Nonexpansive Random Operators Research Articles

Random fixed point results for generalized asymptotically nonexpansive random operators

In this paper, we define an implicit random iterative process with errors for three finite families of generalized asymptotically nonexpansive random operators. We also prove some convergence theorems using this iteration method in separable Banach spaces.

Fixed points of multivalued nonexpansive random operator in banach spaces

In this paper, we show that a multivalued random operator has a fixed point in uniformly convex Banach space. In addition, the presence of a limit point in a multivalued random operator is shown, and a new iteration sequence for determining convergence of a multivalued random operator is introduced.

Convergence of random iterative scheme to a common random fixed points

In the context of uniformly convex separable Banach spaces, we implement an iterations scheme under Fibonacci sequence for approximating common random fixed points of two asymptotically nonexpansive random operators, and define weak and strong convergence results for common random fixed points of asymptotically nonexpansive random operator.

Random fixed point for random Fibonacci Noor iteration scheme

The goal of this paper is to describe a new random iteration scheme known as the Fibonacci Noor random operator iteration. Under Fibonacci sequnce, three measures are included in the new iterative scheme. We prove some convergence, presence in separable Banach spaces that are uniformly convex. In addition, under the Fibonacci series, we generalized the asymptotically non-expansive random operator form.

Stability and convergence theorems of pointwise asymptotically nonexpansive random operator in Banach space

In this paper, we prove the existence of a random fixed point of by using pointwise asymptotically nonexpansive random operator and the stability resultsof two iterative schemes for random operator.

Approximating random fixed points under a new iterative sequence

In this paper we introduce and study Fibonacci-Mann random scheme with random fixed points of a monotone asymptotically nonexpansive random operators under Fibonacci equation in order uniformly convex separable Banach spaces and established weak and strong convergence results for random fixed points of monotone Fibonacci asymptotically nonexpansive random operators.

Distributed Multiagent Convex Optimization Over Random Digraphs

This paper considers an unconstrained collaborative optimization of a sum of convex functions, where agents make decisions using local information in the presence of random interconnection topologies. We recast the problem as minimization of the sum of convex functions over a constraint set defined as the set of fixed-value points of a random operator derived from weighted matrices of random graphs. We show that the derived random operator has nonexpansivity property; therefore, this formulation does not need the distribution of random communication topologies. Hence, it includes random networks with/without asynchronous protocols. As an extension of the problem, we define a novel optimization problem, namely minimization of a convex function over the fixed-value point set of a nonexpansive random operator. We propose a discrete-time algorithm using diminishing step size for converging almost surely and in mean square to the global solution of the optimization problem under suitable assumptions. Consequently, as a special case, it reduces to a totally asynchronous algorithm for the distributed optimization problem. We show that fixed-value point is a bridge from deterministic analysis to random analysis of the algorithm. Finally, a numerical example illustrates the convergence of the proposed algorithm.

Measure of noncompactness and application to stochastic differential equations

In this paper, we study the existence and uniqueness of the solution of stochastic differential equation by means of the properties of the associated condensing nonexpansive random operator. Moreover, by taking account of the results of Diaz and Metcalf, we prove the convergence of Kirk’s process to this solution for small times.

Random fixed point theorem of Krasnoselskii type for the sum of two operators

In this paper, we prove some random fixed point theorem for the sum of a weakly-strongly continuous random operator and a nonexpansive random operator in Banach spaces. Our results are the random versions of some deterministic fixed point theorems of Edmund (Math. Ann. 174:233-239, 1967), O’Regan (Appl. Math. Lett. 9:1-8, 1996) and some known results in the literature.

Convergence of implicit random iteration process with errors for a finite family of asymptotically quasi-nonexpansive random operators

In this paper, we prove that an implicit random iteration process with errors which is generated by a finite family of asymptotically quasi- nonexpansive random operators converges strongly to a common random fixed point of the random operators in uniformly convex Banach spaces.

