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Abstract

This paper treats the question of the existence of solutions of a fourth order boundary value problem having the following form: $$\eqalign{ &{x^{(4)}}(t) + f(t,x(t),x' '(t)) = 0,\ \quad 0 < t < 1,\cr &x(0) = x'(0) = 0,\ \quad {x' '(1) = 0},\ \quad {{x^{(3)}}(1) = 0}. }$$ Boundary value problems of very similar type are also considered. It is assumed that $f$ is a function from the space $C( {[ {0,1} ] \times \mathbb R^2,\mathbb R})$. The main tool used in the proof is the Leray–Schauder nonlinear alternative.