Strong convergence theorem for common solutions to quasi variational inclusion and fixed point problems
Authors
Xianzhi Tang
 Department of of basic courses, Yellow River Conservancy Technical Institute, Kaifeng 475004, China.
Huanhuan Cui
 Department of Mathematics, Luoyang Normal University, Luoyang, 471022, China.
Abstract
In this paper, we consider a problem that consists of finding a common solution to quasi variational
inclusion and fixed point problems. We first present a simple proof to the strong convergence theorem
established by Zhang et al. recently. Next, we propose a new algorithm to solve such a problem. Under
some mild conditions, we establish the strong convergence of iterative sequence of the proposed algorithm.
Share and Cite
ISRP Style
Xianzhi Tang, Huanhuan Cui, Strong convergence theorem for common solutions to quasi variational inclusion and fixed point problems, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 8, 52525258
AMA Style
Tang Xianzhi, Cui Huanhuan, Strong convergence theorem for common solutions to quasi variational inclusion and fixed point problems. J. Nonlinear Sci. Appl. (2016); 9(8):52525258
Chicago/Turabian Style
Tang, Xianzhi, Cui, Huanhuan. "Strong convergence theorem for common solutions to quasi variational inclusion and fixed point problems." Journal of Nonlinear Sciences and Applications, 9, no. 8 (2016): 52525258
Keywords
 Variational inclusion
 fixed point problem
 inverse strongly monotone operator
 nonexpansive mapping
 multivalued maximal monotone mapping.
MSC
 47J25
 47H05
 47H09
 47H04
 47J22
References

[1]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103120

[2]
B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957961

[3]
H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inversestrongly monotone mappings, Nonlinear Anal., 61 (2005), 341350

[4]
A. Moudafi, Viscosity approximation methods for fixedpoints problems, J. Math. Anal. Appl., 241 (2000), 4655

[5]
N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191201

[6]
W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417428

[7]
R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel), 58 (1992), 486491

[8]
H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240256

[9]
H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279291

[10]
L.C. Zeng, J.C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math., 10 (2006), 12931303

[11]
S.S. Zhang, J. H. W. Lee, C. K. Chan, Algorithms of common solutions to quasi variational inclusion and fixed point problems, Appl. Math. Mech. (English Ed.), 29 (2008), 571581