x=A(t)x where A(t)Rnn. Suppose system (1) is uniformly asymptotically stable with respect to the zero solution. Let f be continuous for (t,x) in RRn and for any >0, there is a >0 such that f(t,x)x for x0. Suppose limtb(t)=0. Prove that there exists a T> such that any solution x(t) of x=A(t)x+f(t,x)+b(t) approaches to zero as t if x(T) is small enough. (1\%) State your method and explain why it works. (1\%) Show your proof. (2\%) Generalize the result of Problem I with b(t) replaced by g(t,x) where g(t,x)0 as t uniformly for x in compact sets. (1\%) State your method and explain why it works. (1\%) Show your proof. x=A(t)x where A(t)Rnn. Suppose system (1) is uniformly asymptotically stable with respect to the zero solution. Let f be continuous for (t,x) in RRn and for any >0, there is a >0 such that f(t,x)x for x0. Suppose limtb(t)=0. Prove that there exists a T> such that any solution x(t) of x=A(t)x+f(t,x)+b(t) approaches to zero as t if x(T) is small enough. (1\%) State your method and explain why it works. (1\%) Show your proof. (2\%) Generalize the result of Problem I with b(t) replaced by g(t,x) where g(t,x)0 as t uniformly for x in compact sets. (1\%) State your method and explain why it works. (1\%) Show your proof