M.Sc Thesis | |

M.Sc Student | Shalit Orr |
---|---|

Subject | Guided Dynamical Systems and Applications to Functional and Partial Differential Equations |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Boris Paneah (Deceased) |

In this work we present
the notion of a *guided dynamical system*, and then we exploit this idea
to solve various problems in functional equations and partial differential
equations. The results presented are a sequel to a series of papers by B.
Paneah published in the years 1997-2004.

A *guided dynamical
system *can be thought of as a discrete dynamical system with several
generating maps, with the peculiarity that each of these maps is defined only
on a subset of *X*, rather than on the entire space *X*.

This work is composed of four main parts. The first part is devoted to developing the basic theory of guided dynamical systems, which serves as our central technical tool in the rest of this work.

In the second part we
prove some results regarding the uniqueness and solvability of certain
functional equations in a compact metric space. Our main result in this part is
the equivalence between the unique solvability of the initial value problem in
a *P*-configuration, and the minimality of the *P*-configuration.

In the third part we
treat the more esoteric problem of *over-determinedness*, which is the
problem of determining the set of solutions to a given functional equation when
its domain of validity is altered.

The fourth and last part concerns the second partly characteristic boundary problem, a third order, strictly hyperbolic partial differential equation in the plane. For this problem we prove a necessary and sufficient condition for unique solvability in terms of a guided dynamical system on the boundary of the domain, generated by the characteristics of the differential operator. We also give some explicit conditions for unique solvability.