Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Smoothness property on parameters of periodic systems Min He | He Journal of Inequalities and Applications 2011 2011 106 http content 2011 1 106 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open Access Smoothness property on parameters of periodic systems Min He Correspondence mhe@ Kent State University at Trumbull Warren OH 44483 USA Abstract This work is concerned with periodic systems dependent on parameters and investigates differentiability with respect to parameters of the periodic solutions of the systems. Some challenging situations arise from a hyperbolic type of periodic boundary value problem. Using some auxiliary operators and applying semigroup theory and a fixed point theorem we are able to handle these cases and obtain the results on the existence and differentiability with respect to parameters of periodic solutions. The application of the obtained abstract results to a periodic boundary value problem is discussed at the end of the article. AMS Subject Classification 47D62 45K05 35L20. Keywords C0-semigroup periodic system parameter differentiability 1 Introduction Our recent work 1 studied the following periodic system dependent on parameter I-J A e z t f Í z t at 1 1 z 0 Z0. and obtained a set of results for the existence of periodic solutions and the differentiability with respect to parameter of such solutions. Those results are effectively applied to parabolic type of periodic systems. The operator of such systems generates an analytic C0-semigroup T t which possesses some nice properties such as a the C0-semigroup T t satisfies the contraction condition b the C0-semigroup T t is differentiable with respect to parameter on the entire space see 2 for details . And these two properties are the key for determining the existence and differentiability with respect to parameter of the periodic solution. Hence the conditions of the obtained theorems in 1 are proposed for determining that the system has these two properties. However .
