Terminal value problems of fractional evolution equations (Q2214412)

The authors study the terminal value problems for a class of nonlinear fractional evolution equations with Weyl-Liouville derivative. It is claimed that by using the Fourier transform, the problem is converted into a singular integral equation on infinite interval and this integral equation is termed as mild solution of the fractional differential equation. This representation is similar to the one obtained for Caputo derivative in [\textit{M. M. El-Borai}, Chaos Solitons Fractals 14, No. 3, 433--440 (2002; Zbl 1005.34051)] and it is true when \(A\) is bounded. The difficulty of unbounded case is discussed in [\textit{K. Balachandran, R. Mabel Lizzy} and \textit{J. J. Trujillo}, J. Appl. Nonlinear Dyn. 8, 677--687 (2019)]. Sufficient conditions are obtained to ensure the existence of a mild solution when the semigroup is compact or noncompact. The results are established by using the Schauder and Darbo-Sadovskii fixed point theorems. An example is provided without details to justify an existence result.