Another fixed point theorem for nonexpansive potential operators
Volume 211 / 2012
Studia Mathematica 211 (2012), 147-151
MSC: 47H09, 47H10.
DOI: 10.4064/sm211-2-3
Abstract
We prove the following result: Let $X$ be a real Hilbert space and let $J:X\to \mathbb{R}$ be a $C^1$ functional with a nonexpansive derivative. Then, for each $r>0$, the following alternative holds: either $J'$ has a fixed point with norm less than $r$, or $$ \sup_{\|x\|=r}J(x)=\sup_{\|u\|_{L^2([0,1],X)}=r} \,\int_0^1J(u(t))\,dt. $$
