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Initial value problems of type can be solved by the contraction-mapping principle in case the initial function ϕ belongs to an associated space whose elements satisfy an interior estimate. The present article proves such an interior estimate in the sup-norm for generalized monogenic functions. The proof is based on a representation of generalized monogenic functions by harmonic functions. That way the article is another example for the technique of transforming solutions of simpler partial differential equations into solutions of more general ones (an overview on such methods can be found in Begehr's and Gilbert's two volumes [Begehr, H.G.W. and Gilbert, R.P., 1992/1993, Transformations, Transmutations and Kernel Functions, Vol. I and II (Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc.)].

This article deals with the Runge type approximations problem, the solution of inhomogeneous system and the additive Cousin problem for Clifford-algebra-valued functions ƒ(x, y) which are regular with respect to x and real-analytic in y. Some main results concerning these three fundamental problems from complex analysis are proved for the class of functions mentioned above.

In this paper we study the extension theorems for solutions of inhomogeneous Cauchy-Fueter system. In particular cases, we obtain the extension theorems for regular functions of several quaternionic variables. At the end of the paper we give some applications of these theorems.