Geodesic
Modified fractional integrals and derivatives in the half-axis and differential equations of fractional order in the space of integrable functions
Trudy Instituta matematiki, Tome 15 (2007) no. 1, pp. 68-77 
A. A. Kilbas; N. V. Kniaziuk. Modified fractional integrals and derivatives in the half-axis and differential equations of fractional order in the space of integrable functions. Trudy Instituta matematiki, Tome 15 (2007) no. 1, pp. 68-77. http://geodesic.mathdoc.fr/item/TIMB_2007_15_1_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Fractional integrals and derivatives of positive order, being modifications of the Liouville fractional integrals and derivatives on the half-axis, are introduced. The properties of such modified constructions are investigated in the space of integrable functions. The results obtained are applied to prove the equivalence of the Cauchy-type problem for the nonlinear differential equation of fractional order and of the nonlinear Volterra integral equations. The Banach fixed point theorem is used to prove the existence and uniqueness of a solution of the above Cauchy-type problem. Particular cases are considered and examples are given.

[1] Samko S.G., Kilbas A.A., Marichev O.I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987 | MR | Zbl

[2] Kilbas A.A., Bonilla B., Trukhillo Kh., “Drobnye integraly i proizvodnye, differentsialnye uravneniya drobnogo poryadka v vesovykh prostranstvakh nepreryvnykh funktsii”, Dokl. NAN Belarusi, 44:6 (2000), 18–22 | MR | Zbl

[3] Kilbas A.A., Rivero M., Trujillo J.J., “Existence and uniqueness theorems for differential equations of fractional order in weighted spaces of continuous functions”, Fract. Calc. Appl. Anal., 6:4 (2003), 363–399 | MR | Zbl

[4] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equation, Elsevier, Amsterdam, 2006 | MR | Zbl

[5] Kilbas A.A., Trujillo J.J., “Differential equation of fractional order: Methods, results and problems – 1”, Appl. Anal., 78:1-2 (2001), 153–192 | DOI | MR | Zbl

[6] Oldham B., Spanier J., The Fractional Calculus, Academic Press, New York, 1974 | MR | Zbl

[7] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999 | MR | Zbl

[8] Kolmogorov A.N., Fomin S.V., Osnovy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1988 | Zbl