On a Third-Order Three-Point Boundary Value Problem

where η ∈ 0, 1 , α, β ∈ R, f ∈ C 0, 1 × R,R . The parameters α and β are arbitrary in R such that 1 2α 2βη − 2β / 0. Our aim is to give new conditions on the nonlinearity of f , then using Leray-Schauder nonlinear alternative, we establish the existence of nontrivial solution. We only assume that f t, 0 / 0 and a generalized polynomial growth condition, that is, there exist two nonnegative functions k, h ∈ L1 0, 1 such that ∣f t, x ∣∣ ≤ k t |x| h t , ∀ t, x ∈ 0, 1 × R, 1.3