Existence and uniqueness results for solutions to initial value problems in scales of Banach spaces: a thesis presented in partial fulfilment for the degree of Master of Science in Mathematics at Massey University

Abstract

This thesis addresses existence and uniqueness of solutions to certain classes of initial-value problems with functional differential equations. The technique of scales of Banach spaces is used. A scale of Banach spaces is a collection of Banach spaces varying on a real parameter. A scale consisting of function spaces can be used to suppress one variable in an initial-value problem in a partial differential equation of two independent variables, therefore enabling local existence and uniqueness of a solution to the problem to be shown with the classical method of successive ap­ proximations from the Picard-Lindelof Theorem of ordinary differential equations. Tuschke's presentation (c.f. [7]) of this technique and a related theorem has been adapted in Chapters 1 and 2. Chapters 3 and 4 present original theorems, stating existence and uniqueness of solutions to more general initial-value problems, having a retarded character.