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Deformed super-Halley’s iteration in Banach spaces and its local and semilocal convergence

  • M. PrashanthEmail author
  • Abhimanyu Kumar
  • D. K. Gupta
  • S. S. Mosta
Article
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Abstract

Deformed super-Halley’s iteration for nonlinear equations is studied in Banach spaces with its local and semilocal convergence. The local convergence is established under Hölder continuous first Fréchet derivative. A theorem for the existence and uniqueness of solution is provided and the radii of convergence balls are obtained. For semilocal convergence, the second order Fréchet derivative is Hölder continuous. The Hölder continuous first Fréchet derivative is not used as it leads to lower R-order of convergence. Recurrence relations depending on two parameters are obtained. A theorem for the existence and uniqueness along with the estimation of bounds on errors is also established. The R-order convergence comes out to be \((2+p), p \in (0,1]\). Nonlinear integral equations and a variety of numerical examples are solved to demonstrate our work.

Keywords

Super-Halley’s method Derivative of Fréchet Majorizing function Recurrence relations 

Mathematics Subject Classification

15A09 65F05 65F35 
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Notes

Acknowledgements

Funding was provided by Amrita Vishwa Vidyapeetham University, Indian Institute of Technology Kharagpur.

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Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  • M. Prashanth
    • 1
    Email author
  • Abhimanyu Kumar
    • 2
  • D. K. Gupta
    • 3
  • S. S. Mosta
    • 4
  1. 1.Department of Mathematics, Amrita School of EngineeringAmrita Vishwa VidyapeethamCoimbatoreIndia
  2. 2.Uma Pandey College, Lalit Narayan Mithila UniversitySarmastpurIndia
  3. 3.Department of MathematicsIndian Institute of Technology KharagpurWest BengalIndia
  4. 4.Department of MathematicsUniversity of SwazilandKwaluseniSwaziland

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