On the fractional differential equations. (English) Zbl 0757.34005

El-Sayed, A. M. A.

Appl. Math. Comput. 49, No. 2-3, 205-213 (1992).

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.

Reviewer: S.D.Bajpai (Isa Town)

MSC:

34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
26A33	Fractional derivatives and integrals

Keywords:

semilinear differential equation; continuation; initial value problem; singular integro-differential equations

Citations:

Zbl 0709.34011

References:

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[3]	El-Sayed, A.M.A., Fractional differential equations, Kyungpook math. J., 28, 2, (1988) · Zbl 0709.34011
[4]	Gelfand, I.M.; Shilov, G.E., Generalized functions, vol. I, (1958), Moscow · Zbl 0091.11103
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