Extremal Problems for Mappings with Generalized Parametric Representation in Cn

In this paper we are concerned with the family (S) over tilde (t)(A)(B-n) (t >= 0) of normalized biholomorphic mappings on the Euclidean unit ball B-n in C-n that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators A is an element of <A)over tilde>, where <A)over tilde> is a family of measurable mappings from [0, infinity) into L(C-n) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family (S) over tilde (t)(A)(B-n), where A is an element of <A)over tilde>. We prove that if f(z,t) = V (t) (-1)z + ... is a normal Loewner chain such that V (s)f (., s) is an element of ex SA(B-n) (resp. (s)f(., S) is an element of supp (S) over tilde (s)(A)), then V(t)f(., t) is an element of ex (S) over tilde (s)(A)(B-n), for all t >= s (resp. V(t)f(.,t) E supp (S) over tilde (s)(A) (B-n), for all t >= s), where V(t) is the unique solution on [0, infinity)) of the initial value problem: dV/dt(t) = -A (t) V (t), a.e. t >= 0, V (0) = I-n. Also, we obtain an example of a bounded support point for the family (S) over tilde (t)(A)(B-2), where A is an element of <A)over tilde> is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators A is an element of <A)over tilde>, and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on Useful examples and applications yield that the study of the family (S) over tilde (t)(A) (B-n) for time-dependent operators A is an element of <A)over tilde> is basically different from that in the case of constant time-dependent linear operators.