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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

\[ \frac{d^q}{dt^q}u(t)=A(t)~u(t),\quad 0 < t \leq T,\;0< q < 1 \text{ and } u(0) = u_0 \in X \tag{1} \]

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).

\[ \frac{d^q}{dt^q}u(t)=A(t)~u(t),\quad 0 < t \leq T,\;0< q < 1 \text{ and } u(0) = u_0 \in X \tag{1} \]

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).

Reviewer: Ahmed M. A. El-Sayed (Alexandria)

### MSC:

34A08 | Fractional ordinary differential equations |

34G10 | Linear differential equations in abstract spaces |

34G20 | Nonlinear differential equations in abstract spaces |

47N20 | Applications of operator theory to differential and integral equations |

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\textit{K. Balachandran} and \textit{J. Y. Park}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 10, 4471--4475 (2009; Zbl 1213.34008)

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