Existence and multiplicity of positive solutions for multi-parameter three-point differential equations system
The author studies, via fixed point index theory, the existence and multiplicity of positive solutions of a system of second order differential equations of the type \[ \begin{gathered} u''(t)+\lambda a_{1}(t) f_{1}(u(t),v(t))=0, \; t \in (0,1),\\ v''(t)+\mu a_{2}(t) f_{2}(u(t),v(t))=0, \; t \in (0,1),\\ \end{gathered} \] subject to the boundary conditions \[ \begin{gathered} u(0)=0,\;\; u(1)= \alpha_{1} u(\eta_{1}),\\ v(0)=0,\;\; v(1)= \alpha_{2} u(\eta_{2}).\\ \end{gathered} \] Here, \(\lambda\) and \(\mu\) are positive parameters, \(0 < \eta_{2} \leq \eta_{1}<1\), \(\alpha_{1}, \alpha_{2} \in (0,1),\) and \(a_{1}, a_{2}, f_{1}, f_{2}\) are continuous functions. The author of this paper remarks that, unlike what usually happens in previous results for systems, here the two nonlinearities possess a somewhat different behavior. In fact, the function \(a_{1}\) is positive and \(f_{1}\) is bounded below by a negative constant, whereas \(f_{2}\) is positive and the function \(a_{2}\) is allowed to change sign.
scientific article; zbMATH DE number 5078076