EXISTENCE AND STRUCTURE OF SOLUTION SETS FOR NONLINEAR DISCRETE BOUNDARY VALUE PROBLEMS WITH SPECTRAL PARAMETERS IN BOUNDARY CONDITIONS

By using topological degree methods and bifurcation theory, we prove the existence of the solution and the bifurcation structure at infinity of the second-order difference boundary value problems $ \left\{ \begin{array}{ll} -\Delta^{2}u(k-1)+c(k)u(k)=0,\quad k\in[1,N ]_{\mathbb{Z}}, \\-\Delta u(0)=(\mu_1+\lambda)\sigma(1)u(1)+f(1,u(1))+h(1),\\ \Delta u(N)=(\mu_1+\lambda)\sigma(N+1)u(N+1)+f(N+1,u(N+1))+h(N+1), \end{array} \right. $ where $ \lambda\in\mathbb{R} $ is a constant, $ \mu_1 $ is a spectral parameter, $ \Delta u(k):=u(k+1)-u(k) $ and $ \Delta^{2}u(k-1):=\Delta(\Delta u(k-1)) $ as the forward difference operator and the second difference operator, respectively.