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

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 $$S_r=\{x\in H : \|x\|=r\}\ .$$