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On the fractional differential equations. (English) Zbl 0757.34005

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.

MSC:

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

Citations:

Zbl 0709.34011
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References:

[1] Apostol, T.M., Mathematical analysis, (1974), Addison-Wesley Publishing Company, Inc · Zbl 0126.28202
[2] Curtain, R.F.; Prichard, A.J., Functional analysis in modern applied mathematics A.P., (1977)
[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
[5] Shilove, G.E., Generalized functions and partial differential equations, Mathematics and its applications, (1968), Science Publishers, Inc
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