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Some common fixed point results for three total asymptotically pseudocontractive map**s

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Abstract

In this article, we introduce a mixed-type implicit iterative scheme to approximate the common fixed points of finite families of three uniformly L-Lipschitzian total asymptotically pseudocontractive map**s in Banach spaces. Also, we prove some strong convergence results of the proposed iterative scheme. Our results which are new, improve and generalize the results of many prominent authors exiting in the literature.

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References

  1. Acedo, G.L., and H.K. Xu. 2007. Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67: 2258–2271.

    Article  MathSciNet  MATH  Google Scholar 

  2. Aibinu, M.O., and J.K. Kim. 2019. Convergence Analysis of viscosity implicit rules of nonexpansive map**s Banach spaces. Nonlinear Funct. Anal. Appl. 24 (4): 691–713.

    MATH  Google Scholar 

  3. Aibinu, M.O., and J.K. Kim. 2020. On the rate of convergence of viscosity implicit iterative algorithms. Nonlinear Funct. Anal. Appl. 25 (1): 135–152.

    MATH  Google Scholar 

  4. Alakoya, T.O., A.O. Owolabi, and O.T. Mewomo. 2021. An inertial algorithm with a self-adaptive step size for a split equilibrium problem and a fixed point problem of an infinite family of strict pseudo-contractions. J. Nonlinear Var. Anal. 5: 803–829.

    Google Scholar 

  5. Bello, A.U., M.T. Omojola, and M.O. Nnakwe. 2021. Two methods for solving split common fixed point problems of strict pseudo-contractve map**s in Hilbert spaces with applications. Appl. Set-Valued Anal. Optim. 3: 75–93.

    Google Scholar 

  6. Chidume, C.E., and N. Shahzad. 2005. Strong convergence of an implicit iteration process for a finite family of nonexpansive map**s. Nonlinear Anal. Theory Methods Appl. 62 (6): 1149–1156.

    Article  MathSciNet  MATH  Google Scholar 

  7. Chima, E.E., and M.O. Osilike. 2016. Split common fixed point problem for a class of total asymptotic pseudocontractions. J. Appl. Math.https://doi.org/10.1155/2016/3435078. (Article ID 3435078).

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, C., and J. Quan. 2012. A strong convergence theorem for total asymptotically pseudocontractive map**s in Hilbert spaces. Abstract Appl Anal. 2012: 8 (Article ID 127851).

    Article  MathSciNet  MATH  Google Scholar 

  9. Ishikawa, S. 1976. Fixed points and iteration of a nonexpansive map** in a Banach space. Proc. Am. Math. Soc. 59 (1): 65–71.

    Article  MathSciNet  MATH  Google Scholar 

  10. Jung, J.S. 2019. Strong convergence of general iterative algorithms for pseudocontractive map**s in Hilbert spaces. Nonlinear Funct. Anal. Appl. 24 (2): 389–406.

    MATH  Google Scholar 

  11. Khomphurngson, K., N. Kamyun, and K. Nammanee. 2022. New modified hybrid algorithm for pseudo-contractive map**s in Hilbert spaces. J. Nonlinear Funct. Anal. 2022: 25.

    Google Scholar 

  12. Kim, J.K., M.I. Bhat, and S. Shafi. 2021. Convergence and stability of iterative algorithm of system of generalized implicit variational-like inclusion problems using (\(\theta\),\(\phi\),\(\gamma\))-relaxed cocoercivity. Nonlinear Funct. Anal. Appl. 26 (4): 749–780.

    MATH  Google Scholar 

  13. Marino, G., and H.K. Xu. 2007. Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl. 329: 336–346.

    Article  MathSciNet  MATH  Google Scholar 

  14. Mann, W.R. 1953. Mean value methods in iteration. Proc. Am. Math. Soc. 4: 506–510.

    Article  MathSciNet  MATH  Google Scholar 

  15. Noor, M.A. 2000. New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251 (1): 217–229.

    Article  MathSciNet  MATH  Google Scholar 

  16. Ofem, A.E. 2020. Strong convergence of modified implicit hybrid S-iteration scheme for finite family of nonexpansive and asymptotically generalized \(\Phi\)-hemicontractive map**s. Malaya J. Matematik 8 (4): 1643–1649. https://doi.org/10.26637/MJM0804/0053.

    Article  MathSciNet  Google Scholar 

  17. Ofem, A.E. 2020. Strong convergence of a multi-step implicit iterative scheme with errors for common fixed points of uniformly \(L\)-Lipschitzian total asymptotically strict pseudocontractive map**s. Result. Nonlinear Anal. 3 (2): 100–116.

    Google Scholar 

  18. Ofem, A.E., D.I. Igbokwe, and X.A. Udo-utun. 2020. Implicit iteration process for Lipschitzian \(\alpha\)-hemicontraction semigroup. MathLAB J. 7: 43–52.

    Google Scholar 

  19. Ofem, A.E., and U.E. Udofia. 2021. Iterative solutions for common fixed points of nonexpansive map**s and strongly pseudocontractive map**s with applications, Canad. J. Appl. Math. 3: 18–36.

    Google Scholar 

  20. Ofem, A.E., and D.I. Igbokwe. 2020. An efficient iterative method and its applications to a nonlinear integral equation and a delay differential equation in banach spaces. Turkish J. Ineq. 4: 79–107.