VISCOSITY APPROXIMATION METHODS OF RANDOM FIXED POINT SOLUTIONS AND RANDOM VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper, we construct random iterative processes for nonexpansive random operators and study necessary conditions for these processes. It is shown that these random iterative processes converge to random fixed points of nonexpansive random operators and solve some random variational inequalities. We also proved that an implicit random iterative process converges to the random fixed point and solves these random variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequality theory and also give generalization stochastic version of some results of Xu [23].

COMPOSITE IMPLICIT RANDOM ITERATIONS FOR APPROXIMATING COMMON RANDOM FIXED POINT FOR A FINITE FAMILY OF ASYMPTOTICALLY NONEXPANSIVE RANDOM OPERATORS

In the present work we construct a composite implicit ran- dom iterative process with errors for anite family of asymptotically nonexpansive random operators and discuss a necessary and sufficient condition for the convergence of this process in an arbitrary real Banach space. It is also proved that this process converges to the common random �xed point of thenite family of asymptotically nonexpansive random op- erators in the setting of uniformly convex Banach spaces. The present work also generalizes a recently established result in Banach spaces.

Multi-step random iterations for approximating fixed points of asymptotically nonexpansive random operators in the intermediate sense

In the present work, we discuss a random multi-step fixed point iteration with errors for a finite family of random asymptotically nonexpansive operators in the intermediate sense. The work is in line with the works on constructions of iterations for finding random fixed points of nonlinear random operators. The present work also generalizes some recently established results in Banach spaces.

Random fixed points of asymptotically nonexpansive random operators on unbounded domains

Abstract The aim of this paper is to prove some random fixed point theorems for asymptotically nonexpansive random operator defined on an unbounded closed and starshaped subset of a Banach space.

Random fixed point theorems for a random operator on an unbounded subset of a Banach space

Results regarding the existence of random fixed points of a nonexpansive random operator defined on an unbounded subset of a Banach space are proved.

The convergence theorems for two nonexpansive random operators

In this paper, we concerned with the study of a random iterative scheme involving two nonexpansive random operators. Using the random iterative scheme, we approximate the random common fixed points of these two operators in a uniformly convex separable Banach space under some appropriate conditions. Mathematics Subject Classification: 47H09, 47J05, 47J25

Random Three-Step Iteration Scheme and Common Random Fixed Point of Three Operators

We construct random iterative processes with errors for three asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. The results presented in this paper extend and improve the recent ones announced by I. Beg and M. Abbas (2006), and many others.

Convergence of iterative algorithms to common random fixed points of random operators

We prove the existence of a common random fixed point of two asymptotically nonexpansive random operators through strong and weak convergences of an iterative process. The necessary and sufficient condition for the convergence of sequence of measurable functions to a random fixed point of asymptotically quasi-nonexpansive random operators in uniformly convex Banach spaces is also established.

Random fixed point theorem for multivalued nonexpansive operators in Uniformly nonsquare Banach spaces

Let (Ω, Σ) be a measurable space, with Σ a sigma-algebra of subset of Ω, and let C be a nonempty bounded closed convex and separable subset of a Banach space X, satisfying Dominguez-Lorenzo condition, KC(X) the family of all compact convex subsets of X. We prove that a 1-χ contractive mutivalued nonexpansive random operator from C into KC(X) satisfying an inwardness condition has a random fixed point. Furthermore, we also prove that a uniformly nonsquare Banach spaces with property WORTH has a random fixed point for multivalued nonexpansive non-self random operators.

Iterative procedures for solutions of random operator equations in Banach spaces

We construct random iterative processes for weakly contractive and asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. It is shown that they converge to the random fixed points of these operators in the setting of Banach spaces. We also proved that an implicit random iterative process converges to the common random fixed point of a finite family of asymptotically quasi-nonexpansive random operators in uniformly convex Banach spaces.