Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME FIXED POINT RESULTS ON DOUBLE CONTROLLED CONE METRIC SPACES

  • A. Herminau Jothy (Department of Mathematics, Bharathidasan University) ;
  • P. S. Srinivasan (Department of Mathematics, Bharathidasan University) ;
  • Laxmi Rathour (Department of Mathematics, National Institute of Technology) ;
  • R. Theivaraman (Department of Mathematics, Bharathidasan University) ;
  • S. Thenmozhi (Department of Mathematics, SRM Institute of Science and Technology)
  • Received : 2023.09.21
  • Accepted : 2024.06.16
  • Published : 2024.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this text, we investigate some fixed point results in double-controlled cone metric spaces using several contraction mappings such as the B-contraction, the Hardy-Rogers contraction, and so on. Additionally, we prove the same fixed point results by using rational type contraction mappings, which were discussed by the authors Dass. B. K and Gupta. S. Also, a few examples are included to illustrate the results. Finally, we discuss some applications that support our main results in the field of applied mathematics.

Keywords

Acknowledgement

All the authors express gratitude to The Korean Journal of Mathematics for their assistance in getting this manuscript completed. We would like to express our gratitude to the editors and the reviewers for their thorough reading and giving the opportunity to reset the manuscript in a nice way.

References

  1. Abbas, M., Rhoades, B. E., Fixed and periodic point results in cone metric space, Appl. Math. Lett. 22 (4) (2009), 511-515. https://doi.org/10.1016/j.aml.2008.07.001 
  2. Abdeljawad, T., Mlaiki, N., Aydi, H., Souayah, N. Double controlled metric type spaces and some fixed point results, Mathematics. 6 (2018), 320. https://doi.org/10.3390/math6120320 
  3. Bakhtin, I. A., The contraction mapping principle in almost metric spaces, Funct. Anal., 30 (1989), 26-37. 
  4. Banach, S., 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. Bianchini, R. M. T., Su un problema di S. Reich riguardante la teoria dei punti fissi, Bolletino U.M.I., 4 (5) (1972), 103-106. 
  6. Brouwer, F., The fixed point theory of multiplicative mappings in topological vector spaces, Mathematische Annalen., 177 (1968), 283-301.  https://doi.org/10.1007/BF01350721
  7. Chatterjea, S. K., Fixed point theorems , C. R. Acad. Bulg. Sci., 25 (1972), 727-730. 
  8. Ciric, L. B., Generalized contractions and fixed point theorems, Publ. Inst. Math. (Bulgr). 12 (26) (1971), 19-26. 
  9. Dass, B. K., Gupta, S., An extension of Banach contraction principle through rational expression, Communicated by F.C. Auluck, FNA.,1975. 
  10. Hardy, G. E., Rogers, T.D., A generalization of fixed point theorem of Reich, Can. Math. Bull., 16 (1973), 201-206.  https://doi.org/10.4153/CMB-1973-036-0
  11. Haung, L. G., Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (4) (2007), 1468-1476. https://doi.org/10.1016/j.jmaa.2005.03.087 
  12. Huaping Huang, Stojan Radenovic, Guantie Deng, A sharp generalization on cone b-metric space over Banach algebra, J. Nonlinear Sci. Appl. 10 (2017), 429-435. http://dx.doi.org/10.22436/jnsa.010.02.09 
  13. Jaggi, D. S., Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics, 8 (1977), 223-230. 
  14. Kannan, R., Some results on fixed points, Bull Calcutta Math.Soc, 60 (1968), 71-76. 
  15. Kannan, R., Some results on fixed points II, Am.Math.Mon. 76 (1969), 405-408. 
  16. Karapinar, E., A new non-unique fixed point theorem, Ann. Funct. Annals. 2 (1) (2011), 51-58.  https://doi.org/10.15352/afa/1399900261
  17. Khan, M. S., A fixed point theorems for metric spaces, Rendiconti Dell 'istituto di mathematica dell' Universtia di tresti, 8 (1976), 69-72. 
  18. Khuri, S. A., Louhichi, I., 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 
  19. Kumar. K, Rathour. L, Sharma. M. K, Mishra V. N. Fixed point approximation for suzuki generalized nonexpansive mapping using B(δ,µ) condition, Applied Mathematics 13 (2) (2022), 215-227. https://doi.org/10.4236/am.2022.132017 
  20. Marudai, M., Bright V. S., Unique fixed point theorem weakly B-contractive mappings, Far East journal of Mathematical Sciences (FJMS), 98 (7) (2015), 897-914.  https://doi.org/10.17654/FJMSDec2015_897_914
  21. Mishra. L. N, Dewangan. V, Mishra. V. N, Karateke. S, Best proximity points of admissible almost generalized weakly contractive mappings with rational expressions on b-metric spaces, J. Math. Computer Sci. 22 (2) (2021), 97-109. https://doi.org/10.22436/jmcs.022.02.01 
  22. Mishra. L. N, Dewangan. V, Mishra. V. N, Amrulloh. H, Coupled best proximity point theorems for mixed g-monotone mappings in partially ordered metric spaces, J. Math. Comput. Sci. 11 (5) (2021), 6168-6192. https://doi.org/10.28919/jmcs/6164 
  23. Mishra. L. N, Mishra. V. N, Gautam. P, Negi. K, Fixed point Theorems for Cyclic-Ciric-Reich-Rus contraction mapping in Quasi-Partial b-metric spaces, Scientific Publications of the State University of Novi Pazar Ser. A: Appl. Math. Inform. and Mech. 12 (1) (2020), 47-56. http://dx.doi.org/10.5937/SPSUNP2001047M 
