Nonexpansive Operators Research Articles

Weak convergence of infinite products of operators in Hadamard spaces

We first prove the weak convergence of iterates of both products and convex combinations of strongly nonexpansive operators in complete CAT(0) spaces. Then we establish the weak convergence of infinite products of possibly discontinuous approximations to such operators. Our main focus is on nearest point projections.

A universal operator on the Gurarii space

We construct a nonexpansive linear operator on the Gurarii space that "captures" all nonexpansive linear operators between separable Banach spaces. Some additional properties involving its restrictions to finite-dimensional subspaces describe this operator uniquely up to an isometry.

Approximating Iterations for Nonexpansive and Maximal Monotone Operators

We present two algorithms for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive operator in Hilbert spaces. We show that these two algorithms converge strongly to the minimum norm common element of the zero of the sum of two monotone operators and the fixed point of a nonexpansive operator.

Stochastic Quasi-Fejér Block-Coordinate Fixed Point Iterations with Random Sweeping

This work proposes block-coordinate fixed point algorithms with applications to nonlinear analysis and optimization in Hilbert spaces. The asymptotic analysis relies on a notion of stochastic quasi-Fejer monotonicity, which is thoroughly investigated. The iterative methods under consideration feature random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and they allow for stochastic errors in the evaluation of the operators. Algorithms using quasi-nonexpansive operators or compositions of averaged nonexpansive operators are constructed, and weak and strong convergence results are established for the sequences they generate. As a by-product, novel block-coordinate operator splitting methods are obtained for solving structured monotone inclusion and convex minimization problems. In particular, the proposed framework leads to random block-coordinate versions of the Douglas--Rachford and forward-backward algorithms and of some of their variant...

Compositions and convex combinations of averaged nonexpansive operators

Properties of compositions and convex combinations of averaged nonexpansive operators are investigated and applied to the design of new fixed point algorithms in Hilbert spaces. An extended version of the forward–backward splitting algorithm for finding a zero of the sum of two monotone operators is obtained.

Convergence of a regularization algorithm for nonexpansive and monotone operators in Hilbert spaces

Variational inequality, fixed point and generalized equilibrium problems are investigated via a regularization algorithm. It is proved that the sequence generated in the regularization algorithm converges strongly to a common solution of the three problems in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding ones announced by many authors.

Hybrid Splitting Methods for the System of Operator Inclusions with Monotone Operators1

New algorithms are proposed to solve a system of operator inclusions with monotone operators acting in a Hilbert space. The algorithms are based on three well-known methods: the Tseng forward-backward splitting algorithm and two hybrid algorithms for approximation of fixed points of nonexpansive operators. Theorems on the strong convergence of the sequences generated by the algorithms are proved.

Linear and strong convergence of algorithms involving averaged nonexpansive operators

We introduce regularity notions for averaged nonexpansive operators. Combined with regularity notions of their fixed point sets, we obtain linear and strong convergence results for quasicyclic, cyclic, and random iterations. New convergence results on the Borwein–Tam method (BTM) and on the cyclically anchored Douglas–Rachford algorithm (CADRA) are also presented. Finally, we provide a numerical comparison of BTM, CADRA and the classical method of cyclic projections for solving convex feasibility problems.

On inclusion of F-contractions in ( δ , k ) -weak contractions

We obtain a certain property of F-contractions, which enables us to generalize and extend Wardowski’s (Fixed Point Theory and Applications 2012:94, 2012) result and other fixed point theorems to the class of nonexpansive operators in real Banach spaces. We give illustrative examples to demonstrate nontrivial applicability of the property and use it to prove that F-contractions are closely related to -weak contractions introduced by Berinde (Carpath. J. Math. 19(1):7-22, 2003; Nonlinear Anal. Forum 9(1):43-53, 2004). MSC:47H09, 47H10, 54H25.

On firmly nonexpansive perturbations of the identity in Hilbert spaces

Abstract. The aim of this paper is to present some properties of firmly nonexpansive operators in Hilbert spaces. As a consequence of the principal result, a fixed point theorem about nonexpansive operators is obtained.

Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces

Abstract The purpose of this paper is to characterize the conditions for the convergence of the iterative scheme in the sense of Agarwal et al. (J. Nonlinear Convex. Anal. 8(1): 61-79, 2007), associated with nonexpansive and ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

The split common fixed point problem for infinite families of total quasi-asymptotically nonexpansive operators

The main purpose in this paper is to introduce an original technique, namely, a specific way of choosing the indexes of the involved operators, by which a new iterative algorithm is developed for solving the split common fixed point problem for infinite families of total quasi-asymptotically nonexpansive operators and some strong and weak convergence theorems are established for this class of nonlinear operators in the framework of infinite-dimensional Hilbert spaces. The result is more applicable than those of other authors with related interest.

A general iterative algorithm for an infinite family of nonexpansive operators in Hilbert spaces

In this paper, we introduce a new general iterative algorithm for an infinite family of nonexpansive operators in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to a common point of the sets of fixed points, which solves a variational inequality. Our results improve and extend the corresponding results announced by many others. As applications, at the end of the paper, we apply our results to the split common fixed point problem.

Asymptotic behavior of compositions of under-relaxed nonexpansive operators

In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.

On a functional Fredholm integral equation, via the technique of nonexpansive operators

In this paper we present one result relative to existence of solutions of the functional Fredholm integral equation with deviating argument by nonexpansive operators technique ... in a subset of a Banach space.

Existence of Iterative Cauchy Fractional Differential Equation

Our main aim in this paper is to use the technique of nonexpansive operators in more general iterative and noniterative fractional differential equations (Cauchy type). The noninteger case is taken in sense of the Riemann-Liouville fractional operators. Applications are illustrated.

Bregman strongly nonexpansive operators in reflexive Banach spaces

We present a detailed study of right and left Bregman strongly nonexpansive operators in reflexive Banach spaces. We analyze, in particular, compositions and convex combinations of such operators, and prove the convergence of the Picard iterative method for operators of these types. Finally, we use our results to approximate common zeros of maximal monotone mappings and solutions to convex feasibility problems.

Iterative methods for approximating fixed points of Bregman nonexpansive operators

Diverse notions of nonexpansive type operators have been extended to the more general framework of Bregman distances in reflexive Banach spaces. We study these classes of operators, mainly with respect to the existence and approximation of their (asymptotic) fixed points. In particular, the asymptotic behavior of Picard and Mann type iterations is discussed for quasi-Bregman nonexpansive operators. We also present parallel algorithms for approximating common fixed points of a finite family of Bregman strongly nonexpansive operators by means of a block operator which preserves the Bregman strong nonexpansivity. All the results hold, in particular, for the smaller class of Bregman firmly nonexpansive operators, a class which contains the generalized resolvents of monotone mappings with respect to the Bregman distance.

Fixed points of nonexpansive potential operators in Hilbert spaces

In this paper, we show the impact of certain general results by the author on the topic described in the title. Here is a sample: Let (X, �· , ·� ) be a real Hilbert space and let T : X → X be a nonexpansive potential operator.

Right Bregman nonexpansive operators in Banach spaces

We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.