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

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