Convergence Theorems and Convergence Rates for the General Inertial Krasnosel’skiǐ–Mann Algorithm

Qiao-Li Dong (), Shang-Hong Ke (), Yeol Je Cho () and Themistocles M. Rassias ()
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Qiao-Li Dong: Civil Aviation University of China, Tianjin Key Laboratory for Advanced Signal Processing and College of Science
Shang-Hong Ke: Civil Aviation University of China
Yeol Je Cho: Gyeongsang National University, Department of Mathematics Education
Themistocles M. Rassias: National Technical University of Athens, Zografou Campus, Department of Mathematics

Chapter Chapter 4 in Advances in Metric Fixed Point Theory and Applications, 2021, pp 61-83 from Springer

Abstract: Abstract The authors [13] introduced a general inertial Krasnosel’skiǐ–Mann algorithm: $$ \left\{ \begin{aligned}&y_n=x_n+\alpha _n(x_n-x_{n-1}),\\&z_n=x_n+\beta _n(x_n-x_{n-1}),\\&x_{n+1}=(1-\lambda _n)y_n+\lambda _nT(z_n) \end{aligned} \right. $$ y n = x n + α n ( x n - x n - 1 ) , z n = x n + β n ( x n - x n - 1 ) , x n + 1 = ( 1 - λ n ) y n + λ n T ( z n ) for each $$n\ge 1$$ n ≥ 1 and showed its convergence with the control conditions $$\alpha _n,\beta _n\in [0,1).$$ α n , β n ∈ [ 0 , 1 ) . In this paper, we present the convergence analysis of the general inertial Krasnosel’skiǐ–Mann algorithm with the control conditions $$\alpha _n\in [0,1]$$ α n ∈ [ 0 , 1 ] , $$\beta _n\in (-\infty ,0]$$ β n ∈ ( - ∞ , 0 ] and $$\alpha _n\in [-1,0]$$ α n ∈ [ - 1 , 0 ] , $$\beta _n\in [0,+\infty )$$ β n ∈ [ 0 , + ∞ ) , respectively. Also, we provide the convergence rate for the general inertial Krasnosel’skiǐ–Mann algorithm under mild conditions on the inertial parameters and some conditions on the relaxation parameters, respectively. Finally, we show that a numerical experiment provided compares the choice of inertial parameters.

Keywords: $$\blacksquare; \blacksquare; \blacksquare; $$; ▪; ▪; ▪ (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-33-6647-3_4

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DOI: 10.1007/978-981-33-6647-3_4

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