ON THE SEMILOCAL CONVERGENCE OF A TWO STEP NEWTON METHOD UNDER THE $\GAMMA-$CONDITION

IOANNIS K.  ARGYROS; SANTHOSH  GEORGE

ON THE SEMILOCAL CONVERGENCE OF A TWO STEP NEWTON METHOD UNDER THE $\GAMMA-$CONDITION

Authors

IOANNIS K. ARGYROS Cameron University, Department of Mathematical Sciences, Lawton, OK 73505, USA
SANTHOSH GEORGE Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, India

Keywords:

Two-step Newton method, Newton method, Banach space, α-theory, semilocal convergence, γ-condition, Frechet-derivative

Abstract

We present a semilocal convergence analysis of a two-step Newton method using the $\alpha-$theory in order to approximate a locally unique solution of an equation in a Banach space setting. The new idea uses a combination of center$-\gamma$ as well as a $\gamma-$ condition in the convergence analysis. This convergence criteria are weaker than the corresponding ones in the literature even in the case of the single step Newton method [3, 14, 15, 16, 17, 18, 19, 20]. Numerical examples involving a nonlinear integral equation where the older convergence criteria are not satisfied but the new convergence criteria are satisfied, are also presented in the paper.