Wojciech Kryszewski

AN INFINITE DIMENSIONAL MORSE THEORY WITH APPLICATIONS

In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the onedimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations. 0. Introduction Let E be a real Hilbert space with an inner product 〈. , .〉 and let Φ be a twice continuously differentiable functional. Denote the Frechet derivative and the gradient of Φ at x by Φ′(x) and ∇Φ(x) respectively, where as usual 〈∇Φ(x), y〉 := Φ′(x)y ∀ y ∈ E. Recall that a point x0 ∈ E is said to be critical if Φ′(x0) = 0, or equivalently, if ∇Φ(x0) = 0. The level c ∈ R will be called regular if Φ−1(c) contains no critical points, and critical if ∇Φ(x0) = 0 for some x0 ∈ Φ−1(c). Let a, b, a < b, be two regular levels of Φ. Denote M := Φ−1([a, b]) and consider the restriction of Φ to M . In Morse theory one is interested in the local topological structure of the level sets of Φ|M near a critical point and in the relation between this local structure and the topological structure of the set M . To be more specific, suppose that x0 ∈M is an isolated critical point of Φ. Then one defines a sequence of critical groups of Φ at x0 by setting cq(Φ, x0) := Hq(Φ c ∩ U,Φ ∩ U − {x0}), q = 0, 1, 2, ..., (0.1) where c := Φ(x0), Φ c := {x ∈ E : Φ(x) ≤ c}, Hq is the q-th singular homology group with coefficients in some field F and U is a neighbourhood of x0. Define the Morse index of x0 to be the maximal dimension of a subspace of E on which the quadratic form 〈Φ′′(x0)y, y〉 is negative definite. One shows that if x0 is a nondegenerate critical point, i.e., if Φ′′(x0) : E → E is invertible, then cq(Φ, x0) = F Received by the editors March 20, 1995. 1991 Mathematics Subject Classification. Primary 58E05; Secondary 34C25, 35J65, 35L05, 55N20, 58F05.

Fixed-point index for compositions of set-valued maps with proximally ∞-connected values on arbitrary ANR's

In the paper the integer-valued fixed-point index theory for compositions of set-valued maps having proximally ∞-connected values satisfying all the axioms of a fixed point index is presented. The considered approach is based on the technique of single-valued approximation on the graph.

On the solution sets of differential inclusions and the periodic problem in Banach spaces

Abstract In this paper, the topological structure of the solution set of a constrained semilinear differential inclusion in a Banach space E is studied. It is shown that the set of all mild solutions, with values in a closed and, in general, thin subset D ⊂ E , is an R δ -set provided natural boundary conditions and appropriate geometrical assumptions on D (which hold, e.g. when D is convex) are satisfied. Applications to the periodic problem and to the existence of equilibria are given.

Graph-approximation of set-valued maps on noncompact domains☆

Abstract In the paper we study the existence of the so-called graph-approximations of upper semicontinuous set-valued maps defined on noncompact domains. We prove, in particular, that an arbitrary neighborhood of the graph of a map whose values satisfy a certain UV -property contains the graph of a continuous single-valued map provided the domain is either a finite-dimensional metric space, a locally finite-dimensional polyhedron or an absolute neighborhood retract.

A coincidence theory involving Fredholm operators of nonnegative index

We construct a homotopy invariant appropriate for studying the existence of coincidence points of Fredholm operators of nonnegative index and multivalued admissible maps. Cohomotopy methods are used as a more suitable tool than homological ones. Both finite and infinite dimensional cases are investigated.

On the Solution Sets of Constrained Differential Inclusions with Applications

We study the existence and the structure of solutions to differential inclusions with constraints. We show that the set of all viable solutions to the Cauchy problem for a Carathéodory-type differential inclusion in a closed domain is an Rδ-set provided some mild boundary conditions expressed in terms of functional constraints defining the domain are satisfied. Presented results generalize most of the existing ones. Some applications to the existence of periodic solutions as well as equilibria are given.

Topological structure of solution sets of differential inclusions: the constrained case

We survey and announce some current results on the existence, the viability, and the topological structure of the viable solutions of differential equations and inclusion in Banach spaces under set constraints. Some new results concerning semilinear differential inclusions with state variables constrained to the so-called regular and strictly regular sets, together with their applications, are presented and discussed.

Topological Methods for the Local Controllability of Nonlinear Systems

Local controllability of systems using the topological degree of finite-dimensional set-valued maps is studied. For perturbed linear systems a generalization of the Lee-Markus sufficient condition of local controllability is established. For systems given by a finite family of continuous vector fields first order controllability condition is obtained. In both cases, a set of control functions sufficient for the local controllability is described, which is homeomorphic with a

Equilibrium for perturbations of multifunctions by convex processes

d

Bifurcation invariants for acyclic mappings

-dimensional unit ball, where