published in Neliniini Kolyvannya, Vol. 10, No. 4, pp. 560–573, October–December, 2007.     

Multiple scale perturbation for second-order non-linear differential equations

Asymptotic solutions of a class of second-order non-linear differential equations with variable coefficients are studied. For large values of the parameter, the differential equations are of the singular-perturbation type and approximations are constructed by the generalized method of multiple scales.     

Abstract Cauchy problem for fractional differential equations

In this paper, a generalized Darbo’s fixed-point theorem associated with Hausdorff measure of noncompactness is established. Then we apply this new variant fixed-point theorem to study some fractional differential equations in Banach spaces via the technique of measure of noncompactness. Many novel existence and uniqueness results for solutions are obtained under the more general conditions.     

A high accuracy difference scheme for the singular perturbation problem of the second-order linear ordinary differential equation in conservation form

In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h³.