existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in banach spaces in this paper we study nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings in a banach space we prove that ifsis an amenable semigroup and s tt t s is a nonexpansive semigroup on a closed convex subsetcin a uniformly convex banach spaceesuch that the setf s of common fixed points of s is nonempty then there exists a nonexpansive retractionpfromcontof s such thatptt ttp pfor eacht sandpx co ttx t s for eachx c in this case there exists a net a of finite averages of s such that for eacht sand for each bounded subsetbofc lim a ttx a x 0 and lim tta x a x 0 uniformly forx b also if the norm ofeis frechet differentiable then for eachx c pxis the unique common fixed point in s s co ttsx t s furthermore if is an asymptotically invariant net of means then for eachx c t x converges weakly topx

stochastic hybrid control the objective of this paper is to study the stochastic version of a previous paper of the authors in which hybrid control for deterministic systems was considered the modelling is quite similar to the deterministic case we have a system whose state is composed of a continuous part and a discrete part they are affected by a continuous type control and an impulse control the dynamics is moreover perturbed by noise also a continuous and a discrete noise process the markovian character of the sate process is preserved we develop the model and show how the dynamic programming approach leads to some involved quasi variational inequality

ergodic retractions for amenable semigroups in banach spaces with normal structure abstract in this paper we discuss the existence of nonexpansive retraction onto the set of common fixed points assume that t s s s is an amenable semigroup of nonexpansive mappings on a closed convex subset c in a reflexive banach space e such that the set f of common fixed points of is nonempty among other things it is shown that if either c has normal structure or the t s s are affine then there exists a nonexpansive retraction p from c onto f such that p t t t t p p for each t s and every closed convex invariant subset of c is also p invariant in the case that the mappings are affine p is also affine and p x c o t t x t s for each x c and it is unique regarding the latter property our results extend corresponding results of t suzuki some remarks on the set of common fixed points of one parameter semigroups of nonexpansive mappings in banach spaces with the opial property nonlinear anal 58 2004 441 458 and r e bruck a common fixed point theorem for a commuting family of nonexpansive mappings pacific j math 53 1974 59 71

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

asymptotic behavior of commutative semigroups of nonexpansive mappings in banachspaces on etablit des theoremes de convergence pour des semigroupes generaux d applications non dilatation dans le cadre des espaces de banach on demontre un theoreme ergodique non lineaire de type rode pour des semigroupes commutatifs

asymptotic stabilization of the position of a rigid body with fixed point without velocity measurements abstract the problem of stabilization of a rigid body with a fixed point in reference to its velocity and position is considered it is assumed that the rigid body is controlled by three external torques the proposed controlling torques need neither the measurements of the velocity nor the information about the inertial parameters such a control may be realized by the on line solution of the auxiliary differential equations the stability of the closed loop nonlinear system is established by means of the lyapunov function and barbashin s theorem the proposed control may be applied to stabilize the satellites controlled by the reactive jets

strong convergence of mann s type iterations for nonexpansive semigroups in general banach spaces abstract in this paper we prove a strong convergence theorem of mann s type for commutative nonexpansive semigroups in general banach spaces using this theorem we obtain some strong convergence theorems in general banach spaces

an ergodic theorem for semigroups of nonexpansive mappings in a hilbert space abstract j b baillon c r acad sci paris ser a 280 1975 1511 1514 proved an ergodic theorem for a single nonexpansive mapping in a hilbert space which is a nonlinear version of von neumann s mean ergodic theorem in this paper we study the ergodic behavior of a semigroup of nonexpansive mappings we try to find a sequence of means on the semigroup generalizing the cesaro means on n such that the corresponding sequence of nonexpansive mappings converges to a projection onto the set of common fixed points our method of proof is an appropriate modification of a pazy s proof israel j math 26 1977 197 204 of baillon s theorem

fixed point set of semigroups of non expansive mappings and amenability abstract in this paper we study the fixed point set of a strongly continuous non expansive semigroup s t t t s of a semi topological semigroup s for which cb s is n extremely left amenable also we study the fixed point set of a strongly continuous semigroup of mappings when s is a semigroup which is a sub semigroup of a locally convex topological vector space with addition some applications to harmonic analysis are also provided

