Approximating the minimum-norm fixed point of pseudocontractive mappings

2015 ◽  
Vol 08 (02) ◽  
pp. 1550036
Author(s):  
H. Zegeye ◽  
O. A. Daman
Keyword(s):  
Fixed Point ◽  
Iterative Process ◽  
Convergence Result ◽  
Monotone Mappings ◽  
Minimum Norm ◽  
Nonlinear Operators ◽  
Linear Inverse Problems ◽  
Minimization Problems ◽  
Pseudocontractive Mappings ◽  
Convex Minimization Problems

We introduce an iterative process which converges strongly to the minimum-norm fixed point of Lipschitzian pseudocontractive mapping. As a consequence, convergence result to the minimum-norm zero of monotone mappings is proved. In addition, applications to convexly constrained linear inverse problems and convex minimization problems are included. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

scholarly journals Strong Convergence Theorems for a Common Fixed Point of a Family of Asymptoticallyk-Strict Pseudocontractive Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
H. Zegeye ◽  
N. Shahzad
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Iterative Process ◽  
Finite Family ◽  
Important Class ◽  
Convergence Theorems ◽  
Nonlinear Operators ◽  
Pseudocontractive Mappings

We provide an iterative process which converges strongly to a common fixed point of finite family of asymptoticallyk-strict pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

scholarly journals Systems of Variational Inequalities with Nonlinear Operators

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 338 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
General System ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Nonlinear Operators ◽  
Pseudocontractive Mappings ◽  
Common Fixed Point Problem

In this work, we concern ourselves with the problem of solving a general system of variational inequalities whose solutions also solve a common fixed-point problem of a family of countably many nonlinear operators via a hybrid viscosity implicit iteration method in 2 uniformly smooth and uniformly convex Banach spaces. An application to common fixed-point problems of asymptotically nonexpansive and pseudocontractive mappings and variational inequality problems for strict pseudocontractive mappings is also given in Banach spaces.

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 Strong Convergence Theorems for a Common Fixed Point of a Finite Family of Bregman Weak Relativity Nonexpansive Mappings in Reflexive Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Habtu Zegeye ◽  
Naseer Shahzad
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Important Class ◽  
Reflexive Banach Spaces ◽  
Nonlinear Operators ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive

We introduce an iterative process for finding an element of a common fixed point of a finite family of Bregman weak relatively nonexpansive mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

scholarly journals A Fast Fixed-Point Algorithm for Convex Minimization Problems and Its Application in Image Restoration Problems

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2619
Author(s):  
Panadda Thongpaen ◽  
Rattanakorn Wattanataweekul
Keyword(s):  
Fixed Point ◽  
Image Restoration ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Convex Minimization ◽  
Fixed Point Algorithm ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
Weak Convergence Theorem ◽  
Inertial Technique

In this paper, we introduce a new iterative method using an inertial technique for approximating a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space. The proposed method’s weak convergence theorem was established under some suitable conditions. Furthermore, we applied our main results to solve convex minimization problems and image restoration problems.

scholarly journals Approximation Analysis for a Common Fixed Point of Finite Family of Mappings Which Are Asymptoticallyk-Strict Pseudocontractive in the Intermediate Sense

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
H. Zegeye ◽  
N. Shahzad
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Iterative Process ◽  
Convex Sets ◽  
Common Fixed Points ◽  
Finite Family ◽  
Important Class ◽  
Pseudocontractive Mappings ◽  
Approximation Analysis ◽  
Uniformly Continuous

We introduce an iterative process which converges strongly to a common fixed point of a finite family of uniformly continuous asymptoticallyki-strict pseudocontractive mappings in the intermediate sense fori=1,2,…,N. The projection ofx0onto the intersection of closed convex setsCnandQnfor eachn≥1is not required. Moreover, the restriction that the interior of common fixed points is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

scholarly journals Viscosity Projection Algorithms for Pseudocontractive Mappings in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xiujuan Pan ◽  
Shin Min Kang ◽  
Young Chel Kwun
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Projection Algorithm ◽  
Strong Convergence Theorem ◽  
Minimum Norm ◽  
Projection Algorithms ◽  
Pseudocontractive Mapping ◽  
Pseudocontractive Mappings

An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping.

scholarly journals An Iteration to a Common Point of Solution of Variational Inequality and Fixed Point-Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
H. Zegeye ◽  
N. Shahzad
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Variational Inequality Problem ◽  
Monotone Mapping ◽  
Common Point ◽  
Finite Family ◽  
Monotone Mappings ◽  
Nonlinear Operators ◽  
Asymptotically Nonexpansive

We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for a monotone mapping and fixed point of uniformly Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme that converges strongly to a common zero of finite family of monotone mappings under suitable conditions. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

scholarly journals Solutions of variational inequality problems in the set of fixed points of pseudocontractive mappings

2014 ◽  
Vol 30 (2) ◽  
pp. 257-265
Author(s):  
HABTU ZEGEYE ◽  
◽  
NASSER SHAHZAD ◽  
Keyword(s):  
Variational Inequality ◽  
Fixed Points ◽  
Iterative Process ◽  
Variational Inequality Problems ◽  
Important Class ◽  
Nonlinear Operators ◽  
Pseudocontractive Mappings ◽  
Accretive Mappings

We introduce an iterative process which converges strongly to a solution of the variational inequality problems for η-inverse strongly accretive mappings in the set of fixed points of pseudocontractive mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

scholarly journals An accelerated viscosity forward-backward splitting algorithm with the linesearch process for convex minimization problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Suthep Suantai ◽  
Pachara Jailoka ◽  
Adisak Hanjing
Keyword(s):  
Lipschitz Constant ◽  
Optimization Methods ◽  
Convergence Result ◽  
Convex Minimization ◽  
Signal Recovery ◽  
Splitting Algorithm ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
Continuity Assumption ◽  
Inertial Technique

AbstractIn this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238, 2016] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362, 2019] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.

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