scholarly journals Convergence of Tseng-type self-adaptive algorithms for variational inequalities and fixed point problems

2021 ◽  
Vol 37 (3) ◽  
pp. 541-550
Author(s):  
YONGHONG YAO ◽  
◽  
NASEER SHAHZAD ◽  
JEN-CHIH YAO ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Hilbert Spaces ◽  
Adaptive Algorithm ◽  
Adaptive Algorithms ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Pseudomonotone Operator ◽  
Weakly Continuous ◽  
Self Adaptive

In this paper, we present a Tseng-type self-adaptive algorithm for solving a variational inequality and a fixed point problem involving pseudomonotone and pseudocontractive operators in Hilbert spaces. A weak convergent result for such algorithm is proved under a weaker assumption than sequentially weakly continuous imposed on the pseudomonotone operator. Some corollaries are also included.

scholarly journals A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Francis Akutsah ◽  
Akindele Adebayo Mebawondu ◽  
Hammed Anuoluwapo Abass ◽  
Ojen Kumar Narain
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Hilbert Spaces ◽  
Lipschitz Constant ◽  
Adaptive Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Inertial Method ◽  
Split Variational Inequality ◽  
Monotone Variational Inequality

<p style='text-indent:20px;'>In this work, we propose a new inertial method for solving strongly monotone variational inequality problems over the solution set of a split variational inequality and composed fixed point problem in real Hilbert spaces. Our method uses stepsizes that are generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the operator norm as well as the Lipschitz constant of the operator. In addition, we prove that the proposed method converges strongly to a minimum-norm solution of the problem without using the conventional two cases approach. Furthermore, we present some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature. The results obtained in this paper extend, generalize and improve results in this direction.</p>

scholarly journals On Mann Viscosity Subgradient Extragradient Algorithms for Fixed Point Problems of Finitely Many Strict Pseudocontractions and Variational Inequalities

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 925 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Adrian Petruşel ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Line Search ◽  
Real Hilbert Space ◽  
Fixed Point Problem ◽  
Iteration Algorithm ◽  
Point Problem ◽  
Search Process ◽  
Pseudomonotone Operator ◽  
Common Solution

In a real Hilbert space, we denote CFPP and VIP as common fixed point problem of finitely many strict pseudocontractions and a variational inequality problem for Lipschitzian, pseudomonotone operator, respectively. This paper is devoted to explore how to find a common solution of the CFPP and VIP. To this end, we propose Mann viscosity algorithms with line-search process by virtue of subgradient extragradient techniques. The designed algorithms fully assimilate Mann approximation approach, viscosity iteration algorithm and inertial subgradient extragradient technique with line-search process. Under suitable assumptions, it is proven that the sequences generated by the designed algorithms converge strongly to a common solution of the CFPP and VIP, which is the unique solution to a hierarchical variational inequality (HVI).

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.

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