Frobenius theorem under Carathéodory type conditions

Abstract

This paper is concerned with a variant of the Frobenius theorem on total differential systems under conditions of the Carathkodory type occurring in the theory of ordinary differential equations. The question of the validity of theorems of this type was proposed to me by Professor L. Hurwicz with a view to applications in mathematical economics. I wish to acknowledge helpful conversations and correspondence with him during the writing of this paper. Section 1 deals with a general definition of exterior derivatives; Sections 2 and 5 with functions and differential forms satisfying CarathCodory type conditions and smooth approximations for them; Section 3 is concerned with ordinary differential equations and the dependence of solutions on initial conditions under Carathkodory type conditions; the generalized Frobenius theorem is stated in Section 4 and proved in Sections 5-8; finally, Section 9 gives a partial answer to other questions of Professor Hurwicz concerning the convexity of solutions of a total differential system.