A New Four-Step Iterative Approximation Scheme for Reich-Suzuki-Type Nonexpansive Operators in Banach Spaces

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-22-2024-42
Published: Mar 4, 2024
Cite as:
Dhekra M. Albaqeri, Hasanen A. Hammad, Habib Ur Rehman, Manuel De la Sen, A New Four-Step Iterative Approximation Scheme for Reich-Suzuki-Type Nonexpansive Operators in Banach Spaces, Int. J. Anal. Appl., 22 (2024), 42.

Main Article Content

Dhekra M. Albaqeri, Hasanen A. Hammad, Habib Ur Rehman, Manuel De la Sen

Abstract

In this paper, we present a new four-step iterative scheme namely DH-iterative which is faster than many super algorithms in the literature for numerical reckoning fixed points. Under this algorithm, some fixed point convergence results and ω 2 -stability for contractive-like and Reich-Suzuki-type nonexpansive mappings are proposed. Our results extend and improve several related results in the literature. Finally, some numerical examples are given to study the efficiency and effectiveness of our iterative method.

Article Details

References

  1. H.A. Hammad, M. De La Sen, A Tripled Fixed Point Technique for Solving a Tripled-System of Integral Equations and Markov Process in CCbMS, Adv. Differ. Equ. 2020 (2020), 567. https://doi.org/10.1186/s13662-020-03023-y.
  2. H.A. Hammad, M.D. La Sen, A Technique of Tripled Coincidence Points for Solving a System of Nonlinear Integral Equations in POCML Spaces, J. Inequal. Appl. 2020 (2020), 211. https://doi.org/10.1186/s13660-020-02477-8.
  3. H.A. Hammad, H. Aydi, Y.U. Gaba, Exciting Fixed Point Results on a Novel Space with Supportive Applications, Journal of Function Spaces. 2021 (2021), 6613774. https://doi.org/10.1155/2021/6613774.
  4. R.A. Rashwan, H.A. Hammad, M.G. Mahmoud, Common Fixed Point Results for Weakly Compatible Mappings Under Implicit Relations in Complex Valued G-Metric Spaces, Inf. Sci. Lett. 8 (2019), 111–119. https://doi.org/10.18576/isl/080305.
  5. V. Berinde, On the Approximation of Fixed Points of Weak Contractive Mappings, Carpathian J. Math. 19 (2003), 7–22. https://www.jstor.org/stable/43996763.
  6. C.O. Imoru, M.O. Olatinwo, On the Stability of Picard and Mann Iteration Processes, Carpathian J. Math. 19 (2003), 155–160. https://www.jstor.org/stable/43996782.
  7. T. Suzuki, Fixed Point Theorems and Convergence Theorems for Some Generalized Nonexpansive Mappings, J. Math. Anal. Appl. 340 (2008), 1088–1095. https://doi.org/10.1016/j.jmaa.2007.09.023.
  8. R. Pant, R. Pandey, Existence and Convergence Results for a Class of Nonexpansive Type Mappings in Hyperbolic Spaces, Appl. Gen. Topol. 20 (2019), 281–295. https://doi.org/10.4995/agt.2019.11057.
  9. H.A. Hammad, H.U. Rehman, M. De la Sen, A Novel Four-Step Iterative Scheme for Approximating the Fixed Point With a Supportive Application, Inf. Sci. Lett. 10 (2021), 333–339. https://doi.org/10.18576/isl/100214.
  10. H.A. Hammad, H. ur Rehman, M. De la Sen, A New Four-Step Iterative Procedure for Approximating Fixed Points with Application to 2D Volterra Integral Equations, Mathematics 10 (2022), 4257. https://doi.org/10.3390/math10224257.
  11. T.M. Tuyen, H.A. Hammad, Effect of Shrinking Projection and CQ-Methods on Two Inertial Forward–backward Algorithms for Solving Variational Inclusion Problems, Rend. Circ. Mat. Palermo, II. Ser 70 (2021), 1669–1683. https://doi.org/10.1007/s12215-020-00581-8.
  12. H.A. Hammad, W. Cholamjiak, D. Yambangwai, H. Dutta, A Modified Shrinking Projection Methods for Numerical Reckoning Fixed Points of G-Nonexpansive Mappings in Hilbert Spaces With Graphs, Miskolc Math. Notes. 20 (2019), 941–956. https://doi.org/10.18514/mmn.2019.2954.
  13. W. Chaolamjiak, D. Yambangwai, H.A. Hammad, Modified Hybrid Projection Methods with SP Iterations for Quasi-Nonexpansive Multivalued Mappings in Hilbert Spaces, Bull. Iran. Math. Soc. 47 (2020), 1399–1422. https://doi.org/10.1007/s41980-020-00448-9.
  14. W.R. Mann, Mean Value Methods in Iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.
  15. S. Ishikawa, Fixed Points by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147–150.
  16. M.A. Noor, New Approximation Schemes for General Variational Inequalities, J. Math. Anal. Appl. 251 (2000), 217–229. https://doi.org/10.1006/jmaa.2000.7042.
  17. R.P. Agarwal, D. O’Regan, D.R. Sahu, Iterative Construction of Fixed Points of Nearly Asymptotically Nonexpansive Mappings, J. Nonlinear Convex Anal. 8 (2007), 61–79.
