Abstract
In Geoffroy et al, Acceleration of convergence in Dontchev's iterative method for solving variational inclusions Serdica Math. J. 29 (2003), pp. 45–54] we showed the convergence of a cubic method for solving generalized equations of the form 0 ∈ f(x) +- G(x) where f is a\(\mathcal{C}^{{\text{2}}} \) function and G stands for a set-valued map. We investigate here the stability of such a method with respect to some perturbations. More precisely, we consider the perturbed equation y ∈ f(x) +- G(x) and we show that the pseudo-Lipschitzness of the map (f +- G)−1 is closely tied to the uniformity of our method in the sense that the attraction region does not depend on small perturbations of the parameter y. Finally, we provide an enhanced version of the convergence theorem established by Geoffroy, et al.
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Geoffroy, M.H., Hilout, S. & Piétrus, A. Stability of a Cubically Convergent Method for Generalized Equations. Set-Valued Anal 14, 41–54 (2006). https://doi.org/10.1007/s11228-005-0003-3
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DOI: https://doi.org/10.1007/s11228-005-0003-3