  24. Mishra L. N, Tiwari. S. K, Mishra. V. N, Fixed point theorems for generalized weakly Scontractive mappings in partial metric spaces, Journal of Applied Analysis and Computation 5 (4) (2015), 600-612. https://doi.org/10.11948/2015047 
  25. Mitrovic, Z. D., Radenovic, S., The Banach and Reich contractions in bv(s)-metric spaces, J. Fixed Point Theory Appl. 19 (2017), 3087-3095. http://dx.doi.org/10.1007/s11784-017-0469-2 
  26. Mlaiki, N., Aydi, H., Souayah, N., Abdeljawad, T., Controlled metric type spaces and related contraction principle, Mathematics 6 (10) (2018), 194. https://doi.org/10.3390/math6100194 
  27. Mlaiki, N., Double controlled metric-like spaces, J. Inequal. Appl. 189 2020. https://doi.org/10.1186/s13660-020-02456-z 
  28. Reich, S., Some remarks connecting contraction mappings, Can. Math. Bull. 14 (1971), 121-124. https://doi.org/10.4153/CMB-1971-024-9 
  29. Roshan, J. R., Parvanesh, V., Kadelburg, Z., Hussain, N., New fixed point results in b-rectangular metric spaces, Nonlinear Analalysis: Modelling and control 21 (5) (2016), 614-634. http://dx.doi.org/10.15388/NA.2016.5.4 
  30. Sanatee. A. G, Rathour. L, Mishra. V. N, Dewangan. V Some fixed point theorems in regular modular metric spaces and application to Caratheodory's type anti-periodic boundary value problem, The Journal of Analysis 31 (2023), 619-632. https://doi.org/10.1007/s41478-022-00469-z 
  31. Sanatee. A. G, Ranmanesh. M. Mishra. L. N, Mishra. V. N, Generalized 2-proximal C-contraction mappings in complete ordered 2-metric space and their best proximity points, Scientific Publications of the State University of Novi Pazar Ser. A: Appl. Math. Inform. and Mech, 12 (1) (2020), 1-11. http://dx.doi.org/10.5937/SPSUNP2001001S 
  32. Shahi P, Rathour L, Mishra. V. N Expansive Fixed Point Theorems for tri-simulation functions, The Journal of Engineering and Exact Sciences -jCEC 08 (3) (2022), 14303-01e. https://doi.org/10.18540/jcecvl8iss3pp14303-01e 
  33. Sharma. N, Mishra. L. N, Mishra. V. N, Almusawa. H, Endpoint approximation of standard three-step multi-valued iteration algorithm for nonexpansive mappings, Applied Mathematics and Information Sciences 15 (1) (2021), 73-81. https://doi.org/10.18576/amis/150109 
  34. Sharma. N, Mishra. L. N, Mishra. V. N, Pandey. S, Solution of Delay Differential equation via Nv1 iteration algorithm, European J. Pure Appl. Math. 13 (5) (2020), 1110-1130. https://doi.org/10.29020/nybg.ejpam.v13i5.3756 
  35. Sharma. N, Mishra. L. N, Mishra. S. N, Mishra. V. N, Empirical study of new iterative algorithm for generalized nonexpansive operators, Journal of Mathematics and Computer Science 25 (3) (2022), 284-295. https://dx.doi.org/10.22436/jmcs.025.03.07 
  36. Shateri, T. L., Double controlled cone metric spaces and the related fixed point theorems, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 30 (1) (2023), 1-13. https://doi.org/10.48550/arXiv.2208.06812 
  37. Slobodanka Jankovic, Zoran Kadelburg, Stojan Radenovic, On cone metric spaces: A survey, Nonlinear Analysis 74 (2011) 2591-2601.  https://doi.org/10.1016/j.na.2010.12.014
  38. Stojan Radenovic, Common Fixed Points Under Contractive Condition in Cone Metric Spaces, Computers and Mathematics with applycation 58 (2019),1273-1278. https://doi.org/10.1016/j.camwa.2009.07.035 
  39. Suzana Aleksic, Zoran Kadelburg, Zoran. D. Mitrovic, Stojan Radenovic, A new survey: cone metric spaces, Journal of the international Mathematical Vertiual Institute 9(2019), 93-121. https://api.semanticscholar.org/CorpusID:119572977 
  40. Theivaraman. R, Srinivasan. P. S, Thenmozhi. S, Radenovic. S, Some approximate fixed point results for various contraction type mappings, 13 (9) (2023), 1-20. https://doi.org/10.28919/afpt/8080 
  41. Theivaraman. R, Srinivasan. P. S, Radenovic. S, Choonkil Park, New Approximate Fixed Point Results for Various Cyclic Contraction Operators on E-Metric Space, 27 (3) (2023), 160-179. https://doi.org/10.12941/jksiam.2023.27.160 
  42. Vishnu Narayanan P: B. Deshpande, V.N. Mishra, A. Handa, L.N. Mishra, Coincidence Point Results for Generalized (ψ, θ, φ)-Contraction on Partially Ordered Metric Spaces, Thai J. Math., 19 (1) (2021), 93-112. 
  43. Vishnu Narayanan P, Mishra. L. N, Tiwari. S. K, Mishra. V. N, Khan. I. A; Unique Fixed Point Theorems for Generalized Contractive Mappings in Partial Metric Spaces, Journal of Function Spaces, 2015 (2021), Article ID 960827, 1-8.  https://doi.org/10.1155/2015/960827
  44. Zamfirescu, T., Fixed point theorems in metric spaces, Arch. Math. (Basel) 23(1972), 292-298.  https://doi.org/10.1007/BF01304884
  45. Zoran Kadelburg, Stojan Radenovic, Vladimir Rakocevic, A note on the equivalence of some metric and cone metric fixed point results, Applied Mathematics Letters, 24 (2011), 370-374. https://doi.org/10.1016/j.aml.2010.10.030