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Semilocal Convergence of a Seventh-Order Method in Banach Spaces Under HÖlder Continuity Condition


Affiliations
1 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh - 462003, India
     

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The motive of this article is to analyze the semilocal convergence of a well existing iterative method in the Banach spaces to get the solution of nonlinear equations. The condition, we assume that the nonlinear operator fulfills the Hölder continuity condition which is softer than the Lipschitz continuity and works on the problems in which either second order Frèchet derivative of the nonlinear operator is challenging to calculate or does not hold the Lipschitz condition. In the convergence theorem, the existence of the solution x* and its uniqueness along with prior error bound are established. Also, the R-order of convergence for this method is proved to be at least 4+3q. Two numerical examples are discussed to justify the included theoretical development followed by an error bound expression.

Keywords

Banach Space, Nonlinear Operator, Semilocal Convergence, Hölder Condition, Frèchet Derivative, Recurrence Relation, Error Bound.
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  • Semilocal Convergence of a Seventh-Order Method in Banach Spaces Under HÖlder Continuity Condition

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Authors

Neha Gupta
Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh - 462003, India
J. P. Jaiswal
Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh - 462003, India

Abstract


The motive of this article is to analyze the semilocal convergence of a well existing iterative method in the Banach spaces to get the solution of nonlinear equations. The condition, we assume that the nonlinear operator fulfills the Hölder continuity condition which is softer than the Lipschitz continuity and works on the problems in which either second order Frèchet derivative of the nonlinear operator is challenging to calculate or does not hold the Lipschitz condition. In the convergence theorem, the existence of the solution x* and its uniqueness along with prior error bound are established. Also, the R-order of convergence for this method is proved to be at least 4+3q. Two numerical examples are discussed to justify the included theoretical development followed by an error bound expression.

Keywords


Banach Space, Nonlinear Operator, Semilocal Convergence, Hölder Condition, Frèchet Derivative, Recurrence Relation, Error Bound.

References





DOI: https://doi.org/10.18311/jims%2F2020%2F23248