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Báo cáo toán học: " Existence of positive solutions to discrete secondorder boundary value problems with indefinite weight"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Existence of positive solutions to discrete secondorder boundary value problems with indefinite weight | Gao et al. Advances in Difference Equations 2012 2012 3 http www.advancesindifferenceequations.eom content 2012 1 3 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Existence of positive solutions to discrete second-order boundary value problems with indefinite weight Chenghua Gao Guowei Dai and Ruyun Ma Correspondence gaokuguo@163. com Department of Mathematics Northwest Normal University Lanzhou 730070 P. R. China Springer Abstract Let T 1 be an integer T 1 2 . T . This article is concerned with the global structure of the set of positive solutions to the discrete second-order boundary value problems A1 2u t - 1 rm t f u t 0 t e T u 0 u T 1 0 where r 0 is a parameter m T R changes its sign m t 0 for t e T and f R R is continuous. Also we obtain the existence of two principal eigenvalues of the corresponding linear eigenvalue problems. MSC 2010 39A12 34B18. Keywords discrete indefinite weighted problems positive solutions principal eigenvalue bifurcation existence 1 Introduction Let T 1 be an integer T 1 2 . T . This article is concerned with the global structure of the set of positive solutions to the discrete second-order boundary value problem BVP A2u t - 1 rm t f u t 0 t e T u 0 u T 1 0 1.1 1.2 where r 0 is a parameter f R R is continuous m t 0 for t e T and m T R changes its sign i.e. there exists a proper subset T of T such that m t 0 for t e T and m t 0 for t e T T . BVPs with indefinite weight arise from a selection-migration model in population genetics see Fleming 1 . That an allele A1 holds an advantage over a rival allele A2 at some points and holds an disadvantage over A2 at some other points can be presented by changing signs of m. The parameter r corresponds to the reciprocal of the diffusion. The existence and multiplicity of positive solutions of BVPs for second-order differential equations with indefinite weight has been studied by many authors see for example 2-5 and the references therein. In 2 using .