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A New Four-Step Iterative Procedure for Approximating Fixed Points with Application to 2D Volterra Integral Equations

Author

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  • Hasanen A. Hammad

    (Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia
    Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt)

  • Habib ur Rehman

    (Department of Mathematics, Monglkut’s University of Technology, Bangkok 10140, Thailand)

  • Manuel De la Sen

    (Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, 48940 Leioa, Bizkaia, Spain)

Abstract

This work is devoted to presenting a new four-step iterative scheme for approximating fixed points under almost contraction mappings and Reich–Suzuki-type nonexpansive mappings (RSTN mappings, for short). Additionally, we demonstrate that for almost contraction mappings, the proposed algorithm converges faster than a variety of other current iterative schemes. Furthermore, the new iterative scheme’s ω 2 —stability result is established and a corroborating example is given to clarify the concept of ω 2 —stability. Moreover, weak as well as a number of strong convergence results are demonstrated for our new iterative approach for fixed points of RSTN mappings. Further, to demonstrate the effectiveness of our new iterative strategy, we also conduct a numerical experiment. Our major finding is applied to demonstrate that the two-dimensional (2D) Volterra integral equation has a solution. Additionally, a comprehensive example for validating the outcome of our application is provided. Our results expand and generalize a number of relevant results in the literature.

Suggested Citation

  • Hasanen A. Hammad & Habib ur Rehman & Manuel De la Sen, 2022. "A New Four-Step Iterative Procedure for Approximating Fixed Points with Application to 2D Volterra Integral Equations," Mathematics, MDPI, vol. 10(22), pages 1-26, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4257-:d:972340
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    References listed on IDEAS

    as
    1. Hasanen A. Hammad & Habib ur Rehman & Manuel De la Sen, 2020. "Shrinking Projection Methods for Accelerating Relaxed Inertial Tseng-Type Algorithm with Applications," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-14, November.
    2. Thakur, Balwant Singh & Thakur, Dipti & Postolache, Mihai, 2016. "A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 147-155.
    3. Adamu, A. & Kitkuan, D. & Padcharoen, A. & Chidume, C.E. & Kumam, P., 2022. "Inertial viscosity-type iterative method for solving inclusion problems with applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 445-459.
    4. Junaid Ahmad & Kifayat Ullah & Muhammad Arshad & Zhenhua Ma & Kaleem R. Kazmi, 2021. "A New Iterative Method for Suzuki Mappings in Banach Spaces," Journal of Mathematics, Hindawi, vol. 2021, pages 1-7, February.
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