Iterative Sequences for Asymptotically Quasi-nonexpansive Mappings

By L. Qihou. 2001

202 citations recorded by the source corpus.

Let E be a real Banach space and T a self-mapping in E. T is called . asymptotically quasi-nonexpansive if there exists u 0, , lim n n n Ž . Ž . u 0, such that T x p 1 u x p , x E, p F T n n Ž Ž . . F T denotes the set of fixed points . n n Ž . T is called asymptotically nonexpansive if T x T y 1 u x n y , x, y E. T is called quasi-nonexpansive if Tx p x p , x Ž . E, p F T . T is called nonexpansive if Tx Ty x y , x, y E. Ž . From the above definitions, it follows that if F T is nonempty, a nonexpansive mapping must be quasi-nonexpansive, and an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive. But the converse does not hold. Petryshyn and Williamson 1 , in 1973, proved a sufficient and necessary condition for Mann iterative sequences to converge to fixed points for quasi-nonexpansive mappings. In 1997, Ghosh and Debnath 2 extended the results of 1 and gave the sufficient and necessary condition for Ishikawa iterative sequences to converge to fixed points for quasi-nonexpansive mappings. Using these, they have also obtained some sufficient conditions for Ishikawa iterative sequences converge to fixed points for nonexpansive mappings. In this manuscript, we will extend the above results to asymptotically quasi-nonexpansive mappings and will prove some sufficiency and necessary conditions for Ishikawa iterative sequences of asymptotically quasi-nonexpansive mappings to converge to fixed points; i.e., we will prove the following theorems.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

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Citations in the corpus

Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces (2006, background): “(2) Theorem 3.2 contains as special cases Theorem 1.1, the main result of Qihou [9], together with [9, Corollaries 1 and 2], which are themselves extensions of the results of Ghosh and Debnath [2] and Petryshyn and Williamson [8].”
Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications (2007, background): “Thus Theorem 3 of Qihou [8], Theorem 2 of Khan and Fukhar [4], Theorem 2 of Senter and Dotson [10], Theorem 1 of Maiti and Ghosh [7] and Theorem 2 of Schu [9] are all special cases of Theorem 2 above.”
Strong convergence of a general iteration scheme in C A T ( 0 ) spaces (2011, unknown): “(1) The approximation results about (i) modified Mann iterations in Hilbert spaces [2], (ii) modified Ishikawa iterations in Banach spaces [3,6,17], and (iii) the three-step iteration scheme in uniformly convex Banach spaces from [4,5] are immediate consequences of our results.”
Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces (2008, background): “Our work is a significant generalization of the corresponding results of Khan and Takahashi [10], Qihou [20], Schu [23], Shahzad and Udomene [24], Tan and Xu [28] and Xu and Noor [29].”