Abstract
The aim of this study is the approximation of a solution x ∗ of the generalized equation 0∈f(x)+F(x) in Banach spaces, where f is a single function whose second order Fréchet derivative ∇2 f verifies an Hölder condition, and F stands for a set-valued map with closed graph. Using a fixed point theorem and proceeding by induction under the pseudo-Lipschitz property of F, we obtain a sequence defined by a midpoint formula whose convergence to x ∗ is superquadratic. Taking a weaker condition, we present the result obtained when ∇2 f satisfies a center-Hölder conditioning.
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Cabuzel, C. A Midpoint Method for Generalized Equations Under Mild Differentiability Condition. Acta Appl Math 116, 269–279 (2011). https://doi.org/10.1007/s10440-011-9642-6
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DOI: https://doi.org/10.1007/s10440-011-9642-6