The aim of this work is to propose viscosity-like methods for finding a specific common fixed point of a finite family T={ T_{i} }_{i=1}^{N} of nonexpansive self-mappings of a closed convex subset C of a Hilbert space H.We propose two schemes: one implicit and the other explicit.The implicit scheme determines a set {x_{t} : 0 < t < 1} through the fixed point equation x_{t}= tf (x_{t} ) + (1− t)Tx_{t}, where f : C¡÷C is a contraction.The explicit scheme is the discretization of the implicit scheme and de defines a sequence {x_{n} } by the recursion x_{n+1}=£\\_{n}f(x_{n}) +(1−£\\_{n})Tx_{n} for n ≥ 0, where {£\\_{n} }⊂ (0,1) It has been shown in the literature that both implicit and explicit schemes converge in norm to a fixed point of T (with additional conditions imposed on the sequence {£\ _{n} } in the explicit scheme).We will extend both schemes to the case of a finite family of nonexpansive mappings. Our proposed schemes converge in norm to a common fixed point of the family which in addition solves a variational inequality.

We consider the problem of finding a common fixed point of an infinite family ${T_n}$ of nonlinear self-mappings of a closed convex subset $C$ of a real Hilbert space $H$. Namely, we want to find a point $x$ with the property (assuming such common fixed points exist): [ xin igcap_{n=1}^infty ext{Fix}(T_n). ] We will use the Krasnoselskii-Mann (KM) Type inertial iterative algorithms of the form $$ x_{n+1} = ((1-alpha_n)I+alpha_nT_n)y_n,quad y_n = x_n + eta_n(x_n-x_{n-1}).eqno(*)$$ We discuss the convergence properties of the sequence ${x_n}$ generated by this algorithm (*). In particular, we prove that ${x_n}$ converges weakly to a common fixed point of the family ${T_n}$ under certain conditions imposed on the sequences ${alpha_n}$ and ${eta_n}$.

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a generalized equilibrium problem (for short, GEP) and the set of fixed points of a nonexpansive mapping in a Hilbert space. First, by using the well-known KKM technique we derive the existence and uniqueness of solutions of the auxiliary problems for the GEP. Second, on account of this result and Nadler's theorem, we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the GEP and the set of fixed points of the nonexpansive mapping. Furthermore, it is proven that the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions of the GEP and the set of fixed points of the nonexpansive mapping.

A sequence fxng generates by the formula x_{n+1} =(1- £\\_n)x_n+ £\\_nT_nx_n is called the Krasnosel'skii-Mann algorithm, where {£\\_n} is a sequence in (0,1) and {T_n} is a sequence of nonexpansive mappings. We introduce KM algorithm and prove that the sequence fxng generated by KM algorithm converges weakly. This result is used to solve the split feasibility problem which is to find a point x with the property that x ∈ C and Ax ∈ Q, where C and Q are closed convex subsets form H1 to H2, respectively, and A is a bounded linear operator form H1 to H2. The purpose of this paper is to present some results which apply KM algorithm to solve the split feasibility problem, the multiple-set split feasibility problem and other applications.

In this paper we study through iterative methods the minimization problem min_{x∈C} £K(x) (P) where the set C of constraints is the set of fixed points of a nonexpansive mapping T in a real Hilbert space H, and the objective function £K:H¡÷R is supposed to be continuously Gateaux dierentiable. The gradient projection method for solving problem (P) involves with the projection P_{C}. When C = Fix(T), we provide a so-called hybrid iterative method for solving (P) and the method involves with the mapping T only. Two special cases are included: (1) £K(x)=(1/2)||x-u||^2 and (2) £K(x)=<Ax,x> - <x,b>. The first case corresponds to finding a fixed point of T which is closest to u from the fixed point set Fix(T). Both cases have received a lot of investigations recently.

In this paper we discuss iterative algorithms for solving the split feasibility problem (SFP). We study the CQ algorithm from two approaches: one is an optimization approach and the other is a fixed point approach. We prove its convergence first as the gradient-projection algorithm and secondly as a fixed point algorithm. We also study a relaxed CQ algorithm in the case where the sets C and Q are level sets of convex functions. In such case we present a convergence theorem and provide a different and much simpler proof compared with that of Yang [7].

Une approche fondamentale pour la résolution de problémes de contrôle optimal est basée sur le principe de programmation dynamique. Ce principe conduit aux équations d'Hamilton-Jacobi, qui peuvent être résolues numériquement par des méthodes classiques comme la méthode des différences finies, les méthodes semi-lagrangiennes, ou les schémas antidiffusifs. À cause de la discrétisation de l'espace d'état, la dimension des problèmes de contrôle pouvant être abordés par ces méthodes classiques est souvent limitée à 3 ou 4. Ce phénomène est appellé malédiction de la dimension. Cette thèse porte sur les méthodes numériques max-plus en contôle optimal deterministe et ses analyses de convergence. Nous étudions et developpons des méthodes numériques destinées à attenuer la malédiction de la dimension, pour lesquelles nous obtenons des estimations théoriques de complexité. Les preuves reposent sur des résultats de théorie de Perron-Frobenius non linéaire. En particulier, nous étudions les propriétés de contraction des opérateurs monotones et non expansifs, pour différentes métriques de Finsler sur un cône (métrique de Thompson, métrique projective d'Hilbert). Nous donnons par ailleurs une généralisation du "coefficient d'ergodicité de Dobrushin" à des opérateurs de Markov sur un cône général. Nous appliquons ces résultats aux systèmes de consensus ainsi qu'aux équations de Riccati généralisées apparaissant en contrôle stochastique.

Consider the variational inequality (VI) x* ∈C, ‹Fx*, x - x* ›≥0, x∈C (*) where C is a nonempty closed convex subset of a real Hilbert space H and F : C¡÷ H is a monotone operator form C into H. It is known that if F is strongly monotone and Lipschitzian, then VI (*) is equivalently turned into a fixed point problem of a contraction; hence Banach's contraction principle applies. However, in the case where F is inverse strongly monotone, VI (*) is equivalently transformed into a fixed point problem of a nonexpansive mapping. The purpose of this paper is to present some results which apply iterative methods for nonexpansive mappings to solve VI (*). We introduce Mann's algorithm and Halpern's algorithm and prove that the sequences generated by these algorithms converge weakly and respectively, strongly to a solution of VI (*), under appropriate conditions imposed on the parameter sequences in the algorithms.