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A hybrid class of expansive-contractive mappings in cone b-metric spaces

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Abstract

In this research work, we generalize classical results on the existence of common fixed points of generalized expanding mappings by removing the constraint on the signs of the underlining coefficients in the inequality considered. The hybrid class obtained surprisingly regroups both classes of generalized contractive maps and classes of generalized expansive maps. The research work also provides a clear understanding of the boundary between contractive and expansive mappings in literature and is applied to product metric-type spaces.

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References

  1. Aage, C.T., Salunke, J.N.: Some fixed point theorems for expansion onto mappings on cone metric spaces. Acta Math. Sin. Engl. Ser. 27(6), 1101–1106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abbas, M., Rhoades, B.E., Nazir, T.: Common fixed points for four maps in cone metric spaces. Appl. Math. Comput. 216, 80–86 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Bakhtin, I.A.: The contraction mapping principle in almost metric spaces. Funct. Anal. Gos. Ped. Inst. Unian. 30, 26–37 (1989)

    MathSciNet  MATH  Google Scholar 

  4. Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3, 133–181 (1922)

    MATH  Google Scholar 

  5. Bhashkar, T.G., Lakshmikantham, V.: Fixed point theorems in partially ordered cone metric spaces and applications. Nonlinear Anal. Theory Methods Appl. 65(7), 1379–1393 (2006)

    Article  MATH  Google Scholar 

  6. Ciric, L.B.: Generalized contractions and fixed-point theorems. Publ. l’Inst. Math. (Beograd) (NS) 12(26), 19–26 (1971)

    MathSciNet  MATH  Google Scholar 

  7. Daffer, P.Z., Kaneko, H.: On expansive mappings. Math. Jpn. 37(1992), 733–735 (1992)

    MathSciNet  Google Scholar 

  8. Han, Y., Xu, S.: Some new theorems of expanding mappings without continuity in cone metric spaces. Fixed Point Theory Appl. 3, 1–9 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Huang, H., Xu, S.: Correction: Fixed point theorems of contractive mappings in cone \(b\)-metric spaces and applications. Fixed Point Theory Appl. 2014, 55 (2014)

    Article  MATH  Google Scholar 

  10. Huang, L.G., 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 

  11. Hussain, N., Shah, M.H.: KKM mappings in cone b-metric spaces. Comput. Math. Appl. 62, 1677–1684 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jungck, G., Rhoades, B.E.: Fixed point for set valued functions without continuity. Indian J. Pure Appl. Math. 29(3), 227–238 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Kumar, S., Garg, S.K.: Expansion mapping theorems in metric spaces. Int. J. Contemp. Math. Sci. 4(36), 1749–1758 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Nashine, H.K., Kadelburg, Z., Radenovic, S.: Coupled common fixed point theorems for w*-compatible mappings in ordered cone metric spaces. Appl. Math. Comput. 218, 5422–5432 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Olaleru, J.O.: Some generalizations of fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 1–10 (2009) (Article ID 657914)

  16. Olaleru, J.: Common fixed points of three self-mappings in cone metric spaces. Appl. Math. E-notes 11, 41–49 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Olaoluwa, H., Olaleru, J.O., Chang, S.S.: Coupled fixed point theorems for asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2013, 68 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Olaoluwa, H., Olaleru, J.: Multipled fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2014, 43 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Olaoluwa, H., Olaleru, J.O.: An approach to the approximation of common coupled fixed points of contractive maps in metric-type spaces. Adv. Fixed Point Theory 5(1), 13–31 (2015)

    Google Scholar 

  20. Olaoluwa, H., Olaleru, J.O.: On common fixed points and multipled fixed points of contractive mappings in metric-type spaces (2015). doi:10.1016/j.jnnms.2015.06.001

    Google Scholar 

  21. Rhoades, B.E.: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rhoades, B.E.: Contractive definitions. In: Rassias, T.M. (ed.) Nonlinear Analysis, pp. 513–526. Word Scientific Publishing Company, NJ (1988)

    Chapter  Google Scholar 

  23. Sahin, I., Telci, M.: A theorem on common fixed points of expansion type mappings in cone metric spaces. An. St. Univ. Ovid. Const. 18, 329–336 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Samet, B., Vetro, C.: Coupled fixed point, \(F\)-invariant set and fixed point of \(N\)-order. Ann. Funct. Anal. 1(2), 46–56 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shatanawi, W., Awawdeh, F.: Some fixed and coincidence point theorems for expansive maps in cone metric spaces. Fixed Point Theory Appl. 2012, 19 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, S.-Z., Li, B.-Y., Gao, Z.-M., Lseki, K.: Some fixed points of expansion mappings. Math. Jpn. 29, 631–636 (1984)

    Google Scholar 

  27. Xu, S., Cheng, S., Han, Y.: Correction: some new theorems of expanding mappings without continuity in cone metric spaces. Fixed Point Theory Appl. 2014, 178 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are grateful to the referees for their helpful suggestions contributing to the improvement of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria

    H. Olaoluwa & J. O. Olaleru

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  1. H. Olaoluwa
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  2. J. O. Olaleru
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Correspondence to H. Olaoluwa.

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Olaoluwa, H., Olaleru, J.O. A hybrid class of expansive-contractive mappings in cone b-metric spaces. Afr. Mat. 27, 825–840 (2016). https://doi.org/10.1007/s13370-015-0381-0

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