Lower and upper solutions for a discrete firstorder nonlocal problems at resonance

We discuss the existence of solutions for the discrete firstorder nonlocal problem [ begin{cases} Delta u(t 1) = f(t, u(t)),quad t in {1, 2, ... , T}, u(0) +Sigma_{i=1}^m alpha_iu(xi_i) = 0, end{cases} ] where (f : {1,..., T} times mathbb{R}rightarrow mathbb{R}) is continuous, (T gt; 1) is a fixed natural number, (alpha_i in (infty; 0],, xi_i in {1,...,T}(i = 1,..., m; 1 leq m leq T; m in mathbb{N})) are given constants such that (Sigma_{i=1}^m alpha_i+ 1 = 0). We develop the methods of lower and upper solutions by the connectivity properties of the solution set of parameterized families of compact vector fields.