Document Type

Article

Publication Date

2015

Publication Source

Opuscula Mathematica

Abstract

The existence of bounded solutions, asymptotically stable solutions, and L1 solutions of a Caputo fractional differential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder's fixed point theorem and Liapunov's method have been employed. The existence of bounded solutions are obtained employing Schauder's theorem, and then it is shown that these solutions are asymptotically stable by a definition found in [C. Avramescu, C. Vladimirescu, On the existence of asymptotically stable solution of certain integral equations, Nonlinear Anal. 66 (2007), 472-483]. Finally, the L1 properties of solutions are obtained using Liapunov's method.

Inclusive pages

181-190

ISBN/ISSN

1232-9274

Document Version

Published Version

Comments

This document has been made available for download in accordance with the publisher's open-access policy.

Permission documentation on file.

Publisher

AGH University of Science and Technology

Volume

35

Issue

2

Place of Publication

Krakow, Poland

Peer Reviewed

yes

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