Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the non-local conditions

Tatar

Description

A second-order abstract problem of neutral type with derivatives of non-integer order in the nonlinearity as well as in the nonlocal conditions is investigated. This model covers many of the existing models in the literature. It extends the integer order case to the fractional case in the sense of Caputo. A fixed point theorem is used to prove existence of mild solutions. AMS Subject Classification 26A33, 34K40, 35L90, 35L70, 35L15, 35L07

Sujets

• Cauchy problem
• Fractional calculus

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 4
• Langue: English

Extrait

TatarAdvances in Difference Equations2011,2011:18 http://www.advancesindifferenceequations.com/content/2011/1/18

R E S E A R C HOpen Access Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the nonlocal conditions

Nassereddine Tatar

Correspondence: tatarn@kfupm. edu.sa Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Abstract A secondorder abstract problem of neutral type with derivatives of noninteger order in the nonlinearity as well as in the nonlocal conditions is investigated. This model covers many of the existing models in the literature. It extends the integer order case to the fractional case in the sense of Caputo. A fixed point theorem is used to prove existence of mild solutions. AMS Subject Classification 26A33, 34K40, 35L90, 35L70, 35L15, 35L07 Keywords:Cauchy problem, Cosine family, Fractional derivative, Mild solutions, Neu tral secondorder abstract problem

1 Introduction In this paper, we investigate the following neutral secondorder abstract differential problem  d     Cα  u ( t ) +g(t,u(t),u(t)) =Au(t) +f t,u ( t ) ,D u ( t ) ,t∈I= [0,T dt   0Cβ (1) u ( 0 ) =u+p u,D u(t) ,   1Cγ u ( 0 ) =u+q u,D u ( t )

C with 0≤a,b,g≤1. Here, the prime denotes time differentiation andD,=a,b, gdenotes fractional time differentiation (in the sense of Caputo). The operatorAis the infinitesimal generator of a strongly continuous cosine familyC(t),t≥0 of bounded + linear operators in the Banach spaceXandf,gare nonlinear functions fromR×X× 0 12 XtoX,uanduare given initial data inX. The functionsp: [C(I;X)]®X,q: [C(I; 2 X)]®Xare given continuous functions (see the example at the end of the paper). This problem has been studied in casea,b,gare 0 or 1 (see [18]). Wellposedness has been established using different fixed point theorems and the theory of strongly continuous cosine families in Banach spaces. We refer the reader to [7,9,10] for a good account on the theory of cosine families. Fractional nonlocal conditions are the natural generalization of the integer order nonlocal conditions as studied by Hernandez [5] and others. They include the discrete case where the solution is prescribed at some finite number of times.

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