The index bundle for selfadjoint Fredholm operators and multiparameter bifurcation for Hamiltonian systems

the search for the self in becketts theatre: waiting for godot and endgame

this thesis is based upon the works of samuel beckett. one of the greatest writers of contemporary literature. here, i have tried to focus on one of the main themes in becketts works: the search for the real "me" or the real self, which is not only a problem to be solved for beckett man but also for each of us. i have tried to show becketts techniques in approaching this unattainable goal, base...

On the Structure of the Set of Bifurcation Points of Periodic Solutions for Multiparameter Hamiltonian Systems

This paper deals with periodic solutions of the Hamilton equation ẋ(t) = J∇xH(x(t), λ), where H ∈ C2,0(R2n × Rk,R) and λ ∈ Rk is a parameter. Theorems on global bifurcation of solutions with periods 2π j , j ∈ N, from a stationary point (x0, λ0) ∈ R2n × Rk are proved. ∇xH(x0, λ0) can be singular. However, it is assumed that the local topological degree of ∇xH(·, λ0) at x0 is nonzero. For system...

Dynamical systems method (DSM) for selfadjoint operators

Let A be a selfadjoint linear operator in a Hilbert space H. The DSM (dynamical systems method) for solving equation Av = f consists of solving the Cauchy problem u̇ = Φ(t, u), u(0) = u0, where Φ is a suitable operator, and proving that i) ∃u(t) ∀t > 0, ii) ∃u(∞), and iii) A(u(∞)) = f . It is proved that if equation Av = f is solvable and u solves the problem u̇ = i(A + ia)u − if, u(0) = u0, wher...

Fredholm Operators and the Generalized Index

One of the most fundamental problems in mathematics is to solve linear equations of the form Tf = g, where T is a linear transformation, g is known, and f is some unknown quantity. The simplest example of this comes from elementary linear algebra, which deals with solutions to matrix-vector equations of the form Ax = b. More generally, if V,W are vector spaces (or, in particular, Hilbert or Ban...

Fredholm Operators and the Family Index