|M.Sc Student||Orr Shalit|
|Subject||Guided Dynamical Systems and Applications to Functional|
and Partial Differential Equations
|Department||Department of Mathematics||Supervisor||Professor Emeritus Paneah Boris|
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.