Nonexpansive Mappings Research Articles

Some fixed-point theorems for a pair of Reich-Suzuki-type nonexpansive mappings in hyperbolic spaces

Abstract In this article, we prove some fixed-point results for a pair of Reich-Suzuki-type nonexpansive mappings in uniformly convex W W -hyperbolic spaces. We introduce a new iterative scheme and establish its convergence to the fixed points of a pair of Reich-Suzuki-type nonexpansive mappings. We illustrate our main result with an example, and using Matlab code, it is observed that our iteration converges faster than the iteration defined by Garodia et al. for a pair of Reich-Suzuki-type nonexpansive mappings. An application is given to substantiate our main result.

Generalized Nearly Asymptotically Nonexpansive Mappings and a System of Variational Inclusions: Iterative Schemes and Convergence Analysis

Generalized Nearly Asymptotically Nonexpansive Mappings and a System of Variational Inclusions: Iterative Schemes and Convergence Analysis

On Fixed Point Convergence Results for a General Class of Nonlinear Mappings with a Supportive Application

In this article, we considered the class of generalized α , β -nonexpansive (GABN) mappings that properly includes all nonexpansive, Suzuki nonexpansive (SN), generalized α -nonexpansive (GAN), and Reich–Suzuki nonexpansive (RSN) mappings. We used the iterative scheme JA for finding fixed points of these mappings in a Banach space setting. We provided both weak and strong convergence results under some mild conditions on the mapping, domain, and on the parameters involved in our iterative scheme. To support these results numerically, we constructed a new example of GABN mappings and proved that the JA iterative scheme converges to its fixed point. Moreover, we proved that JA iterative scheme converges faster to the fixed point corresponding to the some other iterative schemes of the literature. Eventually, we carried out an application of our main outcome to solve a split feasibility of problems (SFPs) in the setting of GABN mappings. Thus, our results were new in the literature and improved well-known results of the literature.

Convergence theorems for total asymptotically nonexpansive mappings in operatorname{CAT} (kappa ) spaces

The purpose of this paper is to study the convergence theorems in operatorname{CAT} (kappa ) spaces with k > 0 for total asymptotically nonexpansive mappings which are essentially wider than nonexpansive mappings, asymptotically nonexpansive mapping, and asymptotically nonexpansive mappings in the intermediate sense. Our results generalize, unify, and improve several comparable results in the existing literature.

A new approach to fixed point property of nonexpansive multivalued mappings

A new approach to fixed point property of nonexpansive multivalued mappings

Implicit viscosity iterative algorithm for nonexpansive mapping on Hadamard manifolds

Implicit viscosity iterative algorithm for nonexpansive mapping on Hadamard manifolds

Variational inclusion problem and total asymptotically nonexpansive mapping: graph convergence, algorithms and approximation of common solutions

Variational inclusion problem and total asymptotically nonexpansive mapping: graph convergence, algorithms and approximation of common solutions

Endpoints of generalized Berinde nonexpansive mappings in hyperbolic spaces

Endpoints of generalized Berinde nonexpansive mappings in hyperbolic spaces

Making Decisions about How to Solve Nonlinear Dynamical Systems of the Kannan Type in a New Complex Function Space of Formal Power Series

The sufficient settings of the space generated by absolute type-weighted gamma matrices of rank p in the Nakano complex functions of the formal power series, as well as its associated prequasioperators’ ideal equipped with definite functions, are defined and explained in this paper. Assorted prequasinorms are shown to have the Fatou characteristic. Some of its geometric and topological features and the prequasioperators’ ideal that goes with it are talked about. These structures have a relationship with the fixed points of the Kannan contraction and nonexpansive mappings. By looking at real-world examples and how they are used, it is shown that there are solutions for nonlinear dynamical systems of the Kannan type.

A New Accelerated Algorithm Based on Fixed Point Method for Convex Bilevel Optimization Problems with Applications

A new accelerated common fixed point algorithm is introduced and analyzed for a countable family of nonexpansive mappings and then we apply it to solve some convex bilevel optimization problems. Then, under some suitable conditions, we prove a strong convergence result of the proposed algorithm. As an application, we employ the proposed algorithm for regression and classification problems. Moreover, we compare the performance of our algorithm with others. By numerical experiments, we found that our algorithm has a better performance than the others.

Triple-adaptive subgradient extragradient with extrapolation procedure for bilevel split variational inequality

This paper introduces a triple-adaptive subgradient extragradient process with extrapolation to solve a bilevel split pseudomonotone variational inequality problem (BSPVIP) with the common fixed point problem constraint of finitely many nonexpansive mappings. The problem under consideration is in real Hilbert spaces, where the BSPVIP involves a fixed point problem of demimetric mapping. The proposed rule exploits the strong monotonicity of one operator at the upper level and the pseudomonotonicity of another mapping at the lower level. The strong convergence result for the proposed algorithm is established under some suitable assumptions. In addition, a numerical example is given to demonstrate the viability of the proposed rule. Our results improve and extend some recent developments to a great extent.

On the Karlsson–Nussbaum conjecture for resolvents of nonexpansive mappings

Let \(D\subset \mathbb{R}^{n}\) be a bounded convex domain and \(F\colon D\rightarrow D\) a 1-Lipschitz mapping with respect to the Hilbert metric \(d\) on \(D\) satisfying condition \(d(sx+(1-s)y,sz+(1-s)w)\leq \max \{d(x,z),d(y,w)\}\). We show that if \(F\) does not have fixed points, then the convex hull of the accumulation points (in the norm topology) of the family \(\{R_{\lambda}\}_{\lambda >0}\) of resolvents of \(F\) is a subset of \(\partial D\). As aconsequence, we show a Wolff-Denjoy type theorem for resolvents of nonexpansive mappings acting on an ellipsoid \(D\).

