Existence of positive solutions to discrete second-order boundary value problems with indefinite weight

Gao Chenghua, Dai Guowei, Ma, Ma Ruyun

Description

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 Δ 2 u ( t - 1 ) + r m ( t ) f ( u ( t ) ) = 0 , t ∈ T , u ( 0 ) = u ( T + 1 ) = 0 , where r ≠ 0 is a parameter, m : T → ℝ changes its sign, m ( t ) ≠ 0 for t ∈ 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.

Sujets

• Bifurcation
• Existence

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 6
• Langue: English

Extrait

Gaoet al.Advances in Difference Equations2012,2012:3 http://www.advancesindifferenceequations.com/content/2012/1/3

R E S E A R C HOpen 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

Abstract LetT> 1 be an integer, = 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 u(t−1) +rm(t)f(u(t)) = 0,t∈ T u0 =u T+ 1= 0,

wherer≠0 is a parameter, T → changes its sign,m(t)≠0 for t T andf:ℝ ®ℝis 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 eigen value, bifurcation, existence

1 Introduction LetT> 1 be an integer, 2, ...,= 1,T . This article is concerned with the global struc ture of the set of positive solutions to the discrete secondorder boundary value pro blem (BVP) 2 u t−1 +u trm t f= 0,t∈ T (1:1)

u0 =u T= 0+ 1 (1:2) wherer≠0 is a parameter,f:ℝ®ℝis continuous,m(t)≠0 for t and T → such thatof ,changes its sign, i.e., there exists a proper subsetm(t) > 0 for t∈ T andm(t) < 0 fort∈ T T . BVPs with indefinite weight arise from a selectionmigration model in population genetics, see Fleming [1]. That an alleleA 1 holds an advantage over a rival alleleA 2 at some points and holds an disadvantage overA 2 at some other points can be presented by changing signs ofm. The parameterrcorresponds to the reciprocal of the diffusion. The existence and multiplicity of positive solutions of BVPs for secondorder differen tial equations with indefinite weight has been studied by many authors, see, for exam ple [25] and the references therein. In [2], using CrandallRabinowitz’s Theorem and Rabinowitz’s global bifurcation theorem, Delgado and Suárez obtained the existence © 2012 Gao et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.