ON SEMILOCAL CONVERGENCE OF A MULTIPOINT THIRD ORDER METHOD WITH R-ORDER (2 + p) UNDER A MILD DIFFERENTIABILITY CONDITION

Abstract

The semilocal convergence of a third order iterative method used for solving nonlinear operator equations in Banach spaces is established by using recurrence relations under the assumption that the second Fr´echet derivative of the involved operator satisfies the <TEX>${\omega}$</TEX>-continuity condition given by <TEX>$||F^{\prime\prime}(x)-F^{\prime\prime}(y)||{\leq}{\omega}(||x-y||)$</TEX>, <TEX>$x,y{\in}{\Omega}$</TEX>, where, <TEX>${\omega}(x)$</TEX> is a nondecreasing continuous real function for x > 0, such that <TEX>${\omega}(0){\geq}0$</TEX>. This condition is milder than the usual Lipschitz/H<TEX>$\ddot{o}$</TEX>lder continuity condition on <TEX>$F^{\prime\prime}$</TEX>. A family of recurrence relations based on two constants depending on the involved operator is derived. An existence-uniqueness theorem is established to show that the R-order convergence of the method is (2+<TEX>$p$</TEX>), where <TEX>$p{\in}(0,1]$</TEX>. A priori error bounds for the method are also derived. Two numerical examples are worked out to demonstrate the efficacy of our approach and comparisons are elucidated with a known result.