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Some topological properties and fixed point results in cone metric spaces over Banach algebras

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Abstract

This note is intended as an attempt at presenting some topological properties in cone metric spaces over Banach algebras. Moreover, the corresponding fixed point results are given. In addition, the P property, T-stability of Picard’s iteration, well-posedness of fixed point problems are also displayed. Our results complement and generalize some previous results in the existing literature.

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References

  1. Abbas, M., Parvaneh, V., Razani, A.: Periodic points of \(T\)-Ćirić generalized contraction mappings in ordered metric spaces. Georgian Math. J. 19(4), 597–610 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alnafei, S.H., Radenović, S., Shahzad, N.: Fixed point theorems for mappings with convex diminishing diameters on cone metric spaces. Appl. Math. Lett. 24(12), 2162–2166 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Asadi, M., Rhoades, B.E., Soleimani, H.: Some notes on the paper “The equivalence of cone metric spaces and metric spaces”. Fixed Point Theory Appl. 2012, 87 (2012)

    Article  MATH  Google Scholar 

  4. Azam, A., Fisher, B., Khan, M.: Common fixed point theorems in complex valued metric spaces. Numer. Funct. Anal. Optim. 32(3), 243–253 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bakhtin, I.A.: The contraction mapping principle in almost metric space. Funct. Anal. Ulyanovsk. Gos. Ped. Inst. Ulyanovsk 30, 26–37 (1989). (Russian)

    MathSciNet  MATH  Google Scholar 

  6. Czerwik, S.: Contraction mappings in \(b\)-metric spaces. Acta Math. Inform. Univ. Ostraviensis 1, 5–11 (1993)

    MathSciNet  MATH  Google Scholar 

  7. de Blasi, F.S., Myjak, J.: Well-posedness of fixed point problems. Comptes Rendus de L’Académie Des Sciences Paris 308, 51–54 (1989)

    MATH  Google Scholar 

  8. Du, W.-S.: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 72(5), 2259–2261 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Du, W.-S., Karapinıar, E.: A note on cone \(b\)-metric and its related results: generalizations or equivalence? Fixed Point Theory Appl. 2013, 210 (2013)

    Article  MathSciNet  Google Scholar 

  10. Ercan, Z.: On the end of the cone metric spaces. Topol. Appl. 166, 10–14 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feng, Y., Mao, W.: The equivalence of cone metric spaces and metric spaces. Fixed Point Theory 11(2), 259–264 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Huang, H., Radenović, S.: Common fixed point theorems of generalized Lipschitz mappings in cone \(b\)-metric spaces over Banach algebras and applications. J. Nonlinear Sci. Appl. 8(5), 787–799 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, H., Radenović, S.: Some fixed point results of generalized Lipschitz mappings on cone \(b\)-metric spaces over Banach algebras. J. Comput. Anal. Appl. 20(3), 566–583 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Huang, H., Xu, S., Liu, H., Radenović, S.: Fixed point theorems and \(T\)-stability of Picard iteration for generalized Lipschitz mappings in cone metric spaces over Banach algebras. J. Comput. Anal. Appl. 20(5), 869–888 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Huang, H., Xu, S.: Some new topological properties in cone metric spaces. J. Math. PRC 35(3), 513–518 (2015)

    Google Scholar 

  16. Huang, L., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332(2), 1468–1476 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Janković, S., Kadelburg, Z., Radenović, S.: On cone metric spaces: a survey. Nonlinear Anal. 74(7), 2591–2601 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jovanović, M., Kadelburg, Z., Radenović, S.: Common fixed point results in metric-type spaces. Fixed Point Theory Appl. (2010). https://doi.org/10.1155/2010/978121

    MathSciNet  MATH  Google Scholar 

  19. Kadelburg, Z., Radenović, S., Rakočević, V.: A note on the equivalence of some metric and cone metric fixed point results. Appl. Math. Lett. 24(3), 370–374 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, H., Xu, S.: Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings. Fixed Point Theory Appl. 2013, 320 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Malhotra, S.K., Sharma, J.B., Shukla, S.: Fixed points of \(\alpha \)-admissible mappings in cone metric spaces with Banach algebra. Int. J. Anal. Appl. 9(1), 9–18 (2015)

    MATH  Google Scholar 

  22. Matthews, S.G.: Partial metric topology. Ann. N. Y. Acad. Sci. 728, 183–197 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mustafa, Z., Sims, B.: A new approach to generalized metric space. J. Nonlinear Convex Anal. 7(2), 289–297 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Öner, T.: Some topological properties of fuzzy cone metric spaces. J. Nonlinear Sci. Appl. 9(3), 799–805 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rezapour, Sh, Derafshpour, M., Hamlbarani, R.: A review on topological properties of cone metric spaces. J. Bone Jt. Surg. s2–9(4), 615–632 (2008)

    Google Scholar 

  26. Rezapour, S., Hamlbarani, R.: Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”. J. Math. Anal. Appl. 345(2), 719–724 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shukla, S., Balasubramanian, S., Pavlović, M.: A generalized Banach fixed point theorem. Bull. Malays. Math. Sci. Soc. 39(4), 1529–1539 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank Professor Zoran D. Mitrovic for their valuable comments and suggestions which improved greatly the quality of this paper. The research was partially supported by the National Natural Science Foundation of China (No. 11271045) and by the Science and Technology Research Project of Education Department in Hubei Province of China (No. Q20172505).

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

    Huaping Huang & Guantie Deng

  2. Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120, Beograd, Serbia

    Stojan Radenović

Authors
  1. Huaping Huang
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  2. Guantie Deng
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  3. Stojan Radenović
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Correspondence to Guantie Deng.

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Huang, H., Deng, G. & Radenović, S. Some topological properties and fixed point results in cone metric spaces over Banach algebras. Positivity 23, 21–34 (2019). https://doi.org/10.1007/s11117-018-0590-5

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