Let $I\subseteq \mathbb{R}$ be an interval and $E\subseteq \mathbb{R}^n$ and let $f_1,f_2,...f_n :I\times E \to \mathbb{R} $ be an given continuous functions
consider the system of nonlinear equations
$$y'=f_1(t,x_1,x_2,....x_n)\\y'=f_2(t,x_1,x_2,....x_n)\\.........................\\y'=f_n(t,x_1,x_2,....x_n)$$--------------(1)
Denoting (column) vector $x$ with components $x_1, x_2 ....x_n$ and vector$ f$ with components $f_1, f2..... f_n$, the system of equations (1) assumes the form $x'=f(t,x),x(t_0)=x_0$
how to prove this system of equation have uniqueness solution by Banach fixed point theorem...
can you someone suggest me good book for understand these topics....thank you