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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).

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