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In the 1950s, Mann (Proc. Am. Math. Soc. 4:506–510, 1953) and Krasnosel’skii (Uspiehi Mat. Nauk 10:123–127, 1955) independently laid the groundwork for a highly effective successive iteration method used in constructing fixed points. For a nonexpansive mapping \(T: C \rightarrow C\) , this method is defined as follows: an initial point \(x_1 \in C\) is chosen arbitrarily, and the subsequent points are given by \(x_{k+1}=c_kT(x_k)+(1-c_k)x_k\) , where \(\{c_k\}\) is a sequence of numbers from the interval (0, 1). In this chapter, we adapt the Krasnosel’skii-Mann method to the setting of pointwise Lipschitzian semigroups.

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Generalised Krasnosel’skii-Mann Iteration Processes

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  • 发表日期：

    2026年

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  • Wojciech M. Kozlowski

摘要

In the 1950s, Mann (Proc. Am. Math. Soc. 4:506–510, 1953) and Krasnosel’skii (Uspiehi Mat. Nauk 10:123–127, 1955) independently laid the groundwork for a highly effective successive iteration method used in constructing fixed points. For a nonexpansive mapping \(T: C \rightarrow C\) , this method is defined as follows: an initial point \(x_1 \in C\) is chosen arbitrarily, and the subsequent points are given by \(x_{k+1}=c_kT(x_k)+(1-c_k)x_k\) , where \(\{c_k\}\) is a sequence of numbers from the interval (0, 1). In this chapter, we adapt the Krasnosel’skii-Mann method to the setting of pointwise Lipschitzian semigroups.