Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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APPLICATIONS OF FIXED POINT THEORY IN HILBERT SPACES

  • Kiran Dewangan (Department of Mathematics, Government Dudhadhari Bajrang Girls Postgraduate Autonomous College)
  • Received : 2023.08.24
  • Accepted : 2024.01.14
  • Published : 2024.03.30
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Abstract

In the presented paper, the first section contains strong convergence and demiclosedness property of a sequence generated by Karakaya et al. iteration scheme in a Hilbert space for quasi-nonexpansive mappings and also the comparison between the iteration scheme given by Karakaya et al. with well-known iteration schemes for the convergence rate. The second section contains some applications of the fixed point theory in solution of different mathematical problems.

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References

  1. H. A. Abbas, A. A., Mebawondu and O. T. Mewomo, Some results for a new three-step iteration scheme in Banach space, Bull. Transilvania Uni., Brasov 11 (2) (2018), 1-18.
  2. H. A. Abass, A. A. Mebawondu, O. K. Narain and J. K. Kim, Outer approximation method for zeros of sum of monotone operators and fixed point problems in Banach spaces, Nonlnear Funct. Anal. Appl. 26 (3) (2021), 451-474. https://doi.org/10.22771/nfaa.2021.26.03.02
  3. R. P. Agrawal, D. O'Regan and D. R. Sahu, Iterative constructions of fixed points of nearly asymptotically nonexpansive mappings, J. Convex Anal. 8, (2007), 61-79.
  4. F. Akutsah, O. K. Narain and J. K. Kim, Improved generalized M-iteration for quasi-nonexpansive multi-valued mappings with application in real Hilbert spaces, Nonlinear Funct. Anal. Appl. 27 (1) (2022), 59-82. http://dx.doi.org/10.22771/nfaa.2022.27.01.04
  5. S. Banach, Sur les operations dans ensembles abstraits et leur application aux equations integrales, Fundamenta Mathematicae 3 (1922), 133-181. https://doi.org/10.4064/FM-3-1-133-181
  6. F. E. Browder, Nonexpansive nonlinear operators in Banach space, Proc. Nat. Sci., USA 54 (1965), 1041-1044. https://doi.org/10.1073/pnas.54.4.1041
  7. F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (2) (1967), 197-228. https://doi.org/10.1016/0022-247X(67)90085-6
  8. C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems 20 (1) (2004), 103-120. https://doi.org/10.1088/0266-5611/20/1/006
  9. J. B. Diaz and F.T. Metcalf, On the structure of the set of subsequential limit points of successive approximations, Bull. Amer. Math. Soc. 73 (1967), 516-519. https://doi.org/10.1006/jmaa.1998.5944
  10. W. G. Dotson and H. F. Senter, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380.
  11. M. K. Ghose and L. Debnath, Approximation of the fixed points of quasi-nonexpansive mappings in a uniformly convex Banach space, Appl. Math. Lett. 5 (3) (1992), 47-50. https://doi.org/10.1016/0893-9659(92)90037-A
  12. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
  13. S. Itoh and W. Takahashi, The common fixed point theory of single valued mappings and multivalued mappings, Pac. J. Math. 79 (1978), 493-508. https://doi.org/10.2140/PJM.1978.79.493
  14. X. Jin and M. Tian, Strong convergent result for quasi-nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Appl. 88 (2011), 1-8. https://doi.org/10.1186/1687-1812-2011-88
  15. K. S. Kim, Convergence theorems of mixed type implicit iteration for nonlinear mappings in convex metric spaces, Nonlinear Funct. Anal. Appl. 27 (4), (2022), 903-920. https://doi.org/10.22771/nfaa.2022.27.04.16
  16. V. Karakaya, V. Atalan and K. Dogan, Some fixed point result for a new three-step iteration process in Banach space, Fixed Point Theory and Appl. 18 (2) (2017), 625-640. http://dx.doi.org/10.24193/fpt-ro.2017.2.50
  17. S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory and Appl. 69 (2013), 1-10. https://doi.org/10.1186/1687-1812-2013-69
  18. J. K. Kim, S. Dasputre and W. H. Lim, Approximation of fixed points for multi-valued nonexpansive mappings in Banach spaces, Global J. of Pure and Appl. Math. 12 (6) (2016), 4901-4912.
  19. W. A. Kirk and W. O. Ray, Fixed point theorems for mappings define on unbounded sets in Banach spaces, Studia mathematica 114 (1979), 127-138. https://doi.org/10.4064/SM-64-2-127-138
  20. W. R. Mann, Mean value methods in iteration, Pro. Amer. Math. Soc. 4, (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  21. 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
  22. P. Kocourek, W. Takahashi and J. C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwan. J. Math. 14 (2010), 2497-2511. https://doi.org/10.11650/twjm/1500406086
  23. C. I. Podilchuk and R. J. Mammone, Image recovery by convex projections using a least-squares constraint, J. Optical Soc. Amer. 7 (3) (1990), 517-521. https://doi.org/10.1364/JOSAA.7.000517
  24. J. Schu, Weak and strong convergence of fixed point of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc. 43 (1) (1991), 153-159. https://doi.org/10.1017/S0004972700028884
  25. S. Suantai and A. Bunyawat, Common fixed points of a finite family of multi-valued quasi-nonexpansive mappings in uniformly convex Banach spaces, Bull. Ira. Math. Soc. 39 (6) (2013), 1125-1135.
  26. W. Takahashi, Nonlinear functional analysis, Yokohama publishers, Yokohama (2000).
  27. J. Tiammee, A. Kaewkhao and S. Suantai, On Browder's convergence theorem and Halpern iteration process for G- nonexpansive mappings in Hilbert spaces endowed with graphs, Fixed Point Theory and Appl. 187 (2015), 1-12. https://doi.org/10.1186/s13663-015-0436-9
  28. D. Thakur, B.S. Thakur and M. Postolache, New iteration scheme for numerical reckoning fixed points of nonexpansive mappings, J. Inequal. Appl. 2014 (2014), 1-15. https://doi.org/10.1186/1029-242X-2014-328