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Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations

Author

Listed:
  • Sukhjit Singh

    (National Institute of Technology, Jalandhar 144011, India)

  • Eulalia Martínez

    (Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain)

  • Abhimanyu Kumar

    (Department of Mathematics, L. N. M. U. Darbhanga-Bihar, Darbhanga 846004, India)

  • D. K. Gupta

    (Department of Mathematics, I.I.T Kharagpur, Kharagpur 721302, India)

Abstract

In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in the previous studies. We assume omega continuity condition on second order Fréchet derivative. This fact it is motivated by showing different problems where the nonlinear operators that define the equation do not verify Lipschitz and Hölder condition; however, these operators verify the omega condition established. Then, the semilocal convergence balls are obtained and the R-order of convergence and error bounds can be obtained by following thee main theorem. Finally, we perform a numerical experience by solving a nonlinear Hammerstein integral equations in order to show the applicability of the theoretical results by obtaining the existence and uniqueness balls.

Suggested Citation

  • Sukhjit Singh & Eulalia Martínez & Abhimanyu Kumar & D. K. Gupta, 2020. "Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations," Mathematics, MDPI, vol. 8(3), pages 1-11, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:384-:d:330297
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    References listed on IDEAS

    as
    1. Martínez, Eulalia & Singh, Sukhjit & Hueso, José L. & Gupta, Dharmendra K., 2016. "Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 252-265.
    2. Xiuhua Wang & Jisheng Kou & Chuanqing Gu, 2012. "Semilocal Convergence of a Class of Modified Super-Halley Methods in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 779-793, June.
    3. Cordero, A. & Ezquerro, J.A. & Hernández-Verón, M.A. & Torregrosa, J.R., 2015. "On the local convergence of a fifth-order iterative method in Banach spaces," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 396-403.
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