Existence of solution for a nonlinear three-point boundary value problem

In this paper, we study the existence of nontrivial solution for the fourth-order three-point boundary value problem given as follows
\begin{gather*} u^{(4)}(t)+f(t,u(t))=0,\quad\text 0<t<1,\\ u^{'}(0)-\alpha u^{'}(\eta)=0,\quad u(0)=u^{'''}(0)=0,\quad u^{'}(1)-\beta u^{'}(\eta)=0, \end{gather*}
where $\eta\in(0,1)$, $\alpha, \beta\in\mathbb{R}$, $f\in C([0,1]\times\mathbb{R},\mathbb{R})$. We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. And by using the Leray–Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.