Strong Convergence Theorems Research Articles

Asymptotic Behaviour of -Nonexpansive Sequences and Mappings in Banach Spaces

In this paper, we introduce the notion of -nonexpansive sequences and maps in a Banach space E, and prove an ergodic theorem, a fixed point theorem, and some weak and strong convergence theorems for such sequences and maps in E, with rather mild assumptions on the Banach space E, weaker than those assumed by previous authors for nonexpansive maps. These results give a partial answer to the still open problem raised in 1981 by Djafari Rouhani for nonexpansive sequences in a Banach space.

Strong convergence theorems by an extragradient-like approximation methods for quasi-variational inequalities

Strong convergence theorems by an extragradient-like approximation methods for quasi-variational inequalities

Convergence analysis for solving equilibrium problems and split feasibility problems in Hilbert spaces

In this article, we propose four alternated inertial algorithms for finding a common solution of equilibrium problems and split feasibility problems in Hilbert spaces. We present a variable step size, which is not required to know the operator norm. Furthermore, these algorithms adopt the new convex subset form by a sequence of closed balls instead of half spaces, and it is easy to calculate the projections onto these sets. We establish strong and weak convergence theorems of these algorithms under some proper assumptions and also present a numerical experiment to illustrate the performance and the advantage of the proposed algorithms.

Two fast converging inertial subgradient extragradient algorithms with variable stepsizes for solving pseudo-monotone VIPs in Hilbert spaces

In this work, we propose two new iterative schemes for finding an element of the set of solutions of a pseudo-monotone, Lipschitz continuous variational inequality problem in real Hilbert spaces. The weak and strong convergence theorems are presented. The advantage of the proposed algorithms is that they do not require prior knowledge of the Lipschitz constant of the variational inequality mapping and only compute one projection onto a feasible set per iteration as well as without using the sequentially weakly continuity of the associated mapping. Under additional strong pseudo-monotonicity and Lipschitz continuity assumptions, we obtain also an R-linear convergence rate of the proposed algorithm. Finally, some numerical examples are given to illustrate the effectiveness of the algorithms.

An inertial s-iteration process for a common fixed point of a family of quasi-Bregman nonexpansive mappings

In this paper, an inertial S-iteration iterative process for approximating a common fixed point of a finite family of quasi-Bregman nonexpansive mappings is introduced and studied in a reflexive Banach space. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, extend, and generalize some recent results in the literature.

On the Convergence of Two Iterative Methods for k-strictly Pseudo-contractive Mappings in Cat(0) Spaces

In this paper, we prove the demiclosedness principle for k -strictly pseudo-contractive mappings and establish the Δ - convergence theorem of the cyclic algorithm for such mappings in CAT(0) spaces. Also, we give the strong convergence theorem of the modified Halpern iteration for k -strictly pseudo-contractive mappings in CAT(0) spaces. Our results extend and improve the corresponding recent results announced by many authors in the literature.

Approximation of common solution of a system of equilibrium problem and fixed point of generalized Bregman nonspreading mappings in Banach spaces

In this paper, we introduce a hybrid iterative method for approximating the common solution of a systems of generalized mixed equilibrium problem and the class of [Formula: see text]-generalized Bregman nonspreading mappings in reflexive Banach space. We prove the asymptotic fixed point property of the class of [Formula: see text]-generalized Bregman nonspreading mappings in reflexive Banach spaces and provide a strong convergence theorem for solving the problem under some mild conditions. Furthermore, some numerical results were presented to show the important and computational performance of the proposed method. This result extends and generalized many interesting results in the contemporary literature.

An Inertial Extragradient Direction Method with Self-Adaptive Step Size for Solving Split Minimization Problems and Its Applications to Compressed Sensing

The purpose of this work is to construct iterative methods for solving a split minimization problem using a self-adaptive step size, conjugate gradient direction, and inertia technique. We introduce and prove a strong convergence theorem in the framework of Hilbert spaces. We then demonstrate numerically how the extrapolation factor (θn) in the inertia term and a step size parameter affect the performance of our proposed algorithm. Additionally, we apply our proposed algorithms to solve the signal recovery problem. Finally, we compared our algorithm’s recovery signal quality performance to that of three previously published works.

Weak and Strong Convergence Theorems of Modified Projection-Type Ishikawa Iteration Scheme for Lipschitz α-Hemicontractive Mappings

In this paper, we establish weak and strong convergence theorems of a two-step modified projection-type Ishikawa iterative scheme to the fixed point of α-hemicontractive mappings without any compactness assumption on the operator or the space. Our results extend, improve and generalize several previously known results of the existing literature.

"Viscosity algorithm for solving split generalized equilibrium problem and fixed-point problem"

"This article considers a split generalized equilibrium problem and fixed point problem for infi- nite family of nonexpansive mapping in Hilbert space. We propose an algorithm for finding a common solution of these problems. Under mild assumptions, we establish a strong convergence theorem for the sequence gen- erated by the proposed algorithm using viscosity technique. We present the implementation of the proposed algorithm by considering some numerical illustrations and comparison of the proposed algorithm with an ex- isting algorithm in the literature."

Strong Convergence Theorem for Finding a Common Solution of Convex Minimization and Fixed Point Problems in CAT(0) Spaces

In this paper, we introduce a proximal point algorithm for approximating a common solution of finite family of convex minimization problems and fixed point problems fork-demicontractive mappings in complete CAT(0) spaces. We prove a strong convergence result and obtain other consequence results which generalize and extend some recent results in the literature. We further provide a numerical example to illustrate the convergence behaviour of the sequence generated by our algorithm.

