Advertisement

Cyclic Contractions and Common Fixed Point Results of Integral Type Contractions in Multiplicative Metric Spaces

  • Talat NazirEmail author
  • Sergei Silvestrov
Conference paper
  • 34 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 317)

Abstract

The existence of common fixed points of cyclic contractive mappings satisfying generalized integral contractive conditions in multiplicative metric spaces is studied. The well-posedness of common fixed point results and periodic point results of cyclic contractions are also established. These results establish some of the general common fixed point theorems for self-mappings.

Keywords

Common fixed point Periodic point Well-posedness Contraction mappings Multiplicative metric space 

MSC 2010 Classification

47H09 47H10 54C60 54H25 

Notes

Acknowledgements

Talat Nazir is grateful for support of EU Erasmus Mundus project FUSION and the research environment Mathematics and Applied Mathematics MAM, Division of Applied Mathematics, Mälardalen University for providing an excellent and inspiring environment during his visits.

References

  1. 1.
    Abbas, M., Ali, B., Suleiman, Y.I.: Common fixed points of locally contractive mappings in multiplicative metric spaces with application. Inter. J. Math. Math. Sci., Article ID 218683, 7 pp. (2015)Google Scholar
  2. 2.
    Abbas, M., Nazir, T., Romaguera, S.: Fixed point results for generalized cyclic contraction mappings in partial metric spaces. Rev. R. Acad. Sci. Exactas Fis. Nat. Ser. A Math. 106, 287–297 (2012)Google Scholar
  3. 3.
    Abbas, M., Fisher, B., T, Nazir: Well-posedness and periodic point property of mappings satisfying a rational inequality in an ordered complex valued metric space. Sci. Stud. Res. Ser. Math. Inf. 22(1), 5–24 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Abbas, M., Rhoades, B.E.: Fixed and periodic point results in cone metric spaces. Appl. Math. Lett. 22, 511–515 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Agarwal, R.P., Alghamdi, M.A., Shahzad, N.: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012, 40, 11 pp. (2012)Google Scholar
  6. 6.
    Altun, I., Abbas, M., Simsek, H.: A fixed point theorem on cone metric spaces with new type contractivity. Banach J. Math. Anal. 5(2), 15–24 (2011)Google Scholar
  7. 7.
    Altun, I., Türkoglu, D., Rhoades, B.E.: Fixed points of weakly compatible maps satisfying a general contractive of integral type. Fixed Point Theory Appl. Article ID 17301 (2007)Google Scholar
  8. 8.
    Alsulami, H.H., Karapınar, E., O’Regan, D., Shahi, P.: Fixed points of generalized contractive mappings of integral type. Fixed Point Theory Appl. 2014, Article ID 213 (2014)Google Scholar
  9. 9.
    Akkouchi, M., Popa, V.: Well-posedness of fixed point problem for three mappings under strict contractive conditions. Bull. Math. Inf. Phys. Petroleum-Gash Univ. Ploiesti 61(2), 1–10 (2009)Google Scholar
  10. 10.
    Aydi, H.,Vetro, C., Sintunavarat, W., Kumam, P.: Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl. 2012, 124, 18 pp. (2012)Google Scholar
  11. 11.
    Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3, 133–181 (1922)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Branciari, A.: A Fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Sci. 29(9), 531–536 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bashirov, A.E., Kurpnar, E.M., Ozyapc, A.: Multiplicative calculus and its applications. J. Math. Anal. Appl. 337, 36–48 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bashirov, A.E., Misirli, E., Tandogdu, Y., Ozyapici, A.: On modeling with multiplicative differential equations. Appl. Math. J. Chin. Univ. 26(4), 425–438 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chaipunya, P., Cho, Y.J., Sintunavarat, W., Kumam, P.: Fixed point and common fixed point Theorems for cyclic quasi-contractions in metric and ultrametric spaces. Adv. Pure Math. 2, 401–407 (2012)CrossRefGoogle Scholar
  16. 16.
    Chaipunya, P., Cho, Y.J., Kumam, P.: A remark on the property \(\rho \) and periodic points of order \(\infty \). Math. Vesnik 66(4), 357–363 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Derafshpour, M., Rezapour, S.: On the existence of best proximity points of cyclic contractions. Adv. Dyn. Sys. Appl. 6(1), 33–40 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Djoudi, A., Aliouche, A.: Common fixed point theorems of Greguš type for weakly compatible mappings satisfying contractive conditions of integral type. J. Math. Anal. Appl. 329, 31–45 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Djoudi, A., Merghadi, F.: Common fixed point theorems for maps under a contractive condition of integral type. J. Math. Anal. Appl. 341, 953–960 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Florack, L., Assen, H.V.: Multiplicative calculus in biomedical image analysis. J. Math. Imag. Vis. 42(1), 64–75 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Feckan, M.: Nonnegative solutions of nonlinear integral equations. Comment. Math. Univ. Carol. 36, 615–627 (1995)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gornicki, J., Rhoades, B.E.: A general fixed point theorem for involutions. Indian J. Pure Appl. Math. 27, 13–23 (1996)MathSciNetzbMATHGoogle Scholar
