Existence of positive solution of two-point boundary problem for one nonlinear ODE of the fourth order

In the work sufficient conditions for existence at least one positive solution of two-point boundary problem for one class of strongly nonlinear differential equations of the fourth order are received. The problem is considered on a segment [0,1] (more general case of $ segment [0, a] $ is reduced to considered). On the ends of a segment the solution of $y$ and its second derivative of $y'' $ are equal to zero. Right part of an equation $ f(x, y) $ isn't negative at $ x\geq $ 0 and at all $y$. Performance of sufficient conditions is easily checked. Performance of these conditions is easily checked. In the proof of existence the theory of cones in banach space is used. Also apriori estimates of positive solution, which is possible to use further at numerical construction of the solution are obtained.