Family Of Nonexpansive Mappings Research Articles

A characterization of the family of iterated nonexpansive mappings under every renorming

We characterize the family of iterated nonexpansive mappings that are stable under every renorming. The family of iterated nonexpansive mappings contains the family of nonexpansive mappings, it also contains quasi-nonexpansive and Suzuki's (C)-type mappings with fixed points, among others. We also give the corresponding characterizations for quasi-nonexpansive and some Suzuki's (C)-type mappings with fixed points.

Modified Mann Subgradient-like Extragradient Rules for Variational Inequalities and Common Fixed Points Involving Asymptotically Nonexpansive Mappings

In a real Hilbert space, we aim to investigate two modified Mann subgradient-like methods to find a solution to pseudo-monotone variational inequalities, which is also a common fixed point of a finite family of nonexpansive mappings and an asymptotically nonexpansive mapping. We obtain strong convergence results for the sequences constructed by these proposed rules. We give some examples to illustrate our analysis.

On Mann-Type Subgradient-like Extragradient Method with Linear-Search Process for Hierarchical Variational Inequalities for Asymptotically Nonexpansive Mappings

We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint.

Some Fixed Point Results for Commuting Families of Mappings in Modular Spaces

Assume that ρ is a convex modular satisfying the -type condition and the modular space is either α-uniformly ρ-noncompact convex or it satisfies the strong Opial condition. We prove that the fixed point set of a commuting family of asymptotically non-expansive mappings defined from a ρ-convex weakly compact set C into C is a non-empty nonexpansive retract of C. We show that our results about existence of fixed point for commuting families of nonexpansive mappings also work if we replace the weak topology by some other topologies weaker than the ρ-topology. In particular, we can obtain fixed point results when C is ρ-a.e. compact in a modular function space.

An inertial iterative method for split generalized vector mixed equilibrium and fixed point problems.

In this paper, we introduce an inertial-type algorithm for approximating a common solution of split generalized mixed vector equilibrium and fixed point problems. In the framework of real Hilbert spaces, we state and prove a strong convergence theorem for obtaining a common solution of split generalized mixed vector equilibrium problem and fixed point of a finite family of nonexpansive mappings. Furthermore, we give some consequences of our main result and also report some numerical illustrations to display the performance of our method. The result obtained in this paper unifies and generalizes other corresponding results in the literature.

Multistep hybrid viscosity method for split monotone variational inclusion and fixed point problems in Hilbert spaces

In this paper, we present a multi-step hybrid iterative method. It is proven that under appropriate assumptions, the proposed iterative method converges strongly to a common element of fixed point of a finite family of nonexpansive mappings, the solution set of split monotone variational inclusion problem and the solution set of triple hierarchical variational inequality problem (THVI) in real Hilbert spaces. In addition, we give a numerical example of a triple hierarchical system derived from our generalization.

Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

Hybrid viscosity approximation methods for systems of variational inequalities and hierarchical fixed point problems

In this paper, we propose an implicit iterative method and an explicit iterative method for solving a general system of variational inequalities with a hierarchical fixed point problem constraint for an infinite family of nonexpansive mappings. We show that the proposed algorithms converge strongly to a solution of the general system of variational inequalities, which is a unique solution of the hierarchical fixed point problem.

Iterative Methods for Finding Solutions of a Class of Split Feasibility Problems over Fixed Point Sets in Hilbert Spaces

We consider the split feasibility problem in Hilbert spaces when the hard constraint is common solutions of zeros of the sum of monotone operators and fixed point sets of a finite family of nonexpansive mappings, while the soft constraint is the inverse image of a fixed point set of a nonexpansive mapping. We introduce iterative algorithms for the weak and strong convergence theorems of the constructed sequences. Some numerical experiments of the introduced algorithm are also discussed.

Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems

In this work, let X be Banach space with a uniformly convex and q-uniformly smooth structure, where 1 < q ≤ 2 . We introduce and consider a generalized Mann-like viscosity implicit rule for treating a general optimization system of variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonexpansive mappings in X. The generalized Mann-like viscosity implicit rule investigated in this work is based on the Korpelevich’s extragradient technique, the implicit viscosity iterative method and the Mann’s iteration method. We show that the iterative sequences governed by our generalized Mann-like viscosity implicit rule converges strongly to a solution of the general optimization system.

On iterative solutions of a split feasibility problem with nonexpansive mappings

We analyze iterative solutions of a split feasibility problem with common fixed point constraints of a family of nonexpansive mappings. We present solution theorems of the feasibility problem under some weak assumptions imposed on different mappings and control sequences.

