Upper and Lower Solution Method for Regular Discrete Second-Order Boundary Value Problems

This paper investigates the existence of solutions for regular discrete second-order single-variable boundary value problems with mixed boundary conditions, including a nonhomogeneous Dirichlet boundary condition, of the form: uΔΔ(t − 1) + h(t, u(t), uΔ(t − 1)) = 0 for t ∈ [1, T + 1]; uΔ(0) = 0; u(T + 2) = g(T + 2) where h is continuous on [1, T + 1] × R2 and g : [0, T + 2] → R+ is continuous. Using the concept of upper and lower solutions, we establish conditions under which the boundary value problem admits at least one solution. Our approach involves constructing an auxiliary problem with a modified nonlinearity and applying the Brouwer Fixed Point Theorem to a carefully defined solution operator. We prove that any solution to this auxiliary problem that remains within the bounds of the upper and lower solutions is equivalent to a solution of the original problem.