(1) Department of Applied Mathematics, Osaka Women's University, 590 Sakai, Osaka, Japan;(2) Department of Applied Mathematics, Graduate School, Okayama University of Science, 700 Ridai-cho, Okayama, Japan
Abstract:
Summary We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential equations. We also give an affirmative answer to a conjecture of H. wiatak. A part of the results was announced in the comprehensive paper 8] on our joint works. To prove the regularity of solutions of functional equations is one of the central problems in the theory of functional equations (see 1]).