The generalized Green's function for boundary value problem of second order difference equation

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],α1 2+α2 2≠0,β1 2+β2 2≠0,and in the periodic s-l problem we have p(a)=p(b+1). © 2015 xiaoling han and juanjuan huang.