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MONOTONE GENERALIZED CONTRACTIONS IN ORDERED METRIC SPACES

  • Alam, Aftab (Department of Mathematics Aligarh Muslim University) ;
  • Imdad, Mohammad (Department of Mathematics Aligarh Muslim University)
  • Received : 2014.12.11
  • Published : 2016.01.31

Abstract

In this paper, we prove some existence and uniqueness results on coincidence points for g-monotone mappings satisfying linear as well as generalized nonlinear contractivity conditions in ordered metric spaces. Our results generalize and extend two classical and well known results due to Ran and Reurings (Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435-1443) and Nieto and $Rodr{\acute{i}}guez$-$L{\acute{o}}pez$ (Acta Math. Sin. 23 (2007), no. 12, 2205-2212) besides similar other ones. Finally, as an application of one of our newly proved results, we establish the existence and uniqueness of solution of a first order periodic boundary value problem.

Keywords

ordered metric space;TCC property;termwise monotone sequence;c-bound

References

  1. R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87 (2008), no. 1, 109-116. https://doi.org/10.1080/00036810701556151
  2. A. Alam, A. R. Khan, and M. Imdad, Some coincidence theorems for generalized non-linear contractions in ordered metric spaces with applications, Fixed Point Theory Appl. 2014 (2014), 216, 30 pp. https://doi.org/10.1186/1687-1812-2014-30
  3. I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl. 2010 (2010), 621469, 17 pp.
  4. H. Amann, Order structures and fixed points, Bochum: Mimeographed lecture notes, Ruhr-Universitat, 1977.
  5. A. Amini-Harandi and H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010), no. 5, 2238-2242. https://doi.org/10.1016/j.na.2009.10.023
  6. A. Bjorner, Order-reversing maps and unique fixed points in complete lattices, Algebra Universalis 12 (1981), no. 3, 402-403. https://doi.org/10.1007/BF02483901
  7. D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464. https://doi.org/10.1090/S0002-9939-1969-0239559-9
  8. J. Caballero, J. Harjani, and K. Sadarangani, Contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations, Fixed Point Theory Appl. 2010 (2010), Art. ID 916064, 14 pp.
  9. S. Carl, A monotone iterative scheme for nonlinear reaction-diffusion systems having nonmonotone reaction terms, J. Math. Anal. Appl. 134 (1988), no. 1, 81-93. https://doi.org/10.1016/0022-247X(88)90008-X
  10. S. Carl, An enclosing theorem and a monotone iterative scheme for elliptic systems having nonmonotone nonlinearities, Z. Angew. Math. Mech. 70 (1990), no. 8, 309-313. https://doi.org/10.1002/zamm.19900700806
  11. S. Carl and S. Heikkila, Fixed Point Theory in Ordered Sets and Applications: from differential and integral equations to game theory, Springer, New York, 2011.
  12. L. Ciric, N. Cakic, M. Rajovic, and J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. 2008 (2008), Art. ID 131294, 11 pp.
  13. P. Cousot and R. Cousot, Constructive versions of Tarski's fixed point theorems, Pacific J. Math. 82 (1979), no. 1, 43-57. https://doi.org/10.2140/pjm.1979.82.43
  14. R. DeMarr, Common fixed points for isotone mappings, Colloq. Math. 13 (1964), 45-48. https://doi.org/10.4064/cm-13-1-45-48
  15. R. H. Haghi, S. Rezapour, and N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Anal. 74 (2011), no. 5, 1799-1803. https://doi.org/10.1016/j.na.2010.10.052
  16. J. Harjani and K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71 (2009), no. 7-8, 3403-3410. https://doi.org/10.1016/j.na.2009.01.240
  17. J. Harjani and K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (2010), no. 3-4, 1188-1197. https://doi.org/10.1016/j.na.2009.08.003
  18. S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, Inc., New York, 1994.
  19. J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear Anal. 74 (2011), no. 3, 768-774. https://doi.org/10.1016/j.na.2010.09.025
  20. M. Jleli, V. C. Rajic, B. Samet, and C. Vetro, Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations, J. Fixed Point Theory Appl. 12 (2012), no. 1-2, 175-192. https://doi.org/10.1007/s11784-012-0081-4
  21. N. Jotic, Some fixed point theorems in metric spaces, Indian J. Pure Appl. Math. 26 (1995), no. 10, 947-952.
  22. G. Jungck, Commuting maps and fixed points, Amer. Math. Monthly 83 (1976), no. 4, 261-263. https://doi.org/10.2307/2318216
  23. G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci. 9 (1986), no. 4, 771-779. https://doi.org/10.1155/S0161171286000935
  24. G. Jungck, Common fixed points for noncontinuous nonself maps on non-metric spaces, Far East J. Math. Sci. 4 (1996), no. 2, 199-215.
  25. D. Kurepa, Fixpoints of decreasing mappings of ordered sets, Publ. Inst. Math. (N.S.) 18(32) (1975), 111-116.
  26. G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985.
  27. V. Lakshmikantham and L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009), no. 12, 4341-4349. https://doi.org/10.1016/j.na.2008.09.020
  28. S. Lipschutz, Schaum's Outlines of Theory and Problems of Set Theory and Related Topics, McGraw-Hill, 1964.
  29. A. Mukherjea, Contractions and completely continuous mappings, Nonlinear Anal. 1 (1977), no. 3, 235-247. https://doi.org/10.1016/0362-546X(77)90033-5
  30. H. K. Nashine and I. Altun, A common fixed point theorem on ordered metric spaces, Bull. Iranian Math. Soc. 38 (2012), no. 4, 925-934.
  31. J. J. Nieto and R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), no. 3, 223-239. https://doi.org/10.1007/s11083-005-9018-5
  32. J. J. Nieto and R. Rodriguez-Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 12, 2205-2212. https://doi.org/10.1007/s10114-005-0769-0
  33. D. O'Regan and A. Petrusel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008), no. 2, 1241-1252. https://doi.org/10.1016/j.jmaa.2007.11.026
  34. A. Petrusel and I. A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006), no. 2, 411-418.
  35. A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435-1443. https://doi.org/10.1090/S0002-9939-03-07220-4
  36. K. P. R. Sastry and I. S. R. Krishna Murthy, Common fixed points of two partially commuting tangential selfmaps on a metric space, J. Math. Anal. Appl. 250 (2000), no. 2, 731-734. https://doi.org/10.1006/jmaa.2000.7082
  37. S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 149-153.
  38. A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955), 285-309. https://doi.org/10.2140/pjm.1955.5.285
  39. M. Turinici, Fixed points for monotone iteratively local contractions, Demonstr. Math. 19 (1986), no. 1, 171-180.
  40. J. S. W. Wong, Common fixed points of commuting monotone mappings, Canad. J. Math. 19 (1967), 617-620. https://doi.org/10.4153/CJM-1967-054-4
  41. J. Wu and Y. Liu, Fixed point theorems for monotone operators and applications to nonlinear elliptic problems, Fixed Point Theory Appl. 2013 (2013), 134, 14 pp. https://doi.org/10.1186/1687-1812-2013-14

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