Existence of fractional parametric Cauchy problem (Q2901304)

By employing the concept of holomorphic flow, the author establishes the existence of solution of fractional parametric Cauchy problems in a complex Banach space of the form NEWLINE\[NEWLINED_z^\alpha u_t(z)=f_\alpha (t,z,u_t(z)),\;t\geq 0,\;0\leq \alpha<1,\;u_t(0)=0NEWLINE\]NEWLINE in the sense of Srivastava-Owa fractional operators. Moreover, by using the concept of admissible functions in complex Banach spaces, he shows that the solution remains in the unit disk.NEWLINENEWLINEDuring the last few decades fractional-order differential equations have emerged vigorously. We observe that there is much interest in developing the qualitative theory of such equations (see [\textit{R. P. Agarwal, M. Benchohra} and \textit{S. Hamani}, Acta Appl. Math. 109, No. 3, 973--1033 (2010; Zbl 1198.26004); \textit{R. P. Agarwal, M. Belmekki} and \textit{M. Benchohra}, Adv. Difference Equ. 2009, Article ID 981728 (2009; Zbl 1182.34103); \textit{R. P. Agarwal, B. de Andrade} and \textit{C. Cuevas}, ibid. 2010, Article ID 179750 (2010; Zbl 1194.34007)]). Indeed, this has strongly been motivated by their natural and widespread applicability in several fields of sciences and technology.