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Rational contraction in multiplicative metric spaces

Year 2018, Volume: 2 Issue: 4, 195 - 201, 24.12.2018
https://doi.org/10.31197/atnaa.481995

Abstract

The purpose of this paper is to prove that instead of a rational contraction shown in the papers  Afrah A. N. Abdou, \emph{Fixed point theorems for generalized contraction mappings in multiplicative metric spaces, }J. Nonlinear Sci.
Appl. 9, 2347-2363,  (2016) and N. Sharma, K. Kumar, S. Sharma, R. Jha, \emph{Rational contractive condition
in multiplicative metric space and common fixed point theorem}, International Journal of Innovative Research in Science,
Engineering and Technology, 5, 10473-10480 (2016) a more general contractive condition can be obtained in multiplicative metric spaces, which is equivalent to a contractive condition in metric spaces.

References

  • [1] M. Abbas, B. Ali, Yi Suleiman, \emph{Common fixed points oflocally contractive mappings in multiplicative metric spaces withapplications,} Int. J., Math. Math. Sci. 2015, Article ID 218683, (2015).
  • [2] M. Abbas, M. De La Sen, T. Nazir, \emph{Common fixed pointsof generalized rational type cocyclic mappings in multiplicative metricspaces,} Discrete Dyn. Nat. Soc. 2015, Article Id 532725, (2015).
  • [3] K. Abodayeh, A. Pitea, W. Shatanawi, T. Abdeljawad, \emph{%Remarks on Multiplicative Metric Spaces and Related Fixed Points,}arXiv:1512.03771v1 [math.GN] 11, (2015).
  • [4] Afrah A. N. Abdou, \emph{Fixed point theorems for generalizedcontraction mappings in multiplicative metric spaces, }J. Nonlinear Sci.Appl. 9, 2347-2363, (2016).
  • [5] D.E. Anderson, K.L.Singh, J.H.M. Whitfield, \emph{Common fixed point for family of mappings}, Internat. J. Math. and Math. Sci., 7(1), 1984, 89-95.
  • [6] R. P. Agarwal, E. Karapinar and B. Samet, \emph{An essentialremark on fixed point results on multiplicative metric spaces, }Fixed Point\theory Appl., 2016:21, (2016).
  • [7] S. Banach, \emph{Sur les op\'{e}rations dans les ensemblesabstraits et leur application aux \'{e}quations int\'{e}grales,} Fundam.Math.,3, 133-181, (1922).
  • [8] A. Bashirov, E. Kurpinar, A. Ozyapici, \emph{Multiplicativecalculus and its applications,} J. Math. Anal. Appl. 337 (1), 36-48, (2008).
  • [9] D.W. Boyd, J.S.Wong, \emph{On linear contractions,} Proc.Amer. Maqth. Soc. 20, 458-464, (1969).
  • [10] Lj.B. \'{C}iri\'{c}, \emph{On Common Fixed Points In Uniform Spaces}, Publications de l'Institut Mathématique, 24(38), 39--43, (1978).
