V. V. Filippov

On the Asymptotic Integration of Quasilinear Differential Equations

Using the notion of convergence of solution spaces, we prove theorems about the asymptotic behavior of nearly linear systems. An example of an application of one of these theorems is given.

On Shifts along Trajectories

The extension of the theory of boundary-value problems to ordinary differential equations and inclusions with discontinuous right-hand sides based on the construction of a new version of the method of shifts along trajectories is continued.

On Fučik Spectra and Periodic Solutions

In this paper, we continue extending the theory of boundary-value problems to ordinary differential equations and inclusions with discontinuous right-hand side. To this end, we construct a new version of the method of shifts along trajectories. We compare the results obtained by the new approach and those obtained by the method of Fučik spectra.

Optimal Solution Space and Regularization of Equations with Discontinuities in the Space Variables

The paper points to the fact that properties of optimal solutions can be studied, bypassing the Cauchy problem theory for the equation of optimal control synthesis.

A selection theorem for a new class of set-valued maps

A new class of set-valued maps that includes all upper and lower semicontinuous set-valued maps is introduced. For this class, a selection theorem having applications in the theory of differential inclusions is presented.

The equicontinuity condition for sequences of solution spaces

The verification of an appropriately stated equicontinuity condition for a sequence of solution spaces is one of the two key points in the theory of the Cauchy problem for equations with singular right-hand sides. We obtain a related sufficient condition.

On acyclicity of the set of solutions to the Cauchy problem for differential equations

The acyclicity of sets of solutions to the Cauchy problem is considered from the viewpoint of the axiomatic theory of solutions spaces of ordinary differential equations. We prove that the class of acyclic solution spaces is closed with respect to passing to the limit space.

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.