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Contractive Operators in Relational Metric Spaces

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Handbook of Functional Equations

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 95))

Abstract

In Sect. 1, some fixed point results for altering contractive maps on (amorphous) metric spaces are given, extending the one due to Khan, Swaleh and Sessa [Bull Aust Math Soc 30:1–9, 1984],. In Sect. 2, a class of monotone contractions is analyzed, via coupled fixed point techniques, in the realm of quasi-ordered metric spaces. Note that, a highly unusual feature of the related fixed point techniques is that, in many cases with a practical relevance, no coupled starting point hypothesis for these operators is needed. Finally, in Sect. 3, some fixed point results are given for contractive operators acting on relational metric spaces.

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References

  1. Agarwal, R.P., El-Gebeily, M.A., O’Regan, D.: Generalized contractions in partially ordered metric spaces. Appl. Anal. 87, 109–116 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 3, 133–181 (1922)

    MATH  Google Scholar 

  3. Berinde, V.: Approximating fixed points of weak \(\varphi\)-contractions using the Picard iteration. Fixed Point Theory 4, 131–142 (2003)

    MATH  MathSciNet  Google Scholar 

  4. Berinde, V.: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 9, 43–53 (2004)

    MATH  MathSciNet  Google Scholar 

  5. Berinde, V.: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 74, 7347–7355 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berzig, M.: Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications. J. Fixed Point Theory. Appl. 12, 221–238 (2013)

    Article  MathSciNet  Google Scholar 

  7. Berzig, M., Rus, M.-D.: Fixed point theorems for α-contraction mappings of Meir–Keeler type and applications. Nonlinear Anal. Model. Control 19, 178–198 2013 (Arxiv, 1303-5798-v1)

    MathSciNet  Google Scholar 

  8. Bhaskar, T.G., Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bhaumik, I., Das, K., Metiya, N., Choudhury, B.S.: A coincidence point result by using altering distance function. J. Math. Comput. Sci. 2, 61–72 (2012)

    MathSciNet  Google Scholar 

  10. Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cohen, P.J.: Set Theory and the Continuum Hypothesis. Benjamin, New York (1966)

    MATH  Google Scholar 

  12. Dutta, P.N., Choudhury, B.S.: A generalization of contraction principle in metric spaces. Fixed Point Theory. Appl. Article ID 406368 (Volume 2008)

    Google Scholar 

  13. Hicks, T.L., Rhoades, B.E.: Fixed point theory in symmetric spaces with applications to probabilistic spaces. Nonlinear Anal. (A) 36, 331–344 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhauser, Boston (1998)

    Book  MATH  Google Scholar 

  15. Jachymski, J.: A generalization of the theorem by Rhoades and Watson for contractive type mappings. Math. Jpn. 38, 1095–1102 (1993)

    MATH  MathSciNet  Google Scholar 

  16. Jachymski, J.: Common fixed point theorems for some families of mappings. Indian J. Pure Appl. Math. 25, 925–937 (1994)

    MATH  MathSciNet  Google Scholar 

  17. Jachymski, J.: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 74, 768–774 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Karapinar, E., Berzig, M.: Fixed point results for \((\alpha\psi,\beta\varphi)\)-contractive mappings for a generalized altering distance. Fixed Point Theory. Appl. 2013:205 (2013)

    Google Scholar 

  19. Kasahara, S.: On some generalizations of the Banach contraction theorem. Publ. Res. Inst. Math. Sci. Kyoto Univ. 12, 427–437 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  20. Khan, M.S., Swaleh, M., Sessa, S.: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30, 1–9 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kirk, W.A., Srinivasan, P.S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4, 79–89 (2003)

    MATH  MathSciNet  Google Scholar 

  22. Leader, S.: Fixed points for general contractions in metric spaces. Math. Jpn. 24, 17–24 (1979)

    MATH  MathSciNet  Google Scholar 

  23. Maia, M.G.: Un’osservazione sulle contrazioni metriche. Rend. Sem. Mat. Univ. Padova 40, 139–143 (1968)

    MATH  MathSciNet  Google Scholar 

  24. Matkowski, J.: Integrable solutions of functional equations. Diss. Math. 127, 1–68 (1975)

    MathSciNet  Google Scholar 

  25. Matthews, S.G.: Partial metric topology. Proceedings of 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 728, 183–197 (1994)

    Google Scholar 

  26. Meir, A., Keeler, E.: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  27. Nashine, H.K., Samet, B.: Fixed point results for mappings satisfying \((\psi,\varphi)\)-weakly contractive condition in partially ordered metric spaces. Nonlinear Anal. 74, 2201–2209 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  28. Nieto, J.J., Rodriguez-Lopez, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. O’Regan, D., Petruşel, A.: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 341, 1241–1252 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Pathak, H.K., Shahzad, N.: Fixed point results for set-valued contractions by altering distances in complete metric spaces. Nonlinear Anal. 70, 2634–2641 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Popescu, O.: Fixed points of \((\psi,\varphi)\)-weak contractions. Appl. Math Lett. 24, 1–4 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Ran, A.C.M., Reurings, M.C.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  34. Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)

    Google Scholar 

  35. Rus, M.-D.: Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric. Nonlinear Anal. 74, 1804–1813 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Rus, M.-D.: The fixed point problem for systems of coordinate-wise uniformly monotone operators and applications. Mediterr. J. Math. 11, 109–122 (2013)

    Article  MathSciNet  Google Scholar 

  37. Samet, B., Turinici, M.: Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications. Commun. Math. Anal. 13, 82–97 (2012)

    MATH  MathSciNet  Google Scholar 

  38. Samet B., Vetro, C., Vetro, P.: Fixed point theorems for \(\alpha-\psi\)-contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. Sastry, K.P.R., Babu, G.V.R.: Some fixed point theorems by altering distances between the points. Indian J. Pure. Appl. Math. 30, 641–647 (1999)

    MATH  MathSciNet  Google Scholar 

  40. Turinici, M.: Fixed points in complete metric spaces. Proc. Inst. Math. Iaşi (Romanian Academy, Iaşi Branch), pp. 179–182, Editura Academiei R.S.R., Bucureşti (1976)

    Google Scholar 

  41. Turinici, M.: Abstract comparison principles and multivariable Gronwall–Bellman inequalities. J. Math. Anal. Appl. 117, 100–127 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  42. Turinici, M.: Ran–Reurings theorems in ordered metric spaces. J. Indian Math. Soc. 78, 207–214 (2011)

    MATH  MathSciNet  Google Scholar 

  43. Turinici, M.: Linear contractions in product ordered metric spaces. Ann. Univ. Ferrar. 59, 187–198 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. “A. Myller” Mathematical Seminar, “A. I. Cuza” University, 700506, Iaşi, Romania

    Mihai Turinici

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  1. Mihai Turinici
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Correspondence to Mihai Turinici .

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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Turinici, M. (2014). Contractive Operators in Relational Metric Spaces. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 95. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1246-9_18

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