tailieunhanh - Báo cáo hóa học: " Solvability of boundary value problems with Riemann-Stieltjes Δ-integral conditions for second-order dynamic equations on time scales at resonance"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài : Solvability of boundary value problems with Riemann-Stieltjes Δ-integral conditions for second-order dynamic equations on time scales at resonance | Li and Shu Advances in Difference Equations 2011 2011 42 http content 2011 1 42 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Solvability of boundary value problems with Riemann-Stieltjes A-integral conditions for second-order dynamic equations on time scales at resonance Yongkun Li and Jiangye Shu Correspondence yklie@. cn Department of Mathematics Yunnan University Kunming Yunnan 650091 People s Republic of China Springer Abstract In this paper by making use of the coincidence degree theory of Mawhin the existence of the nontrivial solution for the boundary value problem with Riemann-Stieltjes A-integral conditions on time scales at resonance xAA t f t x t xA t e t . t e 0 T t T xA 0 0 x T f x s Ag s 0 is established where f 0 T t xIRxR R satisfies the Carathéodory conditions and e 0 T t R is a continuous function and g 0 T t R is an increasing function with f0TT Ag s 1 .An example is given to illustrate the main results. Keywords boundary value problem with Riemann-Stieltjes A A -integral conditions resonance time scales 1 Introduction Hilger 1 introduced the notion of time scales in order to unify the theory of continuous and discrete calculus. The field of dynamical equations on time scales contains links and extends the classical theory of differential and difference equations besides many others. There are more time scales than just R corresponding to the continuous case and N corresponding to the discrete case and hence many more classes of dynamic equations. An excellent resource with an extensive bibliography on time scales was produced by Bohner and Peterson 2 3 . Recently existence theory for positive solutions of boundary value problems BVPs on time scales has attracted the attention of many authors Readers are referred to for example 4-11 and the references therein for the existence theory of some two-point BVPs and 12-17 for three-point BVPs on time scales. For the .