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On the midpoint method for solving generalized equations. (English) Zbl 1226.65048
Summary: We approximate a locally unique solution of a generalized equations in a Banach space setting using a new midpoint method. An existence/convergence theorem and a radius of convergence are given under Lipschitz and center-Lipschitz conditions on the first order Fréchet derivative and Lipschitz-like continuity property of set-valued mappings. We show that our method is locally quadratically convergent using a fixed points theorem. Motivated by optimization considerations related to the resolution on nonlinear equations, a smaller ratio and a larger radius of convergence are also provided. Our methods extend the midpoint method related to the resolution of nonlinear equations.
MSC:
65J15Equations with nonlinear operators (numerical methods)
47J05Equations involving nonlinear operators (general)
47J25Iterative procedures (nonlinear operator equations)
65Q30Numerical methods for recurrence relations