Differential equations and Optimal control

On continuous solutions of the Cauchy problem for equations of fractional order

P. P. Zabreiko, S. Ponomareva

Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

Abstract: 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.

UDC: 517.955:519.622

Received: 11.06.2018