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 ⊆ R an interval and Ψ : X × I → ] − ∞ , + ∞ ] . Assume that the function Ψ ( x , ⋅ ) is lower semicontinuous and quasi-concave in I for all x ∈ X , while the function Ψ ( ⋅ , q ) has compact sublevel sets and one local minimum at most for each q in a dense subset of I. Then, one has sup q ∈ I inf x ∈ X Ψ ( x , q ) = inf x ∈ X sup q ∈ I Ψ ( x , q ) .