Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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ON SEMILOCAL CONVERGENCE OF A MULTIPOINT THIRD ORDER METHOD WITH R-ORDER (2 + p) UNDER A MILD DIFFERENTIABILITY CONDITION

  • Parida, P.K. (Center for Applied Mathematics, Central University of Jharkhand) ;
  • Gupta, D.K. (Department of Mathematics, Indian Institute of Technology) ;
  • Parhi, S.K. (Department of Mathematics, Indian Institute of Information Technology)
  • Received : 2012.05.13
  • Accepted : 2012.12.01
  • Published : 2013.05.30
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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 ${\omega}$-continuity condition given by $||F^{\prime\prime}(x)-F^{\prime\prime}(y)||{\leq}{\omega}(||x-y||)$, $x,y{\in}{\Omega}$, where, ${\omega}(x)$ is a nondecreasing continuous real function for x > 0, such that ${\omega}(0){\geq}0$. This condition is milder than the usual Lipschitz/H$\ddot{o}$lder continuity condition on $F^{\prime\prime}$. 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+$p$), where $p{\in}(0,1]$. 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.

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References

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