On the convergence of Newton's method under $\omega ^\star $-conditioned second derivative

Tom 38 / 2011

Ioannis K. Argyros, Saïd Hilout Applicationes Mathematicae 38 (2011), 341-355 MSC: 65G99, 65J15, 47H17, 49M15. DOI: 10.4064/am38-3-5


We provide a new semilocal result for the quadratic convergence of Newton's method under $\omega ^\star $-conditioned second Fréchet derivative on a Banach space. This way we can handle equations where the usual Lipschitz-type conditions are not verifiable. An application involving nonlinear integral equations and two boundary value problems is provided. It turns out that a similar result using $\omega $-conditioned hypotheses can provide usable error estimates indicating only linear convergence for Newton's method.


  • Ioannis K. ArgyrosDepartment of Mathematical Sciences
    Cameron University
    Lawton, OK 73505, U.S.A.
  • Saïd HiloutLaboratoire de Mathématiques et Applications
    Université de Poitiers
    Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179
    86962 Futuroscope Chasseneuil Cedex, France

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