    Google Scholar 

  21. Ofem, A.E., and D.I. Igbokwe. 2021. A new faster four step iterative algorithm for Suzuki generalized nonexpansive map**s with an application. Adv. Theory Nonlinear Anal. Appl. 5: 482–506. https://doi.org/10.31197/atnaa.869046.a.

    Article  Google Scholar 

  22. Ofem, A.E., et al. 2022. A new iterative approximation scheme for Reich-Suzuki type nonexpansive operators with an application. J. J. Inequal. Appl. 1: 1–26.

    MathSciNet  MATH  Google Scholar 

  23. Ofem, A.E., U.E. Udofia, and D.I. Igbokwe. 2021. A robust iterative approach for solving nonlinear Volterra delay integro-differential equations. Ural Math. J. 7 (2): 59–85.

    Article  MathSciNet  MATH  Google Scholar 

  24. Osilike, M.O. 2004. Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. J. Math. Anal. Appl. 294 (1): 73–81.

    Article  MathSciNet  MATH  Google Scholar 

  25. Osilike, M.O., and B.G. Akuchu. 2004. Common fixed points of finite family of asymptotically pseudocontractive maps. Fixed Point Theory Appl. 2004: 81–88.

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen, R.D., Y.S. Song, and H. Zhou. 2006. Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive map**s. J. Math. Anal. Appl. 314: 701–706.

    Article  MathSciNet  MATH  Google Scholar 

  27. Qin, X., S.Y. Cho, and J.K. Kim. 2010. Convergence theorems on asymptotically pseudocontractive map**s in the intermediate sense. Fixed Point Theory Appl. 2010: 14 (Article ID 186874).

    Article  MathSciNet  MATH  Google Scholar 

  28. Qin, X., S.Y. Cho, and S.M. Kang. 2011. Weak convergence theorem for total asymptotically pseudocontractive map**s in Hilbert spaces. Fixed Point Theory Appl. 2011: 11 (Article ID 859795).

    Article  MathSciNet  MATH  Google Scholar 

  29. Saluja, G.S. 2014. Convergence of the explicit iteration method for strictly asymptotically pseudocontractive map**s in the intermediate sense. Novi Sad J. Math. 44 (1): 75–90.

    MathSciNet  MATH  Google Scholar 

  30. Schu, J. 1991. Iterative construction of fixed points of asymptotically nonexpansive map**s. J. Math. Anal. Appl. 158 (2): 407–413.

    Article  MathSciNet  MATH  Google Scholar 

  31. Schu, J. 1991. Weak and strong convergence to fixed points of asymptotically nonexpansive map**s. Bull. Aust. Math. Soc. 43 (1): 153–159.

    Article  MathSciNet  MATH  Google Scholar 

  32. Sun, Z.H. 2003. Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive map**s. J. Math. Anal. Appl. 286: 351–358.

    Article  MathSciNet  MATH  Google Scholar 

  33. Thakur, B.S., R. Dewangan, and M. Postolache. 2014. General composite implicit iteration process for a finite family of asymptotically pseudo-contractive map**s General composite implicit iteration process for a finite family of asymptotically pseudo-contractive map**s. Fixed Point Theory Appl. 2014: 90.

    Article  MATH  Google Scholar 

  34. Rhoades, B.E. 1976. Comments on two fixed point iteration methods. J Math. Anal Appl. 56 (3): 741–750.

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, Y., and C. Wang. 2013. Convergence of a new modified Ishikawa type iteration for common fixed points of total asymptotically strict pseudocontractive semigroups. Abstract Appl. Anal. 2013: 7. https://doi.org/10.1155/2013/319241. (Article ID 319241).

    Article  MathSciNet  MATH  Google Scholar 

  36. Xu, H.K. 1991. Inequalities in Banach spaces with applications. Nonlinear Anal. 16: 1127–1138.

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, H.K., and R.G. Ori. 2001. An implicit iteration process for nonexpansive map**s. Numer. Funct. Anal. Optim. 22: 767–773.

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhou, Y., and S.S. Chang. 2002. Convergence of implicit iteration process for a finite family of asymptotically nonexpansive map**s in Banach spaces. Numer. Funct. Anal. Optim. 23: 911–921.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Akwa Ibom state University, Ikot Akpaden, Mkpat Enin, Nigeria

    M. O. Udo

  2. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    A. E. Ofem

  3. Department of Mathematics, University of Calabar, Calabar, Nigeria

    J. Oboyi, C. F. Chikwe & S. E. Ekoro

  4. Department of Mathematics, University of Cross River state, Calabar, Nigeria

    F. A. Adie

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  1. M. O. Udo
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  2. A. E. Ofem
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  3. J. Oboyi
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  4. C. F. Chikwe
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  5. S. E. Ekoro
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Correspondence to A. E. Ofem.

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Udo, M.O., Ofem, A.E., Oboyi, J. et al. Some common fixed point results for three total asymptotically pseudocontractive map**s. J Anal 31, 2005–2022 (2023). https://doi.org/10.1007/s41478-023-00548-9

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  • DOI: https://doi.org/10.1007/s41478-023-00548-9

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