Multiplicity of global minima for parametrized functions

Abstract

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 ${\lambda }^{\ast }\in I$ such that $\Psi (\cdot ,{\lambda }^{\ast })$ has at least two global minima. An application involving the integral functional of the calculus of variations is also presented.