On linear homogeneous almost periodic systems that satisfy the Favard condition

We prove the existence of a linear homogeneous almost periodic system of differential equations that has nontrivial bounded solutions and is such that all systems from a certain neighborhood of it have no nontrivial almost periodic solutions. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3. pp. 409–413, March, 1998.     

A stability criterion for periodic solutions of two-dimensional discontinuous dynamical systems

We establish stability conditions for periodic solutions of two-dimensional systems of ordinary differential equations with pulse influence. We study the properties of the jump operator for such systems. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1262–1266, September, 1999.     

Periodic solutions of a weakly nonlinear system of partial differential equations with pulse influence

We establish conditions of the existence of solutions periodic in t with period T for a weakly nonlinear system of partial differential equations with pulse influence. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 601–605, April, 1997.     

The correlation matrix of random solutions of a dynamical system with Markov coefficients

For dynamical systems which are described by systems of differential or difference equations dependent on a finite-valued Markov process, we suggest a new form of equations for moments of their random solution. We derive equations for a correlation matrix of random solutions. Kiev Economic Institute, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 338–348, March, 1999.     

On the stability of a trivial invariant torus of one class of impulsive systems

We consider the problem of asymptotic stability of the trivial invariant torus of one class of impulsive systems. Sufficient criteria of asymptotic stability are obtained by the method of freezing in one case, and by the direct Lyapunov method for the investigation of stability of solutions of impulsive systems in another case. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 338–349, March, 1998.     

A criterion for the solvability of A linear boundary-value problem for A system of the second order

We obtain necessary and sufficient conditions for the solvability of a two-point boundary-value problem for systems of linear differential equations of the second order in the critical case where the corresponding homogeneous boundary-value problem has nontrivial solutions. We construct the general solution of the considered boundary-value problem. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 861–864, June, 2000.     

On Periodic Solutions of One Class of Systems of Differential Equations

We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the existence of periodic solutions of two-dimensional nonlinear differential systems with Hamiltonian linear part and their construction. The results obtained are generalized to systems of higher orders. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 483–495, April, 2005.     

Periodic Solutions of Classical Hamiltonian Systems without Palais-Smale Condition

In this paper, we study the existence of periodic solutions for classical Hamiltonian systems without the Palais-Smale condition. We prove that the information of the potential function contained in a finite domain is sufficient for the existence of periodic solutions. Moreover, we establish the existence of infinitely many periodic solutions without any symmetric condition on the potential function V.     

Asymptotic Stability of Large-amplitude Oscillations in Systems with Hysteresis

We study depending on a parameter periodic systems with the main linear part and a hysteresis nonlinearity; the linearized at infinity system has a one-dimensional subspace of periodic solutions for the critical parameter value. We prove theorems on a number, localization, and asymptotic stability of large-amplitude periodic solutions for the nonlinear system. Received March 1998     

Periodic solutions for Hamiltonian systems under strong constraining forces

We consider periodic solutions of Hamiltonian systems in Euclidean spaces whose motion is constrained to a submanifold M. We prove that under some nondegeneracy assumptions, periodic solutions persist when the constraint is replaced by a strong restoring potential.     

Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

We use the bifurcation method of dynamical systems to study the (2+1)‐dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow‐up wave solutions, periodic smooth wave solutions, periodic blow‐up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow‐up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

一类时滞反应扩散方程的周期解和概周期解

通过构造上、下控制函数，结合上、下解方法及相应的单调迭代方法研究了一类时滞反应扩散方程，证明了在反应项非单调时，如果一雏边值问题存在一对周期（或概周期）上、下解，则方程一定存在唯一的周期（或概周期）解．并给出了二维边值问题周期（或概周期）解存在唯一性的充分条件．推广了已有的一些结果。     

On axiomatizations of Boolean algebras

We construct some new axiomatic systems for the Boolean algebra. In particular, an axiomatic system for disjunction and logical negation consists of three axioms. We prove the independence of the axiomatic systems proposed. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 937–942, July, 1997.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Periodic solutions of abstract functional differential equations with state‐dependent delay

In this paper, we are concerned with the existence of solutions of systems determined by abstract functional differential equations with infinite and state‐dependent delay. We establish the existence of mild solutions and the existence of periodic solutions. Our results are based on local Lipschitz conditions of the involved functions. We apply our results to study the existence of periodic solutions of a partial differential equation with infinite and state‐dependent delay. Copyright © 2016 John Wiley & Sons, Ltd.     

On the existence of periodic solutions for large-scale systems

In recent ten years or more, many scholars have engaged in the investigation concerning the stability of large-scale systems, but up to the present, the problem on the existence of periodic solutions for large-scale systems has yet been seldomly touched upon in the literature.In this paper, by means of the method of constructing Lyapunov function. We study the problem on the existence of periodic solutions for linear and nonlinear large-scale systems, and obtain several sufficient conditions which guarantee the existence of periodic solutions.     

Random periodic solutions of random dynamical systems

In this paper, we give the definition of the random periodic solutions of random dynamical systems. We prove the existence of such periodic solutions for a C¹ perfect cocycle on a cylinder using a random invariant set, the Lyapunov exponents and the pullback of the cocycle.     

Periodic solutions of a periodically forced and undamped beam resting on weakly elastic bearings

We study the problem of existence of periodic solutions to a partial differential equation modelling the behavior of an undamped beam subject to an external periodic force. We assume that the ordinary differential equation associated to the first two modes of vibration of the beam has a symmetric homoclinic solution. By using methods borrowed by dynamical systems theory we prove that, if the period is non resonant with the (infinitely many) internal periods of the PDE, the equation has a weak periodic solution of the same period as the external force. In particular we obtain continua of periodic solutions for the undamped beam in absence of external forces. This result may be considered as an infinite dimensional analogue of a result obtained in [16] concerning accumulation of periodic solutions to homoclinic orbits in finite dimensional reversible systems. Matteo Franca: Partially supported by G.N.A.M.P.A. – INdAM (Italy).     

Solution spaces of homogeneous linear difference systems with coefficient matrices from commutative groups

We analyse the solution spaces of limit periodic homogeneous linear difference systems, where the coefficient matrices of the considered systems are taken from a commutative group which does not need to be bounded. In particular, we study such systems whose fundamental matrices are not asymptotically almost periodic or which have solutions vanishing at infinity. We identify a simple condition on the matrix group which guarantees that the studied systems form a dense subset in the space of all considered systems. The obtained results improve previously known theorems about non-almost periodic and non-asymptotically almost periodic solutions. Note that the elements of the coefficient matrices are taken from an infinite field with an absolute value and that the corresponding almost periodic case is treated as well.     

Semi-Fredholm problems on periodic solutions of functional-differential equations of neutral type

We consider the problem on periodic solutions for linear systems of functionaldifferential equations of neutral type with periodic coefficients and periodic deviations of the argument. By reduction to associated functional equations, we derive necessary and sufficient conditions under which the problem on periodic solutions for such a system is Fredholm or semi-Fredholm.