The generalized Green's function for boundary value problem of second order difference equation (Q492505)

Summary: Let \(b>a+2\) and \([a+1,b+1]=\{a+1,a+2,\dots,b+1\}\). In this paper, by building the generalized Green's function for the problems, we study the solvability of the S-L problem \(Lx=\Delta[p(t-1)\Delta x(t-1)] +[q(t)+\lambda r(t)]x(t)=-f(t)\), \(U_1(x)=\alpha_1x(a)+\alpha_2\Delta x(a)=0\), \(U_2(x)=\beta_1x(b+1)+\beta_2\Delta x(b+1)=0\), and the periodic S-L problem \(Lx=\Delta[p(t-1)\Delta x(t-1)]+[q(t)+\lambda r(t)]x(t)=-f(t)\), \(U_3(x)=x(a)-x(b+1)=0\), \(U_4(x)=\Delta x(a)-\Delta x(b+1)=0\), where the parameter \(\lambda\) is an eigenvalue of the linear problem \(Lx=0\), \(U_1(x)=0\), \(U_2(x)=0\) or the problem \(Lx=0\), \(U_3(x)=0\), \(U_4(x)=0\), and \(p\colon[a,b+1]\to (0,+\infty)\), \(r\colon[a+1,b+1]\to (0,+\infty)\), \(q(t)\) is defined and real valued on \([a+1,b+1]\), \(\alpha^2_1+\alpha^2_2\neq 0\), \(\beta^2_1+\beta^2_2\neq 0\), and in the periodic S-L problem we have \(p(a)=p(b+1)\).