Let H be a real Hilbert space and consider the abstract Cauchy problem $ \left\{ {{*{20}c} {\frac{{{\text{du}}}} {{{\text{dt}}}}\, + \,{\text{Au}}\,{\text{ = }}\,{\text{0}}\,\,\,\,\,{\text{t}}\,{\text{ > }}\,{\text{0}}} \\ {{\text{u}}\left( 0 \right) = {\text{x}}} \\ } \right. $ \left\{ {\begin{array}{*{20}c} {\frac{{{\text{du}}}} {{{\text{dt}}}}\, + \,{\text{Au}}\,{\text{ = }}\,{\text{0}}\,\,\,\,\,{\text{t}}\,{\text{ > }}\,{\text{0}}} \\ {{\text{u}}\left( 0 \right) = {\text{x}}} \\ \end{array} } \right. ((1.1)) where u(t) is a H valued function and A is an operator from H into H. The equation in (1.1) may hold in one of many different senses but for the moment this makes no difference.

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An ergodic theorem for semigroups of nonlinear contractions having precompact trajectories in a Banach space is proved.

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Let f ∈ L 1 ( 0 , ∞ ) , δ > 0 f \in {L^1}(0,\infty ),\delta > 0 and ( G δ f ) ( t ) = δ − 1 ∫ t ∞ e ( t − s ) / δ f ( s ) d s ({G_\delta }f)(t) = {\delta ^{ - 1}}\smallint _t^\infty {e^{(t - s)/\delta }}f(s)ds . Given a partition P = { 0 = t 0 > t 1 > ⋯ > t i > t i + 1 > ⋯ } P = \{ 0 = {t_0} > {t_1} > \cdots > {t_i} > {t_{i + 1}} > \cdots \} of [ 0 , ∞ ) [0,\infty ) where t i → ∞ {t_i} \to \infty , we approximate f by the step function A P f {A_P}f defined by \[ A P f ( t ) = ( G δ i G δ i − 1 ⋯ G δ i f ) ( 0 ) for t i − 1 ⩽ t > t i , {A_P}f(t) = ({G_{{\delta _i}}}{G_{{\delta _{i - 1}}}} \cdots {G_{{\delta _i}}}f)(0)\quad {\text {for}}\;{t_{i - 1}} \leqslant t > {t_i}, \] where δ i = t i − t i − 1 {\delta _i} = {t_i} - {t_{i - 1}} . The main results concern several properties of this process, with the most important one being that A P f → f {A_P}f \to f in L 1 ( 0 , ∞ ) {L^1}(0,\infty ) as μ ( P ) = sup i δ i → 0 \mu (P) = {\sup _i}{\delta _i} \to 0 . An application to difference approximations of evolution problems is sketched.

: This note gives a simple unified presentation of some recent ergodic results for semigroups of nonexpansive mappings in Hilbert space.

This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inL p , 1<p<∞, while the sufficient condition in the other characterization is meaningful in the case of nonlinear operators.

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SynopsisThe asymptotic behaviour of semigroups of nonlinear contractions which have a set of fixed points containing a ball of finite codimension is studied. It is shown that the ω-limit sets of such semigroups are finite dimensional tori, and that an analogue of the classical Kronecker-Weil theorem holds for such semigroups.

LetT be a nonexpansive mapping on a closed convex subsetC of a real Hilbert spaceH. In the present note we deal with the weak convergence of the sequenceT n x and the sequenceS n x of the arithmetical means of the sequenceT n x, asn → ∞. We also give some results concerning the strong convergence ofS n x.

A Trotter-Lie type formula for semigroups of nonlinear operators on a Banach space X into itself, which are generated by compact operators, is proved.

In this paper we prove existence, and study the asymptotic behavior of mild solutions of a class of semi-linear equations of evolution which are characterized by the fact that the associated homogeneous linear problem “generates” a strongly continuous semi-group of compact operators. We also prove a regularity result for which the associated homogeneous linear problem generates a holomorphic semi-group.