Geodesic
Extending the convergence domain of Newton’s method for generalized equations
Serdica Mathematical Journal, Tome 43 (2017) no. 1, pp. 065-078.

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We present semi-local convergence results for Newton’s method to solve generalized equations. Using a combination of Lipschitz and center Lipschitz conditions on the operators involved instead of just Lipschitz conditions we show that our Newton-Kantorovich criteria are weaker than earlier sufficient conditions for the convergence of Newton’s method. In particular, we provide finer error bounds and a better information on the location of the solution. Our results apply to solve generalized equations involving single as well as multivalued operators, which include variational inequalities, nonlinear complementarity problems and non smooth convex minimization problems. Numerical examples validate the theoretical results by showing that equations that could not be solved before can be solved using our new approach.
Keywords: Hilbert space, generalized equation, Newton’s method, Lipschitz conditions, Newton–Kantorovich hypothesis, local-semilocal convergence theorems, coercivity, multivalued maximal monotone operator, radius of convergence, 65B05, 65G99, 65N35, 47H17, 49M15 
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     title = {Extending the convergence domain of {Newton{\textquoteright}s} method for generalized equations},
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Argyros, Ioannis K.; George, Santhosh. Extending the convergence domain of Newton’s method for generalized equations. Serdica Mathematical Journal, Tome 43 (2017) no. 1, pp. 065-078. http://geodesic.mathdoc.fr/item/SMJ2_2017_43_1_a4/