Opuscula Mathematica
Opuscula Math. 35, no. 2 (), 181-190
Opuscula Mathematica

Bounded, asymptotically stable, and L1 solutions of Caputo fractional differential equations

Abstract. The existence of bounded solutions, asymptotically stable solutions, and \(L^1\) 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 \(L^1\) properties of solutions are obtained using Liapunov's method.
Keywords: Caputo fractional differential equations, Volterra integral equations, weakly singular kernel, Schauder fixed point theorem, Liapunov's method.
Mathematics Subject Classification: 34K20, 45J05, 45D05.
Cite this article as:
Muhammad N. Islam, Bounded, asymptotically stable, and L1 solutions of Caputo fractional differential equations, Opuscula Math. 35, no. 2 (2015), 181-190, http://dx.doi.org/10.7494/OpMath.2015.35.2.181
Download this article's citation as:
a .bib file (BibTeX), a .ris file (RefMan), a .enw file (EndNote)
or export to RefWorks.

RSS Feed

horizontal rule

ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
Copyright © 2003−2017 OPUSCULA MATHEMATICA
Contact: opuscula@agh.edu.pl
Made by Tomasz Zabawa

horizontal rule

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.