Properties of the Set of Extreme Solutions for a Class of Nonlinear Second-Order Evolution Inclusions

We establish the existence of extreme solutions for a class of nonlinear second-order evolution inclusions with a nonconvex right-hand side defined on an evolution triple of Banach spaces. Then we show that extreme solutions which belong to the solution set of the original system are in fact dense and codense in the solution set of a system with a convexified right-hand side. The necessary and sufficient conditions for closedness of the solution set for the original system in an appropriate spaces of functions are given as well. Finally, an example of a nonlinear hyperbolic distributed parameter system is worked out in detail.     

Strongly Exposed Points of Decomposable Sets in Spaces of Bochner Integrable Functions

A characterization of strongly exposed points of a decomposable bounded closed convex set $\Gamma \subset L_p (T,X)$ , where $1 \leqslant p < \infty $ , in terms of strongly exposed points of values of the set-valued representation $F:T \to 2^X $ of $\Gamma $ is given. As a corollary, necessary conditions characterizing strongly exposed points of the unit ball in $L_p (T,X) $ , where $1 \leqslant p < \infty $ , in terms of strongly exposed points of the unit ball in X are obtained.     

Subdifferential inclusions with unbounded perturbation: Existence and relaxation theorems

The paper studies an evolution inclusion in a separable Hilbert space whose right-hand side contains the subdifferential of a proper convex lower semicontinuous function of time and a set-valued perturbation. Together with this inclusion, an inclusion with convexified perturbation values is considered. The existence and density of the solution set of the initial inclusion in the closure of the solution set of the inclusion with convexified perturbation are proved. This property is usually called relaxation. Traditional assumptions for relaxation theorems are the compactness property of the convex function and the boundedness of the perturbation. In the present paper, such assumptions are not made. Assumptions for subdifferential inclusions described by polyhedral sweeping processes and variational inequalities with time-dependent obstacles and constraints are specified.     

Lp-continuous extreme selectors of multifunctions with decomposable values: Existence theorems

The existence theorems of L _p -continuous selectors that values are extreme points are proved for a class of multivalued maps. Applications to multivalued maps appearing in multivalued differential equations are presented.Supported in part by RFFI Grant 93-011-264.     

Existence and properties of solutions of a control system with hysteresis effect

We consider a control system described by two ordinary nonlinear differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values, which depends on the phase variables. One of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable. The equation containing the subdifferential describes an input-output relation of hysteresis type.Along with the original control constraint, we also consider the convexified control constraint and the constraint consisting of the extremal points of the convexified control constraint.We prove the existence of solutions of our control system with various control constraints and establish certain relationships between corresponding solution sets.     

Continuous selections for a family of nonconvex-valued mappings with noncompact domain

The research was financially supported by the Russian Foundation for Fundamental Research (Grant 93-011-264).