Jumps of the fundamental solution matrix in discontinuous systems and applications

This paper deals with differential equations with discontinuous right-hand side. The concept of a solution for a discontinuous system is defined on the basis of differential inclusions using Filippov’s method. We study in particular the behaviour of solutions crossing a discontinuity surface transversally. A formula characterizing jumps of the fundamental solution matrix is derived. As an application of it, the concept of Poincaré mapping is defined for such systems.     

Some results for fractional boundary value problem of differential inclusions

In this paper, we study fractional differential inclusions with Dirichlet boundary conditions. We prove the existence of a solution under both convexity and nonconvexity conditions on the multi-valued right-hand side. The proofs rely on nonlinear alternative Leray–Schauder type, Bressan–Colombo selection theorem and Covitz and Nadler’s fixed point theorem for multi-valued contractions. The compactness of the set solutions and relaxation results is also established. In the last section we consider the fractional boundary value problem with infinite delay.     

On periodic solutions of ordinary differential equations with discontinuous right-hand side

A new version of the method of translation along trajectories, which does not require the uniqueness of the solution of the Cauchy problem, is applied to the proof of the existence theorem for vector-valued periodic solutions of ordinary differential equations of first and second order. This result is applicable to equations and differential inclusions with discontinuous right-hand side. Several applications of the theorems proved in this paper are considered in cases which are not covered by the classical theory of ordinary differential equations with continuous right-hand side and equations with right-hand side satisfying the Carathéodory conditions.     

The Upper and Lower Solutions Method for Stochastic Inclusions with Discontinuous Multivalued Mappings

Abstract

In this article, we consider a stochastic integral inclusion driven by semimartingale with discontinuous multivalued right hand side. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered sets. The presented studies extend some recent results both for deterministic differential inclusions and stochastic differential equations for increasing operators.     

The invariance principle for nonautonomous differential equations with discontinuous right-hand side

We study limit differential inclusions for nonautonomous differential equations with discontinuous right-hand side and Filippov solutions. Using Lyapunov functions with derivatives of constant sign, we establish an analog of LaSalle’s invariance principle. We study differential equations with either measurable or piecewise continuous right-hand side.     

Singular Dirichlet second-order BVPs with impulses

The paper deals with the existence of solutions to singular second-order differential equations with impulse effects and with the Dirichlet boundary conditions. The right-hand side of the differential equation can be singular in its phase variable.     

On the Application of the Averaging Principle in Stochastic Differential Equations of Hyperbolic Type

We prove a theorem on the application of the Bogolyubov–Mitropol'skii averaging principle to stochastic partial differential equations of the hyperbolic type.     

Existence and relaxation of solutions to differential inclusions with unbounded right-hand side in a Banach space

In a separable Banach space we consider a differential inclusion whose values are nonconvex, closed, but not necessarily bounded sets. Along with the original inclusion, we consider the inclusion with convexified right-hand side. We prove existence theorems and establish relations between solutions to the original and convexified differential inclusions. In contrast to assuming that the right-hand side of the inclusion is Lipschitz with respect to the phase variable in the Hausdorff metric, which is traditional in studying this type of questions, we use the (ρ–H) Lipschitz property. Some example is given.     

Solution tube method for impulsive periodic differential inclusions of first order

The aim of this paper is to obtain an existence result for impulsive differential inclusions of first order with boundary conditions in Hilbert spaces under a hypothesis of integrability in the Henstock-Lebesgue sense for the multifunction on the right-hand side. The proof is based on the assumption that there exists a solution tube for the inclusion taken under consideration (this novel concept which generalizes the extensively used notions of upper and lower solution was adapted to the present setting). Finally, a compactness property is proved.     

Singularly perturbed differential inclusions: An averaging approach

We consider nonlinear, singularly perturbed differential inclusions and apply the averaging method in order to construct a limit differential inclusion for slow motion. The main approximation result states that the existence and regularity of the limit differential inclusion suffice to describe the limit behavior of the slow motion. We give explicit approximation rates for the uniform convergence on compact time intervals. The approach works under controllability or stability properties of fast motion.     

Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients

We prove an existence theorem for weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients. A weak solution of an equation is understood as a weak solution of a stochastic differential inclusion constructed on the basis of the equation. We derive conditions providing the absence of blow-up in weak solutions.     

Impulsive boundary value problems with nonlinear boundary conditions

In this paper, the upper and lower solution method and Schauder’s fixed point theorem are employed in the study of boundary value problems for a class of second-order impulsive ordinary differential equations with nonlinear boundary conditions. We prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem.     

A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation

We study a nonlinear controlled functional operator equation in an ideal Banach space. We establish sufficient conditions for the global solvability for all controls from a given set, and obtain a pointwise estimate for solutions. Using upper and lower estimates of the functional component in the right-hand side of the initial equation (with a fixed operator component), we obtain majorant and minorant equations. We prove the stated theorem, assuming the monotonicity of the operator component in the right-hand side and the global solvability of both majorant andminorant equations. We give examples of the reduction of controlled initial boundary value problems to the equation under consideration.     

Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form

We consider the Cauchy problem for a semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle, the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We also prove regularity properties and approximation results for solutions of the problem.     

Brownian motion and exposed solutions of differential inclusions

We present a research program designed by A. Bressan and some partial results related to it. First, we construct a probability measure supported on the space of solutions to a planar differential inclusion, where the right-hand side is a Lipschitz continuous segment. Such measure assigns probability one to solutions having derivatives a.e. equal to one of the endpoints of the segment. Second, for a class of planar differential inclusions with Hölder continuous right-hand side F, we prove existence of solutions whose derivatives are exposed points of F. Finally, we complete the research program if the right-hand side of the differential inclusion does not depend on the state and prove a result on the Lipschitz continuity of an auxiliary map. The proofs rely on basic properties of Brownian motion.     

A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral type

We consider a regularization for a class of discontinuous differential equations arising in the study of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to so-called “breaking points”, where the solution derivative is again discontinuous. Consequently, the problem of continuing the solution in a right neighborhood of a breaking point is equivalent to a Cauchy problem for an ode with a discontinuous right-hand side (see e.g. Bellen et al., 2009 [4]). Therefore a classical solution may cease to exist.The regularization is based on the replacement of the vector-field with its time average over an interval of length ε>0. The regularized solution converges as ε→⁰⁺ to the classical Filippov solution (Filippov, 1964, 1988 [13] and [14]). Several properties of the solutions corresponding to small ε>0 are presented.     

Functional differential inclusions generated by functional differential equations with discontinuities

Given a functional differential equation with a discontinuity, a construction of its extension in the shape of a functional differential inclusion is offered. This construction can be regarded as a generalization of the famous Filippov approach to study ordinary differential equations with discontinuities. Some basic properties of the solutions of the introduced functional differential inclusions are studied. The developed approach is applied to analysis of gene regulatory networks with general delays.     

Almost-periodic solutions to systems of differential equations with fast and slow time in the degenerate case

We consider a system of ordinary first-order differential equations. The right-hand sides of the system are proportional to a small parameter and depend almost periodically on fast time and periodically on slow time. With this system, we associate the system averaged over fast time. We assume that the averaged system has a structurally unstable periodic solution. We prove a theorem on the existence and stability of almost periodic solutions of the original system. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 451–456, March, 1998.     

Existence of periodic solutions

An apparatus for proving existence theorems for periodic solutions of equations with discontinuous right-hand side and differential inclusions is developed. Translated fromMatematischeskie Zametki, Vol. 61, No. 5, pp. 769–784, May, 1997. Translated by V. N. Dubrovsky     

Extension of E.A. Barbashin’s and N.N. Krasovskii’s stability theorems to controlled dynamical systems

The known theorems by E.A. Barbashin and N.N. Krasovskii (1952) about the asymptotic and global stability of an equilibrium state for an autonomous system of differential equations are extended to nonautonomous differential inclusions with closed-valued (but not necessarily compact-valued) right-hand sides, where the equilibrium state is a weakly invariant (with respect to solutions of the inclusion) set. The statements are formulated in terms of the Hausdorff-Bebutov metric, the dynamical system of translations corresponding to the right-hand side of the differential inclusion, and the weakly invariant set corresponding to the inclusion.