Common Fixed Point Theorems for Six Self-Mappings on S- Metric Spaces

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Thangjam Bimol Singh, G. A. Hirankumar Sharma, Y. Mahendra Singh, M. Ranjit Singh

Abstract

In this paper, we introduce the concepts of common property - (E.A) and common limit range property for six self-mappings and prove common fixed point theorems of such mappings satisfying (ψ, φ)-weak contraction on an S-metric space. Examples are given to illustrate our results.

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References

  1. M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl., 270 (2002), 181-188.
  2. M. Abbas, D. Doric, Common fixed point theorem for four mappings satisfying generalized weak contractive conditions, Filomat. 24(2) (2010), 1-10.
  3. M. Abbas, G. Jungck, Common fixed point results for non-commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008), 416-420.
  4. M. Abbas, M. S. Khan, Common fixed point theorem of two mappings satisfying a generalized weak contractive condition, Int. J. Math. Math. Sci. 2009 (2009), 131068.
  5. I.Y. Alber, S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert space, in: I. Gohberg and Y. Lyubich, (Eds.): New Results in Operator Theory and its Appl., Birkhnuser, Basel, Switzerland, 98 (1997), 7-22.
  6. I. Beg, M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theory Appl. 2006 (2006), 74503.
  7. V. Berinde, Approximating fixed points of weak φ-contractions, Fixed Point Theory. 4 (2003), 131-142.
  8. Y. J. Cho, P. P. Murthy, G. Jungck, A common fixed point theorems of Meir-Keeler type, Int. J. Math. Math. Sci. 16 (4) (1993), 669-674.
  9. B.S. Choudhury, P. Konor, B.E. Rhoades, N. Metiya, Fixed point theorems for generalized weakly contractive mapping, Nonlinear Anal.: Theory Meth. Appl. 74 (2011), 2116-2126.
  10. B.C. Dhage, Generalized metric space and mapping with fixed point, Bull. Cal. Math. Soc. 84 (1992), 329-336.
  11. B.C. Dhage, Generalized metric space and topological structure I, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S) 46 (2000), 3-24.
  12. B.C. Dhage, On generalized metric spaces and topological structure II, Pure. Appl. Math. Sci. 40 (1994), 37-41.
  13. H. Ding, Z. Kadelburg, E. Karapinar, S. Radenovic, Common fixed points of weak contractions in cone metric spaces, Abstr. Appl. Anal. 2012 (2012), 793862.
  14. N.V. Dung, N.T. Hieu, S. Radojevic, Fixed point theorems for g-monotone maps on partially ordered S-metric spaces, Filomat, 28 (9) (2014), 1885-1898.
  15. P.N. Dutta, B.S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl. 2008 (2008), 406368.
  16. J.X. Fang, Y. Gao, Common fixed point theorems under strict contractive conditions in Menger spaces, Nonlinear Anal.: Theory Meth. Appl. 70 (1) (2009), 184-193.
  17. M. Imdad, B.D. Pant, S. Chauhan, Fixed point theorems in menger spaces using the (CLRST ) property and applications, J. Nonlinear Anal. Optim. 3 (2) (2012), 225-237.
  18. G. Jungck, B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math. 29 (3) (1998), 227-238.
  19. Y. Liu, J. Wu, Z. Li, Common fixed points of single-valued and multi-valued maps, Int. J. Math. Math. Sci. 19 (2005), 3045-3055.
  20. Y. Mahendra Singh, G.A. Hirankumar Sharma, M. R. Singh, Common fixed point theorems for (ψ, φ)- weak contractive conditions in metric spaces, Hacet. J. Math. Stat. 48 (5) (2019), 1398-1408.
  21. Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006), 289-297.
  22. R.P. Pant, R- weakly commutativity and common fixed points, Soochow J. Math. 25 (1999), 37-42.
  23. H.K. Pathak, S.S. Chang, Y.J. Cho, Fixed point theorems for compatible mappings of type (P), Indian J. Math. 36 (2) (1994), 151-166.
  24. H.K. Pathak, Y.J. Cho, S.M. Kang, B. Madharia, Compatible mappings of type (C) and common fixed point theorem of Greguˇs type, Demonstr. Math. 31 (3) (1998), 499-517.
  25. H.K. Pathak, M.S. Khan, Compatible mappings of type (B) and common fixed point theorems of Greguˇs type, Czechoslovak Math. J. 45 (120) (1995), 685-698.
  26. S. Sedghi, N. Shobe, A. Aliouche, A generalization of fixed point theorems in S-metric spaces, Mat. Vesnik, 64 (3) (2012), 258-266.
  27. S. Sedghi, N. Shobe, H. Zhou, A common fixed point theorem in D*-metric spaces, Fixed Point Theory Appl. 2007 (2007), 27906.
  28. S. Sedghi, N. Shobkolaei, M. Shahraki, T. Doˇsenovic, Common fixed point of four maps in S-metric spaces, Math. Sci. 12 (2018), 137-143.
  29. M.R. Singh, Th. Bimol Singh, Some results for α-(ψ, φ)- contractive mappings in S-metric spaces, J. Adv. Math. Stud. 14 (2) (2021), 279-293.
  30. M.R. Singh, G.A. Hirankumar Sharma, Y. Mahendra Singh, Common fixed points for weak contraction occasionally weakly biased mappings, Adv. Fixed Point Theory, 7 (4) (2017), 458-467.
  31. M.R. Singh, Y. Mahendra Singh, Compatible mappings of type (E) and common fixed point theorems of Meir-Keeler type, Int. J. Math. Sci. Engg. Appl. 1 (2) (2007), 299-315.
  32. M.R. Singh, Y. Mahendra Singh, On various types of compatible maps and common fixed point theorems for non-continuous maps, Hacet. J. Math. Stat., 40 (4) (2011), 503-513.
  33. W. Sintunavarat, P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, J. Appl. Math. 2011 (2011), 637958.