# A more complete version of a minimax theorem

@inproceedings{Ricceri2021AMC, title={A more complete version of a minimax theorem}, author={Biagio Ricceri}, year={2021} }

In this paper, we present a more complete version of the minimax theorem established in [7]. As a consequence, we get, for instance, the following result: Let X be a compact, not singleton subset of a normed space (E, ‖ · ‖) and let Y be a convex subset of E such that X ⊆ Y . Then, for every convex set S ⊆ Y dense in Y , for every upper semicontinuous bounded function γ : X → R and for every λ > 4 supX |γ| diam(X) , there exists y ∈ S such that the function x→ γ(x) + λ‖x− y‖ has at least two…

## References

SHOWING 1-10 OF 17 REFERENCES

Convexity of Chebyshev Sets

- Mathematics
- 2001

In Theorem 3.5 we saw that every closed convex subset of a Hilbert space is a Chebyshev set. In this chapter we will study the converse problem of whether or not every Chebyshev subset of a Hilbert…

On a minimax theorem: an improvement, a new proof and an overview of its applications

- Mathematics
- 2016

Theorem 1 of [14], a minimax result for functions $f:X\times Y\to {\bf R}$, where $Y$ is a real interval, was partially extended to the case where $Y$ is a convex set in a Hausdorff topological…

A Class of Equations with Three Solutions

- Mathematics, Physics
- 2020

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q \lambda_1$ and for every convex set $S\subseteq L^{\infty}(\Omega)$…

A class of functionals possessing multiple global minima

- Physics, Mathematics
- 2020

We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $\Omega\subset {\bf R}^n$ ($n\geq 2$) be a smooth bounded domain and let $\Phi:{\bf R}^2\to {\bf R}$ be…

An Invitation to the Study of a Uniqueness Problem

- Mathematics
- 2020

In this very short paper, we provide a strong motivation for the study of the following problem: given a real normed space $E$, a closed, convex, unbounded set $X\subseteq E$ and a function $f:X\to…

An alternative theorem for gradient systems.

- Mathematics, Physics
- 2020

Here is one of the result obtained in this paper: Let $\Omega\subset {\bf R}^2$ be a smooth bounded domain and let $F, G : {\bf R}\to {\bf R}$ be two $C^1$ functions satisfying the following…

Miscellaneous Applications of Certain Minimax Theorems II

- Mathematics
- 2020

In this paper, we present new applications of our general minimax theorems. In particular, one of them concerns the multiplicity of global minima for the integral functional of the Calculus of…

Another multiplicity result for the periodic solutions of certain systems

- Mathematics
- 2019

In this paper, we deal with a problem of the type $$\cases{(\phi(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , \hskip 3pt u'(0)=u'(T)\ ,\cr}$$ where, in particular, $\phi$ is a…

Geometric properties of Banach spaces and the existence of nearest and farthest points

- Mathematics
- 2005

The aim of this paper is to present some generic existence results
for nearest and farthest points in connection with some geometric
properties of Banach spaces.