Nonlinear Contraction Semigroups Research Articles

The asymptotic behaviour of semigroups of nonlinear contractions having large sets of fixed points

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.

On the convergence of difference approximations to nonlinear contraction semigroups in Hilbert spaces

Convergence properties of the difference schemes (S) \[ h − 1 ∑ j = 0 k α j u n + j + ∑ j = 0 k β j A u n + j = 0 , n ⩾ 0 , {h^{ - 1}}\sum \limits _{j = 0}^k {{\alpha _j}{u_{n + j}}} + \sum \limits _{j = 0}^k {{\beta _j}A{u_{n + j}}} = 0,\quad n \geqslant 0, \] , for evolution equations (E) \[ d u ( t ) d t + A u ( t ) = 0 , t ⩾ 0 ; u ( 0 ) = u 0 ∈ D ( A ) ¯ \frac {{du(t)}}{{dt}} + Au(t) = 0,\quad t \geqslant 0;\quad u(0) = {u_0} \in \overline {D(A)} \] are studied. Here A is a nonlinear, maximally monotone operator in a real Hilbert space. It is shown, in particular, that if the scheme (S) is consistent and stable for the test equation x ′ = λ x x\prime = \lambda x for λ ∈ C − K \lambda \in {\text {C}} - K , where K is a compact subset of the right half-plane, then (S) is convergent as h ↓ 0 h \downarrow 0 , with suitable initial values, for (E), on compact intervals [0, T]. Moreover, the convergence is uniform on the half-axis t ⩾ 0 t \geqslant 0 , if the solution u ( t ) u(t) tends strongly to a constant as t → ∞ t \to \infty . We also show that under weaker stability conditions one can construct conditionally convergent methods.

Asymptotic behavior of semigroups of nonlinear contractions in Banach spaces

Asymptotic behavior of semigroups of nonlinear contractions in Banach spaces

On approximation by semigroups of nonlinear contractions. II

This note is a sequel to the preceding one with the same title published in this Journal ( 13, 205–217 (1975)). The contents of the first paper are assumed to be known. References are in alphabetical order in each paper, but they, as well as the sections, are numbered consecutively.

On approximation by semigroups of nonlinear contractions, I

On approximation by semigroups of nonlinear contractions, I

Asymptotic convergence of nonlinear contraction semigroups in Hilbert space

Let S be a contraction semigroup on a closed convex subset C of a Hilbert space. If the generator of S satisfies a strengthened monotonicity condition then the weak limt → ∞ S(t)x exists for all x in C. As one consequence, the method of steepest descent converges weakly for convex functions in Hilbert space; and it converges strongly for even convex functions.

Generation of semi-groups of nonlinear contractions

Generation of semi-groups of nonlinear contractions

Asymptotic behavior of nonlinear contraction semigroups

The asymptotic behavior of weak solutions of the equation {u(t) + Au(t) ϵ ƒ(t) (A maximal monotone in Hilbert space) is determined via a characterization of ω-limit sets of the contraction semigroup generated by − A.

Semigroups of nonlinear contractions on convex sets

Semigroups of nonlinear contractions on convex sets

On the generation of semigroups of nonlinear contractions

On the generation of semigroups of nonlinear contractions

Regularity Properties of Flows Through Porous Media: A Counterexample

Regularity Properties of Flows Through Porous Media: A Counterexample

Semi-groups of nonlinear contractions and dissipative sets

Semi-groups of nonlinear contractions and dissipative sets

Note on Nonlinear Contraction Semigroups

Note on Nonlinear Contraction Semigroups

Note on nonlinear contraction semigroups

1. Let X be a Banach space and let { F(£); 0^£< » } be a family of nonlinear operators from X into itself satisfying the following conditions: (i) T(0)=I (the identity) and T{£+7]) = T{£)T{v) for ?, t?^0. (ii) ||r(£)x-7X£);y||=g||x-3,|| for x, y EX. (iii) limj_o+ ||F(£)x — x|| =0 for xEX. It is clear that for any fixed xEX, T{ii)x is strongly continuous in ?2;0. Such a family JF(£); 0:S£< °o ] is called a nonlinear contraction semigroup. Then we define the infinitesimal generator A of {F(£);0^<~} by