Oscillation of Nonlinear Hyperbolic Differential Equations with Impulses

We study oscillatory properties of solutions of nonlinear impulsive hyperbolic differential equations and find new necessary and sufficient conditions for the existence of oscillations.__________Published in Neliniini Kolyvannya, Vol. 7, No. 4, pp. 439–445, October–December, 2004.     

On Some Nonlinear Fractional Equations Involving the Bessel Operator

Under different assumptions on the potential functions b and c, we study the fractional equation \(\left( I-\varDelta \right) ^{\alpha } u = \lambda b(x) |u|^{p-2}u+c(x)|u|^{q-2}u\) in \(\mathbb {R}^N\). Our existence results are based on compact embedding properties for weighted spaces.     

Iteration Method for Systems of Nonlinear Differential Equations with Delay and Restrictions

We establish a consistency condition for systems of nonlinear differential equations with delay and restrictions and justify the applicability of the iteration method to these problems.     

Solution of Ordinary Differential Equations with a Large Lie Symmetry Group

The group method of solving ODEs without any quadrature goes back to Lie. In order to apply it, the number of symmetries admitted by a given ODE has to be greater by one than the number of arbitraryconstants in the general solution of the equation. In this paper, we use thetechnique of canonical Lie–Bäcklund symmetries that makes theproof of the statement concerning integrals of ODEs more evident. The methodis extended to the solution of system of ODEs with a small parameter of arbitrary order.     

Asymptotic Solution of Some Systems of Nonlinear Differential-Difference Equations

Asymptotic methods developed by Feshchenko, Shkil', and their disciples for the integration of differential equations with slowly varying coefficients are used for the solution of certain nonlinear differential-difference equations.     

Generalized Euler Method for Nonlinear First-Order Partial Differential Equations

Classical solutions of nonlinear first-order partial differential equations are approximated by solutions of quasilinear systems of difference equations. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on the comparison method. Nonlinear estimates of the Perron type are assumed for increment functions.This new approach to the numerical solution of nonlinear equations is based on the method of quasilinearization for mixed problems. Numerical examples are given.     

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.     

On Periodic Solutions of a System of Nonlinear Functional Partial Differential Equations

We obtain sufficient conditions for the existence of periodic solutions of a system of nonlinear functional partial differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 154–158, April–June, 2005.     

Solutions of Weakly Nonlinear Systems of Ordinary Differential Equations Bounded on R

We obtain conditions for the existence of solutions bounded on the entire axis R for weakly nonlinear systems of ordinary differential equations in the case where the corresponding unperturbed homogeneous linear differential system is exponentially dichotomous on the semiaxes R ₊ and R _–.     

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ (ＢＶＰ)ｕ″ ｇ(ｔ)ｆ(ｕ) =0 ,　　 0 <ｔ<1 ,αｕ(0 ) -βｕ′(0 ) =0 ,　　γｕ(1 ) δｕ′(1 ) =0 ,(1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,ρ:=βγ αγ αδ>0 ,ｆ∈Ｃ([0 ,∞ ) ,[0 ,∞ ) ) ,ｇｍａｙｂｅｓｉｎｇｕｌａｒａｔｔ=0ａｎｄ/ｏｒｔ=1 .Ｔｈｉｓｐｒｏｂｌｅｍａｒｉｓｅｓｎａｔｕｒａｌｌｙｉｎｔｈｅｓｔｕｄｙｏｆｒａｄｉａｌｌｙｓｙｍｍｅｔ…     

On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

Lyapunov Stability and Adaptive Regulation of a Class of Nonlinear Nonautonomous Second-Order Differential Equations

This paper presents an adaptive regulation scheme for a class ofordinary nonlinear nonautonomous second-order differential equationswhich includes as particular cases a number of particular differentialequations which occur in applications. The unforced reference model isproposed to be a stable differential parametrization within the generalclass dealt with. Therefore, some sufficient Lyapunov's stabilityconditions for such a class are previously investigated which can beused, in particular, to set an appropriate reference model. Theresulting closed-loop adaptive scheme is proved to be stable and itinvolves a parameter estimation scheme of least-squares type which isproved to possess all suitable properties in terms of estimatesboundedness and asymptotic convergence of the estimates to finite limitsas well as time-integrability of the squared adaptation error.     

Stability and Boundedness Results for Solutions of Certain Third Order Nonlinear Vector Differential Equations

In this paper, we investigate the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. The results are proved using Lyapunov’s second (or direct method). Our results include and improve some well known results existing in the literature.     

Asymptotic Integration of Weakly Nonlinear Systems with Unstable Spectrum

We construct an asymptotic solution of the Cauchy problem for a weakly nonlinear system of ordinary differential equations with turning point. The linear part of this system contains an almost diagonal matrix.     

On the Asymptotic Behavior of Solutions of a System of Nonlinear Differential Equations

We investigate the asymptotic behavior of a system of nonlinear differential equations of a special form at infinity. We also propose a method for the reduction of more general systems of nonlinear differential equations to this form, which enables one to study their asymptotic properties.     

Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perrons method. Then, we prove the smoothness of these invariant manifolds.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Construction of a General Solution of a Degenerate Linear System of Differential Equations with Irregular Singular Point

We investigate the asymptotic behavior of the general solution of the linear system of differential equations with irregular singular point

Oscillations in Differential Equations with State-Dependent Delays

We study the existence of oscillating solutions of delay differential equations with delay depending directly on a state. Necessary and sufficient conditions for oscillations are established.     

Projection-Iterative Method for Differential Equations with Restrictions

We propose and justify a new projection-iterative method for differential equations with restrictions.     

Uniform Convergence Series to Solve Nonlinear Partial Differential Equations: Application to Beam Dynamics

Extended trigonometric series of uniform convergence are proposed as a method to solve the nonlinear dynamic problemsgoverned by partial differential equations. In particular, the method isapplied to the solution of a uniform beam supported at its ends withnonlinear rotational springs and subjected to dynamic loads. The beam isassumed to be both material and geometrically linear and the end springs are of the Duffing type. The action may be a continuous load q = q(x, t) within a certain range and/or concentrated dynamic moments at the boundaries. The adopted solution satisfies the differential equation, the initial conditions, andthe nonlinear boundary conditions. It has been previously demonstrated that, due to the uniform convergence of the series, the method yieldsarbitrary precision results. An illustration example shows theefficiency of the method.