Positive solutions of some nonlocal boundary value problems
For two 4-point BVP \[ u''(t)+g(t)f(u(t))=0\quad \text{a.e. on }[0,1], \] \[ u'(0)=0,\quad u(1)=\alpha_1u(\eta_1)+\alpha_2u(\eta_2), \] or \[ u(0)=0,\quad u(1)=\alpha_1u(\eta_1)+\alpha_2u(\eta_2), \] the authors determine a region in the \((\alpha_1,\,\alpha_2)\)-plane which ensures the existence of positive solutions. Further, they conclude that one can obtain the existence of positive solutions for an \(m\)-point boundary value problem under the weaker assumption that all parameters occurring in the boundary conditions are not required to be positive. Hence, their results allow more general behavior on \(f\) than being either sub- or superlinear.
scientific article; zbMATH DE number 2163631