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Approximation of fixed points for a class of mappings satisfying property (CSC) in Banach spaces

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Abstract

In this research article, we connect the \(K^{*}\) iterative process with the class of mappings having property (CSC). We provide some weak and strong convergence theorems regarding the iterative scheme for mappings endowed with property (CSC) in uniformly convex Banach spaces. An example of mappings endowed with property (CSC) is provided which does not satisfy property (C). The \(K^{*}\) iteration process and many other iterative processes are connected with this example to support the theoretical outcome. Our results improve and extend the corresponding well-known results of the current literature.

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References

  1. Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3, 133–181 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  2. Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Nat. Acad. Sci. USA 54, 1041–1044 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kirk, W.A.: A fixed point theorem for mappings which do not increase distance. Am. Math. Mon. 72, 1004–1006 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cholamjiak, W., Suantai, S., Suparatulatorn, R., Kesornprom, S., Cholamjiak, P.: Viscosity approximation methods for fixed point problems in Hilbert spaces endowed with graphs. J. Appl. Numer. Optim. 1, 25–38 (2019)

    Google Scholar 

  5. Khan, S.H., Fukhar-ud-din, H.: Approximating fixed points of \(\rho\)-nonexpansive mappings by RK-iterative process in modular function spaces. J. Nonlinear Var. Anal. 3, 107–114 (2019)

    MATH  Google Scholar 

  6. Yambangwai, D., Aunruean, S., Thianwan, T.: A new modified three step iteration method for \(G\)-nonexpansive mappings in Banach spaces with a graph. Numer. Algorithm (2019). https://doi.org/10.1007/s11075-019-00768-w

    MATH  Google Scholar 

  7. Pandey, R., Pant, R., Rakocevic, V., Shukla, R.: Approximating fixed points of a general class of nonexpansive mappings in Banach spaces with applications. Results Math. 74, 1–24 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sahin, A., Basarïr, M.: Some convergence results for nonexpansive mappings in uniformly convex hyperbolic spaces. Creat. Math. Inform. 26, 331–338 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Basarïr, M., Sahin, A.: Some results of the new iterative scheme in hyperbolic space. Commun. Korean Math. Soc. 32, 1009–1024 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mapping. J. Math. Anal. Appl. 340, 1088–1095 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dhomphongsa, S., Inthakon, W., Kaewkhao, A.: Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 350, 12–17 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Phuengrattana, W.: Approximating fixed points of Suzuki generalized nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 5, 583–590 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Abkar, A., Eslamian, M.: A fixed point theorem for generalized nonexpansive multivalued mappings. Fixed Point Theory 12(2), 241–246 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Basarir, M., Sahin, A.: On the strong and \(\Delta\)-convergence of S-iteration process for generalized nonexpansive mappings on CAT(0) spaces. Thai J. Math. 12, 549–559 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Nanjaras, B., Panyanak, B., Phuengrattana, W.: Fixed point theorems and convergence theorems for Suzuki generalized nonexpansive mappings in CAT(0) spaces. Nonlinear Anal. Hybrid Syst. 4, 25–31 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ullah, K., Arshad, M.: New iteration process and numerical reckoning fixed point in Banach spaces. U.P.B. Sci. Bull. (Ser. A) 79(4), 113–122 (2017)

    MathSciNet  Google Scholar 

  17. Uddin, I., Imdad, M.: On certain convergence of S-iteration scheme in CAT(0) spaces. Kuwait J. Sci. 42(2), 93–106 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Uddin, I., Imdad, M.: Convergence of SP-iteration for generalized nonexpansive mapping in Hadamard spaces. Hacet J. Math. Stat. 47(6), 1595–1604 (2018)

    MathSciNet  Google Scholar 

  19. Thakur, B.S., Thakur, D., Postolache, M.: A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings. Appl. Math. Comput. 275, 147–155 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Ullah, K., Arshad, M.: New three-step iteration process and fixed point approximation in Banach spaces. J. Linear Topol. Algebra 7(2), 87–100 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Karapïnar, E., Tas, K.: Generalized (C)-conditions and related fixed point theorems. Comput. Math. Appl. 61, 3370–3380 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  24. Abbas, M., Nazir, T.: A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesnik 66, 223–234 (2014)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Agarwal, R.P., O’Regan, D., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8(1), 61–79 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Clarkson, J.A.: Uniformly convex spaces. Trans. Am. Math. Soc. 40, 396–414 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  28. Takahashi, W.: Nonlinear Functional Analysis. Yokohoma Publishers, Yokohoma (2000)

    MATH  Google Scholar 

  29. Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications Series. Topological Fixed Point Theory and Its Applications, vol. 6. Springer, New York (2009)

    MATH  Google Scholar 

  30. Opial, Z.: Weak and strong convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MATH  Google Scholar 

  31. Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc. 43, 153–159 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Senter, H.F., Dotson, W.G.: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375–380 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was conducted while Dr. J.U. and Prof. Dr. M.A. were visiting the ULM, Lakki Marwat. We would like to thank Dr. K.U. for giving a wonderful hospitality.

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Authors and Affiliations

  1. Department of Mathematics, International Islamic University, H-10, Islamabad, 44000, Pakistan

    Junaid Ahmad & Muhammad Arshad

  2. Department of Mathematics, University of Lakki Marwat, Lakki Marwat, Khyber Pakhtunkhwa, 28420, Pakistan

    Kifayat Ullah

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  1. Junaid Ahmad
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  2. Kifayat Ullah
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  3. Muhammad Arshad
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All the the three authors contributed equally to this research article.

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Correspondence to Junaid Ahmad.

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Ahmad, J., Ullah, K. & Arshad, M. Approximation of fixed points for a class of mappings satisfying property (CSC) in Banach spaces. Math Sci 15, 207–213 (2021). https://doi.org/10.1007/s40096-021-00407-3

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  • DOI: https://doi.org/10.1007/s40096-021-00407-3

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