  • [11] T. Do\v{s}enovi\'{c}, M. Postolache, S. Radenovi\'{c}, \emph{On multiplicative metric spaces: Survey}, Fixed Point TheoryAppl., 2016:92, (2016).
  • [12] T. Do\v{s}enovi\'{c}, S. Radenovi\'{c}, \emph{Some criticalremarks on the paper: ''An essential remark on fixed point results onmultiplicative metric spaces''}, J. Adv. Math. Stud.,10(1), 20-24, (2017).
  • [13] M. Edelstein, \emph{On fixed and periodic points under contractive mappings}, J. London Math. Soc., 37, 74-79, (1962).
  • [14] X. He, M. Song and D. Chen, \emph{Common fixed points for weakcommutative mappings on a multiplicative metric space,} Fixed Point TheoryAppl., 2014:48, (2009).
  • [15] G. Jungck, \emph{Compatible mappings and common fixed points}, Internat. J. Math. Math.Sci., 9, 771 - 779, (1986).
  • [16] G. Jungck, B. E. Rhoades, \emph{Fixed point for set-valued functions without continuity,} Indian J. Pure Appl. Math., 29 (1998), 227–238.
  • [17] R. Kannan, \emph{Some results on fixed points}, Bull. Cal. Math., 60, 71-76, (1968).
  • [18] S. M. Kang, P. Kumar, S. Kumar, P. Nagpal, S.K Garg, \emph{%Common fixed points for compatible mappings and its variants inmultiplicative metric spaces}, Int. J. Pure Appl. Math. 102 (2), 383-406,(2015).
  • [19] C. Mongkolkeha, W. Shatanawi, \emph{Best proximity points formultiplicative proximal contraction mapping on multiplicative metric spaces,}J. Nonlinear Sci. Appl. 8, 1134-1140, (2015).
  • [20] M. Sarwar, B.-e-Rome, \emph{Some Unique Fixed PointTheorems in Multiplicative Metric Space,} arXiv:1410.3384v2 [math.GM] 29,(2014).
  • [21] M. \"{O}zavsar, A. C. Cevikel, \emph{Fixed pointsof multiplicative contraction mappings on multiplicative metric spacers,}arXiv:1205.5131v1 [math.GM] 23, (2012).
  • [22] S. Shukla, \emph{Some critical remarks on the multiplicativemetric spaces and fixed point results}, to appear in J. Adv. Math. Studies,(2016).
  • [23] D. Stanley, \emph{A multiplicative calculus,} Primus IX (4) 310326, (1999).
  • [24] O. Yamaod, W. Sintunavarat, \emph{Some fixed point resultsfor generalized contraction mappings with cyclic (}$\alpha ,\beta $\emph{%)-admissible mapping in multiplicative metric spaces,} J. Inequal.Appl.,, 2014:488, (2014).
  • [25] O. Zamaod, W. Sintunavarat, \emph{Some fixed point results forgeneralized contraction mappings with cyclic }$\left( \alpha ,\beta \right)- $\emph{admissible mappings in multiplicative metric spaces,} J. Inequal.Appl., 2014:488, ( 2014).
Year 2018, Volume: 2 Issue: 4, 195 - 201, 24.12.2018
https://doi.org/10.31197/atnaa.481995

