Tzanko Donchev

Discrete Approximation of Impulsive Differential Inclusions

The article deals with the approximation of the solution set and the reachable sets of an impulsive differential inclusion with variable times of impulses. It is strongly connected to [11] and is its continuation. We achieve order of convergence 1 for the Euler approximation under Lipschitz assumptions on the set-valued right-hand side and on the functions describing the jump surfaces and jumps themselves. Another criterion prevents the beating phenomena and generalizes available conditions. Several test examples illustrate the conditions and the practical evaluation of the jump conditions.

Filippov---Pliss lemma and m-dissipative differential inclusions

In the paper we prove a variant of the well known Filippov–Pliss lemma for evolution inclusions given by multivalued perturbations of m-dissipative differential equations in Banach spaces with uniformly convex dual. The perturbations are assumed to be almost upper hemicontinuous with convex weakly compact values and to satisfy one-sided Peron condition. The result is then applied to prove the connectedness of the solution set of evolution inclusions without compactness and afterward the existence of attractor of autonomous evolution inclusion when the perturbations are one-sided Lipschitz with negative constant.

Higher order Runge–Kutta methods for impulsive differential systems

Abstract This paper studies higher order approximations of solutions of differential equations with non-fixed times of impulses. We assume that the right-hand side is sufficiently smooth. Using a Runge–Kutta method of higher order and natural assumptions on the impulsive surfaces and the impulses, we calculate good approximations of the jump times, which enables us to extend the classical results for higher order of convergence of Runge–Kutta methods to more complicated systems.

Numerical Approximations of Impulsive Delay Differential Equations

The article deals with numerical approximations of impulsive delay differential equations with a non-fixed time of impulses. The right-hand side of the approximation is assumed to be Lipschitz with respect to the norm of the measurable functions, which allows us to estimate the distance between functions with different times of jumps. Illustrative examples are provided.

Nonlocal evolution inclusions under weak conditions

We study evolution inclusions given by multivalued perturbations of m-dissipative operators with nonlocal initial conditions. We prove the existence of solutions. The commonly used Lipschitz hypothesis for the perturbations is weakened to one-sided Lipschitz ones. We prove an existence result for the multipoint problems that cover periodic and antiperiodic cases. We give examples to illustrate the applicability of our results.

Generalized Solutions of Semilinear Evolution Inclusions

In this paper we study different types of (generalized) solutions for semilinear evolution inclusions in general Banach spaces, called limit and weak solutions, which are extensions of the weak solutions studied by T. Donchev [Nonlinear Anal., 16 (1991), pp. 533--542] and the directional solutions studied by J. Tabor [Set-Valued Anal., 14 (2006), pp. 121--148]. Under appropriate assumptions, we show that the set of the limit solutions is compact

Runge-Kutta Methods for Differential Equations with Variable Time of Impulses

R_delta

Value Function and Optimal Control of Differential Inclusions

. When the right-hand side satisfies the one-sided Perron condition, a variant of the well-known lemma of Filippov--Plis, as well as a relaxation theorem, are proved.

Global character of a host-parasite model

In this article, Runge-Kutta methods of order p are used to approximate the solutions of differential equations with variable times of impulses. It is proved that these methods have order of approximation O(h p ) w.r.t. the defined measure of distance. Illustrative examples with different strategies to find the events are provided.

Discrete Approximations and Optimization of Evolution Inclusions

Abstract Optimal control system described by differential inclusion with continuous and one sided Perron right-hand side in a finite dimensional space is studied in the paper. We prove that the value function is the unique solution of a proximal Hamilton-Jacobi inequalities.