The Cauchy problem for certain generalized differential equations of second order

The present paper concerns some generalized differential equations of second order for mappings from a subset of Banach space into a Banach space. The subject matter refers to studies of generalized differential equations of the first order submitted in [4]. Let X,Y be Banach spaces over the field R and let U, V be open subsets of X and V, respectively. Let h1, h2 be mappings from U into X. In this paper we study the Cauchy problem D2f(x)(h2(x)) = F{x, Df(x)(h1(x))), x € U for mappings from a subset of a Banach space into a Banach space, which are defined in C class, at a neighbourhood of nonsingular point (that is, at a neighbourhood of such point xo for whichh1(xo)= 0 and h2(xo= 0).