Existence of solutions for a fully nonlinear fourth-order two-point boundary value problem

In this paper, we investigate the existence of solutions of a fully nonlinear fourth-order differential equation $$x^{(4)}=f(t,x,x',x'',x'''),\quad t\in 0,1]$$ with one of the following sets of boundary value conditions; $$x'(0)=x(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0,$$ $$x(0)=x'(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0.$$ By using the Leray-Schauder degree theory, the existence of solutions for the above boundary value problems are obtained. Meanwhile, as an application of our results, an example is given.