A hybrid method for solving variational inequalities over the common fixed point sets of infinite families of nonexpansive mappings in Banach spaces

ABSTRACT In this paper, we introduce a hybrid method, a combination of the steepest-descent method and the Krasnosel'skii-Mann one, for solving a variational inequality over the set of common fixed points of an infinite family of nonexpansive mappings in Banach spaces under two different conditions on the Banach space, either a uniformly smooth Banach space or a reflexive and strictly convex one with a uniformly Gâteaux differentiable norm, without imposing the sequential weak continuity of the normalized duality mapping. The method is an improvement and extension of some other published results. We also give a numerical example to illustrate the convergence analysis of the proposed method.

Iterative methods for fixed points and zero points of nonlinear mappings with applications

ABSTRACT The purpose of this article is to investigate a descent-type iterative algorithm for finding a common element of fixed-point sets of a finite family of nonexpansive mappings and zero-point sets of pseudomonotone mappings in Hilbert spaces. With suitable assumptions, we derive necessary and sufficient conditions for the strong convergence of the algorithm. Numerical examples and applications to signal processing problems are provided to support the main results.

Fixed point results in convex metric spaces

The purpose of this paper is to establish and prove a strong convergence theorem of Ishikawa iterative for finding a common fixed point of the combination of a finite family of nonexpansive mappings in a convex metric space. Moreover, our result generalizes and modifies the other result, Phuengrattana and Suantai’s result Phuengrattana and Suantai (Ind J Pure Appl Math 45(1):121–136, 2014).

An Iterative Method for Finding Common Solution of the Fixed Point Problem of a Finite Family of Nonexpansive Mappings and a Finite Family of Variational Inequality Problems in Hilbert Space

In this paper, a hybrid iterative algorithm is proposed for finding a common element of the set of common fixed points of finite family of nonexpansive mappings and the set of common solutions of the variational inequality for an inverse strongly monotone mapping on the real Hilbert space. We establish the strong convergence of the proposed method for approximating a common element of the above defined sets under some suitable conditions. The results presented in this paper extend and improve some well-known corresponding results in the earlier and recent literature.

Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

In this paper, we study a general system of variational inequalities with a hierarchical variational inequality constraint for an infinite family of nonexpansive mappings. We introduce general implicit and explicit iterative algorithms. We prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the studied problems.

AMENABLE SEMIGROUPS OF NONLINEAR OPERATORS IN UNIFORMLY CONVEX BANACH SPACES

In 1965, Browder proved the existence of a common fixed point for commuting families of nonexpansive mappings acting on nonempty bounded closed convex subsets of uniformly convex Banach spaces. The purpose of this paper is to extend this result to left amenable semigroups of nonexpansive mappings.

The Product Formula and Evolution Families of Nonexpansive Mappings

In this work we obtain a product formula type for a two-parameters commuting family of nonexpansive mappings on $${\mathbb {D}}$$ . This is established by following the techniques used by Simeon Reich and David Shoikhet in the study of one-parameter semigroups of holomorphic and nonexpansive self-mappings in $${\mathbb {D}}$$ . Also, we stablish such a formula for the family of non-linear resolvent of a strongly $$\rho $$ -monotone functions on $${\mathbb {D}}$$ and its relation with evolution families of nonexpansive mappings on $${\mathbb {D}}$$ . It is worthy mentioning that the product formula is linked with semigroup of linear and nonlinear operators. Also it is associated with the study of vector fields and flows, but in the literature it is established a product formula for time independent flow.

Convergence theorems for split feasibility problems on a finite sum of monotone operators and a family of nonexpansive mappings

In this paper, we present two iterative algorithms for approximating a solution of the split feasibility problem on zeros of a sum of monotone operators and fixed points of a finite family of nonexpansive mappings. Weak and strong convergence theorems are proved in the framework of Hilbert spaces under some mild conditions. We apply the obtained main result for the problem of finding a common zero of the sum of inverse strongly monotone operators and maximal monotone operators, for finding a common zero of a finite family of maximal monotone operators, for finding a solution of multiple sets split common null point problem, and for finding a solution of multiple sets split convex feasibility problem. Some applications of the main results are also provided.

Modified Proximal Point Algorithms for Solving Constrained Minimization and Fixed Point Problems in Complete CAT(0) Spaces

In this paper, we propose a new modified proximal point algorithm for a countably infinite family of nonexpansive mappings in complete CAT(0) spaces and prove strong convergence theorems for the proposed process under suitable conditions. We also apply our results to solving linear inverse problems and minimization problems. Several numerical examples are given to show the efficiency of the presented method.