Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the non-local conditions | Tatar Advances in Difference Equations 2011 2011 18 http content 2011 1 18 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the non-local conditions Nasser-eddine Tatar Correspondence tatarn@kfupm. Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals Dhahran 31261 Saudi Arabia SpringerOpen0 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 Neutral second-order abstract problem 1 Introduction In this paper we investigate the following neutral second-order abstract differential problem u t g t u t m t Au t f t u t CDau t e I 0 T u 0 u0 p u CD@u t 1 u 0 u 1 q u CDru t with 0 a b g 2 1. Here the prime denotes time differentiation and CDK K a b g denotes fractional time differentiation in the sense of Caputo . The operator A is the infinitesimal generator of a strongly continuous cosine family C t t 0 of bounded linear operators in the Banach space X and f g are nonlinear functions from R X X X X to X u0 and u1 are given initial data in X. The functions p C I X 2 X q C I X 2 X are given continuous functions see the example at the end of the paper . This problem has been studied in case a b g are 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 .
