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This paper treats the question of the existence of solutions of a fourth order boundary value problem having the following form: $x^{\left(4\right)}\left(t\right)+f(t,x\left(t\right),x^{\text{'}\text{'}}\left(t\right))=0$, 0 < t < 1, x(0) = x’(0) = 0, x”(1) = 0, $x^{\left(3\right)}\left(1\right)=0$. Boundary value problems of very similar type are also considered. It is assumed that f is a function from the space C([0,1]×ℝ²,ℝ). The main tool used in the proof is the Leray-Schauder nonlinear alternative.