A Cauchy-type problem with a sequential fractional derivative in the space of continuous functions

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A Cauchy-type problem with a sequential fractional derivative in the space of continuous functions Khaled M Furati Correspondence: kmfurati@kfupm. edu.sa Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

Abstract A Cauchy-type nonlinear problem for a class of fractional differential equations with sequential derivatives is considered in the space of weighted continuous functions. Some properties and composition identities are derived. The equivalence with the associated integral equation is established. An existence and uniqueness result of the global continuous solution is proved. AMS Subject Classification: 26A33; 34A08; 34A34; 34A12; 45J08. Keywords: fractional derivatives, Riemann-Liouville fractional derivative, sequential fractional derivative, fractional differential equation

1 Introduction We consider a Cauchy-type problem associated with the equation   Dαa (x − a)r Dβa y(x) = f (x, y),

x > a,

0 < α < 1,

0 ≤ β ≤ 1,

r < α,

(1)

where Dαa and Dβa are the Riemann-Liouville fractional derivatives. In recent years there has been a considerable interest in the theory and applications of fractional differential equations. As for the theory, the investigations include the existence and uniqueness of solutions, asymptotic behavior, stability, etc. See for example the books [1-3] and the articles [4-10] and the references therein. As for the applications, fractional models provide a tool for capturing and understanding complex phenomena in many fields. See for example the surveys in [1,11] and the collection of applications in [12]. Some recent applications include control systems [13,14], viscoelasticity [15-18], and nanotechnology [19]. Also fractional models are used to model a vibrating string [20], and anomalous transport [21], anomalous diffusion [22-24]. Another field of applications is in random walk and stochastic processes [25-27] and its applications in financial modeling [28-30]. Other physical and engineering processes are given in [31,32] In a series of articles, [33-35], Glushak studied the uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator. He also proposed a criterion for the uniform correctness of unbounded operator.

© 2012 Furati; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Furati Boundary Value Problems 2012, 2012:58 http://www.boundaryvalueproblems.com/content/2012/1/58

Page 2 of 14

In this article we prove an existence and uniqueness result for a nonlinear Cauchytype problem associated with the Equation (1) in the space of weighted continuous functions. We start with some preliminaries in Section 2. In Section 3 we define the sequential derivative and develop some properties and composition identities. I