A viscosity approximation method for weakly relatively nonexpansive mappings by the sunny nonexpansive retractions in Banach spaces

作者：Pang Chin Tzong; Naraghirad Eskandar^*; Wen Ching Feng

来源：JOURNAL OF INEQUALITIES AND APPLICATIONS, 2015, 2015(1): 36.

DOI：10.1186/s13660-014-0546-7

• viscosity approximation method
• fixed point
• weak relatively nonexpansive mapping
• strong convergence

In this paper, we introduce a new viscosity approximation method by using the shrinking projection algorithm to approximate a common fixed point of a countable family of nonlinear mappings in a Banach space. Under quite mild assumptions, we establish the strong convergence of the sequence generated by the proposed algorithm and provide an affirmative answer to an open problem posed by Mainge (Comput. Math. Appl. 59: 74-79, 2010) for quasi-nonexpansive mappings. In contrast with related processes, our method does not require any demiclosedness principle condition imposed on the involved operators belonging to the wide class of quasi-nonexpansive operators. As an application, we also introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of quasi-nonexpansive mappings and the set of solutions of a mixed equilibrium problem in a real Banach space. We prove a strong convergence theorem by using the proposed algorithm under some suitable conditions. Our results improve and generalize many known results in the current literature.

• 出版日期2015-2-3