Pietro Zecca

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

Mathematical models for hysteresis

The various existing classical models for hysteresis, Preisach, Ishlinskii, Duhem–Madelung, are surveyed, as well a more modern treatments by contemporary workers. The emphasis is on a clear mathematical description of the formulation and properties of each model. In addition the authors try to make the reader aware of the many open questions in the study of hysteresis.

Nonlinear boundary value problems in Banach spaces for multivalue differential equations on a non-compact interval

where A and L are linear operators and M is a continuous, generally nonlinear operator. We want to show that, under suitable hypotheses, the problem (1.1) has a solution defined on an interval J = [a, b), x < a < b d + xc;. The analogous problem for ordinary differential equations on a compact interval has been treated by Scrucca [ 11. The linear boundary value problem has been studied by Cecchi et al. [Z] in the univoque case and by Anichini and Zecca [3,4], in the multivoque one. For a wide bibliography and exposition on boundary value problems for differential equations see Conti [5]. 2. NOTATIONS AND HYPOTHESES

CNN model for studying dynamics and travelling wave solutions of FitzHugh-Nagumo equation

In this paper, a cellular neural network (CNN) model of FitzHugh-Nagumo equation is introduced. Dynamical behavior of this model is investigated using harmonic balance method. For the CNN model of FitzHugh-Nagumo equation, propagation of solitary waves have been proved.

On the topological structure of the solution set for a semilinear ffunctional-differential inclusion in a Banach space

In this paper we show that the set of all mild solutions of the Cauchy problem for a functional-differential inclusion in a separable Banach space E of the form x′(t) ∈ A(t)x(t) + F (t, xt) is an Rδ-set. Here {A(t)} is a family of linear operators and F is a Caratheodory type multifunction. We use the existence result proved by V. V. Obukhovskĭi [22] and extend theorems on the structure of solutions sets obtained by N. S. Papageorgiou [23] and Ya. I. Umanskĭi [32]. Introduction. Beginning in the seventies the multivalued Cauchy problem in abstract spaces has been studied by many authors; we mention the existence theorems obtained by Chow and Schuur [6], Muhsinov [20], De Blasi [8], Anichini and Zecca [3], Sentis [26], Pavel and Vrabie [25], Tostonogov [27] and [28] and Kisielewicz [16]. The first approach to the structure of the solution set was by Davy [7] in the finite dimensional case. He proved that the set S of solution is a continuum in C([0, T ],R). Later Lasry and Robert [18] showed that S is acyclic whenever F has compact and convex values and is Hausdorff 1991 Mathematics Subject Classification: Primary 34A60; Secondary 34K30, 93B52, 93C20. The paper is in final form and no version of it will be published elsewhere.

On the solvability of systems of inclusions involving noncompact operators

We consider the solvability of a system y E F(x, y), X E G(x, y) of set-valued maps in two different cases. In the first one, the map (x, y) -o F(x, y) is supposed to be closed graph with convex values and condensing in the second variable and (x, y) -o G(x, y) is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case F is as before with compact, not necessarily convex, values and G is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set x -o S(x) of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.

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 in the problem of the motion of a viscoelastic fluid

We study an optimization problem for the feedback control system emerging as a regularized model for the motion of a viscoelastic fluid subject to the Jeffris-Oldroyd rheological relation. The approach includes systems governed by the classical Navier-Stokes equation as a particular case. Using the topological degree theory for condensing multimaps we prove the solvability of the approximating problem and demonstrate the convergence of approximate solutions to a solution of a regularized one. At last we show the existence of a solution minimizing a given convex, lower semicontinuous functional.

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.

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.