Algorithm for Solving a Generalized Mixed Equilibrium Problem with Perturbation in a Banach Space
- PDF / 402,963 Bytes
- 23 Pages / 600.05 x 792 pts Page_size
- 73 Downloads / 191 Views
Research Article Algorithm for Solving a Generalized Mixed Equilibrium Problem with Perturbation in a Banach Space Lu-Chuan Ceng,1 Sangho Kum,2 and Jen-Chih Yao3 1
Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China 2 Department of Mathematics Education, Chungbuk National University, Cheongju 361763, Republic of Korea 3 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan Correspondence should be addressed to Jen-Chih Yao, [email protected] Received 22 February 2010; Accepted 11 April 2010 Academic Editor: Wataru Takahashi Copyright q 2010 Lu-Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let B be a real Banach space with the dual space B ∗ . Let φ : B → R ∪ {∞} be a proper functional and let Θ : B × B → R be a bifunction. In this paper, a new concept of η-proximal mapping of φ with respect to Θ is introduced. The existence and Lipschitz continuity of the η-proximal mapping of φ with respect to Θ are proved. By using properties of the η-proximal mapping of φ with respect to Θ, a generalized mixed equilibrium problem with perturbation for short, GMEPP is introduced and studied in Banach space B. An existence theorem of solutions of the GMEPP is established and a new iterative algorithm for computing approximate solutions of the GMEPP is suggested. The strong convergence criteria of the iterative sequence generated by the new algorithm are established in a uniformly smooth Banach space B, and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived in B H a Hilbert space.
1. Introduction Let X be a real Banach space with norm · and let X ∗ be its dual space. The value of f ∈ B∗ at x ∈ B will be denoted by f, x . The normalized duality mapping J from B into the family of nonempty by Hahn-Banach theorem weak-star compact subsets of its dual space B∗ is defined by Jx f ∈ B∗ : f, x f2 x2 ,
∀x ∈ B.
1.1
2
Fixed Point Theory and Applications
It is known that the norm of B is said to be Gateaux differentiable and B is said to be smooth if
lim
t→0
x ty − x t
1.2
exists for every x, y in U {x ∈ B : x 1}, the unit sphere of B. It is said to be uniformly Gateaux differentiable if for each y ∈ U, this limit is attained uniformly for x ∈ U. The norm of B is said to be uniformly Frechet differentiable and B is said to be uniformly smooth if the limit in 1.2 is attained uniformly for x, y ∈ U × U. Every uniformly smooth Banach space B is reflexive and has a uniformly Gateaux differentiable norm. Recall also that if B is smooth, then J is single-valued and continuous from the norm topology of B to the weak star topology of B∗ , that is, norm-to-weak∗ continuous. It is also well known that if B has a uniformly Gatea
Data Loading...
