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Strong convergence of an inertial Halpern type algorithm in Banach spaces

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Abstract

In this article, we obtain the strong convergence of the new modified Halpern iteration process

$$\begin{aligned} x_{n+1} = \alpha _{n}u + (1-\alpha _{n})T_{n}P(x_{n} + \theta _{n}(x_{n} - x_{n-1})), \ \ \ \ \ \ n=1,2,3,\ldots , \end{aligned}$$

to a common fixed point of \( \{ T_{n} \}\), where \(\{ T_{n}\}_{n=1}^{\infty }\) is a family of nonexpansive mappings on the closed and convex subset C of a Banach space X, \(P: X \longrightarrow C\) is a nonexpansive retraction, \(\{\alpha _n\} \subset [0, 1]\) and \(\{\theta _n\}\subset R^+\). Some applications of this result are also presented.

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Acknowledgements

The author is grateful to the referees for their valuable comments and suggestions.

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  1. Department of Mathematics, Higher Education Center of Eghlid, Eghlid, Iran

    Sajad Ranjbar

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  1. Sajad Ranjbar
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Ranjbar, S. Strong convergence of an inertial Halpern type algorithm in Banach spaces. Rend. Circ. Mat. Palermo, II. Ser 72, 1517–1526 (2023). https://doi.org/10.1007/s12215-022-00749-4

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