Multiplicity of global minima for parametrized functions (Q966207)

Summary: Let \(X\) be a topological space, \(I\) a real interval and \(\Psi\) a real-valued function on \(X\times I\). In this paper, we prove that if \(\Psi\) is lower semicontinuous and inf-compact in \(X\), quasiconcave and continuous in \(I\) and satisfies \(\sup_I \inf_X \Psi< \inf_X\sup_I \Psi\), then there exists \(\lambda^*\in I\) such that \(\Psi(\cdot,\lambda^*\)) has at least two global minima. An application involving the integral functional of the calculus of variations is also presented.