# Proving the existence and uniqueness of partial integro-differential equation

I am working on a type (shown below) of nonlinear partial integro-differential equation with conformable fractional derivative,

$$Τ_t^α u(x,t)=g(x,t)+u(x,t)+\int_0^t \int_0^x (f(y,s))^p/y^{1-α} s^{1-α} dyds,$$

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

$$T_t^α(f)(x)=\lim_{ε→0} \frac{f(t+εt^{1-α} )-f(t)}{ε}.$$
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

• Can you maybe edit your question and write down the mathematical definition of $T^\alpha$ and $I^\alpha$? I think not everyone is aware how these are defined. – Fritz May 16 at 13:19
• @Fritz I edited the question as you said, but I don't really know how to make a better code for the expressions involved like the $lim$ and $y^(1-α)$. – A.Z May 19 at 19:30
• @Fritz thank you for the help :) – A.Z May 20 at 2:53