A General Inertial Viscosity Type Method for Nonexpansive Mappings and Its Applications in Signal Processing

In this paper, we propose viscosity algorithms with two different inertia parameters for solving fixed points of nonexpansive and strictly pseudocontractive mappings. Strong convergence theorems are obtained in Hilbert spaces and the applications to the signal processing are considered. Moreover, some numerical experiments of proposed algorithms and comparisons with existing algorithms are given to the demonstration of the efficiency of the proposed algorithms. The numerical results show that our algorithms are superior to some related algorithms.

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

Symmetry ◽

10.3390/sym14010123 ◽

2022 ◽

Vol 14 (1) ◽

pp. 123

Author(s):

Vasile Berinde

Keyword(s):

Fixed Point ◽

Strong Convergence ◽

Fixed Points ◽

Hilbert Spaces ◽

Convergence Theorem ◽

Numerical Experiments ◽

Nonexpansive Mappings ◽

Strong Convergence Theorem ◽

Fixed Point Algorithm ◽

Mann Algorithm

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.

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Computing the Fixed Points of Strictly Pseudocontractive Mappings by the Implicit and Explicit Iterations

Abstract and Applied Analysis ◽

10.1155/2012/315835 ◽

2012 ◽

Vol 2012 ◽

pp. 1-13

Author(s):

Yeong-Cheng Liou

Keyword(s):

Fixed Point ◽

Inverse Problems ◽

Fixed Points ◽

Hilbert Spaces ◽

Nonexpansive Mappings ◽

Minimum Norm ◽

Pseudocontractive Mappings ◽

Special Cases ◽

Implicit And Explicit ◽

Strictly Pseudocontractive Mappings

It is known that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings have been constructed in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.

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Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators

Annals of West University of Timisoara - Mathematics and Computer Science ◽

10.2478/awutm-2018-0013 ◽

2018 ◽

Vol 56 (2) ◽

pp. 13-27 ◽

Cited By ~ 1

Author(s):

Vasile Berinde

Keyword(s):

Hilbert Space ◽

Strong Convergence ◽

Fixed Points ◽

Iterative Algorithm ◽

Hilbert Spaces ◽

Weak And Strong Convergence ◽

Convergence Theorems ◽

Pseudocontractive Mappings ◽

Strictly Pseudocontractive Mappings ◽

Nonlinear Mappings

AbstractIn this paper, we introduce and study the class of enriched strictly pseudocontractive mappings in Hilbert spaces and extend some convergence theorems, i.e., Theorem 12 in [Brow-der, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228] and Theorem 3.1 in [Marino, G., Xu, H.-K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), no. 1, 336–346], from the class of strictly pseudocontractive mappings to that of enriched strictly pseudocontractive mappings and thus include many other important related results from literature as particular cases.

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ON STRONG CONVERGENCE OF HALPERN’S METHOD FOR QUASI-NONEXPANSIVE MAPPINGS IN HILBERT SPACES

Mathematical Modelling and Analysis ◽

10.3846/13926292.2016.1132787 ◽

2016 ◽

Vol 21 (1) ◽

pp. 63-82 ◽

Cited By ~ 2

Author(s):

Jesus Garcia Falset ◽

Enrique Llorens-Fuster ◽

Giuseppe Marino ◽

Angela Rugiano

Keyword(s):

Hilbert Space ◽

Nonexpansive Mapping ◽

Strong Convergence ◽

Fixed Points ◽

Hilbert Spaces ◽

Iterative Scheme ◽

Nonexpansive Mappings ◽

Common Fixed Points ◽

Type Method ◽

Numerical Example

In this paper, we introduce a Halpern’s type method to approximate common fixed points of a nonexpansive mapping T and a strongly quasi-nonexpansive mappings S, defined in a Hilbert space, such that I − S is demiclosed at 0. The result shows as the same algorithm converges to different points, depending on the assumptions of the coefficients. Moreover, a numerical example of our iterative scheme is given.

