Guided Dynamical Systems and Applications to Functional and Partial Differential Equations

TL;DR

This thesis introduces guided dynamical systems to address functional equations and PDEs, providing conditions for unique solutions and analyzing overdeterminedness in these mathematical problems.

Contribution

It develops new theoretical conditions for the solvability of functional and boundary value problems using guided dynamical systems.

Findings

Characterization of unique solvability for initial value functional equations

Proof of overdeterminedness in a broad class of functional equations

Necessary and sufficient conditions for boundary value problems in third order hyperbolic PDEs

Abstract

In this thesis we introduce the concept of a guided dynamical system, and exploit this idea to solve various problems in functional equations and PDE's. Our main results are 1) a necessary and sufficient condition for unique-solvability of an initial value functional equation, 2) a proof of the overdeterminedness of a large class of functional equations and 3) a necessary and sufficient condition for unique-solvability of a boundary value problem for a third order, hyperbolic PDE.