Geodesic
Existence of solution for a nonlinear three-point boundary value problem
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1120-1134.

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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 01,\\ 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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Z. Bekri; S. Benaicha. Existence of solution for a nonlinear three-point boundary value problem. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1120-1134. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a98/

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