On the fractional differential equations (Q1194409)

The author deals with the semilinear differential equation \(d^ \alpha x(t)/dt^ \alpha=f(t,x(t))\), \(t>0\), where \(\alpha\) is any positive real number. In [Kyungpook Math. J. 28, No. 2, 119-122 (1988; Zbl 0709.34011)] the author has proved the existence, uniqueness, and some properties of the solution of this equation when \(0<\alpha<1\). Here he mainly studies (besides the other properties) the continuation of the solution of this equation to the solution of the corresponding initial value problem when \([\alpha]=k\), \(k=1,2,3,\dots\;\). Applications of singular integro- differential equations are considered.