Mikhail Kamenskii

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 of Weak Solutions to Stochastic Evolution Inclusions

Abstract We prove the existence of a weak mild solution to the Cauchy problem for the semilinear stochastic differential inclusion in a Hilbert space where W is a Wiener process, A is a linear operator that generates a C 0-semigroup, F and G are multifunctions with convex compact values satisfying some growth condition, and with respect to the second variable, a condition weaker than the Lipschitz condition. The weak solution is constructed in the sense of Young measures.

Existence of Periodic Solutions of an Autonomous Damped Wave Equation in Thin Domains

For a nonlinear autonomous damped wave equation in a thin domain we provide conditions ensuring the existence of periodic solutions in time. Our approach uses both methods developed by Hale and Raugel and methods based on the topological degree theory together with some results on the functionalization of parameter.

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.

Small parameter perturbations of nonlinear periodic systems

In this paper we consider a class of nonlinear periodic differential systems perturbed by two nonlinear periodic terms with multiplicative different powers of a small parameter ? > 0. For such a class of systems we provide conditions that guarantee the existence of periodic solutions of given period T > 0. These conditions are expressed in terms of the behaviour on the boundary of an open bounded set U of of the solutions of suitably defined linearized systems. The approach is based on the classical theory of the topological degree for compact vector fields. An application to the existence of periodic solutions to the van der Pol equation is also presented.

An averaging method for singularly perturbed systems of semilinear differential inclusions with analytic semigroups

We consider a system of two semilinear parabolic inclusions depending on a small parameter e > 0 which is present both in front of the derivative in one of the two inclusions and in the nonlinear terms to model high-frequency inputs.The aim is to provide conditions in order to guarantee, for e > 0 sufficiently small, the existence of periodic solutions and in order to study their behaviour as e tends to zero. Our assumptions permit the definition of upper semicontinuous, convex valued, compact vector operators whose fixed points represent the sought-after periodic solutions. The existence of fixed points is shown by using topological degree theory arguments.

Bifurcation and multiplicity results for periodic solutions of a damped wave equation in a thin domain

We study the bifurcation problem for periodic solutions of a nonautonomous damped wave equation dened in a thin domain. Here the bifurcation parameter is represented by the thinness >0 of the considered domain. This study has as starting point the existence result of periodic solutions already stated by the authors for this equation and it makes use of the condensivity properties of the associated Poincare map and its linearization around these solutions. We establish sucient conditions to guarantee that =0 is or not a bifurcation point and a related multiplicity result. These results are in the spirit of those given by Krasnosel’skii and they are obtained by using the topological degree theory for k-condensing operators. c 2000 Elsevier Science B.V. All rights reserved.

Boundary value problems for semilinear differential inclusions of fractional order in a Banach space

In the present paper, we show that the solution set of a fractional order semilinear differential inclusion in a Banach space has the topological structure of an -set. This result allows to apply a fixed point result for condensing multimaps to the translation multioperator along the trajectories of such inclusion and to prove the existence of solutions satisfying periodic and anti-periodic boundary value conditions. An example concerning with a fractional order feedback control system is presented.

A bifurcation problem for a class of periodically perturbed autonomous parabolic equations

The paper deals with the problem of the existence of a branch of T-periodic solutions originating from the isolated limit cycle of an autonomous parabolic equation in a Banach space when it is perturbed by a nonlinear T-periodic term of small amplitude.We solve this problem by first introducing a novel integral operator, whose fixed points are T-periodic solutions of the considered equation and vice versa. Then we compute the Malkin bifurcation function associated to this integral operator and we provide conditions under which the well-known assumption of the existence of a simple zero of the Malkin bifurcation function guarantees the existence of the branch.MSC:35K58, 35B10, 35B20, 35B32.

On perturbations of systems with multidimensional degeneration

Bifurcation conditions are found for the periodic solutions in systems of differential equations with the perturbation (small disturbance) in the case of existence of joined Floke solutions in a linearized nonperturbed system. For this case a multidimensional analog of the Malkin bifurcation function is built up.