Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

NEW APPROXIMATE FIXED POINT RESULTS FOR VARIOUS CYCLIC CONTRACTION OPERATORS ON E-METRIC SPACES

  • R. THEIVARAMAN (DEPARTMENT OF MATHEMATICS, BHARATHIDASAN UNIVERSITY) ;
  • P. S. SRINIVASAN (DEPARTMENT OF MATHEMATICS, BHARATHIDASAN UNIVERSITY) ;
  • S. RADENOVIC (FACULTY OF MECHANICAL ENGINEERING, UNIVERSITY OF BELGRADE) ;
  • CHOONKIL PARK (RESEARCH INSTITUTE OF NATURAL SCIENCES, HANYANG UNIVERSITY)
  • Received : 2023.04.11
  • Accepted : 2023.08.19
  • Published : 2023.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate the existence and diameter of the approximate fixed point results on E-metric spaces (not necessarily complete) by using various cyclic contraction mappings, including the B-cyclic contraction, the Bianchini cyclic contraction, the Hardy-Rogers cyclic contraction, and so on. Additionally, we prove the approximate fixed point results for rational type cyclic contraction mappings, which were discussed mainly in [35] and [37], in the setting of E-metric space. Also, a few examples are provided to demonstrate our findings. Subsequently, we discuss some applications of approximate fixed point results in the field of applied mathematics rigorously.

Keywords

Acknowledgement

The first author wishes to thank Bharathidasan University for its financial support under the URF scheme. Also, all the authors express gratitude to The Journal of the Korean Society for Industrial and Applied Mathematics Management for their unwavering assistance in getting this manuscript finished. We would like to express our gratitude to the editor and reviewers for their thorough reading. Once again, we thank the editor for giving us the opportunity to reset the manuscript in a nice way.