A Continuous Method for Finding a Generalized Fixed Point of a Nonexpansive Mapping on a Set in a Hilbert Space

We introduce the concept of a generalized fixed point of a nonexpansive operator on a convex closed set in a Hilbert space. To find this point, we construct a regularizing algorithm in the form of the Cauchy problem for a first-order differential equation and establish sufficient conditions for the strong convergence of the resulting approximations to a normal generalized fixed point under approximate specification of the nonexpansive operator and the convex closed set on which the desired generalized fixed point of the operator is located. Examples of parametric functions are given that ensure the convergence of the approximations in the norm of the Hilbert space to a normal generalized fixed point of the operator on the convex closed set in this space.

Two Convergence Results for Inexact Infinite Products of Non-Expansive Mappings

We analyze the asymptotic behavior of infinite products of non-linear operators which take a non-empty, closed subset of a complete metric space into the space, taking into account summable computational errors. Our results can be applied in methods for solving convex feasibility and optimization problems.

On distal flows and common fixed point theorems in Banach spaces

A semigroup S consisting of self-mappings defined on a set B of a Banach space X is said to be a distal flow on B provided0∉{Sx−Sy:S∈S}‾,for allx≠y∈B. In this paper, we show that if T is a nonexpansive self-mapping defined on a nonempty compact convex set C of X so that S≡{Tn:n∈N} forms a distal flow, then T:C→C is a surjective isometry; in particular, if C is of finite dimension, then T is a surjective affine isometry. Consequently, i) every distal S of nonexpansive mappings on a convex compact set C consists of surjective self-isometries on C; ii) all S in the distal S have a common fixed point in the Chebyshev center of C; iii) if, in addition, C is finite dimensional, then every S in S is a surjective affine self-isometry on C. Therefore, it can also be understood as a refinement of a recent common fixed point theorem due to Wiśnicki. We also show that the conclusion i) is not true if C is not compact. Finally, we give a localized setting of Lim-Lin's common fixed point theorem for isometries.

An intermixed method for solving the combination of mixed variational inequality problems and fixed-point problems

In this paper, we introduce an intermixed algorithm with viscosity technique for finding a common solution of the combination of mixed variational inequality problems and the fixed-point problem of a nonexpansive mapping in a real Hilbert space. Moreover, we propose the mathematical tools related to the combination of mixed variational inequality problems in the second section of this paper. Utilizing our mathematical tools, a strong convergence theorem is established for the proposed algorithm. Furthermore, we establish additional conclusions concerning the split-feasibility problem and the constrained convex-minimization problem utilizing our main result. Finally, we provide numerical experiments to illustrate the convergence behavior of our proposed algorithm.

Computational comparative analysis of fixed point approximations of generalized $ \alpha $-nonexpansive mappings in hyperbolic spaces

<abstract><p>In this article, we use the Picard-Thakur hybrid iterative scheme to approximate the fixed points of generalized $ \alpha $-nonexpansive mappings. For generalized $ \alpha $-nonexpansive mappings in hyperbolic spaces, we show several weak and strong convergence results. It is proved numerically and graphically that the Picard-Thakur hybrid iterative scheme converges more faster than other well-known hybrid iterative methods for generalized $ \alpha $-nonexpansive mappings. We also present an application to Fredholm integral equation.</p></abstract>

Iterative approximation of fixed points of generalized $ \alpha _{m} $-nonexpansive mappings in modular spaces

<abstract><p>Our aim of this work is to approximate the fixed points of generalized $ \alpha _{m} $-nonexpansive mappings employing $ AA $-iterative scheme in the structure of modular spaces. The results of fixed points for generalized $ \alpha _{m} $-nonexpansive mappings is proven in this context. Moreover, the stability of the scheme and data dependence results are given for $ m $-contraction mappings. In order to demonstrate that the $ AA $-iterative scheme converges faster than some other schemes for generalized $ \alpha_{m} $-nonexpansive mappings, numerical examples are shown at the end.</p></abstract>

Iterative schemes for numerical reckoning of fixed points of new nonexpansive mappings with an application

<abstract><p>The goal of this manuscript is to introduce a new class of generalized nonexpansive operators, called $ (\alpha, \beta, \gamma) $-nonexpansive mappings. Furthermore, some related properties of these mappings are investigated in a general Banach space. Moreover, the proposed operators utilized in the $ K $-iterative technique estimate the fixed point and examine its behavior. Also, two examples are provided to support our main results. The numerical results clearly show that the $ K $-iterative approach converges more quickly when used with this new class of operators. Ultimately, we used the $ K $-type iterative method to solve a variational inequality problem on a Hilbert space.</p></abstract>

Strong convergence theorems for split variational inequality problems in Hilbert spaces

<abstract><p>In this paper, we consider the variational inequality problem and the split common fixed point problem. Considering the common fixed points of an infinite family of nonexpansive mappings, instead of just the fixed point of one nonexpansive mapping, we generalize the results of Tian and Jiang. By removing a projection operator, we improve the efficiency of our algorithm. Finally, we propose a very simple modification to the extragradient method, which gives our algorithm strong convergence properties. We also provide some numerical examples to illustrate our main results.</p></abstract>