On three theorems of lees for numerical treatment of semilinear two-point boundary value problems

This paper is concerned with semilinear tow-point boundary value problems of the form -(p(x)u′)′ + f(x, u) = 0, a ≤ x ≤ b, α₀u(a) - α₁u′(a) = α, β₁u′(b) + β,₁u′(b) = β, α_i ≥ 0, β_i≥ 0, i = 0, 1, α₀+α₁ > 0, β₀+β ₁ > 0, α₀+β₀ > 0. Under the assumption inf f_u > -λ₁, where λ₁ is the smallest eigenvalue of u = -(pu′)′ with the boundary conditions, unique existence theorems of solution for the continuous problem and a discretized system with not necessarily uniform nodes are given as well as error estimates. The results generalize three theorems of Lees for u″ = f(x, u), 0 ≤ x ≤ 1, u(0) = α, u(1) = β.