  18. M. Abbas, T. Nazir, A New Faster Iteration Process Applied to Constrained Minimization and Feasibility Problems, Mat. Vesn. 66 (2014), 223–234.
  19. R. Chugh, V. Kumar, S. Kumar, Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces, Amer. J. Comput. Math. 2 (2012), 345–357. https://doi.org/10.4236/ajcm.2012.24048.
  20. D.R. Sahu, A. Petru¸sel, Strong Convergence of Iterative Methods by Strictly Pseudocontractive Mappings in Banach Spaces, Nonlinear Anal.: Theory Methods Appl. 74 (2011), 6012–6023. https://doi.org/10.1016/j.na.2011.05.078.
  21. F. Gürsoy, V. Karakaya, A Picard-S Hybrid Type Iteration Method for Solving a Differential Equation With Retarded Argument, arXiv:1403.2546 [math.FA], (2014). http://arxiv.org/abs/1403.2546.
  22. B.S. Thakur, D. Thakur, M. Postolache, A New Iterative Scheme for Numerical Reckoning Fixed Points of Suzuki’s Generalized Nonexpansive Mappings, Appl. Math. Comput. 275 (2016), 147–155. https://doi.org/10.1016/j.amc.2015.11.065.
  23. K. Ullah, M. Arshad, Numerical Reckoning Fixed Points for Suzuki’s Generalized Nonexpansive Mappings via New Iteration Process, Filomat. 32 (2018), 187–196. https://doi.org/10.2298/fil1801187u.
  24. J. Ahmad, K. Ullah, M. Arshad, Z. Ma, A New Iterative Method for Suzuki Mappings in Banach Spaces, J. Math. 2021 (2021), 6622931. https://doi.org/10.1155/2021/6622931.
  25. H.A. Hammad, H. Ur Rehman, M. Zayed, Applying Faster Algorithm for Obtaining Convergence, Stability, and Data Dependence Results With Application to Functional-Integral Equations, AIMS Math. 7 (2022), 19026–19056. https://doi.org/10.3934/math.20221046.
  26. V. Berinde, Picard Iteration Converges Faster Than Mann Iteration for a Class of Quasicontractive Operators, Fixed Point Theory Appl. 2004 (2004), 97–105. https://doi.org/10.1155/S1687182004311058.
  27. H.F. Senter, W.G. Dotson, Approximating Fixed Points of Nonexpansive Mappings, Proc. Amer. Math. Soc. 44 (1974), 375–380. https://doi.org/10.1090/s0002-9939-1974-0346608-8.
  28. X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc. 113 (1991), 727–731.
  29. J. Schu, Weak and Strong Convergence to Fixed Points of Asymptotically Nonexpansive Mappings, Bull. Aust. Math. Soc. 43 (1991), 153–159.
  30. K. Ullah, J. Ahmad, M. Arshad, Z. Ma, Approximating Fixed Points Using a Faster Iterative Method and Application to Split Feasibility Problems, Computation. 9 (2021), 90. https://doi.org/10.3390/computation9080090.
  31. A.M. Harder, Fixed Point Theory and Stability Results for Fixed-Point Iteration Procedures, Ph.D. Thesis, University of Missouri-Rolla, Missouri, (1987).
  32. D. Albaqeri, R. Rashwan, The Comparably Almost (S, T)−Stability for Random Jungck-Type Iterative Schemes, Facta Univ. Ser. Math. Inf. 34 (2019), 175–192. https://doi.org/10.22190/fumi1902175a.
  33. D. Albaqeri, H.A. Hammad, Convergence and Stability Results for New Random Algorithms in Separable Banach Spaces, Appl. Math. Inf. Sci. 16 (2022), 441–456. https://doi.org/10.18576/amis/160306.
  34. S. Hassan, M. De la Sen, P. Agarwal, Q. Ali, A. Hussain, A New Faster Iterative Scheme for Numerical Fixed Points Estimation of Suzuki’s Generalized Nonexpansive Mappings, Math. Probl. Eng. 2020 (2020), 3863819. https://doi.org/10.1155/2020/3863819.
  35. H. Iqbal, M. Abbas, S.M. Husnine, Existence and Approximation of Fixed Points of Multivalued Generalized α-Nonexpansive Mappings in Banach Spaces, Numer. Algor. 85 (2019), 1029–1049. https://doi.org/10.1007/s11075-019-00854-z.
  36. A.E. Ofem, H. I¸sık, F. Ali, J. Ahmad, A New Iterative Approximation Scheme for Reich–suzuki-Type Nonexpansive Operators With an Application, J. Inequal. Appl. 2022 (2022), 28. https://doi.org/10.1186/s13660-022-02762-8.
  37. G.A. Okeke, M. Abbas, M. de la Sen, Approximation of the Fixed Point of Multivalued Quasi-Nonexpansive Mappings via a Faster Iterative Process with Applications, Discr. Dyn. Nat. Soc. 2020 (2020), 8634050. https://doi.org/10.1155/2020/8634050.
  38. V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, (2007).
  39. I. Timi¸s, On the Weak Stability of Picard Iteration for Some Contractive Type Mappings, Ann. Univ. Craiova, Math. Comput. Sci. Ser. 37 (2010), 106–114.
  40. I. Timi¸s, On the Weak Stability of Picard Iteration for Some Contractive Type Mappings and Coincidence Theorems, Int. J. Comput. Appl. 37 (2012), 9–13.
  41. T. Cardinali, P. Rubbioni, A Generalization of the Caristi Fixed Point Theorem in Metric Spaces, Fixed Point Theory. 11 (2010), 3–10.