For any $\beta>0$ we denote the Riemann-Liouville fractional integral operator of order $\beta$ by
$$ I^{\beta}(f)(t)=\frac{1}{\Gamma(\beta)} \int_{0}^{t}(t-\tau)^{\beta-1} f(\tau) d \tau, \quad f \in L^{1}(0, T), \text { a.e. } t \in(0, T),$$
where $T>0$ and $\Gamma(\beta)=\int_{0}^{\infty} t^{\beta-1} e^{-t} d t$ is the Euler Gamma function.
The Caputo fractional derivative of order $\alpha \in(1,2)$ is given by $$ \partial_{t}^{\alpha} f(t)=\frac{1}{\Gamma(2-\alpha)} \int_{0}^{t}(t-\tau)^{1-\alpha} \frac{d^{2} f}{d \tau^{2}}(\tau) d \tau $$
Let the following equation:
$$ \partial_{t}^{\alpha} u(t,x) + A(u(t,x)) = f(t,x)$$ where $A$ is a differential operator non linear.
The question is: How to prove the existence and uniqueness of this equation