Another fixed point theorem for nonexpansive potential operators

Volume 211 / 2012

Biagio Ricceri Studia Mathematica 211 (2012), 147-151 MSC: 47H09, 47H10. DOI: 10.4064/sm211-2-3


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 $$ \sup_{\|x\|=r}J(x)=\sup_{\|u\|_{L^2([0,1],X)}=r} \,\int_0^1J(u(t))\,dt. $$


  • Biagio RicceriDepartment of Mathematics
    University of Catania
    Viale A. Doria 6
    95125 Catania, Italy

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