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
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. ) 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.