Positive solutions of nonlinear three-point boundary-value problems
The authors study the existence of positive solutions to the nonlinear three-point boundary value problem \[ u''(t)+a(t)u'(t)+b(t)u(t)+h(t)f(u)=0,\quad t\in (0,1),\quad u(0)=0,\quad \alpha u(\eta)=u(1), \] where \(0<\eta<1,\) \(f\in C([0,\infty),[0,\infty)),\) \(h\in C([0,1],[0,\infty))\) and there exists \(x_{0}\in [0,1]\) such that \(h(x_{0})>0,\) \(a\in C[0,1]\) and \(b\in C([0,1],(-\infty,0)).\) They prove the existence of at least one positive solution if \(f\) is either superlinear or sublinear by applying the fixed-point theorem in cones.
second-order multi-point boundary value problem
scientific article; zbMATH DE number 1915744