scholarly journals Solving Split Variational Inclusion Problem and Fixed Point Problem for Nonexpansive Semigroup without Prior Knowledge of Operator Norms

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Haitao Che ◽  
Meixia Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Point Problem ◽  
Split Variational Inclusion Problem ◽  
Variational Inclusion Problem ◽  
Nonexpansive Semigroups ◽  
Operator Norms

We introduce an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups. Moreover, numerical results demonstrate the performance and convergence of our result, which may be viewed as a refinement and improvement of the previously known results announced by many other researchers.

scholarly journals Split Variational Inclusion Problem and Fixed Point Problem for a Class of Multivalued Mappings in CAT(0) Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 749 ◽  
Author(s):  
Mujahid Abbas ◽  
Yusuf Ibrahim ◽  
Abdul Rahim Khan ◽  
Manuel De la Sen
Keyword(s):  
Fixed Point ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Multivalued Mappings ◽  
Strong Convergence Theorem ◽  
Inclusion Problem ◽  
Point Problem ◽  
Strictly Pseudocontractive Mapping ◽  
Split Variational Inclusion Problem ◽  
Variational Inclusion Problem

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in C A T ( 0 ) spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of C A T ( 0 ) spaces.

scholarly journals Convergence Theorems for Common Solutions of Split Variational Inclusion and Systems of Equilibrium Problems

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 255
Author(s):  
Yan Tang ◽  
Yeol Cho
Keyword(s):  
Equilibrium Problems ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Point Problem ◽  
Step Size ◽  
Convergence Theorems ◽  
Common Solution ◽  
Variational Inclusion Problem

In this paper, the split variational inclusion problem (SVIP) and the system of equilibrium problems (EP) are considered in Hilbert spaces. Inspired by the works of Byrne et al., López et al., Moudafi and Thukur, Sobumt and Plubtieng, Sitthithakerngkiet et al. and Eslamian and Fakhri, a new self-adaptive step size algorithm is proposed to find a common element of the solution set of the problems SVIP and EP. Convergence theorems are established under suitable conditions for the algorithm and application to the common solution of the fixed point problem, and the split convex optimization problem is considered. Finally, the performances and computational experiments are presented and a comparison with the related algorithms is provided to illustrate the efficiency and applicability of our new algorithms.

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.

scholarly journals Implicit Extragradient-Like Method for Fixed Point Problems and Variational Inclusion Problems in a Banach Space

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 998
Author(s):  
Sun Young Cho
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Infinite Family ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Symmetric System ◽  
Inclusion Problem ◽  
Point Problem ◽  
Variational Inclusion Problem ◽  
Common Fixed Point Problem

In a uniformly convex and q-uniformly smooth Banach space with q ∈ ( 1 , 2 ] , one use VIP to indicate a variational inclusion problem involving two accretive mappings and CFPP to denote the common fixed-point problem of an infinite family of strict pseudocontractions of order q. In this paper, we introduce a composite extragradient implicit method for solving a general symmetric system of variational inclusions (GSVI) with certain VI and CFPP. We then investigate its convergence analysis under some weak conditions. Finally, we consider the celebrated LASSO problem in Hilbert spaces.

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