Geodesic
On continuous solutions of the Cauchy problem for equations of fractional order
Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2018), pp. 39-45

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It is studied the nonlocal conditions of solving Cauchy-type problem for fractional differential equations with Riemann – Liouville derivatives in some special function space. The Cauchy problem is reduced to a the finding fixed point of an integral operator A, then is constructed an invariant set for $A$ (the «shift» of a ball from the space of continuous functions, and then it is applied the Schauder anf Banach – Caccioppoli fixed point principles. As a result, the conditions of solvability and unique solvability for the Cauchy problem under consideration are given.
Keywords: Cauchy problem; fractional Riemann – Liouville derivative; the Schauder fixed point principle; the Banach – Сaccioppoli fixed point principle. 
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     author = {P. P. Zabreiko and S. Ponomareva},
     title = {On continuous solutions of the {Cauchy} problem for equations of fractional order},
     journal = {Journal of the Belarusian State University. Mathematics and Informatics},
     pages = {39--45},
     publisher = {mathdoc},
     volume = {3},
     year = {2018},
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     url = {http://geodesic.mathdoc.fr/item/BGUMI_2018_3_a4/}
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P. P. Zabreiko; S. Ponomareva. On continuous solutions of the Cauchy problem for equations of fractional order. Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2018), pp. 39-45. http://geodesic.mathdoc.fr/item/BGUMI_2018_3_a4/