Extremal Problems for Mappings with Generalized Parametric Representation in $${{mathbb {C}}}^{n}$$

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