Nonlocal Cauchy problem for abstract fractional semilinear evolution equations (Q838042)

In the first part of the paper, the authors study the existence of a unique solution of the problem \[ \frac{d^q}{dt^q}u(t)=A(t)~u(t),\quad 0 < t \leq T,\;0< q < 1 \text{ and } u(0) = u_0 \in X \tag{1} \] where (HA): \(A(t)\) is a bounded linear operator on a Banach space \(X\) for each \(t \in J=[0,T]\). The function \(t\rightarrow A(t)\) is continuous in the uniform operator topology. They claim that problem (1) is equivalent to the integral equation \[ u(t)=u_0+I^{q} A(t)u(t). \tag{2} \] Unfortunately, this claim is not true. This is due to the fact that, since the derivative of \(u(t)\) in (2), \(\frac{d}{dt}u(t)\) does not exist, consequently, the fractional order derivative \(\frac{d^q}{dt^q}u(t)\) does not exist. So, there is no solution for problem (1). The second part contains the same error for the nonlinear case of (1).