## Positive solutions of a three-point boundary value problem.(English)Zbl 1144.34311

Summary: We study the existence of positive solutions of the three-point boundary value problem $\begin{cases} u''+g(t)f(u)=0,\;t\in (0, 1), \\ u'(0)=0,\;u(1)=\alpha u(\eta), \end{cases}$ where $$\eta \in (0, 1)$$, and $$\alpha \in R$$ with $$0<\alpha<1$$. The existence of positive solutions is studied by means of fixed point index theory under conditions concerning the first eigenvalue with respect to the relevant linear operator. The results essentially extend and improve the result of J. R. L. Webb [Nonlinear Anal., Theory Methods Appl. 47, No. 7, 4319–4332 (2001; Zbl 1042.34527)].

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations

Zbl 1042.34527