The nonlinear difference equation $Py(t - k) = f(t,y(t)$ with $(j,n - j)$-conjugate boundary conditions is considered, where $Py(t - k) = 0$ is an nth-order linear difference equation and k is a fixed integer, $0 \leq k < n$. Peterson considered this type of problem for the cases $j = n - 1$ and $j = 1$. This paper extends his results to the $(j,n - j)$-problem. A comparison theorem for solutions of related linear inequalities is obtained, leading to some disconjugacy results. Then a shooting method type of proof is used to prove existence and uniqueness theorems for certain boundary value problems where f satisfies a two-sided Lipschitz condition.