Nonlinear Analysis and Differential Equations An Introduction

iii Preface The subject of Differential Equations is a well established part of mathematics and its systematic development goes back to the early days of the development of Calculus. Many recent advances in mathematics, paralleled by a renewed and flourishing interaction between mathematics, the sciences, and engineering, have again shown that many phenomena in the applied sciences, modelled by differential equations will yield some mathematical explanation of these phenomena (at least in some approximate sense). The intent of this set of notes is to present several of the important existence theorems for solutions of various types of problems associated with differential equations and provide qualitative and quantitative descriptions of solutions. At the same time, we develop methods of analysis which may be applied to carry out the above and which have applications in many other areas of mathematics, as well. As methods and theories are developed, we shall also pay particular attention to illustrate how these findings may be used and shall throughout consider examples from areas where the theory may be applied. As differential equations are equations which involve functions and their derivatives as unknowns, we shall adopt throughout the view that differential equations are equations in spaces of functions. We therefore shall, as we progress, develop existence theories for equations defined in various types of function spaces, which usually will be function spaces which are in some sense natural for the given problem.