The asymptotic behavior of nonlinear semigroups and invariant means (Q1076325)

The author proves several results concerning the asymptotic behavior of nonlinear semigroups in reflexive Banach spaces. A typical result is as follows: Let A be an accretive (multivalued) nonlinear operator in a reflexive Banach space E, let \(R(I+rA)\supset cl(D(A))\) for \(r>0\), let S be the semigroup generated by \(-A,\) and let \(d=d(0,R(A))\). Then \(d=d(0,\overline{co}\{(x-S(t)x)/t\})\) for every \(x\in cl(D(A))\) and there exists an element \(x_ 0\) with \(| x_ 0| =d\) such that \(x_ 0\in \overline{co}\{(x-S(t)/t\}\) for every \(x\in cl(D(A))\). Other results concern necessary and sufficient conditions for \(\{\) S(t)x\(\}\) to be bounded for some \(x\in cl(D(A))\), and similar results for expansive mappings in reflexive Banach spaces. The results generalize results of S. Reich to the case in which the Banach space E is not necessarily strictly convex.