A CONVERGENCE THEOREM FOR COMMON ELEMENTS OF EQUILIBRIUM PROBLEMS AND MAPPINGS SATISFYING CONDITION (Φ-Eµ) IN UNIFORMLY CONVEX AND UNIFORMLY SMOOTH BANACH SPACES
Keywords:
mapping satisfying (Φ-Eµ), equilibrium problem, hybrid iteration, uniformly convex and uniformly smooth Banach space
Abstract
In this paper, we propose a new hybrid iteration for finding a common element of solution set of equilibrium problems and the fixed point set of mappings satisfying condi- tion (Φ-Eµ), and establish the convergence of this iteration in uniformly convex and uniformly smooth Banach spaces. From this theorem, we get
a corollary for the convergence for equilibrium problems and mappings satisfying condition (Eµ) in real Hilbert spaces. In addition, an example is provided to illustrate for the convergence of equilibrium problems and mappings satisfying condition (Φ-Eµ). These results are the generations and improvements of some existing results in the literature
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References
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inequalities to equilibrium problems. Math Stud.
1994;63.
[2] Takahashi W, Zembayashi K. Strong and weak
convergence theorems for equilibrium problems and
relatively nonexpansive mappings in Banach spaces.
Nonlinear Anal. 2009;70.
[3] Qin X, Cho YJ, Kang SM. Convergence theorems
of common elements for equilibrium problems and
fixed point problems in Banach spaces. J Comput
Appl Math. 2009;225.
[4] Alizadeh S, Moradlou F. A strong convergence
theorem for equilibrium problems and generalized
hybrid mappings. Mediterr J Math. 2016;13(1).
[5] Trương Cẩm Tiên, Nguyễn Trung Hiếu. Sự hội tụ
của dãy lặp hỗn hợp cho bài toán cân bằng và ánh xạ
thỏa mãn điều kiện (Φ-Eµ) trong không gian Banach
trơn đều và lồi đều. Tạp chí Khoa học Đại học Đồng
Tháp. 2017;27(8)
[6] Alber A. Metric and generalized projection operators
in Banach spaces: properties and applications. In:
Kartosator A G, editor. Theory and Applications of
Nonlinear Operators of Accretive and Monotone Type.
vol. 178. Dekker, New York; 1996;15-50. .
[7] Kamimura S, Takahashi W. Strong convergence of a
proximal-type algorithm in a Banach space. SIAM J
Optim. 2002;13(3).
[8] Marino G, Xu H. Weak and strong convergence
theorems for strict pseudo-contractions in Hilbert
spaces. J Math Anal Appl. 2007;329.
[9] Garcia-Falset J, Llorens-Fuster E, Suzuki T. Fixed
point theory for a class of generalized nonexpansive
mappings. J Math Anal Appl. 2011;375(1).
[10] Qin X, Cho YJ, Kang SM, Zhou H. Convergence of a
modified Halpern-type iteration algorithm for quasi-
Φ-nonexpansive mappings. Appl Math Lett. 2009;22.
Published
01-June-2018
How to Cite
1.
Nguyen H, Truong T. A CONVERGENCE THEOREM FOR COMMON ELEMENTS OF EQUILIBRIUM PROBLEMS AND MAPPINGS SATISFYING CONDITION (Φ-Eµ) IN UNIFORMLY CONVEX AND UNIFORMLY SMOOTH BANACH SPACES. journal [Internet]. 1Jun.2018 [cited 22Dec.2024];8(30):67-. Available from: https://journal.tvu.edu.vn/tvujs_old/index.php/journal/article/view/21
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