SOME FIXED POINT THEOREMS FOR ORDERED CONTRACTIONS IN PARTIAL B-METRIC SPACES

Abstract

In this paper, some fixed point theorems in a partial b-metric space endowed with a partial order are proved. The results of this paper generalize and extend the Banach contraction principle and some other known results in partial b-metric spaces endowed with a partial order. Some examples are given which illustrate the cases when new results can be applied while old ones cannot.

Keywords

• Partial order,partial b-metric space; b-metric space; contraction; fixed point.

References

1. bibitem{Aa} A. Aghajani, M. Abbas, and J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Mathematica Slovaka, 64(4) (2014), 941-960.
2. bibitem{Ami1} A. Amini-Harandi, H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to
3. ordinary differential equations, Nonlinear Anal. 72 (2010), 2238-2242.
4. bibitem{3} A. Mukheimer, $alpha$-$psi$-$phi$-contractive mappings in ordered partial b-metric spaces, J. Nonlinear Sci. Appl., 7 (2014), 168-179.
5. bibitem{Ran} A. C. M. Ran and M. C. B Reurings, A fixed point theorem in
6. partially ordered sets and some application to matrix equations, Proc.
7. Amer. Math. Sco., 132 (2004), 1435-1443.
8. bibitem{Sto} D. DH ori'{c}, Z. Kadelburg, S. Radenovi'{c} and Poom Kumam, A note on fixed point results without monotone property

9. in partially ordered metric space, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas (2013) DOI 10.1007/s13398-013-0121-y
10. bibitem{Dor} D. DH ori'{c}, Z. Kadelburg, S. Radenovi'{c}, Coupled fixed point results for mappings without mixed monotone property,
11. Appl. Math. Lett. 25 (2012), 1803-1808.
12. bibitem{Ore} D. O'Regan, A. Petru¸sel, Fixed point theorems for generalzied contractions in ordered metric spaces, J. Math. Anal. Appl. 341
13. (2008), 1241-1252.
14. bibitem{Bak} I.A. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal., Unianowsk Gos. Ped. Inst. 30 (1989), 26-37.
15. bibitem{Cab} J. Caballero, J. Harjani, K. Sadarangani, Contractive-like mapping principles in ordered metric spaces and application to
16. ordinary differential equations, Fixed Point Theory Appl. (2010), Article ID 916064, 14 pages, doi:10.1155/2010/916064.
17. bibitem{Harj2} J. Harjani, K. Sadarangani, Fixed point theorems for monotone generalized contractions in partially ordered metric spaces
18. and applications to integral equations, J. Convex Anal. 19 (2012), 853-864.
19. bibitem{Harj} J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets,
20. Nonlinear Anal. 71 (2009), 3403-3410.
21. bibitem{Harj1} J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary
22. differential equations, Nonlinear Anal. 72 (2010), 1188-1197.
23. bibitem{Nie1} J.J. Nieto, R. Rodr'{i}guez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary
24. differential equations, Order 22 (2005), 223-239.
25. bibitem{Nie2} J.J. Nieto, R. Rodr'{i}guez-Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to
26. ordinary differential equations, Acta Math. Sinica, English series 23 (2007), 2205-2212.
27. bibitem{Nie3} J.J. Nieto, R. Rodr'{i}guez-Lopez, Existence of extremal solutions for quadratic fuzzy equations, Fixed Point Theory Appl.
28. (2005), 321-342.
29. bibitem{Nie4} J.J. Nieto, R. Rodr'{i}guez-Lopez, Fixed point theorems in ordered abstract spaces, Proc. Am. Math. Soc. 135 (2007), 2505-2517.
30. bibitem{Aaa} J.R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei and W. Shatanawi, Common fixed points of almost generalized $(psi,phi)_s$-contractive mappings in ordered b-metric spaces, Fixed Point Theory and Applications 2013 2013:159.
31. bibitem{Cir} Lj. '{C}iri'{c}, N. Caki'{c}, M. Rajovi'{c}, J.S. Ume, Monotone generalized nonlinear contractions in partially ordered
32. metric spaces, Fixed Point Theory Appl. (2008), Article ID 131294, doi:10.1155/2008/131294.
33. bibitem{1} N. Hussain, J.R. Roshan, V. Parvaneh, A. Latif, A unification of G-Metric, partial metric, and b-metric spaces, Abstract and Applied Analysis, Volume 2014 (2014), Article ID 180698, 14 pages http://dx.doi.org/10.1155/2014/180698
34. bibitem{2} N. Hussain, J.R. Roshan, V. Parvaneh, Z. Kadelburg, Fixed points of contractive mappings in b-metric-like spaces, The Scientific World Journal, Volume 2014 (2014), Article ID 471827, 15 pages, http://dx.doi.org/10.1155/2014/471827
35. bibitem{Ma} M.A. Khamski, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal., 73(9)(2010), 3123-3129.
36. bibitem{Abba} M. Abbas, V. Parvaneh, A Razani, Periodic points of T -Ciric generalized contraction mappings in ordered metric spaces, Georgian Math. J. (2012) DOI 10.1515/gmj-2012-0036
37. bibitem{Bor} M. Boriceanu, M. Bota, A. Petrusel, Mutivalued fractals in b-metric spaces, Cen. Eur. J. Math. 8(2)(2010), 367-377.
38. bibitem{Bot} M. Bota, A. Molnar, V. Csaba, On Ekeland's variational principle in b-metric spaces, Fixed Point Theory, 12(2011), 21-28.
39. bibitem{Mcb} M.C.B. Reurings, Contractive maps on normed linear spaces and their applications to nonlinear matrix equations,
40. Lin. Algebra Appl. 418 (2006), 292-311.
41. bibitem{Jov} M. Jovanovi'{c}, Z. Kadelburg, and S. Radenovi'{c}, Common fixed point results in metric type spaces, Fixed Point Theory Appl. Vol. 2010, Article ID 315398.
42. bibitem{Kan} R. Kannan, Some results on fixed points, Amer. Math. Monthly, 76, (1969),405-408.
43. bibitem{Aga} R.P. Agarwal, W. Sintunavarat, P. Kumam, Coupled coincidence point and common coupled fixed point theorems with
44. lacking the mixed monotone property, Fixed Point Theory Appl. 2013:22 (2013), doi:10.1186/1687-1812-2013-22.
45. bibitem{Cz1} S. Czerwik, Contraction Mappings in b-metric Spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1(1993), 5-11.
46. bibitem{Cz} S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Univ. Modena, 1998, 46, 263-276.
47. bibitem{Mat} S.G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Application, in: Ann. New York Acad. Sci., vol. 728, (1994) pp. 183-197.
48. bibitem{Shu} S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterr. J. Math., May 2014, Volume 11, Issue 2, pp 703-711. DOI 10.1007/s00009-013-0327-4
49. bibitem{Shur} S. Shukla, Reich type contractions on cone rectangular metric spaces endowed with a graph, Theory and Applications of Mathematics & Computer Science 4(1) (2014), 14-25.
50. bibitem{Rad} S. Radenovi'{c}, Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces, Comput. Math. Appl. 60 (2010), 1776-1783.
51. bibitem{4} Z. Mustafa, J.R. Roshan, V. Parvaneh, Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, Journal of Inequalities and Applications 2013, 2013:562.
52. bibitem{Xun} Xun Ge, Shou Lin, A note on partial b-metric spaces, Mediterr. J. Math. (2016) 13: 1273. doi:10.1007/s00009-015-0548-9