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Ceng, Lu-Chuan; Wen, Ching-Feng

Hybrid method with perturbation for Lipschitzian pseudocontractions. (English) Zbl 1251.65082

J. Appl. Math. 2012, Article ID 250538, 20 p. (2012).
Summary: Assume that \(F\) is a nonlinear operator which is Lipschitzian and strongly monotone on a nonempty closed convex subset \(C\) of a real Hilbert space \(H\). Assume also that \(\Omega\) is the intersection of the fixed point sets of a finite number of Lipschitzian pseudocontractive self-mappings on \(C\). By combining hybrid steepest-descent method, Mann’s iteration method and projection method, we devise a hybrid iterative algorithm with perturbation \(F\), which generates two sequences from an arbitrary initial point \(x_0 \in H\). These two sequences are shown to converge in norm to the same point \(P_\Omega x_0\) under very mild assumptions.

Cited in 1 Document

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

Lipschitzian nonlinear operator; hybrid steepest-descent method
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\textit{L.-C. Ceng} and \textit{C.-F. Wen}, J. Appl. Math. 2012, Article ID 250538, 20 p. (2012; Zbl 1251.65082)
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