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

Abstract EN

Let b>a+2 and [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 problem Lx=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 problem Lx=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 parameter λ is an eigenvalue of the linear problem Lx=0, U1(x)=0, U2(x)=0 or the problem Lx=0, U3(x)=0, U4(x)=0, and p:[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 have p(a)=p(b+1).