Taiwanese Journal of Mathematics
- Taiwanese J. Math.
- Volume 15, Number 3 (2011), 1341-1367.
Strong and Weak Convergence Theorems for Generalized Mixed Equilibrium Problem with Perturbation and Fixed Pointed Problem of Infinitely Many Nonexpansive Mappings
Very recently, Plubtieng and Kumam [S. Plubtieng, P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings, J. Comput. Appl. Math. 224 (2009) 614-621] proposed an iterative algorithm for finding a common solution of a variational inequality problem for an inverse-strongly monotone mapping and a fixed point problem of a countable family of nonexpansive mappings, and obtained a weak convergence theorem. In this paper, based on Plubtieng-Kumam's iterative algorithm we introduce a new iterative algorithm for finding a common solution of a generalized mixed equilibrium problem with perturbation and a fixed point problem of a countable family of nonexpansive mappings in a Hilbert space. We first derive a strong convergence theorem for this new algorithm under appropriate assumptions and then consider a special case of this new algorithm. Moreover, we establish a weak convergence theorem for this special case under some weaker assumptions. Such a weak convergence theorem unifies, improves and extends Plubtieng-Kumam's weak convergence theorem. It is worth pointing out that the proof method of strong convergence theorem is very different from the one of weak convergence theorem.
Taiwanese J. Math., Volume 15, Number 3 (2011), 1341-1367.
First available in Project Euclid: 18 July 2017
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 49J30: Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 47H17 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.
generalized mixed equilibrium problem with perturbation fixed point problem variational inequality nonexpansive mapping demiclosedness principle inverse-strongly monotone mapping strong and weak convergence
Ceng, L. C.; Hu, Hui-Ying; Wong, M. M. Strong and Weak Convergence Theorems for Generalized Mixed Equilibrium Problem with Perturbation and Fixed Pointed Problem of Infinitely Many Nonexpansive Mappings. Taiwanese J. Math. 15 (2011), no. 3, 1341--1367. doi:10.11650/twjm/1500406303. https://projecteuclid.org/euclid.twjm/1500406303