nonexpansive retractions and fixed point properties abstract among other results we prove that if a dual banach space e has the weak fixed point property then e has the weak fixed point property for commuting separable semitopological semigroups of nonexpansive mappings and study the existence of ergodic retractions for right amenable semigroups and commuting families of nonexpansive mappings this answers an open problem of lau and mah 1976 12 for commuting separable semi topological semigroups

iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings abstract we introduce iterative algorithms for finding a common element of the set of solutions of a system of equilibrium problems and of the set of fixed points of a finite family and a left amenable semigroup of nonexpansive mappings in a hilbert space we prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality which is the optimality condition for a minimization problem our results extend for example the recent result of v colao g marino h k xu an iterative method for finding common solutions of equilibrium and fixed point problems j math anal appl 344 2008 340 352 to systems of equilibrium problems

controlled diffusion processes with markovian switchings for modeling dynamical engineering systems abstract in this paper the intrinsic complex nature of engineering systems under control is treated by introducing an approach based on controlled stochastic differential equations with markovian switchings in short csdems technical conditions for the existence and uniqueness of the solutions of the csdems are provided in this context it is not unusual to deal with non linear csdems that cannot be solved analytically therefore we develop a new two step predictor corrector method for finding numerical approximations to solutions of csdems this method utilizes the euler maruyama method an illustrative application to the biochemical engineering area is presented to highlight the usefulness of our approach as a simulation tool

a mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense abstract let x be a uniformly convex banach space such that its dual x has the kadec klee property c a bounded closed convex subset of x and t t t t 0 a one parameter nonlinear semigroup which is asymptotically nonexpansive in the intermediate sense we show that every continuous almost orbit of t is weakly almost convergent to a common fixed point of t a discrete version of our main result is also included

convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in banach spaces abstract the purpose of this paper is to study hybrid iterative schemes of halpern types for a semigroup t s s s of relatively nonexpansive mappings on a closed and convex subset c of a banach space with respect to a sequence n of asymptotically left invariant means defined on an appropriate invariant subspace of l s we prove that given a certain sequence n in 0 1 x c we can generate an iterative sequence x n which converges strongly to f x where f x is the generalized projection from c onto the fixed point set f our main result is even new for the case of a hilbert space

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

some experiments on numerical simulations of stochastic differential equations and a new algorithm in this paper we compare three second order methods for the numerical integration of ito stochastic differential equations one of these methods is new we consider stability of implicit vs explicit methods and compare the effects of step size sample size and type of increment used to approximate the brownian motion

nonexpansive retractions and nonlinear ergodic theorems in banach spaces on donne des theoremes ergodiques non lineaires pour des semigroupes commutatifs d applications non dilatation sans utiliser le concept de moyenne compacte

optimal control of switching diffusions with application to flexible manufacturing systems a controlled switching diffusion model is developed to study the hierarchical control of flexible manufacturing systems the existence of a homogeneous markov nonrandomized optimal policy is established by a convex analytic method using the existence of such a policy the existence of a unique solution in a certain class to the associated hamilton jacobi bellman equations is established and the optimal policy is characterized as a minimizing selector of an appropriate hamiltonian

fixed points of non expansive mappings associated with invariant means in a banach space in this paper we study the fixed point set of the non expansive mapping t for a banach space with uniformly g teaux differentiable norm when is a multiplicative left invariant mean on l s as an application we establish nonlinear ergodic properties for an extremely amenable semigroup of non expansive mappings in a banach space with uniformly g teaux differentiable norm furthermore we improve a recent result of atsushiba and takahashi s atsushiba w takahashi weak and strong convergence theorems for non expansive semigroups in a banach spaces satisfying opial s condition sci math jpn in press on the fixed point set of non expansive mappings associated with a left invariant mean on a left amenable semigroup

the retractions onto the common fixed points of some families and semigroups of mappings abstract assume that c is a closed convex subset of a reflexive banach space e and t i i i is a family of self mappings on c of type such that f the common fixed point set of is nonempty from our results in this paper it can be derived that a if i i f t i is contained in a 3 dimensional subspace of e then f is a nonexpansive retract of c b if is commutative there exists a retraction r of type from c onto f such that r t i t i r r i and every closed convex invariant subset of c is r invariant the same result holds for a non commutative right amenable semigroup under some additional assumptions moreover the existence of a t i ergodic retraction r of type from c x i l e x i c i i onto f in l e for the family is discussed we also apply some of our results to find ergodic retractions for nonexpansive affine mappings