Convergence and certain control conditions for hybrid viscosity approximation methods |
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Authors: | Lu-Chuan Ceng Jen-Chih Yao |
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Affiliation: | a Department of Mathematics, Shanghai Normal University, Shanghai 200234, China b Scientific Computing Key Laboratory of Shanghai Universities, China c Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan |
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Abstract: | Very recently, Yao, Chen and Yao [20] proposed a hybrid viscosity approximation method, which combines the viscosity approximation method and the Mann iteration method. Under the convergence of one parameter sequence to zero, they derived a strong convergence theorem in a uniformly smooth Banach space. In this paper, under the convergence of no parameter sequence to zero, we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping, which solves a variational inequality. An appropriate example such that all conditions of this result are satisfied and their condition βn→0 is not satisfied is provided. Furthermore, we also give a weak convergence theorem for their method involving a nonexpansive mapping in a Hilbert space. |
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Keywords: | 47H10 47H09 47H17 |
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