TY  - JOUR
ID  - Islam2015
LB  - Islam2015
AU  - Islam, Muhammad N.
TI  - Bounded, asymptotically stable, and L<sup>1</sup> solutions of Caputo fractional differential equations
ST  - Bounded, asymptotically stable, and L<sup>1</sup> solutions of Caputo fractional differential equations
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2015
VL  - 35
IS  - 2
SP  - 181
EP  - 190
DO  - http://dx.doi.org/10.7494/OpMath.2015.35.2.181
UR  - http://dx.doi.org/10.7494/OpMath.2015.35.2.181
AB  - 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.
KW  - Caputo fractional differential equations
KW  - Volterra integral equations
KW  - weakly singular kernel
KW  - Schauder fixed point theorem
KW  - Liapunov's method
SN  - 1232-9274
ER  -