References

  1. H. Poincare, Memoire sur les courbes definies par une equation differentielle (I) , J. Math. Appl., 7 (1881), 375-422. 
  2. Jesper Lutzen, Joseph Loiuvile's contribution to the the theory of integral equations, Historica Mathematica., 9 (1982), 373-391.  https://doi.org/10.1016/0315-0860(82)90104-5
  3. L. E. Brouwer, Uber abbildung von mannigfaltigkeiten , Math. Ann., 71(1) (1911), 97-115.  https://doi.org/10.1007/BF01456931
  4. S. Banach. Surles operations dans les ensembles abstract et leur application aux equation integrals, Fund. Math., 3 (1922), 133-181.  https://doi.org/10.4064/fm-3-1-133-181
  5. R. M. T. Bianchini, Su un problema di S. Reich riguardante la teoria dei punti fissi, Boll. U. M. I., 4(50), (1972), 103-106. 
  6. LB. Ciric, Generalized contractions and fixed point theorems, Publ.Inst.Math.(Bulgr), 12(26) (1971), 19-26. 
  7. G. E. Hardy and T. D. Rogers, A generalization of fixed point theorem of Reich, Can. Math. Bull, 16 (1973), 201-206.  https://doi.org/10.4153/CMB-1973-036-0
  8. S. Reich, Some remarks connecting contraction mappings, Can. Math. Bull., 14 (1971), 121-124.  https://doi.org/10.4153/CMB-1971-024-9
  9. T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. (Basel), 23 (1972), 292-298.  https://doi.org/10.1007/BF01304884
  10. L. G. Haung and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332(2) (2007), 1468-1476.  https://doi.org/10.1016/j.jmaa.2005.03.087
  11. R. Kannan, Some results on fixed points, Bull. Cal. Math. Soc., 60 (1968), 71-76. 
  12. S.K. Chatterjea, Fixed point theorems. C.R. Acad.Bulg. Sci., 25 (1972), 727-730. 
  13. A. Al-Rawashdeh, W. Shatanawi, and M. Khandaqji, Normed ordered and E-metric spaces. Int. J. Math. Sci., (272137-1-272137-11), 2012. 
  14. A. Azam and N. Mehmood. Multivalued fixed points in TVS-cone metric spaces. Fixed point theory. Appl., 1 (2013), 184. 
  15. K. Deimling, Nonlinear functional analysis, Cour. Corp., New York, 2010. 
  16. M. Frchet, Sur quelques questions de calcul des variations, Bull. Soc. Math. Fr., 33 (1905), 73-78. 
  17. A. Granas and J. Dugundji, Fixed point theory, Springer, Berlin, 2013. 
  18. H. Haung, Topological properties of E-metric spaces with application to fixed point theory, Mathematics, 7 (2019), 1222. 
  19. N. Mehmood, A. Al-Rawashdeh, and S. Radenovic, New fixed point results for E-metric spaces, Positivity, 23 (2019), 1101-1111.  https://doi.org/10.1007/s11117-019-00653-9
  20. N. Mehmood, A. Azam, and S. Aleksic, Topological vector space valued cone Banach spaces, Int. J. Anal. Appl., 6(2) (2014), 205-219. 
  21. N. Mehmood, A. Azam, and L. D. Kocinac, Multivalued fixed point results in cone metric spaces, Topol. Appl., 179 (2015), 156-170.  https://doi.org/10.1016/j.topol.2014.07.011
  22. Z. Pales and I. R. Petre, Iterative fixed point theorems on E-metirc spaces, Acta Math. Hung., 140 (2013), 134-144.  https://doi.org/10.1007/s10474-012-0274-8
  23. S. Rezapour and R. Halmbarani, Some notes on the proper cone metric spaces and fixed point theorem of contractive mappings, J. Anal. Appl., 354(2) (2008), 719-724.  https://doi.org/10.1016/j.jmaa.2008.04.049
  24. B. Rzepecki, On fixed point theorems of maia type, Publ. Inst. Math., 28(42) (1980), 179-186. 
  25. M. Berinde, Approximate fixed point theorems, Stud. Univ. "Babes-Bolyai", Mathematica Libertica, 1 (2006), 11-25. 
  26. V. Berinde, Iterative approximation of fixed points, Editura Efemeride, Baia Mare, 2002. 
  27. V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math, 19(1) (2003), 7-22. 
  28. D. Dey and M. Saha, Approximate fixed point of Reich operator, Acta Mathematica Universitatis Comenianae, 2012. 
  29. S. A. M. Mohsenalhosseini and Saheli. M, Some family of contractive type maps and approximate common fixed point. J. Fixed Point Theory, 25, 2021:2. 
  30. S. A. M. Mohsenialhosseini, Aproximate fixed point theorems of cyclical contraction mappings, Mathematica Moravica, 21(1) (2017), 125-137.  https://doi.org/10.5937/MatMor1701125M
  31. S. Tijs, A. Torre, and P. Branzei, Approximate fixed point theorems, Libertas Mathematica, 23 (2003), 35-39. 
  32. W. A. Kirk, P. S. Srinivasan, and P. Veeramani, Fixed point for mappings satisfying cyclical contractive conditions, Fixed point theory, 4 (2003), 79-89. 
  33. K. Tijani and S. Olayemi, Approximate fixed point results for rational-type contraction mappings, International Journal of Analysis and Optimization: Theory and Applications, (2021), 76-86. 
  34. I. A. Bakhtin. The contraction mapping principle in almost metric spaces, Func. Anal., 30 (1989), 26-37. 
  35. B. K. Dass and S. Gupta, An extension of Banach contraction principle through rational expression, Communicated by F.C. Auluck, FNA, 1975. 
  36. Komal Goyal and Bhagwati Prasad, Approximate fixed points and summable almost stability, Global Journal of Pure and Applied Mathematics, (2017), 6029-6040. 
  37. D. S. Jaggi, Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics, 8 (1977), 223-230. 
  38. M. S. Khan, A fixed point theorems for metric spaces, Rendiconti Dell 'istituto di mathematica dell' Universtia di tresti, 8 (1976), 69-72. 
  39. M. Marudai and Bright V. S, Unique fixed point theorem weakly B-contractive mappings, Far East journal of Mathematical Sciences (FJMS), 97(7) (2015), 897-914.  https://doi.org/10.17654/FJMSDec2015_897_914
  40. V. Berinde and M. Pacurar, Fixed point theorems for enriched Ciri c-Reich-Rus contraction in Banach spaces and convex metric spaces, Carpathian J. Math., 37(2) (2021), 173-184.  https://doi.org/10.37193/CJM.2021.02.03
  41. P. Debnath, N. Konwar, and S. Radenovic, Metric fixed point theory, applications in Science, Engineering and Behavioural Sciences, Forum for Inter disciplinary Mathematics, Springer, (2021). 
  42. S. A. Khuri and I. Louhichi, A novel Ishikawa-Green's fixed point scheme for the solution of BVPs, Appl. Math. Lett., 82 (2018), 50-57. https://doi.org/10.1016/j.aml.2018.02.016