  23. 23.
    He, X., Song, M., Chen, D.: Common fixed points for weak commutative mappings on a multiplicative metric space. Fixed Point Theory Appl. 2014, 48, 9 pp. (2014)Google Scholar
  24. 24.
    Jachymski, J.: Remarks on contractive conditions of integral type. Nonlinear Anal. 71, 1073–1081 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jeong, G.S., Rhoades, B.E.: Maps for which \(F(T)=F(T^{n})\). Fixed Point Theory 6, 87–131 (2005)Google Scholar
  26. 26.
    Karapinar, E.: Fixed point theory for cyclic weak \(\phi \)-contraction. Appl. Math. Lett. 24, 822–825 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kumam, P., Rahimi, H., Rad, G.S.: The existence of fixed and periodic point Theorems in cone metric type spaces. J. Nonlinear Sci. Appl. 7, 255–263 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kirk, W.A., Srinivasan, P.S., Veeramini, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4(1), 79–89 (2003)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Liu, Z., Li, X., Kang, S.M., Cho, S.Y.: Fixed point theorems for mappings satisfying contractive conditions of integral type and applications. Fixed Point Theory Appl. 2001, 64 (2011)Google Scholar
  30. 30.
    Liu, Z., Zou, X., Kang, S.M., Ume, J.S.: Common fixed points for a pair of mappings satisfying contractive conditions of integral type. J. Inequal. Appl. 2014, 394 (2014)Google Scholar
  31. 31.
    Murthy, P.P., Kumar, S., Tas, K.: Common fixed points of self maps satisfying an integral type contractive condition in fuzzy metric spaces. Math. Commun. 15, 521–537 (2010)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Nashine, H.K., Sintunavarat, W., Kumam, P.: Cyclic generalized contractions and fixed point results with applications to an integral equation. Fixed Point Theory Appl. 2012, 217, 13 pp. (2012)Google Scholar
  33. 33.
    Nashine, H.K., Samet, B., Vetro, C.: Fixed point theorems in partially ordered metric spaces and existence results for integral equations. Numer. Funct. Anal. Optim. 33(11), 1304–1320 (2012)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ozavsar, M., Cevikel, A.C.: Fixed point of multiplicative contraction mappings on multiplicative metric space (2012). arXiv:1205.5131v1 [matn.GN]
  35. 35.
    Păcurar, M., Rus, I.A.: Fixed point theory for cyclic \(\phi \)-contractions. Nonlinear Anal. 72(3–4), 1181–1187 (2010)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Piatek, B.: On cyclic Meir-Keeler contractions in metric spaces. Nonlinear Anal. 74, 35–40 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Rahimia, H., Rhoades, B.E., Radenović, S., Rad, G.S.: Fixed and periodic point Theorems for \(T\)-contractions on cone metric spaces. Filomat 27(5), 881–888 (2013)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Reich, S. Zaslawski, A.J.: Well posedness of fixed point problems. Far East J. Math. Sci. Special Volume Part 3, 393–401 (2001)Google Scholar
  39. 39.
    Rhoades, B.E., Abbas, M.: Maps satisfying a contractive condition of integral type for which fixed point and periodic point coincidence. Int. J. Pure Appl. Math. 45(2), 225–231 (2008)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Rus, I.A.: Cyclic Representation and fixed points. In: Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, vol. 3, 171–178 (2005)Google Scholar
  41. 41.
    Sintunavarat, W., Kumam, P.: Gregus-type common fixed point theorems for tangential multi-valued mappings of integral type in metric spaces. Int. J. Math. Math. Sci. 2011, Article ID 923458 (2011)Google Scholar
  42. 42.
    Singh, K.L.: Sequences of iterates of generalized contractions. Fund. Math. 105, 115–126 (1980)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Yamaod, O., Sintunavarat, W.: Some fixed point results for generalized contraction mappings with cyclic (\(\alpha ,\beta \))-admissible mapping in multiplicative metric spaces. J. Inequal. Appl. 2014, 488, 15 pp. (2014)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsCOMSATS Institute of Information TechnologyAbbottabadPakistan
  2. 2.Division of Applied Mathematics, School of Education, Culture and CommunicationMälardalen UniversityVästeråsSweden

Personalised recommendations