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Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Caratheodory operator of differential forms | Tang and Zhu Journal of Inequalities and Applications 2011 2011 88 http www.journalofinequalitiesandapplications.eom content 2011 1 88 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Caratheodory operator of differential forms Zhaoyang Tang and Jianmin Zhu Correspondence tzymath@gmail. com Department of Mathematics and System Science National University of Defense Technology Changsha PR China Springer Abstract This article is devoted to extensions of some existing results about the Caratheodory operator from the function sense to the differential form situation. Similarly as the function sense we obtain the convergence of sequences of differential forms defined by the Caratheodory operator. The main result in this article is the continuity and mapping property from one space of differential forms to another under some dominated conditions. Keywords differential forms Caratheodory operator continuity of operator 1 Introduction It is well known that differential forms are generalizations of differentiable functions in RN and have been applied to many fields such as potential theory partial differential equations quasi-conformal mappings nonlinear analysis electromagnetism and control theory 1-12 . One of the important work in the field of differential forms is to develop various kinds of estimates and inequalities for differential forms under some conditions. These results have wide applications in the A-harmonic equation which implies more versions of harmonic equations for functions 1 5 6 . The Caratheodory operator arose from the extension of Peano theorem about the existence of solutions to a first-order ordinary differential equation which says that this kind equation has a solution under relatively mild conditions. It is very interesting to characterize equivalently the Caratheodory s conditions and the continuity of Car-atheodory operator which form classic examples to discuss boundedness and continuity of nonlinear .
