The Nonlocal Cauchy Problem In Banach Space

The nonlocal Cauchy problem in Banach space is an important branch of functional analysis and the theory of functional differential equations.Because of its wide application in biotechnology,physics,optimal control and other fields,the research on nonlocal problems has been rapid development.In recent years,the research of nonlocal Cauchy problem mainly focused on the existence of mild solution under Lipschitz continuous and the compactness conditions,while there is little research on the regularity of the mild solution.In this paper,the study mainly focus on semilinear nonlocal differential equations in Banach spaces (?)Where A is a closed linear operator on Banach space X,f:[0,T]× X ? X and g:C([0,T],X)?X are given maps.In the case of A generating CO semigroups(t>0,T(t)is continuous under?·?),we uses the Leray-Schauder fixed point theory to discuss the existence and regularity of mild solutions in analytic semigroup.