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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

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  • $\begingroup$ 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. $\endgroup$ – Fritz May 16 at 13:19
  • $\begingroup$ @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 $y^(1-α)$. $\endgroup$ – A.Z May 19 at 19:30
  • $\begingroup$ @Fritz thank you for the help :) $\endgroup$ – A.Z May 20 at 2:53

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