I want to Construct a function $f(t,x(t),I_{x}(t),J_{x}(t))$ on $t\in I=[0,T_{0}],$ where, $$I_x(t)=\int_{0}^{t}a(t,s,x(s))ds,$$ $$J_x(t)=\int_{0}^{T_{0}}b(t,s,x(s))ds$$ under the restriction that
(1) f is Lipschitz with respect to spatial coordinate and $$\|I_x(t)-I_y(t)\|\leq M_a\|x-y\|$$ $$\|J_x(t)-J_y(t)\|\leq M_b\|x-y\|$$ where $M_a,M_b$ are positive constants.
(2)For all $u,v,w\in X,$ the function is strongly measurable on I and $f(t,.,.,)\in C(X\times X\times X,X)$ and there exists a function $m\in L(I, R^{+})$such that ${{{s_i}^{D}}_{t}}^{-\alpha}m\in C(J_i,R^{+})$ and $\lim_{t\rightarrow {s_{i}}^{+}}{{{s_i}^{D}}_{t}}^{-\alpha}m(t)=0$ $J_i=[s_i,t_{i+1}] and
$$\|f(t,u,v,w)\|\leq m(t).$$ Here $ {{{s_i}^{D}}_{t}}^{-\alpha}m(t)$ represent the Riemann Liouville integral of order $\alpha$,$ (0<\alpha<1)$ and $(X,\|\|)$ is a Banach space and $$ 0=t_0=s_0<t_1\leq s_1<...<t_{N+1}=T_0$$ are fixed number.