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: Abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm | Wang et al. Advances in Difference Equations 2011 2011 25 http content 2011 1 25 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm Rong-Nian Wang Jun Liu and De-Han Chen Correspondence rnwang@mail. Department of Mathematics NanChang University NanChang JiangXi 330031 People s Republic of China Springer Abstract In the present paper we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions ensuring the existence of mild solutions are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A Krasnoselkii s fixed point theorem and Schauder s fixed point theorem. An application to a fractional partial integrodifferential equation with nonlocal initial condition is also considered. Mathematics subject classification 2000 26A33 34G10 34G20 Keywords Cauchy problem of abstract fractional evolution equation Nonlocal initial condition Fixed point theorem Mild solution a-norm 1 Introduction Let A D A be the infinitesimal generator of a compact analytic semigroup of bounded linear operators T t i 0 on a real Banach space X . and 0 e r A . Denote by Xa the Banach space D Aa endowed with the graph norm u a Aau for u e Xa. The present paper concerns the study of the Cauchy problem for abstract fractional integro-differential equation involving nonlocal initial condition Du t Au t F t u t u Ki t - t K t - s G s u s u 2 s ds t e 0 T u 0 H u u0 in Xa where cDf 0 b 1 stands for the Caputo fractional derivative of order b and K 0 T R 1 . 2 0 T 0 T F G 0 T X Xa X Xa X H C 0 T Xa Xa are given functions to be specified later. As can .
