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In this paper we study the third-order nonlinear boundary value problem
(Φ(u''))'(t) + f(t, u(t), u'(t), u''(t)) = 0 a.e. t ∈ [0,1],
u(0) = 0, g(u'(0), u''(0)) = A, h(u'(1), u''(1)) = B,
where A, B ∈ ℝ, f : [0,1] × ℝ3 → ℝ is a Carathéodory function, g, h ∈ C0(ℝ2,ℝ) and Φ ∈ C0(ℝ, ℝ). Using apriori estimates, the Nagumo condition, upper and lower solutions and the Schauder fixed point theorem, we are able to prove existence of solutions of this problem.