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

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.

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

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 288 ◽  
Author(s):  
Yinglin Luo ◽  
Meijuan Shang ◽  
Bing Tan
Keyword(s):  
Signal Processing ◽  
Strong Convergence ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Nonexpansive Mappings ◽  
Type Method ◽  
Pseudocontractive Mappings ◽  
Inertia Parameters ◽  
Strictly Pseudocontractive Mappings

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.

scholarly journals Some Algorithms for Finding Fixed Points and Solutions of Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Jong Soo Jung
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Minimum Norm ◽  
Strictly Pseudocontractive Mapping ◽  
Pseudocontractive Mapping ◽  
Unique Minimum ◽  
Implicit And Explicit

We introduce new implicit and explicit algorithms for finding the fixed point of ak-strictly pseudocontractive mapping and for solving variational inequalities related to the Lipschitzian and strongly monotone operator in Hilbert spaces. We establish results on the strong convergence of the sequences generated by the proposed algorithms to a fixed point of ak-strictly pseudocontractive mapping. Such a point is also a solution of a variational inequality defined on the set of fixed points. As direct consequences, we obtain the unique minimum-norm fixed point of ak-strictly pseudocontractive mapping.

scholarly journals Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
C. E. Chidume ◽  
C. O. Chidume ◽  
N. Djitté ◽  
M. S. Minjibir
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Hilbert Spaces ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Convergence Theorems ◽  
Strictly Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Iteration Sequence ◽  
Strictly Pseudocontractive Mappings

LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Suppose thatT:K→2Kis a multivalued strictly pseudocontractive mapping such thatF(T)≠∅. A Krasnoselskii-type iteration sequence{xn}is constructed and shown to be an approximate fixed point sequence ofT; that is,limn→∞d(xn,Txn)=0holds. Convergence theorems are also proved under appropriate additional conditions.

scholarly journals Finding Minimum Norm Fixed Point of Nonexpansive Mappings and Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Xue Yang ◽  
Yeong-Cheng Liou ◽  
Yonghong Yao
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Minimum Norm ◽  
New Methods

We construct two new methods for finding the minimum norm fixed point of nonexpansive mappings in Hilbert spaces. Some applications are also included.

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

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
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.

scholarly journals Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition

2020 ◽  
Vol 36 (1) ◽  
pp. 27-34 ◽  
Author(s):  
VASILE BERINDE
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Displacement Condition ◽  
Nonlinear Mappings

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.

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

Symmetry ◽  
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.

scholarly journals Iterative Schemes for Fixed Point Computation of Nonexpansive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Rudong Chen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Minimum Norm ◽  
Practical Applications ◽  
Fixed Point Computation ◽  
The Common ◽  
Special Case

Fixed point (especially, the minimum norm fixed point) computation is an interesting topic due to its practical applications in natural science. The purpose of the paper is devoted to finding the common fixed points of an infinite family of nonexpansive mappings. We introduce an iterative algorithm and prove that suggested scheme converges strongly to the common fixed points of an infinite family of nonexpansive mappings under some mild conditions. As a special case, we can find the minimum norm common fixed point of an infinite family of nonexpansive mappings.

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

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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