Geodesic
Nontrivial solution for a three-point boundary-value problem
Electronic journal of differential equations, Tome 2004 (2004)
In this paper, we study the existence of nontrivial solutions for the second-order three-point boundary-value problem

$\displaylines{ u''+f(t,u)=0,\quad 0 less than t less than 1, \cr u'(0)=0,\quad u(1)=\alpha u'(\eta). }$

where $\eta \in (0,1), \alpha \in \mathbb{R}, f\in C([0,1]\times \mathbb{R},\mathbb{R})$. Under certain growth conditions on the nonlinearity $f$ and by using Leray-Schauder nonlinear alternative, sufficient conditions for the existence of nontrivial solution are obtained. We illustrate the results obtained with some examples.
Classification : 34B10, 34B15
Keywords: three-point boundary-value problem, nontrivial solution, Leray-Schauder nonlinear alternative 
@article{EJDE_2004__2004__a103,
     author = {Sun,  Yong-Ping},
     title = {Nontrivial solution for a three-point boundary-value problem},
     journal = {Electronic journal of differential equations},
     year = {2004},
     volume = {2004},
     zbl = {1076.34010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a103/}
}
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%A Sun,  Yong-Ping
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Sun,  Yong-Ping. Nontrivial solution for a three-point boundary-value problem. Electronic journal of differential equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a103/