Multiple positive solutions for nonlinear \(m\)-point boundary value problems. (Q1415278)

The paper deals with an \(m\)-point boundary value problems of the form \[ (p(t)u')'-q(t)u+f(t,u)=0, \] \[ a u(0)-b p(0)u'(0)=\sum_{i=1}^{m-2} \alpha_i u(\xi_i), \quad c u(1)+d p(1) u'(1) = \sum_{i=1}^{m-2} \beta_iu(\xi_i), \] where \(a,b,c,d \in [0,\infty)\) with \(ac+ad+bc>0\), \(\xi_i \in (0,1)\), \(\alpha_i, \beta_i \in [0,\infty)\) for \(i=1,\dots,m-2\). The function \(f\) is supposed to be continuous and nonnegative on \([0,1]\times[0,\infty)\), the functions \(p\in C^1[0,1]\) and \(q\in C[0,1]\) are positive. The paper provides the existence and multiplicity of positive solutions for the above problem. The main tool of the paper is the Guo-Krasnoselskij fixed-point theorem on cones in Banach spaces.