An improvement of a saddle point theorem and some of its applications

TL;DR

This paper improves a saddle point theorem by removing a weak lower semicontinuity assumption and explores its applications, including a general local result in Hilbert spaces involving $C^{1,1}$ functions.

Contribution

It presents an enhanced saddle point theorem without the weak lower semicontinuity condition and applies it to derive a new local result in Hilbert spaces.

Findings

Established an improved saddle point theorem removing the weak lower semicontinuity assumption.

Derived a general local result for $C^{1,1}$ functions in Hilbert spaces.

Identified two unique points with specific properties related to the function $\Phi$.

Abstract

In this paper, we establish an improved version of a saddle point theorem ([4]) removing a weak lower semicontinuity assumption at all. We then revisit some of the applications of that theorem in the light of such an improvement. For instance, we obtain the following very general result of local nature: Let $(H,\langle\cdot,\cdot\rangle)$ be a real Hilbert space and $\Phi:B_{\rho}\to H$ a $C^{1,1}$ function, with $\Phi(0)\neq 0$. Then, for each $r>0$ small enough, there exist only two points points $x^*, u^*\in S_r$, such that $$\max\{\langle \Phi(x^*),x^*-x\rangle, \langle \Phi(x),x^*-x\rangle\}< 0\ ,$$ for all $x\in B_r\setminus \{x^*\}$, $$\|\Phi(u^*)-u^*\|=dist(\Phi(u^*),B_r)$$ and $$\|\Phi(x)-u^*\|<\|\Phi(x)-x\|$$ for all $x\in B_r\setminus \{u^*\}$, where $$B_r=\{x\in H : \|x\|\leq r\}$$ and $$S_r=\{x\in H : \|x\|=r\}\ .$$