scholarly journals Hierarchical Fixed Point Problems in Uniformly Smooth Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Problems ◽  
Approximation Methods ◽  
Viscosity Approximation ◽  
Uniformly Smooth ◽  
Hierarchical Fixed Point ◽  
Viscosity Approximation Methods ◽  
Uniformly Smooth Banach Spaces ◽  
Hierarchical Fixed Point Problems

We propose some relaxed implicit and explicit viscosity approximation methods for hierarchical fixed point problems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces. These relaxed viscosity approximation methods are based on the well-known viscosity approximation method and hybrid steepest-descent method. We obtain some strong convergence theorems under mild conditions.

scholarly journals Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-18
Author(s):  
Lu-Chuan Ceng ◽  
Abdul Latif ◽  
Abdullah E. Al-Mazrooei
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Approximation Methods ◽  
Uniformly Convex ◽  
Viscosity Approximation ◽  
Uniformly Smooth ◽  
Viscosity Approximation Methods

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

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