Existence, Uniqueness, and Construction of the Solution of a System of Ordinary Functional Differential Equations, with Application to the Design of Perfectly Focusing Symmetric Lenses

Abstract

In the design of perfectly focusing symmetric lenses, one is led, in a natural way, to a set offunctional differential equations; that is, differential equations involving composites of unknown functions, with initial conditions prescribed on the lens axis. This paper concentrateson those features of the equations which make them uniquely solvable. They are: (i) a contractiveness property of the equations near the axis; (ii) a uniform retardation in the arguments of thecomposite functions away from the axis. The second and third sections of this paper generalize and formalize these properties and provide proofs of existence, uniqueness, and continuous dependence on the data for solutions of such generalized systems of functional differential equations. Because of the lens context which motivates our study, the problem in which the contractiveness property (i) above holds is called the ‘local’ problem, and the problem in which the arguments of composite functions are uniformly retarded is called the ‘global’ problem. In the final section of the paper we apply the general results of the preceding sections to prove existence and uniqueness of perfectly focusing symmetric lenses up to distances from the lens axis at which various types of breakdown, discussed in the text, may occur.