Proof regarding the Picard iterative process for ODE

Well, first of all, this is my first question and i hope that someone can help me and my classmates with this problem

Show if $\left|\varphi_0(t)- \varphi(t) \right| \leq c \left|t-t_0\right|^d,$ $a \leq t,$ $t_0 \leq b$, $c>0$, $d>0$, then $$\left| \varphi_n(t)- \varphi(t) \right| \leq c \left|t-t_0 \right|^{d+1} \left\{ \dfrac{K}{d+1} + \dfrac{ K^2}{(d+1)(d+2)}+ \cdots \right\}$$

Always when $\varphi(t)$ is in $$G^1=\left\{ (t,x)| \quad \left|t-t_0 \right| \leq c, \quad\left|x-x_0 \right| \leq M\left|t-t_0 \right| \right\}$$ i.e. a close and bounded domain in $G= \left\{(t,x)| \quad \left|t-t_0 \right| \leq a, \quad \left|x-x_0\right| \leq b \right\}$\

Edit: the problem is $$\dot{x}(t)=f(t,x) \quad x(t_0)=x_0$$ With $f$ a Lipschitz function with constant $K$, and $$\varphi_{n+1}=x_0 + \int_{t_0}^t f(s, \varphi_n(s))ds$$ $$\varphi_0=x_0$$

• How do you define $\phi_n(t)$? – iamvegan Aug 22 '16 at 0:34
• I haved edited!, is the sucesion of picard – Rogelio Arancibia Bustos Aug 22 '16 at 0:44
• What are your thoughts or attempts on this problem? – Cyclohexanol. Aug 22 '16 at 1:11
• I passed almost two weeks trying to show this result , and well , I'm starting to thinking that problem is wrong , so, I came her for help – Rogelio Arancibia Bustos Aug 22 '16 at 1:18
• I'm not 100%, but Gronwalls inequality might help. – Mattos Aug 22 '16 at 1:48