Abstract:
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.
Keywords:Green's function, Nontrivial solution, Leary-Schauder nonlinear alternative, Fixed point theorem, Boundary value problem.
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