Abstract

References

  • [1] M. Abbas, B. Ali, Yi Suleiman, \emph{Common fixed points oflocally contractive mappings in multiplicative metric spaces withapplications,} Int. J., Math. Math. Sci. 2015, Article ID 218683, (2015).
  • [2] M. Abbas, M. De La Sen, T. Nazir, \emph{Common fixed pointsof generalized rational type cocyclic mappings in multiplicative metricspaces,} Discrete Dyn. Nat. Soc. 2015, Article Id 532725, (2015).
  • [3] K. Abodayeh, A. Pitea, W. Shatanawi, T. Abdeljawad, \emph{%Remarks on Multiplicative Metric Spaces and Related Fixed Points,}arXiv:1512.03771v1 [math.GN] 11, (2015).
  • [4] Afrah A. N. Abdou, \emph{Fixed point theorems for generalizedcontraction mappings in multiplicative metric spaces, }J. Nonlinear Sci.Appl. 9, 2347-2363, (2016).
  • [5] D.E. Anderson, K.L.Singh, J.H.M. Whitfield, \emph{Common fixed point for family of mappings}, Internat. J. Math. and Math. Sci., 7(1), 1984, 89-95.
  • [6] R. P. Agarwal, E. Karapinar and B. Samet, \emph{An essentialremark on fixed point results on multiplicative metric spaces, }Fixed Point\theory Appl., 2016:21, (2016).
  • [7] S. Banach, \emph{Sur les op\'{e}rations dans les ensemblesabstraits et leur application aux \'{e}quations int\'{e}grales,} Fundam.Math.,3, 133-181, (1922).
  • [8] A. Bashirov, E. Kurpinar, A. Ozyapici, \emph{Multiplicativecalculus and its applications,} J. Math. Anal. Appl. 337 (1), 36-48, (2008).
  • [9] D.W. Boyd, J.S.Wong, \emph{On linear contractions,} Proc.Amer. Maqth. Soc. 20, 458-464, (1969).
  • [10] Lj.B. \'{C}iri\'{c}, \emph{On Common Fixed Points In Uniform Spaces}, Publications de l'Institut Mathématique, 24(38), 39--43, (1978).
  • [11] T. Do\v{s}enovi\'{c}, M. Postolache, S. Radenovi\'{c}, \emph{On multiplicative metric spaces: Survey}, Fixed Point TheoryAppl., 2016:92, (2016).
  • [12] T. Do\v{s}enovi\'{c}, S. Radenovi\'{c}, \emph{Some criticalremarks on the paper: ''An essential remark on fixed point results onmultiplicative metric spaces''}, J. Adv. Math. Stud.,10(1), 20-24, (2017).
  • [13] M. Edelstein, \emph{On fixed and periodic points under contractive mappings}, J. London Math. Soc., 37, 74-79, (1962).
  • [14] X. He, M. Song and D. Chen, \emph{Common fixed points for weakcommutative mappings on a multiplicative metric space,} Fixed Point TheoryAppl., 2014:48, (2009).
  • [15] G. Jungck, \emph{Compatible mappings and common fixed points}, Internat. J. Math. Math.Sci., 9, 771 - 779, (1986).
  • [16] G. Jungck, B. E. Rhoades, \emph{Fixed point for set-valued functions without continuity,} Indian J. Pure Appl. Math., 29 (1998), 227–238.
  • [17] R. Kannan, \emph{Some results on fixed points}, Bull. Cal. Math., 60, 71-76, (1968).
  • [18] S. M. Kang, P. Kumar, S. Kumar, P. Nagpal, S.K Garg, \emph{%Common fixed points for compatible mappings and its variants inmultiplicative metric spaces}, Int. J. Pure Appl. Math. 102 (2), 383-406,(2015).
  • [19] C. Mongkolkeha, W. Shatanawi, \emph{Best proximity points formultiplicative proximal contraction mapping on multiplicative metric spaces,}J. Nonlinear Sci. Appl. 8, 1134-1140, (2015).
  • [20] M. Sarwar, B.-e-Rome, \emph{Some Unique Fixed PointTheorems in Multiplicative Metric Space,} arXiv:1410.3384v2 [math.GM] 29,(2014).
  • [21] M. \"{O}zavsar, A. C. Cevikel, \emph{Fixed pointsof multiplicative contraction mappings on multiplicative metric spacers,}arXiv:1205.5131v1 [math.GM] 23, (2012).
  • [22] S. Shukla, \emph{Some critical remarks on the multiplicativemetric spaces and fixed point results}, to appear in J. Adv. Math. Studies,(2016).
  • [23] D. Stanley, \emph{A multiplicative calculus,} Primus IX (4) 310326, (1999).
  • [24] O. Yamaod, W. Sintunavarat, \emph{Some fixed point resultsfor generalized contraction mappings with cyclic (}$\alpha ,\beta $\emph{%)-admissible mapping in multiplicative metric spaces,} J. Inequal.Appl.,, 2014:488, (2014).
  • [25] O. Zamaod, W. Sintunavarat, \emph{Some fixed point results forgeneralized contraction mappings with cyclic }$\left( \alpha ,\beta \right)- $\emph{admissible mappings in multiplicative metric spaces,} J. Inequal.Appl., 2014:488, ( 2014).
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Tatjana Dosenovıc 0000-0002-3236-4410

Stojan Radenovıc 0000-0002-7417-1342

Publication Date December 24, 2018
Published in Issue Year 2018 Volume: 2 Issue: 4

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