Certain Fixed-Point Results via DS-Weak Commutativity Condition in Neutrosophic Metric Spaces With Application to Non-linear Fractional Differential Equations

Main Article Content

Umar ishtiaq, Muhammad Saeed, Khaleel Ahmad, Ilsa Shokat, Manuel De la Sen


This study demonstrates that, for the non-linear contractive conditions in Neutrosophic metric spaces, a common fixed-point theorem may be proved without requiring the continuity of any mappings. A novel commutativity requirement for mappings weaker than the compatibility of mappings is used to demonstrate the conclusion. We provide several examples to illustrate our major idea. Also, we provide an application to the non-linear fractional differential equation to show the validity of our main result.

Article Details


  1. M.S. El Naschie, Wild Topology, Hyperbolic Geometry and Fusion Algebra of High Energy Particle Physics, Chaos, Solitons Fractals. 13 (2002), 1935-1945. https://doi.org/10.1016/s0960-0779(01)00242-9.
  2. M.S. El Naschie, From Experimental Quantum Optics to Quantum Gravity via a Fuzzy Kähler Manifold, Chaos Solitons Fractals. 25 (2005), 969-977. https://doi.org/10.1016/j.chaos.2005.02.028.
  3. J.H. Park, Intuitionistic Fuzzy Metric Spaces, Chaos Solitons Fractals. 22 (2004), 1039–1046. https://doi.org/10.1016/j.chaos.2004.02.051.
  4. K.T. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets Syst. 20 (1986), 87-96. https://doi.org/10.1016/s0165-0114(86)80034-3.
  5. A. George, P. Veeramani, On Some Results in Fuzzy Metric Spaces, Fuzzy Sets Syst. 64 (1994), 395–399. https://doi.org/10.1016/0165-0114(94)90162-7.
  6. C. Alaca, D. Turkoglu, C. Yildiz, Fixed Points in Intuitionistic Fuzzy Metric Spaces, Chaos Solitons Fractals. 29 (2006), 1073–1078. https://doi.org/10.1016/j.chaos.2005.08.066.
  7. Ivan Kramosil; Jiří Michálek, Fuzzy Metrics and Statistical Metric Spaces, Kybernetika, 11 (1975), 336-344. http://dml.cz/dmlcz/125556.
  8. M. Grabiec, Fixed Points in Fuzzy Metric Spaces, Fuzzy Sets Syst. 27 (1988), 385-389. https://doi.org/10.1016/0165-0114(88)90064-4.
  9. S. Banach, Theoriles operations. Linearies Manograie Mathematyezne, Warsaw, Poland, 1932.
  10. M. Edelstein, On Fixed and Periodic Points Under Contractive Mappings, J. Lond. Math. Soc. s1-37 (1962), 74–79. https://doi.org/10.1112/jlms/s1-37.1.74.
  11. G. Jungck, Compatible Mappings and Common Fixed Points, Int. J. Math. Math. Sci. 9 (1986), 771–779. https://doi.org/10.1155/s0161171286000935.
  12. D. Turkoglu, C. Alaca, Y.J. Cho, C. Yildiz, Common Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces, J. Appl. Math. Comput. 22 (2006), 411–424. https://doi.org/10.1007/bf02896489.
  13. V. Gregori, S. Romaguera, P. Veeramani, A Note on Intuitionistic Fuzzy Metric Spaces, Chaos Solitons Fractals. 28 (2006), 902–905. https://doi.org/10.1016/j.chaos.2005.08.113.
  14. R. Sadati, J.H. Park, On the Intuitionistic Topological Spaces, Chaos Solutions Fractals, 27 (2006), 331-344.
  15. J. Rodrı́guez-López, S. Romaguera, The Hausdorff Fuzzy Metric on Compact Sets, Fuzzy Sets Syst. 147 (2004), 273–283. https://doi.org/10.1016/j.fss.2003.09.007.
  16. S. Sessa, On a Weak Commutativity Condition of Mappings in Fixed Point Considerations, Publ. Inst. Math.32 (1982), 149–153.
  17. S.N. Mishra, N. Sharma, S.L. Singh, Common Fixed Points of Maps on Fuzzy Metric Spaces, Int. J. Math. Math. Sci. 17 (1994), 253–258.
  18. N. Simsek, M. Kirişci, Fixed Point Theorems in Neutrosophic Metric Spaces, Sigma J. Eng. Nat. Sci. 10 (2019), 221-230.
  19. H.K. Pathak, Y.J. Cho, S.M. Kang, Remarks on R-weakly Commuting Mappings and Common Fixed Point Theorems, Bull. Korean Math. Soc. 34 (1997), 247–257.
  20. S.N. Ješić, N.A. Babačev, Common Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces and L-Fuzzy Metric Spaces With Nonlinear Contractive Condition, Chaos Solitons Fractals. 37 (2008), 675–687. https://doi.org/10.1016/j.chaos.2006.09.048.
  21. S. Sharma, B. Deshpande, Common Fixed Point Theorems for Non-Compatible Mappings and Meir–Keeler Type Contractive Condition in Fuzzy Metric Spaces, Int. Rev. Fuzzy Math. 1(2006), 147–159.
  22. D.W. Boyd, J.S.W. Wong, On Nonlinear Contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464.
  23. M. Kirişci, N. Şimşek, Neutrosophic Metric Spaces, Math. Sci. 14 (2020), 241–248. https://doi.org/10.1007/s40096-020-00335-8.
  24. S. Sowndrarajan, M. Jeyarama, F. Smarandache, fixed point Results for Contraction Theorems in Neutrosophic Metric Spaces, Neutrosophic Sets Syst. 36 (2020), 23.
  25. B. Schweizer, A. Sklar, Statistical Metric Spaces, Pac. J. Math. 10 (1960), 314–334.
  26. B. Deshpande, Fixed Point and (DS)-Weak Commutativity Condition in Intuitionistic Fuzzy Metric Spaces, Chaos Solitons Fractals. 42 (2009), 2722–2728. https://doi.org/10.1016/j.chaos.2009.03.178.