Nonlinear ergodic theorems for almost nonexpansive curves over commutative semigroups
The author proves the following theorem: Let \(S\) be a commutative semigroup of nonlinear operators on a real Hilbert space \(H\) with identity. Let \(u\) be an almost nonexpansive curve (that is, \(u\) is a function from \(S\) into \(H\) such that there exists a real valued function \(\varepsilon(\cdot,\cdot)\) on \(S\times S\) such that \(|u(s+h)- u(t+h)|^2\leq|u(s)- u(t)|^2+ \varepsilon(s,t)\), where \(\lim_{s,t\to\infty} \varepsilon(s,t)=0\)), let \(\mu\) be an invariant mean on a subspace \(X\) of \(\ell^\infty(S)\), (that is, \(\mu\in X^*\) and \(|\mu|=\mu(1)=1\) where 1 is the constant function in \(\ell^\infty(S)\)), let \(P\) be the metric projection of \(H\) onto \(F_\mu(u)\), where \(F_\mu(u)\) is the set of generalized fixed points (that is, \(q\in F_\mu(u)\) if and only if \(|u(t)-q|^2\leq|u(s)-q|^2+ \varepsilon(s)\), where \(\varepsilon(s)= \mu\varepsilon(s,t)\)). Then \(Pu(t)\) converges strongly to \(u(\mu)\), which is the \(\mu\)-asymptotic center of \(u\) in \(H\).
commutative semigroup of nonlinear operators
scientific article; zbMATH DE number 856955