Topological Vector-Space Valued Cone Banach Spaces

Main Article Content

Nayyar Mehmood
Akbar Azam
Suzana Aleksic

Abstract

In this paper we introduce the notion of tvs-cone normed spaces, discuss related topological concepts and characterize the tvs-cone norm in various directions. We construct generalize locally convex tvs generated by a family of tvs-cone seminorms. The class of weak contractions properly includes large classes of highly applicable contractions like Banach, Kannan, Chatterjea and quasi etc. We prove fixed point results in tvs-cone Banach spaces for nonexpansive self mappings and self/non-self weak contractive mappings. We discuss the necessary conditions for T -stability of Picard iteration. To ensure the novelty of our work we establish an application in homotopy theory without the assumption of normality on cone and many non-trivial examples.

Article Details

References

  1. Beg, I, Azam, A, Arshad, M: Common fixed points for maps on topological vector space valued cone metric spaces. Int. J. Math. Math. Sci. 2009 (2009), Article ID 15.
  2. Huang, L, Zhang, X: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332(2007), 1468-1476.
  3. Azam, A, Beg, I, Arshad, M: Fixed Point in Topological Vector Space-Valued Cone Metric Spaces. Fixed Point Theory and Appl. 2010 (2010), Article ID 604084.
  4. Azam, A, Mehmood, N: Multivalued Fixed Point Theorems in tvs-Cone Metric Spaces. Fixed Point Theory and Applications. 2013 (2013), Article ID 184.
  5. Radenovi ´c, S, Kadelburg Z, Jankovi ´c, S: On cone metric spaces. A survey, Nonlinear Anal. 74 (2011), 2591-260.
  6. Rezapour, SH, Khandani, H, Vaezpour, SM: Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions. Rendiconti del Circolo Matematico di Palermo. 59 (2010), 185-197.
  7. Rudin, W: Functional Analysis. McGraw-Hill, Inc. USA. 1973.
  8. Schaefer, H, H, Wolff, M, P: Topological vector spaces, 2nd Edition. 1999 Springer-Verlag New York, Inc.
  9. Arshad, M, Azam, A and Vetro, P, Some common fixed point results in cone metric spaces, Fixed Point Theory Appl. 2009 (2009), Article ID 493965.
  10. Azam, A, Arshad, M, Beg, I: Common fixed points of two maps in cone metric spaces. Rend. Circ. Mat. Palermo 57 (2008), 433-441.
  11. Haghi, RH, Rezapour, S, Shahzad, N: Some fixed point generalizations are not real generalizations. Nonlinear Analysis: Theory, Methods & Applications. 74(5)(2011), 1799-1803.
  12. Khani, M, Pourmahdian, M: On the metrizability of cone metric spaces. Topology Appl. 158(2)(2011), 190-193.
  13. Rezapour, SH: Best Approximations in Cone Metric Spaces. Mathematica Moravica. 11(2007), 85-88.
  14. Rezapour, SH, Hamlbarani, R: Some notes on paper ”Cone metric spaces and fixed point theorems of contractive mappings”. J. Math. Anal. Appl. 345(2008), 719-724.
  15. Al-Rawashdeh, A., Shatanawi, W. and Khandaqji, M: Normed Ordered and E-Mertic Spaces, International Journal of Mathematics and Mathematical Sciences, 2012(2012), Article ID 272137.
  16. Karapinar, E: Fixed Point Theorems in Cone Banach Spaces. Fixed Point Theory and Applications. 2009(2009), Article ID 609281.
  17. Abdeljawad, T, Karapinar, E, Tas, K: Common fixed point theorems in cone Banach space. Hacettepe Journal of Mathematics and Statistics, Volume 40 (2) (2011), 211 - 217.
  18. Mutlu, A, Yolcu, N: Fixed point theorems for φp -operator in cone Banach spaces. Fixed Point Theory and Applications 2013(2013), Article ID 56.
  19. Yousefi, B, Yadegarnejad, A, Kenary, HA, Park, C: Equivalence of semistability of Picard, Mann, Krasnoselskij and Ishikawa iterations. Fixed Point Theory and Applications, 2014(2014), Article ID 5.
  20. Berinde, V: On the approximation of fixed points of weak contractive mappings. Carpathian J. Math, 19(1)(2003), 7-22.
  21. Alghamdi, M A, Berinde, V, Shahzad, N: Fixed points of multivalued nonself almost contractions. Journal of Applied Mathematics, 2013.
  22. Assad, N. A: A fixed point theorem in Banach space. Publications de l'Institut Math ´ematique (Beograd)(NS), 47(61)(1990), 137-140.
  23. Rhoades, B.E: A fixed point theorem for some non-self-mappings, Math. Japon. 23 (1978), 457-459.
  24. Berinde, V: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Analysis Forum. Vol. 9. 2004.
  25. Berinde, V: Iterative approximation of fixed points. Berlin: Springer, (2007)
  26. Berinde, V: A convergence theorem for some mean value fixed point iteration procedures. Dem Math, 38(1)(2005), 177-184.
  27. Berinde, V: A convergence theorem for Mann iteration in the class of Zamfirescu operators. Univ. Vest Timi. Ser. Mat.-Inform, 45(2007), 33-41.
  28. Kadelburg, Z, Radenovi ´c, S, Rakoˇcevi ´c, V: Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems. Fixed Point Theory and Applications, 2010(2010), Article ID 170253.
  29. Radenovi ´c. S, Kadelburg Z: Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 5 (2011), no. 1, 38-50.
  30. Asadi, M., Soleimani, H.,BE, R: On T-stability of picard iteration in cone metric spaces. Fixed Point Theory and Applications, 2009.