Necessary and Sufficient Conditions for Associated Differential Operators in Quaternionic Analysis and Applications To Initial Value Problems

Abstract

This paper deals with the initial value problem of type in the space of generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation where is the time variable, x runs in a bounded and simply connected domain in is a real number, and is the Cauchy-Fueter operator. We prove necessary and sufficient conditions on the coefficients of the operator under which is associated with the operator , i.e. transforms the set of all solutions of the differential equation into solutions of the same equation for fixedly chosen t. This criterion makes it possible to construct operators for which the initial value problem is uniquely soluble for an arbitrary initial generalized regular function u (0) by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also generalized regular for each t.