chapter  9
63 Pages

Chapter 9

Boundary Value Problems Nonlinear Systems

Definition 9.1.1. Let E be a real vector space. A genemlized norm on E is a mapping ll·llo: E-+ R~ denoted by llullo = (o1(u),· · ·,on(u)) such that

(i) llullo ;::: 0, i.e. o;(u) ;::: 0 for all i (ii) llullo = 0 if and only if u = 0, i.e. o;(u) = 0 for all i if and only if u = 0 (iii) 11-Xullo = 1-XIIIullo, i.e. o;(.Xu) = 1-XIo;(u) for all i (iv) llu +vile~ llullo + llvllo, i.e. o;(u + v) ~ o;(u) + o;(v) for all i.