Positive solutions for second-order four-point boundary value problems with alternating coefficient (Q1001874)

The paper deals with the special problem of existence of positive solutions for second-order four-point boundary value problem \[ u''(t)+\lambda\, a(t)\,f(u(t))=0\,, \quad t\in (0,1)\,, \] \[ u(0)= \alpha \,u(\xi)\, \,, \qquad u(1)=\beta \,u(\eta)\, , \] where \(\,\lambda \,\) is a positive real parameter, \(0\leq\alpha,\, \,\beta<1\,, \) \(\,0<\xi <\eta <1\,,\) and where \(\,f \,\) is a nondecreasing function and \(a\) is an alternating function. Under some conditions and by using the Krasnoselski fixed point theorem the existence of positive solutions is given.