I am working on a type (shown below) of nonlinear partial integro-differential equation with conformable fractional derivative,
$$Τ_t^α u(x,t)=g(x,t)+u(x,t)+\int_0^t \int_0^x \dfrac{(f(y,s))^p}{y^{1-α} s^{1-α}} dyds,$$
where $(x,t)∈[0,1]×[0,L],$ $α∈(1/2,1),$ with the initial condition $u(x,0)=b(x),$ where $x,t$ are independent variables, $u(x,t)$ is an unkown function, $g(x,t), b(x)$ are a known functions, $p≥1$ is a positive integer, $T^α$ is the conformable fractional derivative of order $α$ (which is defined below).
Def: Given a function $f:[0,∞)→R$. Then the conformable fractional derivative of order α is defined by
$$T_t^α(f)(t)=\lim_{ε→0} \frac{f(t+εt^{1-α} )-f(t)}{ε}.$$
for all $t>0,α∈(0,1)$.
I want to prove the existence and uniqueness of this equation via the Banach fixed point theorem and my main question here is what norm should I take to make the proof.
any help will be appreciated