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In the paper piecewise continuous analogues of Liapunov's functions have been introduced for systems of differential equations with impulses. They serve to study the problems for uniqueness, continuity and boundedness of the solutions of nonlinear systems with impulses. The asymptotic behaviour of the bounded solutions has also been considered.

The present paper is the first to find sufficient efficient conditions for stability of solutions of singularly perturbed systems with impulse effect.

The paper justifies the averaging method for nonlinear integro‐differential equations of the type (x ∈ Rn, ε > 0 is a small parameter, ω is a constant), experiencing impulse effect.

In this paper the asymptotic and globally asymptotic stability of the zero solution of systems with impulses is investigated. For this purpose piecewise continuous auxiliary functions are used which are an analogue to Lyapunov's functions. The theorem of Marachkov on the asymptotic stability of systems without impulses is generalized. The results obtained are formulated in four theorems.

The paper considers integral surfaces of systems of differential equations with impulse perturbations at fixed moments of time. Sufficient conditions have been obtained for the existence of integral surfaces with definite properties and the behaviour of the solutions has been studied with initial conditions outside these surfaces.

Investigates the global stability of the zero solution of an impulsive system of differential‐difference equations with variable impulsive perturbations. By means of piecewise continuous functions which are analogues of Lyapunov’s functions, and of the comparison principle, sufficient conditions for global stability of the zero solution of the systems considered are found.

This paper aims to present how to use the difference equations for analysis of flyback converter circuit.

Switching circuits have variable structural topologies. In every switched-mode, they have different dynamics and different equations. First, the exact equivalent circuit of flyback converter, then, set of difference equations are obtained. The flyback converter has a nonlinear structure; however, the presented technique allows the circuit equations to be linear. The transient-state and steady-state analysis of flyback converter, one of popular switching circuits, are carried out by using difference-equations.

The proposed analysis method does not contain any numerical approximation and the results are in the form of exact solution. Another superiority of the method is that the desired instantaneous values can be obtained directly, the simulation does not need to be started from the beginning. Numerical results agree well with the theoretical results of flyback converter. The simulation results obtained by using the proposed method and Matlab-based results are compared.

This paper contributes a different mathematical background for analysis of switching converters to the literature.

This paper is concerned with the existence of mild solutions for a class of fractional semilinear integro-differential equations having non-instantaneous impulses. The result is obtained by using noncompact semigroup theory and fixed point theorem. The obtained result is illustrated by an example at the end.