Minimax theorems for limits of parametrized functions having at most one local minimum lying in a certain set

In this paper, we establish some minimax theorems, of purely topological nature, that, through the variational methods, can be usefully applied to nonlinear differential equations. Here is a (simplified) sample: Let X   be a Hausdorff topological space, I⊆RI⊆R an interval and Ψ:X×I→]−∞,+∞]. Assume that the function Ψ(x,⋅)Ψ(x,⋅) is lower semicontinuous and quasi-concave in I   for all x∈Xx∈X, while the function Ψ(⋅,q)Ψ(⋅,q) has compact sublevel sets and one local minimum at most for each q in a dense subset of I. Then, one hassupq∈Iinfx∈XΨ(x,q)=infx∈Xsupq∈IΨ(x,q).