Solvability of a three-point nonlinear boundary-value problem. (English) Zbl 1206.34033

Guezane-Lakoud, Assia; Kelaiaia, Smail

Electron. J. Differ. Equ. 2010, Paper No. 139, 9 p. (2010).

Summary: Using the Leray Schauder nonlinear alternative, we prove the existence of a nontrivial solution for the three-point boundary-value problem
\[ u''+f(t,u)= 0,\quad 0<t<1 \]
\[ u(0)= \alpha u'(0),\quad u(1)=\beta u'(\eta ), \]
where \(\eta \in (0,1)\), \(\alpha ,\beta \in \mathbb{R}\), \(f\in C([0,1] \times\mathbb{R},\mathbb{R})\). Some examples are given to illustrate the results obtained.

MSC:

34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

fixed point theorem; three-point boundary-value problem; non trivial solution