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

Abstract

The existence of bounded solutions, asymptotically stable solutions, and L 1 solu- tions of a Caputo fractional dierential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional dierential 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 the- orem 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 asymp- totically 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.