The methods of lower and upper solutions for a second order three-point boundary value problem at resonance
u″(t)=f(t, u(t), u′(t)), t∈[0, 1],
u′(0)=0, u(1)=u(η)
were developed by using the connectivity properties of the solution sets of parameterized families of compact vector fields, where η∈(0, 1), f:[0, 1]×R2→R is continuous and satisfies the Nagumo condition.