Nonlinear Functional Analysis and Applications

The Basic Sciences Research Institute, Kyungnam University (경남대학교 기초과학연구소)
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DISCUSSION ON b-METRIC SPACES AND RELATED RESULTS IN METRIC AND G-METRIC SPACES

  • Received : 2021.05.29
  • Accepted : 2021.11.08
  • Published : 2022.06.08
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Abstract

In the present manuscript, we employ the concepts of Θ-map and Φ-map to define a strong (𝜃, 𝜙)s-contraction of a map f in a b-metric space (M, db). Then we prove and derive many fixed point theorems as well as we provide an example to support our main result. Moreover, we utilize our results to obtain many results in the settings of metric and G-metric spaces. Our results improve and modify many results in the literature.

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References

  1. K. Abodayeh, T. Qawasmeh, W. Shatanawi and A. Tallafha, ϵ𝜑-contraction and some fixed point results via modified ω-distance mappings in the frame of complete quasi metric spaces and applications, Inter. J. Electrical Comp. Eng., 10(4) (2020), 3839-3853. https://doi.org/10.11591/ijece.v10i4.pp3839-3853
  2. H. Aydi, E. Karapinar and M. Postolache, Tripled coincidence point theorems for weak phi-contractions in partially ordered metric spaces, Fixed Point Theory and Appl., 44 (2012).
  3. H. Aydi, M. Postolache and W. Shatanawi, Coupled fixed point results for (Ψ, Φ)-weakly contractive mappings in ordered G-metric spaces, Comput. Math. Appl., 63 (2012), 298-309. https://doi.org/10.1016/j.camwa.2011.11.022
  4. H. Aydi, W. Shatanawi, M. Postolache, Z. Mustafa, and N. Tahat, Theorems for Boyd-Wong-Type Contractions in Ordered Metric Spaces, Abstr. Appl. Anal., 2012, Article ID 359054, (2012), 14 pages.
  5. H. Aydi, W. Shatanawi and C. Vetro, On generalized weak G-contraction mapping in G-metric spaces, Comput. Math. Appl., 62 (2011), 4223-4229.
  6. I.A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., Gos. Ped. Inst., Unianowsk, 30 (1989), 26-37.
  7. S. Banach, Sur Les operations dans les ensembles abstraits et leur application aux equations integrals, Fund. Math., 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181
  8. A. Bataihah, W. Shatanawi, T. Qawasmeh and R. Hatamleh, On H-Simulation Functions and Fixed Point Results in the Setting of wt-Distance Mappings with Application on Matrix Equations, Mathematics, 8(5) (2020), 837; https://doi.org/10.3390/math8050837.
  9. A. Bataihah, A. Tallafha and W. Shatanawi, Fixed point results with Ω-distance by utilizing simulation functions, Ital. J. Pure Appl. Math., 43, (2020), 185-196.
  10. A. Bataihah, W. Shatanawi and A. Tallafha, Fixed point results with simulation functions, Nonlinear Funct. Anal. App., 25(1), (2020), 13-23.
  11. V. Berinde, Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011), 7347-7355. https://doi.org/10.1016/j.na.2011.07.053
  12. S. Chandok and M. Postolache, Fixed point theorem for weakly Chatterjea-type cyclic contractions, Fixed Point Theory Appl., 2013 (2013).
  13. L.B. Ciric, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267-273. https://doi.org/10.1090/S0002-9939-1974-0356011-2
  14. S. Czerwik, Contraction mappings in b-metric spaces, Acta mathematica et informatica universitatis ostraviensis, 1(1) (1993), 5-11.
  15. H.S. Ding and E. Karapinar, Meir-Keeler type contractions in partially ordered G-metric spaces, Fixed Point Theory Appl., 2013, Article ID 2013:35, (2013), 1-10. https://doi.org/10.1186/1687-1812-2013-1
  16. M. Jleli and B. Samet, Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory Appl., Article ID 210, (2012).
  17. R. Kannan, Some results on fixed point, Bull. Calcutta Math. Soc., 60 (1968), 71-76.
  18. J.K. Kim, M. Kumar, P. Bhardwaj and M. Imdad, Common fixed point theorems for generalized 𝜓∫𝜑-weakly contractive mappings in G-metric spaces, Nonlinear Funct. Anal. Appl., 26(3) (2021), 565-580. https://doi.org/10.22771/NFAA.2021.26.03.08
  19. N.V. Luong and N.X. Thuan, Coupled fixed point theorems in partially ordered G-metric spaces, Math. Comput. Model., 55 (2012), 1601-1609. https://doi.org/10.1016/j.mcm.2011.10.058
  20. Z. Mustafa, M. Khandaqji and W. Shatanawi, Fixed point results on complete G-metric spaces, Stud. Sci. Math. Hung., 48 (2011), 304-319. https://doi.org/10.1556/SScMath.48.2011.3.1170
  21. Z. Mustafa, H. Obiedat and F. Awawdeh, Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl., Article ID 189870, (2008), 1-12.
  22. Z. Mustafa and B. Sims, Fixed point theorems for contractive mappings in complete G-metric spaces, Fixed Point Theory Appl. , Article ID 917175, (2009), 1-10.
  23. Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7(2) (2006), 289-297.
  24. J. Oudetallah, M. Rousan and I. Batiha, On D-Metacompactness In Topological Spaces, J. Appl. Math. & Informatics, 39 (2021), 919-926. https://doi.org/10.14317/JAMI.2021.919
  25. T. Qawasmeh, W. Shatanawi, A. Bataihah and A. Tallafha, Common Fixed Point Results for Rational (α, β)𝜑-mω Contractions in Complete Quasi Metric Spaces, Mathematics, 7(5) (2019), 392. https://doi.org/10.3390/math7050392
  26. T. Qawasmeh, A. Tallafha and W. Shatanawi, Fixed and common fixed point theorems through modified ω-distance mappings, Nonlinear Funct. Anal. Appl., 24(2) (2019), 221-239.
  27. T. Qawasmeh, A. Tallafha and W. Shatanawi, Fixed Point Theorems through Modified ω-Distance and Application to Nontrivial Equations, Axioms, 8(2) (2019), 57. https://doi.org/10.3390/axioms8020057
  28. W. Shatanawi, Common Fixed Points for Mappings under Contractive Conditions of (α, β, 𝜓)-Admissibility Type, Mathematics, 6 (2018).
  29. W. Shatanawi and M. Postolache, Common fixed point results for mappings under nonlinear contraction of cyclic form in ordered metric spaces, Fixed Point Theory and Applications, 2013 (2013).
  30. T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008), 1861-1869. https://doi.org/10.1090/S0002-9939-07-09055-7
  31. J. Yadav, M. Kumar, Reena, M. Imdad and S. Arora, Fixed point theorems for (ξ, β)-expansive mapping in g-metric space using control function, Nonlinear Funct. Anal. Appl., 26(5) (2021), 949-959. https://doi.org/10.22771/NFAA.2021.26.05.06
  32. D. Zheng, Fixed point theorems for generalized 𝜃 - 𝜙-contractions in G-metric spaces, J. Funct. Spaces, (2018), 8 pages.
  33. D. Zheng, Z. Cai and P. Wang, New fixed point theorems for 𝜃-𝜙-contraction in complete metric spaces, J. Nonlinear Sci. Appl., 10(5) (2017), 2662-2670. https://doi.org/10.22436/jnsa.010.05.32