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Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. (English) Zbl 1213.34008

In the first part of the paper, the authors study the existence of a unique solution of the problem

d q dt q u(t)=A(t)u(t),0<tT,0<q<1andu(0)=u 0 X(1)

where (HA): A(t) is a bounded linear operator on a Banach space X for each tJ=[0,T]. The function tA(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)·(2)

Unfortunately, this claim is not true. This is due to the fact that, since the derivative of u(t) in (2), d dtu(t) does not exist, consequently, the fractional order derivative 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).

34A08Fractional differential equations
34G10Linear ODE in abstract spaces
34G20Nonlinear ODE in abstract spaces
47N20Applications of operator theory to differential and integral equations