Valeri Obukhovskii

Condensing multivalued maps and semilinear differential inclusions in Banach spaces

Multivalued maps: general properties * Measures of noncompactness and condensing multimaps * Topological degree theory for condensing multifields * Semigroups and measures of noncompactness * Semilinear differential inclusions: initial problem * Semilinear inclusions: periodic problems

EXISTENCE AND GLOBAL BIFURCATION OF PERIODIC SOLUTIONS TO A CLASS OF DIFFERENTIAL VARIATIONAL INEQUALITIES

In this paper, by using the topological degree theory for multivalued maps and the method of guiding functions, the existence and global bifurcation for periodic solutions of a class of differential variational inequalities are studied.

Method of guiding functions in problems of nonlinear analysis

1 Background.- 2 MGF in Finite-Dimensional Spaces.- 3 Guiding Functions in Hilbert Spaces.- 4 Second-Order Differential Inclusions.- 5 Nonlinear Fredholm Inclusions.

Optimal feedback control for a semilinear evolution equation

In this paper, we consider a minimization problem of a cost functional associated to a nonlinear evolution feedback control system with a given boundary condition which includes the periodic one as a particular case. Specifically, by using an existence result for a system of inclusions involving noncompact operators (see Ref. 1), we first prove that the solution set of our problem is nonempty. Then, from the topological properties of this set, we derive the existence of a solution of the minimization problem under consideration.

Decay solutions for a class of fractional differential variational inequalities

Abstract Our aim is to study a new class of differential variational inequalities involving fractional derivatives. Using the fixed point approach, the existence of decay solutions to the mentioned problem is proved.

On boundary value problems for degenerate differential inclusions in Banach spaces

We consider the applications of the theory of condensing set-valued maps, the theory of set-valued linear operators, and the topological degree theory of the existence of mild solutions for a class of degenerate differential inclusions in a reflexive Banach space. Further, these techniques are used to obtain the solvability of general boundary value problems for a given class of inclusions. Some particular cases including periodic problems are considered.

On coincidence index for multivalued perturbations of nonlinear Fredholm maps and some applications

We define a nonoriented coincidence index for a compact, fundamentally restrictible, and condensing multivalued perturbations of a map which is nonlinear Fredholm of nonnegative index on the set of coincidence points. As an application, we consider an optimal controllability problem for a system governed by a second-order integro-differential equation.

An oriented coincidence index for nonlinear Fredholm inclusions with nonconvex-valued perturbations

We suggest the construction of an oriented coincidence index for nonlinear Fredholm operators of zero index and approximable multivalued maps of compact and condensing type. We describe the main properties of this characteristic, including applications to coincidence points. An example arising in the study of a mixed system, consisting of a first-order implicit differential equation and a differential inclusion, is given.

On a class of fractional order differential inclusions with infinite delays

Our aim is to study fractional order differential inclusions with infinite delays in Banach spaces. We impose the regularity condition on multivalued nonlinearity in terms of measures of noncompactness to get the existence result. Some properties of the solution map are proved.

On Semilinear Integro-Differential Equations with Nonlocal Conditions in Banach Spaces

We study the abstract Cauchy problem for a class of integrodifferential equations in a Banach space with nonlinear perturbations and nonlocal conditions. By using MNC estimates, the existence and continuous dependence results are proved. Under some additional assumptions, we study the topological structure of the solution set.