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A modified Halpern-type iteration algorithm for totally quasi-\(\phi \)-asymptotically nonexpansive mappings with applications. (English) Zbl 1252.65097

Authors’ abstract: The purpose of this article is to modify the Halpern-type iteration algorithm for total quasi-\(\phi \)-asymptotically nonexpansive mappings to have strong convergence under a limit condition only in the framework of Banach spaces. The results presented improve and extend the corresponding results of X. L. Qin et al. [Appl. Math. Lett. 22, No. 7, 1051–1055 (2009; Zbl 1179.65061)], Z. Wang [J. Comput. Appl. Math. 235, No. 8, 2364–2371 (2011; Zbl 1213.65082)], Y. Su, H.-K. Xu and X. Zhang [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 12, 3890–3906 (2010; Zbl 1215.47091)], C. Martinez-Yanes and H.-K. Xu [Nonlinear Anal., Theory Methods Appl. 64, No. 11, A, 2400–2411 (2006; Zbl 1105.47060)] and others.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J05 Equations involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators
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References:

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