﻿ 一类分数阶微分方程解的性质探讨

# 一类分数阶微分方程解的性质探讨Exploration on the Nature of Solutions for a Differential Equation of Fractional Order

Abstract: We prove existence and uniqueness of the solution of a nonlinear differential equation of fractional order. The differential operator is the Caputo fractional derivative. For the solvability of the equation is equivalent to a class of Volterra integral equation, we study the existence and uniqueness of the integral equation. We prove the existence of the solution of integral equation by Schau- der fixed point theorem and the uniqueness of the solution by contraction mapping principle.

[1] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.

[2] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.

[3] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach. Gordon and Breach Science Publishers, Yverdon.

[4] West, B.J., Bologna, M. and Grigolini, P. (2003) Physics of Fractal Operators. Springer, New York.
http://dx.doi.org/10.1007/978-0-387-21746-8

[5] Daftardar-Gejji, V. and Babakhani, A. (2004) Analysis of a System of Fractional Differential Equations. Journal of Mathematical Analysis and Applications, 293, 511-522.
http://dx.doi.org/10.1016/j.jmaa.2004.01.013

[6] Diethelm, K. and Ford, N.J. (2002) Analysis of Fractional Differential Equations. Journal of Mathematical Analysis and Applications, 265, 229-248.
http://dx.doi.org/10.1006/jmaa.2000.7194

[7] Delbosco, D. and Rodino, L. (1996) Existence and Uniqueness for a Nonlinear Fractional Differential Equation. Journal of Mathematical Analysis and Applications, 204, 609-625.
http://dx.doi.org/10.1006/jmaa.1996.0456

[8] El-Sayed, A.M.A. (1988) Fractional Differential Equations. Kyungpook Math. J, 28, 22-28.

[9] Kosmatov, N. (2009) Integral Equations and Initial Value Problems for Nonlinear Differential Equations of Fractional Order. Nonlinear Analysis: Theory, Methods & Applications, 70, 2521-2529.
http://dx.doi.org/10.1016/j.na.2008.03.037

[10] Caputo, M. (1967) Linear Models of Dissipation Whose Q Is Almost Frequency Independent (Part II). Geophysical Journal International, 13, 529-539.
http://dx.doi.org/10.1111/j.1365-246X.1967.tb02303.x

Top