Generalized relaxed inertial method with regularization for solving split feasibility problems in real Hilbert spaces

2021 ◽  
pp. 2250106
Author(s):  
A. A. Mebawondu ◽  
L. O. Jolaoso ◽  
H. A. Abass ◽  
O. K. Narain
Keyword(s):  
Hilbert Spaces ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Inertial Method ◽  
Split Feasibility

In this paper, we propose a new modified relaxed inertial regularization method for finding a common solution of a generalized split feasibility problem, the zeros of sum of maximal monotone operators, and fixed point problem of two nonlinear mappings in real Hilbert spaces. We prove that the proposed method converges strongly to a minimum-norm solution of the aforementioned problems without using the conventional two cases approach. In addition, we apply our convergence results to the classical variational inequality and equilibrium problems, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

scholarly journals A Unified Iterative Treatment for Solutions of Problems of Split Feasibility and Equilibrium in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Young-Ye Huang ◽  
Chung-Chien Hong
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Zero Point ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Special Cases ◽  
Split Feasibility

We at first raise the so called split feasibility fixed point problem which covers the problems of split feasibility, convex feasibility, and equilibrium as special cases and then give two types of algorithms for finding solutions of this problem and establish the corresponding strong convergence theorems for the sequences generated by our algorithms. As a consequence, we apply them to study the split feasibility problem, the zero point problem of maximal monotone operators, and the equilibrium problem and to show that the unique minimum norm solutions of these problems can be obtained through our algorithms. Since the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems.

scholarly journals Multiple-sets split feasibility problem and split equality fixed point problem for firmly quasi-nonexpansive or nonexpansive mappings

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tongxin Xu ◽  
Luoyi Shi
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Split Feasibility Problem ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Projection Operators ◽  
Nonexpansive Operators ◽  
Split Feasibility

AbstractIn this paper, we propose a new iterative algorithm for solving the multiple-sets split feasibility problem (MSSFP for short) and the split equality fixed point problem (SEFPP for short) with firmly quasi-nonexpansive operators or nonexpansive operators in real Hilbert spaces. Under mild conditions, we prove strong convergence theorems for the algorithm by using the projection method and the properties of projection operators. The result improves and extends the corresponding ones announced by some others in the earlier and recent literature.

scholarly journals An Inertial Iterative Algorithm with Strong Convergence for Solving Modified Split Feasibility Problem in Banach Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Huijuan Jia ◽  
Shufen Liu ◽  
Yazheng Dang
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Maximal Monotone ◽  
Split Feasibility Problem ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Mild Assumption ◽  
Inertial Technique ◽  
Split Feasibility

In this paper, we propose an iterative scheme for a special split feasibility problem with the maximal monotone operator and fixed-point problem in Banach spaces. The algorithm implements Halpern’s iteration with an inertial technique for the problem. Under some mild assumption of the monotonicity of the related mapping, we establish the strong convergence of the sequence generated by the algorithm which does not require the spectral radius of A T A. Finally, the numerical example is presented to demonstrate the efficiency of the algorithm.

scholarly journals Shrinking Projection Methods for Split Common Fixed-Point Problems in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Hilbert Spaces ◽  
Split Feasibility Problem ◽  
Projection Methods ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Split Common Fixed Point

Inspired by Moudafi (2011) and Takahashi et al. (2008), we present the shrinking projection method for the split common fixed-point problem in Hilbert spaces, and we obtain the strong convergence theorem. As a special case, the split feasibility problem is also considered.

scholarly journals Linear Convergence of Split Equality Common Null Point Problem with Application to Optimization Problem

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1836
Author(s):  
Yaqian Jiang ◽  
Rudong Chen ◽  
Luoyi Shi
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Linear Convergence ◽  
Monotone Operators ◽  
Null Point ◽  
Point Problem ◽  
Bounded Linear ◽  
Zero Points

The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in Hilbert spaces. We introduce the concept of bounded linear regularity for the SECNP and construct several sufficient conditions to ensure the linear convergence of the algorithm. Moreover, some numerical experiments are given to test the validity of our results.

scholarly journals Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Kamonrat Nammanee ◽  
Suthep Suantai ◽  
Prasit Cholamjiak
Keyword(s):  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Convex Banach Space ◽  
Point Problem ◽  
Convergence Theorems ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive

We introduce hybrid-iterative schemes for solving a system of the zero-finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a system of convex minimization problems.

scholarly journals The zeros of monotone operators for the variational inclusion problem in Hilbert spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pattanapong Tianchai
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Feasibility Problem ◽  
Strong Convergence Theorem ◽  
Inclusion Problem ◽  
Point Problem ◽  
Variational Inclusion Problem

AbstractIn this paper, we introduce a regularization method for solving the variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters, which allow us to obtain a short proof of another strong convergence theorem for this problem. We also apply our main result to the fixed point problem of the nonexpansive variational inequality problem, the common fixed point problem of nonexpansive strict pseudocontractions, the convex minimization problem, and the split feasibility problem. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique.

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