People, could you help me with this question about Picard's iteration?
Show that if $f(t,y)$ and $\frac{\partial f}{\partial y}$ are continuous in the rectangle $$R=\{(t,y)\in \mathbb{R}^{2}: \alpha<t<\beta,\, \delta <y<\gamma\},$$ and $a,b$ are positive constants such that $$ |f(t,y)|\leq b,~~ |f(t,y)-f(t,z)| \leq a|y-z| $$ for $\alpha<t<\beta$, $\delta <y,z<\gamma$, then there exists $\alpha', \beta'$, with $\alpha \leq \alpha'<t_{0}<\beta\leq \beta'$ such that the sequence $$ y_{0}(t)=y_{0},~~ y_{n}(t)=y_{0}+\int_{t_{0}}^{t}f(s,y_{n-1}(s)ds,~ n=1,2,... $$ satisfies $\delta < y_{n} < \gamma$, whenever $\alpha'<t<\beta'$.
Hint: Show that, $$ |y_{n}(t)-y_{0}| \leq \left(\frac{b}{a}-1\right)e^{a|t-t_{0}|} . $$
Why does exist $\alpha', \beta'$ and how to use the hint? Please helpe me! Thanks!
Note that on any solution $$ |y'(t)|\le |f(t,y_0)|+|f(t,y(t))-f(t,y_0)|\le b+a\,|y(t)-y_0| $$ Now compare to the solutions of $$ \pm u'=b+a\,u. $$