Fixed Point Theorem of Modified S-Iteration Process for Ciric Quasi Contractive Operator in CAT(0) spaces

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G. S. Saluja


The aim of this paper is to study the strong convergence of modified S-iteration process for Ciric quasi contractive operator in the framework of CAT(0) spaces. Also we give an application of our result with supporting example. Our result improves and extends some corresponding previous result from the existing literature (see, e.g., [3, 29] and many others).

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  1. R.P. Agarwal, D. O'Regan, D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8(1) (2007), 61-79.
  2. A. Abkar and M. Eslamian, Common fixed point results in CAT(0) spaces, Nonlinear Anal.: Theory, Method and Applications, 74 (2011), no.5, 1835-1840.
  3. V. Berinde, Iterative approximation of fixed points, Baia Mare: Efemeride, 2000.
  4. V. Berinde, Iterative approximation of fixed points, Springer-Verlag, Berlin Heidelberg, 2007.
  5. A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 29 (2002), 531-536.
  6. M.R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, 319 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999.
  7. K.S. Brown, Buildings, Springer, New York, NY, USA, 1989.
  8. F. Bruhat and J. Tits, ”Groups reductifs sur un corps local”, Institut des Hautes Etudes Scientifiques. Publications Mathematiques, 41 (1972), 5-251.
  9. P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl. 320 (2006), no.2, 983-987.
  10. S.K. Chatterjee, Fixed point theorems compactes, Rend. Acad. Bulgare Sci. 25 (1972), 727- 730.
  11. L.B. Ciric, A generalization of Banach principle, Proc. Amer. Math. Soc. 45 (1974), 727-730.
  12. S. Dhompongsa, A. Kaewkho and B. Panyanak, Lim's theorems for multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl. 312 (2005), no.2, 478-487.
  13. S. Dhompongsa and B. Panyanak, On 4-convergence theorem in CAT(0) spaces, Comput. Math. Appl. 56 (2008), no.10, 2572-2579.
  14. R. Espinola and A. Fernandez-Leon, CAT(k)-spaces, weak convergence and fixed point, J. Math. Anal. Appl. 353 (2009), no.1, 410-427.
  15. K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, 83 of Monograph and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, NY, USA, 1984.
  16. N. Hussain and M.A. Khamsi, On asymptotic pointwise contractions in metric spaces, Nonlinear Anal.: Theory, Method and Applications, 71 (2009), no.10, 4423-4429.
  17. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.
  18. R. Kannan, Some results on fixed point theorems, Bull. Calcutta Math. Soc. 10 (1968), 71-76.
  19. M.A. Khamsi and W.A. Kirk, An introduction to metric spaces and fixed point theory, Pure Appl. Math, Wiley-Interscience, New York, NY, USA, 2001.
  20. S.H. Khan and M. Abbas, Strong and 4-convergence of some iterative schemes in CAT(0) spaces, Comput. Math. Appl. 61 (2011), no.1, 109-116.
  21. A.R. Khan, M.A. Khamsi and H. Fukhar-ud-din, Strong convergence of a general iteration scheme in CAT(0) spaces, Nonlinear Anal.: Theory, Method and Applications, 74 (2011), no.3, 783-791.
  22. W.A. Kirk, Fixed point theory in CAT(0) spaces and R-trees, Fixed Point and Applications, 2004 (2004), no.4, 309-316.
  23. W.A. Kirk, Geodesic geometry and fixed point theory, in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), 64 of Coleccion Abierta, 195-225, University of Seville Secretary of Publications, Seville, Spain, 2003.
  24. W.A. Kirk, Geodesic geometry and fixed point theory II, in International Conference on Fixed point Theory and Applications, 113-142, Yokohama Publishers, Yokohama, Japan, 2004.
  25. W. Laowang and B. Panyanak, Strong and ∆-convergence theorems for multivalued mappings in CAT(0) spaces, J. Inequal. Appl. 2009 (2009), Article ID 730132, 16 pages.
  26. L. Leustean, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl. 325 (2007), no. 1, 386-399.
  27. W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
  28. Y. Niwongsa and B. Panyanak, Noor iterations for asymptotically nonexpansive mappings in CAT(0) spaces, Int. J. Math. Anal. 4 (2010), no.13, 645-656.
  29. B.E. Rhoades, Fixed point iteration using infinite matrices, Trans. Amer. Math. Soc. 196 (1974), 161-176.
  30. S. Saejung, Halpern's iteration in CAT(0) spaces, Fixed Point Theory and Applications, 2010 (2010), Article ID 471781, 13 pages.
  31. N. Shahzad, Fixed point results for multimaps in CAT(0) spaces, Topology and its Applications, 156 (2009), no.5, 997-1001.
  32. T. Zamfirescu, Fixed point theorems in metric space, Arch. Math. (Basel), 23 (1972), 292-298.