Existence results for impulsive hyperbolic differential inclusions

In this article we investigate the existence of solutions for second order impulsive hyperbolic differential inclusions in separable Banach spaces. By using suitable fixed point theorems, we study the case when the multi-valued map has convex and non-convex values.     

An infinite-time relaxation theorem for differential inclusions

The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wazewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is presented, which provides approximations on infinite intervals, but does not guarantee that the approximation and the reference trajectory satisfy the same initial condition.

     

Existence theorems for hyperbolic differential inclusions in Banach algebras

In this paper, an existence theorem for a certain hyperbolic differential inclusions in Banach algebras is proved under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions for Carathéodory as well as discontinuous hyperbolic differential inclusions is also proved under certain monotonicity conditions.     

Existence theory for perturbed impulsive hyperbolic differential inclusions with variable times

In this paper, the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is proved under the mixed generalized Lipschitz and Carathéodory's conditions.     

N. N. Bogolyubov's research in mathematics and theoretical physics

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1156–1164, September, 1989.     

Continuous version of Filippov’s theorem for fractional differential inclusions

Using Bressan-Colombo results, concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values, we prove a continuous version of Filippov’s theorem for a fractional differential inclusion.     

On solutions of hyperbolic differential inclusions with nonconvex right-hand side

The existence of a generalized solution with continuous derivativesu _x ,u _y is proved for the differential inclusionu _xy F(x, y, u) with a nonconvex right-hand side satisfying the Lipschitz conditioninx, y, andu.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 531–534, April, 1995.     

Generalization of the second Bogolyubov's theorem for non-almost periodic systems

The article is devoted to the generalization of the second Bogolyubov's theorem to non-almost periodic dynamical systems. We prove the analog of the second Bogolyubov's theorem for recurrent or pseudorecurrent dynamical systems in Banach spaces. Namely, we obtain the relation between a recurrent dynamical system and its averaged dynamical system. We also study existence of recurrent and pseudorecurrent motions (including special cases of periodic, quasi-periodic and almost periodic motions) in related nonautonomous systems.     

A non-existence theorem of lacunas for hyperbolic differential operators with constant coefficients

We prove a theorem on non-existence of lacunas of the fundamental solution for hyperbolic differential operators with constant coefficients.     

Extremal problems for discrete and differential inclusions of hyperbolic type with distributed parameters

For the first time, necessary and sufficient conditions for an extremum are proved for the first mixed problem for discrete and differential inclusions of hyperbolic type. Some of the results are generalized to the multidimensional case of a second order elliptic operator in bounded cylindrical domains.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1641–1649, December, 1990.     

Set-valued solutions to the Cauchy problem for hyperbolic systems of partial differential inclusions

We prove the existence of global set-valued solutions to the Cauchy problem for partial differential equations and inclusions, with either single-valued or set-valued initial conditions. The method is based on the equivalence between this problem and problem of finding viability tubes of the associated characteristic system of ordinary differential equations. As an application we construct the value function of the Mayer problem arising in control theory. Received August 25, 1995     

On N. N. Bogolyubov's works in classical and quantum statistical mechanics

A review of N. N. Bogolyubov's works in classical and quantum statistical mechanics is presented.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 155–201, February, 1993.     

An implicit-function theorem for inclusions

We consider the question of the solvability of an inclusion F(x, σ) ∈ A, i.e., of determining a mapping (implicit function) σ ? x(σ) defined on a set such that F(x(σ), σ) ∈ A for any σ from this set. Results of this kind play a key role in the different branches of analysis and, especially, in the theory of extremal problems, where they are the main tool for deriving conditions for an extremum.     

On the set of solutions of some nonconvex nonclosed hyperbolic differential inclusions

We consider a class of nonconvex and nonclosed hyperbolic differential inclusions and we prove the arcwise connectedness of the solution set.