Continuous solutions of nonlinear functional difference equations and their properties

We establish conditions for the existence of continuous bounded solutions of systems of nonlinear functional difference equations and study their properties.     

Global solutions of systems of nonlinear difference equations and their properties

We establish sufficient conditions for the existence and uniqueness of a global solution of one class of systems of nonlinear difference equations and investigate its properties. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 291–297, July–September, 2007.     

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.     

Existence of positive solutions of systems of Volterra nonlinear difference equations

In this paper, a classification scheme for eventually positive solutions of a class of two-dimensional Volterra nonlinear difference equations is given in terms of asymptotic magnitudes. Some necessary as well as sufficient conditions for the existence of such solutions are provided without any monotonicity conditions on the nonlinear term. Published in Neliniini Kolyvannya, Vol. 9, No. 1, pp. 37–47, January–March, 2006.     

On the structure of the set of continuous solutions of functional difference equations with linearly transformed argument

We investigate the structure of sets of continuous solutions of linear functional difference equations with linearly transformed argument.     

Oscillation of a class of nonlinear partial difference equations with continuous variables

We consider a class of nonlinear partial difference equations with continuous variables. Some oscillation criteria are obtained using an integral transformation and inequalities.     

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 stability and boundedness of solutions to third order nonlinear differential equations with retarded argument

This paper investigates stability and boundedness of solutions to third order nonlinear differential equation with retarded argument: $$\begin{array}{l}x'''(t)+\varphi\bigl(x(t-r),x'(t-r),x''(t-r)\bigr)x''(t)\\\qquad{}+\psi\bigl(x'(t-r)\bigr)+h\bigl(x(t-r)\bigr)\\\quad=p\bigl(t,x(t),x(t-r),x'(t),x'(t-r),x''(t)\bigr).\end{array}$$ By the use of the Lyapunov functional, sufficient conditions for stability and boundedness of solutions to the considered equations are obtained. Examples are introduced throughout the paper for illustrations.     

Asymptotic properties of solutions of systems of nonlinear functional differential equations of neutral type

We establish sufficient conditions for systems of nonlinear functional differential equations of neutral type to have solutions that are continuously differentiable and bounded for t ∈ ℝ (together with their first derivatives) and investigate the asymptotic properties of these solutions. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 20–26, January–March, 2009.     

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.     

Investigation of the structure of the set of continuous solutions of systems of linear functional difference equations

We establish conditions for the existence of continuous solutions of systems of linear functional difference equations with linearly transformed argument and develop a method for the construction of these solutions.     

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.     

Solving systems of nonlinear difference equations by the multiple scales perturbation method

In this paper, we apply an improved version of the multiple scales perturbation method to a system of weakly nonlinear, regularly perturbed ordinary difference equations. Such systems arise as a result of the discretization of a system of nonlinear differential equations, or as a result in the stability analysis of nonlinear oscillations. In our procedure, asymptotic approximations of the solutions of the difference equations will be constructed which are valid on long iteration scales.     

Systems of nonlinear differential equations with bounded solutions

We establish sufficient conditions for the existence of solutions bounded on ℝ for the equation {fx168-01}, in a finite-dimensional Banach space {ie168-01}. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 160–167, April–June, 2008.     

Self-similar solutions of some nonlinear equations

The method of group invariance under an infinitesimal transformation is applied to a class of nonlinear partial differential equations. Besides yielding a large number of known forms of self-similar solutions, the method also gives new types of solutions.