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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Multiple unbounded solutions for a boundary value problem on infinite intervals | Lian and Geng Boundary Value Problems 2011 2011 51 http www.boundaryvalueproblems.eom content 2011 1 51 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Multiple unbounded solutions for a boundary value problem on infinite intervals Hairong Lian and Fengjie Geng Correspondence lianhr@126.com Schoolof Information Engineering China University of Geoscience Beijing 100083 People s Republic of China Springer Abstract This paper is concerned with the existence of multiple unbounded solutions for a Sturm-Liouville boundary value problem on the half-line. By assuming the existence of two pairs of unbounded upper and lower solutions the existence of at least three solutions is obtained using the degree theories. Nagumo condition plays an important role in the nonlinear term involved in the first-order derivative. It is an interesting point that the method of unbounded upper and lower solutions is extended to obtain conditions for the existence of multiple solutions. Mathematics Subject Classification 2000 34B10 34B40 Keywords infinite interval problem multiplicity unbounded upper solutions unbounded lower solutions degree theory 1 Introduction In this paper we will employ the method of unbounded upper and lower solutions to study the existence of Sturm-Liouville boundary value problem on the half-line i u t 0 t f t u t u t 0 t e 0 rc u 0 - au 0 B u ix C 1 where j 0 ro 0 ro f 0 X R2 R are continuous a 0 B C e R. The method of upper and lower solutions is a powerful tool to prove the existence of differential equation subject to certain boundary conditions. It is well known that nonlinear problems always have at least one solution in the ordered interval defined by one pair of well-ordered upper and lower solutions. To show this kind of result we can employ the topological degree theory or monotone iterative technique etc see 1-5 and the reference therein. Boundary value problems to differential equations on the half-line arise naturally in the study of