Periodic solutions of weakly nonlinear systems with impulse interaction

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 622–626, May, 1989.     

Periodic solutions of nonlinear integro-differential equations with an impulse effect

The paper applies a numerical-analytical method for finding periodic solutions of the system of integro-differential equations $$\begin{gathered} \dot x = f(t,x,\mathop \smallint \limits_0^t \varphi (t,s,x(s))ds), t \ne t_i (x), \hfill \\ \Delta x|_{t = t_i (x)} = I_i (x). \hfill \\ \end{gathered} $$ Two theorems for existence of periodic solutions are proved for the cases whent = t _i andt = t _i(x).     

Periodic solutions of singularly-perturbed systems of differential equations with impulse effect

Summary The present paper considers the problem for existence, uniqueness and asymptotic representation of periodic solutions of some singularly-perturbed systems of differential equations with impulses. The study is carried out with the help of a modification of the boundary functions method, adequately worked out to match the specific issues of the considered problem.

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Zusammenfassung Wir betrachten das Problem der Existenz, Eindeutigkeit und der asymptotischen Darstellung periodischer Lösungen singulär gestörter Systeme von Differentialgleichungen mit Impulsen. Die Studie wird mittels speziell dem Problem angepaßter Modifikation der Randwertfunktionsmethode ausgeführt.

     

Existence of periodic solutions of nonlinear systems of differential equations with impulse effect

Sufficient conditions for the existence of periodic solutions of nonlinear systems of differential equations with impulses in a canonic domain have been found in the paper.     

Periodic solutions of nonlinear equations generalizing logistic equations with delay

Mathematical Notes -     

Periodic solutions of nonlinear wave equations with damping

A known existence theorem for doubly periodic solutions of nonlinear wave equations with linear damping is being proved in a direct manner by an approach which has been developed by the authors in [5, 6, 7] for hyperbolic problems, when the kernel of the underlying linear operator is infinite dimensional.     

Periodic solutions of nonlinear integral equations

The existence of a continuous periodic solution of the system

Periodic solutions of nonlinear delay equations

Existence and uniqueness of 2π-periodic solutions of $\text{d}{}^{\mathrm{j}}\text{x(t)}\text{dt}{}^{\mathrm{j}}\text{+}\text{grad}\text{G(x(t ? τ)) = e(t, x(t), x(t ? τ)) (j = 1, 2)}$, where x(t) is in $\text{R}$ⁿ and e(t, u, v) is a given vector function, 2π-periodic in t, are shown under conditions on the spectrum of the Hessian of G. The equation is studied using a fixed point theorem in the space L²(0, 2π). One feature of this approach is that no relationship between the delay and the period is necessary.     

Periodic and asymptotically periodic solutions of systems of nonlinear difference equations with infinite delay

In this paper we study the existence of periodic and asymptotically periodic solutions of a system of nonlinear Volterra difference equations with infinite delay. By means of fixed point theory, we furnish conditions that guarantee the existence of such periodic solutions.     

Periodic solutions of nonlinear differential equations with double resonance

Summary We prove the existence of periodic solutions of a second order nonlinear ordinary differential equation whose nonlinearity is at resonance with two successive eigenvalues of the associated linear operator and satisfies some Landesman-Laser type conditions at both of them.     

Stability of solutions of pseudolinear differential equations with impulse action

In this paper, we develop a new approach to constructing piecewise differentiable Lyapunov functions for certain classes of nonlinear differential equations with impulse action. This approach is based on the method of “frozen” coefficients, and the required function is constructed as a pseudoquadratic form. For the case under consideration, stability conditions in the sense of Lyapunov are obtained. The proposed approach can be used to study the stability of the critical equilibrium states of systems of differential equations with impulse action.