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Modified CQ-Algorithms for G-Nonexpansive Mappings in Hilbert Spaces Involving Graphs

New Mathematics and Natural Computation ◽

10.1142/s1793005720500064 ◽

2020 ◽

Vol 16 (01) ◽

pp. 89-103

Author(s):

W. Cholamjiak ◽

D. Yambangwai ◽

H. Dutta ◽

H. A. Hammad

Keyword(s):

Hilbert Space ◽

Fixed Point ◽

Strong Convergence ◽

Common Fixed Point ◽

Rate Of Convergence ◽

Hilbert Spaces ◽

Numerical Experiments ◽

Nonexpansive Mappings ◽

Iterative Schemes ◽

Cq Method

In this paper, we introduce four new iterative schemes by modifying the CQ-method with Ishikawa and [Formula: see text]-iterations. The strong convergence theorems are given by the CQ-projection method with our modified iterations for obtaining a common fixed point of two [Formula: see text]-nonexpansive mappings in a Hilbert space with a directed graph. Finally, to compare the rate of convergence and support our main theorems, we give some numerical experiments.

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A Hybrid Forward–Backward Algorithm and Its Optimization Application

Mathematics ◽

10.3390/math8030447 ◽

2020 ◽

Vol 8 (3) ◽

pp. 447

Author(s):

Liya Liu ◽

Xiaolong Qin ◽

Jen-Chih Yao

Keyword(s):

Signal Processing ◽

Strong Convergence ◽

Hilbert Spaces ◽

Numerical Experiments ◽

Sparse Reconstruction ◽

Convergence Solution

In this paper, we study a hybrid forward–backward algorithm for sparse reconstruction. Our algorithm involves descent, splitting and inertial ideas. Under suitable conditions on the algorithm parameters, we establish a strong convergence solution theorem in the framework of Hilbert spaces. Numerical experiments are also provided to illustrate the application in the field of signal processing.

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A Strong Convergence Theorem for a Common Fixed Point of Two Sequences of Strictly Pseudocontractive Mappings in Hilbert Spaces and Applications

Abstract and Applied Analysis ◽

10.1155/2010/876819 ◽

2010 ◽

Vol 2010 ◽

pp. 1-17 ◽

Cited By ~ 2

Author(s):

Kasamsuk Ungchittrakool

Keyword(s):

Fixed Point ◽

Strong Convergence ◽

Common Fixed Point ◽

Hilbert Spaces ◽

Convergence Theorem ◽

Common Element ◽

Strong Convergence Theorem ◽

Pseudocontractive Mappings ◽

Strict Pseudocontraction ◽

Strictly Pseudocontractive Mappings

We prove a strong convergence theorem for a common fixed point of two sequences of strictly pseudocontractive mappings in Hilbert spaces. We also provide some applications of the main theorem to find a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem in Hilbert spaces. The results extend and improve the recent ones announced by Marino and Xu (2007) and others.

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Tikhonov Regularization Terms for Accelerating Inertial Mann-Like Algorithm with Applications

Symmetry ◽

10.3390/sym13040554 ◽

2021 ◽

Vol 13 (4) ◽

pp. 554

Author(s):

Hasanen A. Hammad ◽

Habib ur Rehman ◽

Hassan Almusawa

Keyword(s):

Variational Inequality ◽

Strong Convergence ◽

Fixed Points ◽

Tikhonov Regularization ◽

Hilbert Spaces ◽

Variational Inequality Problem ◽

Nonexpansive Mappings ◽

Inclusion Problem ◽

Monotone Inclusion ◽

Monotone Inclusion Problem

In this manuscript, we accelerate the modified inertial Mann-like algorithm by involving Tikhonov regularization terms. Strong convergence for fixed points of nonexpansive mappings in real Hilbert spaces was discussed utilizing the proposed algorithm. Accordingly, the strong convergence of a forward–backward algorithm involving Tikhonov regularization terms was derived, which counts as finding a solution to the monotone inclusion problem and the variational inequality problem. Ultimately, some numerical discussions are presented here to illustrate the effectiveness of our algorithm.

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Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family ofki-Strictly Pseudocontractive Mapping in Hilbert Spaces

Journal of Applied Mathematics ◽

10.1155/2012/416476 ◽

2012 ◽

Vol 2012 ◽

pp. 1-23

Author(s):

Haitao Che ◽

Meixia Li ◽

Xintian Pan

Keyword(s):

Hilbert Spaces ◽

Equilibrium Problems ◽

Iterative Scheme ◽

Nonexpansive Mappings ◽

Infinite Family ◽

Common Element ◽

Strictly Pseudocontractive Mapping ◽

Pseudocontractive Mappings ◽

Definition Of ◽

Strictly Pseudocontractive Mappings

We first extend the definition of Wnfrom an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family ofki-strictly pseudocontractive mappings in Hilbert spaces. The results obtained in this paper extend and improve the recent ones announced by many others. Furthermore, a numerical example is presented to illustrate the effectiveness of the proposed scheme.

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