Summary.
The Kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields. This theorem was weakened recently by Argyros who used a combination of Lipschitz and center-Lipschitz conditions in place of the Lipschitz conditions of the Kantorovich theorem. In the present paper we use the Argyros theorem to formulate a generalized Kantorovich theorem that enables us deduce the solvability of equations and the convergence of Newton’s method with minimal assumptions.
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Manuscript received: May 17, 2007 and, in final form, March 9, 2008.
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Uko, L.U., Argyros, I.K. A generalized Kantorovich theorem on the solvability of nonlinear equations. Aequ. math. 77, 99–105 (2009). https://doi.org/10.1007/s00010-008-2950-x
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DOI: https://doi.org/10.1007/s00010-008-2950-x