Opuscula Math. 35, no. 2 (2015), 181-190
http://dx.doi.org/10.7494/OpMath.2015.35.2.181

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.

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1. F.M. Atici, P.W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), 981-989.
2. C. Avramescu, C. Vladimirescu, On the existence of asymptotically stable solution of certain integral equations, Nonlinear Anal. 66 (2007), 472-483.
3. L.C. Becker, Resolvents and solutions of weakly singular linear Volterra integral equations, Nonlinear Anal. 74 (2011), 1892-1912.
4. T.A. Burton, Liapunov Theory for Integral Equations with Singular Kernels and Fractional Differential Equations, CreateSpace Independent Publishing Platform, 2012.
5. T.A. Burton, Bo Zhang, $$L^p$$-solutions of fractional differential equations, Nonlinear Stud. 19 (2012), 307-324.
6. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, New York, 2004.
7. C.M. Kirk, W.E. Olmstead, Blow-up solutions of the two-dimensional heat equation due to a localized moving source, Anal. Appl. 3 (2005), 1-16.
8. V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamic System, Cambridge Scientific Publishers, Cambridge, 2009.
9. W.R. Mann, F. Wolf, Heat transfer between solids and gases under nonlinear boundary conditions, Quart. Appl. Math. 9 (1951), 163-184.
10. R.K. Miller,Nonlinear Volterra Integral Equations, Benjamin, New York, 1971.
11. R.S. Nicholson, I. Shain, Theory of stationary electrode polography, Analytical Chemistry 36 (1964), 706-723.
12. H.F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods, Blasidell, New York, 1965.
• University of Dayton, Department of Mathematics, Dayton, OH 45469-2316 USA
• Communicated by Theodore A. Burton.
• Revised: 2014-03-14.
• Accepted: 2014-05-14.
• Published online: 2014-11-18.