The author deals with the semilinear differential equation $d\sp \alpha x(t)/dt\sp \alpha=f(t,x(t))$, $t>0$, 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.