Differential equations on convex sets

where the function $g$ is only defined on a set of the form $C\times[0, a]$ for some convex set $C$ in a Banach space. The methods used are not new (see, e. g., [3], [8]), but the main result seems to have gone unnoticed and serves to clarify some of the theory of semi-groups of nonlinear transformations and the related theory of accretive mappings in Banach spaces. Simple (but basic) existence theorems for (1) are established in Section 1. Section 2 contains applications of these results to the theory of nonlinear pseudo-contractive and accretive operators. For aesthetic reasons, applications to the semi-group theory (where one must deal with ”multi-valued” mappings) are not given here.