The Generalized Green’s Function for Boundary Value Problem of Second Order Difference Equation

Abstract

Letb>a+2and[a+1,b+1]={a+1,a+2,…,b+1}. In this paper, by building the generalized Green’s function for the problems, we study the solvability of the S-L problemLx=Δ[p(t-1)Δx(t-1)]+[q(t)+λr(t)]x(t)=-f(t),U1(x)=α1x(a)+α2Δx(a)=0,U2(x)=β1x(b+1)+β2Δx(b+1)=0, and the periodic S-L problemLx=Δ[p(t-1)Δx(t-1)]+[q(t)+λr(t)]x(t)=-f(t),U3(x)=x(a)-x(b+1)=0,U4(x)=Δx(a)-Δx(b+1)=0,where the parameterλis an eigenvalue of the linear problemLx=0,U1(x)=0,U2(x)=0or the problemLx=0,U3(x)=0,U4(x)=0,andp:[a,b+1]→(0,+∞),r:[a+1,b+1]→(0,+∞),q(t)is defined and real valued on[a+1,b+1],α12+α22≠0,β12+β22≠0,and in the periodic S-L problem we havep(a)=p(b+1).