tailieunhanh - Báo cáo nghiên cứu khoa học: "Periodic solutions of some linear evolution systems of natural differential equations on 2-dimensional tore"

Tuyển tập báo cáo nghiên cứu khoa học trường đại học quốc gia hà nội: Periodic solutions of some linear evolution systems of natural differential equations on 2-dimensional tore. | VNU Journal of Science Mathematics - Physics 26 2010 17-27 Periodic solutions of some linear evolution systems of natural differential equations on 2-dimensional tore Dang Khanh Hoi Department of General Education Hoa Binh University Tu Liem Hanoi Vietnam Received 3 December 2009 Abstract. In this paper we study periodic solutions of the equation 1 d A 4 aA i u x t fG u 1 with conditions II I ut b ị u x ĩ 1 dx 0 2 over a Riemannian manifold X. where Gu x i j g x y u y dy is an integral operator u x i is a differential form on X A i d- -ỗ is a natural differential operator in X. We consider the case when X is a tore n2. It is shown that the set of parameters ỉ b for which the above problem admits a unique solution is a measurable set of complete measure in c X . Keyworks and phrases Natural differential operators small denominators spectrum of compact operators. 1. Introduction Beside authors as from . Dezin see 1 considered the linear differential equations on manifolds in which includes the external differential operators. At research of such equations appear so named the small denominators so such equations is incorrect in the classical space. There is extensive literature on the different types of the equations in which appear small denominators. We shall note in particular work of . Ptashnika. see 2 This work further develops part of the authors result in 3 on the problem on the periodic solution to the equation in the space of the smooth functions on the multidimensional tore nra. We shall consider one private event when the considered manifold is 2-dimension tore n2 and the considered space is space of the smooth differential forms on n2. E-mail dangkhanhhoi@ 17 18 . Hoi VNU Journal of Science Mathematics - Physics 26 2010 17-27 We shall note that X-n-dimension Riemannian manifold of the class C00 is always expected oriented and close. Let Ệ OỆP OAP X 0C is the complexified cotangent bundle of manifolds X is the space of smooth differential