# Convergence analysis of new modified iterative approximating processes for two finite families of total asymptotically nonexpansive nonself mappings in hyperbolic spaces

Volume 10, Issue 4, pp 1407--1423

Publication Date: 2017-04-20

http://dx.doi.org/10.22436/jnsa.010.04.11

### Authors

Ting-jian Xiong - Department of Mathematics, Sichuan University of Science & Engineering, 643000 Zigong, Sichuan, P. R. China.
Heng-you Lan - Department of Mathematics, Sichuan University of Science & Engineering, 643000 Zigong, Sichuan, P. R. China.

### Abstract

In this paper, we introduce and study a class of new modified iterative approximation processes for two finite families of total asymptotically nonexpansive nonself mappings in hyperbolic spaces. By using generalization of Schu’s lemma and Tan-Xu’s inequality, some important related properties of this modified iterative approximation are proposed and analyzed. Further, based on the related properties, we prove $\Delta$-convergence and strong convergence of the modified iterative approximating process in hyperbolic spaces. Because a total asymptotically nonexpansive nonself mapping in hyperbolic spaces includes asymptotically nonexpansive mapping, (generalized) nonexpansive mapping of all normed linear spaces, Hadamard manifolds and CAT(0) spaces as special cases, the results presented in this paper improve and generalize the corresponding results in the literature.

### Keywords

Convergence analysis, new modified iterative approximating process, $\Delta$-convergence and strong convergence, total asymptotically nonexpansive nonself mapping, hyperbolic space.

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