SELF ADAPTIVE VISCOSITY-TYPE INERTIAL EXTRAGRADIENT ALGORITHMS FOR SOLVING VARIATIONAL INEQUALITIES WITH APPLICATIONS

In this paper, we introduce two new inertial extragradient algorithms with non-monotonic stepsizes for solving monotone and Lipschitz continuous variational inequality problems in real Hilbert spaces. Strong convergence theorems of the suggested iterative schemes are established without the prior knowledge of the Lipschitz constant of the mapping. Finally, some numerical examples are provided to illustrate the efficiency and advantages of the proposed algorithms and compare them with some related ones.

An asynchronous inertial algorithm for solving convex feasibility problems with strict pseudo-contractions in Hilbert spaces

AbstractFinding a common point in a finite intersection of sets, say, $C=\cap _{i=1}^{n} F(T_i)$, where each $T_i$ is a non-expansive-type mapping, is a central task in mathematics as it cuts across different areas of application, such as road design and medical image reconstruction. There are many algorithms for approximating solutions of such problems. Of particular interest in the implementation of these algorithms are cost and speed. This is due to the large computations to be performed at each step of the iterative process. One of the most efficient methods that optimizes the time of computation and cost of implementation is the asynchronous-parallel algorithm method. In this paper, we prove a weak convergence theorem for the asynchronous sequential inertial (ASI) algorithm (introduced by Heaton and Censor in [H. Heaton and Y. Censor, Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems, J. Glob. Optim. 74 (2019), 95–119.] ) for strictly pseudo-contractive mappings in Hilbert spaces. Under additional mild conditions, we also obtain a strong convergence theorem. Finally, we apply the ASI algorithm to solving convex minimization problems and Hammerstein integral equations.

"A modified Halpern extragradient method for equilibrium and fixed point problems in CAT(0) space"

"In this paper, we introduce a modified Halpern extragradient-type algorithm for approximating an element in the intersection of the set of common solutions of equilibrium problems and common fixed points of family of nonexpansive mappings in a complete CAT(0) space. We establish strong convergence theorem that improve and generalize recently announced results in the literature."

Strong convergence of Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations under local Lipschitz conditions of state variables

AbstractThis paper develops strong convergence of the Euler–Maruyama (EM) schemes for approximating McKean–Vlasov stochastic differential equations (SDEs). In contrast to the existing work, a novel feature is the use of a much weaker condition—local Lipschitzian in the state variable, but under uniform linear growth assumption. To obtain the desired approximation, the paper first establishes the existence and uniqueness of solutions of the original McKean–Vlasov SDE using a Euler-like sequence of interpolations and partition of the sample space. Then, the paper returns to the analysis of the EM scheme for approximating solutions of McKean–Vlasov SDEs. A strong convergence theorem is established. Moreover, the convergence rates under global conditions are obtained.

A Regularized Generalized Popov’s Method to Solve the Hierarchical Variational Inequality Problem with Generalized Lipschitzian Mappings

In this article, we introduce a new inertial multi-step regularized generalized Popov’s extra-gradient method to solve the hierarchical variational inequality problem (HVIP). We extend the previous Lipschitzian and strongly monotone mapping to a hemicontinuous, generalized Lipschitzian and strongly monotone mapping. We also obtain a strong convergence theorem about the new Popov’s algorithm. Furthermore, we utilize some numerical experiments to highlight the feasibility and effectiveness of our method.

A Modified Krasnosel’skiǐ–Mann Iterative Algorithm for Approximating Fixed Points of Enriched Nonexpansive Mappings

For approximating the fixed points of enriched nonexpansive mappings in Hilbert spaces, we consider a modified Krasnosel’skiǐ–Mann algorithm for which we prove a strong convergence theorem. We also empirically compare the rate of convergence of the modified Krasnosel’skiǐ–Mann algorithm and of the simple Krasnosel’skiǐ fixed point algorithm. Based on the numerical experiments reported in the paper we conclude that, for the class of enriched nonexpansive mappings, it is more convenient to work with the simple Krasnosel’skiǐ fixed point algorithm than with the modified Krasnosel’skiǐ–Mann algorithm.

On numerical approximation of common fixed points for enriched Kannan semigroups and its experiment

In this paper, we present a concept of semigroups of enriched Kannan mappings, which is based on enriching contractive type mappings that have been effectively used in many recent studies. We investigate some weak and strong convergence theorems for the Mann iterative process to approximate common fixed points of enriched Kannan semigroups in uniformly convex Banach spaces. In the end, by using MATLAB, we present a numerical example to support our results.

Self-adaptive algorithms for the split problem of the quasi-pseudocontractive operators in Hilbert spaces

<abstract><p>Self-adaptive algorithms are presented for solving the split common fixed point problem of quasi-pseudocontractive operators in Hilbert spaces. Weak and strong convergence theorems are given under some mild assumptions.</p></abstract>

An inertial iterative method for solving split equality problem in Banach spaces

<abstract><p>In this paper, a new self-adaptive algorithm with the inertial technique is proposed for solving the split equality problem in $ p $-uniformly convex and uniformly smooth Banach spaces. Under some mild control conditions, a strong convergence theorem for the proposed algorithm is established. Furthermore, the results are applied to split equality fixed point problem and split equality variational inclusion problem. Finally, numerical examples are provided to illustrate the convergence behaviour of the algorithm. The main results in this paper improve and generalize some existing results in the literature.</p></abstract>