Title: A Weak Contraction Principle in Partially Ordered Cone Metric Space with Three Control Functions
Author(s): Binayak S. Choudhury, L. Kumar, T. Som, N. Metiya
Pages: 18-27
Cite as:
Binayak S. Choudhury, L. Kumar, T. Som, N. Metiya, A Weak Contraction Principle in Partially Ordered Cone Metric Space with Three Control Functions, Int. J. Anal. Appl., 6 (1) (2014), 18-27.

Abstract


In this paper we utilize three functions to define a weak contraction in a cone metric space with a partial order and establish that this contraction has necessarily a fixed point either under the continuity assumption or an order condition which we state here. The uniqueness of the fixed point is also derived under some additional assumptions. The result is supported with an example. The methodology used is a combination of order theoretic and analytic approaches.

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References


  1. Ya. I. Alber and S. Guerre-Delabriere, Principles of weakly contractive maps in Hilbert spaces, Oper. Theory Adv. Appl. 98 (1997), 7-22.

  2. I. Altun, B. Damjanovi´c, D. Djori´c, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. 23 (2010), 310-316.

  3. I. Altun, V. Rakoˇcevi´c, Ordered cone metric spaces and fixed point results, Comput. Math. Appl. 60 (2010), 1145-1151.

  4. B. S. Choudhury, A common unique fixed point result in metric spaces involving generalised altering distances, Math. Commun. 10 (2005), 105-110.

  5. B. S. Choudhury, K. Das, A coincidence point result in Menger spaces using a control function, Chaos Solitons Fractals 42 (2009), 3058-3063.

  6. B. S. Choudhury, N. Metiya, Fixed points of weak contractions in cone metric spaces, Nonlinear Anal. 72 (2010), 1589-1593.

  7. B. S. Choudhury, N. Metiya, The point of coincidence and common fixed point for a pair of mappings in cone metric spaces, Comput. Math. Appl. 60 (2010), 1686-1695.

  8. B. S. Choudhury, P. Konar, B. E. Rhoades, N. Metiya, Fixed point theorems for generalized weakly contractive mappings, Nonlinear Anal. 74 (2011), 2116-2126.

  9. B. S. Choudhury, A. Kundu, (ψ, α, β) - Weak contractions in partially ordered metric spaces, Appl. Math. Lett. 25 (2012), 6 - 10.

  10. B. S. Choudhury, N. Metiya, Coincidence point and fixed point theorems in ordered cone metric spaces, J. Adv. Math. Stud. 5 (2012), 20-31.

  11. B. S. Choudhury, N. Metiya, Fixed point and common fixed point results in ordered cone metric spaces, An. St. Univ. Ovidius Constanta 20 (2012), 55-72.

  12. L. Ciri´c, M. Abbas, R. Saadati, N. Hussain, Common fixed points of almost generalized ´ contractive mappings in ordered metric spaces, Appl. Math. Comput. 217 (2011), 5784-5789.

  13. K. Deimling, Nonlinear Functional Analysis, Springer-Verlage, 1985.

  14. D. Dori´c, Common fixed point for generalized (ψ, ϕ)-weak contractions, Appl. Math. Lett. 22 (2009), 1896-1900.

  15. W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261.

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

  17. A. A. Harandi, M. Fakhar, Fixed point theory in cone metric spaces obtained via the scalarization method, Comput. Math. Appl. 59 (2010), 3529-3534.

  18. J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71 (2009), 3403-3410.

  19. L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468-1476.

  20. D. Ili´c, V. Rako´cevi´c, Common fixed point for maps on cone metric space, J. Math. Anal. Appl. 341 (2008), 876-882.

  21. S. Jankovi´c, Z. Kadelburg, S. Radenovi´c, B. E. Rhoades, Assad-Kirk-Type fixed point theorems for a pair of nonself mappings on cone metric spaces, Fixed Point Theory Appl. 2009 (2009), Article ID 761086.

  22. S. Jankovi´c, Z. Kadelburg, S. Radenovi´c, On cone metric spaces: A survey, Nonlinear Anal. 74 (2011), 2591-2601.

  23. Z. Kadelburg, M. Pavlovi´c, S. Radenovi´c, Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comput. Math. Appl. 59 (2010), 3148-3159.

  24. Z. Kadelburg, S. Radenovic, V. Rakocevic, A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett. 24 (2011), 370-374.

  25. M. S. Khan, M. Swaleh, S. Sessa, Fixed points theorems by altering distances between the points, Bull. Austral. Math. Soc. 30 (1984), 1-9.

  26. S. V. R. Naidu, Some fixed point theorems in metric spaces by altering distances, Czechoslovak Math. J. 53 (2003), 205-212.

  27. J. J. Nieto, R. Rodroguez-Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica 23 (2007), 2205- 2212.

  28. O. Popescu, Fixed points for (ψ, φ) - weak contractions, Appl. Math. Lett. 24 (2011), 1-4.

  29. A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443.

  30. B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (200l), 2683- 2693.

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

  32. Q. Zhang, Y. Song, Fixed point theory for generalized φ− weak contractions, Appl. Math. Lett. 22 (2009), 75-78.