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

Furati

Description

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.

Sujets

• Fractional calculus

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 8
• Langue: English

Extrait

FuratiBoundary Value Problems2012,2012:58 http://www.boundaryvalueproblems.com/content/2012/1/58

R E S E A R C HOpen Access 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   α r β D(x−a)D y(x) =f(x,y),x>a, 0< α <1, 0≤β≤1,r< α, (1) a a α β where D a and D a 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 exam ple 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 under standing 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 Cau chytype problem with two fractional derivatives and bounded operator. He also pro posed a criterion for the uniform correctness of unbounded operator.

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