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Let C be a closed, bounded, convex subset of a uniformly convex Banach space, and let \(\{T_s\}\) be an asymptotic pointwise Lipschitzian semigroup of nonlinear mappings acting within C. In this chapter, we consider the implicit iteration process defined by the sequence of equations \( x_{k+1} = c_k T_{s_{k+1}}(x_{k+1}) + (1 - c_k) x_k, \) where each \(c_k \in (0,1)\) , and the initial point \(x_0 \in C\) is arbitrarily chosen. In this context, we examine the conditions under which the sequence \(\{x_k\}\) converges, whether weakly or strongly, to a common fixed point of the semigroup \(\{T_s\}\) .

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Implicit Iteration Processes

  • 章节

  • 发表日期：

    2026年

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  • Wojciech M. Kozlowski

摘要

Let C be a closed, bounded, convex subset of a uniformly convex Banach space, and let \(\{T_s\}\) be an asymptotic pointwise Lipschitzian semigroup of nonlinear mappings acting within C. In this chapter, we consider the implicit iteration process defined by the sequence of equations \( x_{k+1} = c_k T_{s_{k+1}}(x_{k+1}) + (1 - c_k) x_k, \) where each \(c_k \in (0,1)\) , and the initial point \(x_0 \in C\) is arbitrarily chosen. In this context, we examine the conditions under which the sequence \(\{x_k\}\) converges, whether weakly or strongly, to a common fixed point of the semigroup \(\{T_s\}\) .