Some convergence and stability results for two new Kirk and Jungck-multi step type iterative processes

 
 
 
  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract


    In this work two new iterative processes called the “Jungck-Kirk generalized multi-step” and “Jungck-Kirk multi-step” are introduced and some convergence and stability results are proved for these iterative process. The results include results of almost stability and summable almost stability. Since these new iterative processes are more general than other ones extant in literature, some results of this work partially generalize results already proved in the existing literature.


  • Keywords


    Fixed point; Jungck type iterative process; Kirk-multistep iteration; Metric spaces; Stability of iterative processes.

  • References


      [1] V. Berinde, “Summable almost stability of fixed point iteration procedures”, Carpathian J. Math., 19 (2003) 81-88.

      [2] A. O. Bosede, B. E. Rhoades, “Stability of Picard and Mann iteration for a general class of functions”, J. Adv. Math. Studies, 3 (2) (2010) 23-25.

      [3] R. Chugh, V. Kumar, “Stability of hybrid fixed point iterative algorithms of Kirk-Noor type in normed linear space for self and nonself operators”, Int. J. Contemp. Math. Sciences, 7 (24) (2012), 1165-1184.

      [4] F. Gursoy, V. Karakaya, B. E. Rhoades, “Some convergence and stability results for the Kirk multistep and Kirk-SP fixed point iterative algorithms”, Abstract and Applied Analysis (2014) 1-12.

      [5] F. Gursoy, V. Karakaya, “Some convergence and stability results for two new Kirk type hybrid fixed point iterative algorithms”, Journal of Function Spaces (2014) 1-8.

      [6] A. M. Harder, T. L. Hicks, “Stability results for fixed point iteration procedures”, Math. Jap. 33, (5) (1988) 693-706.

      [7] O. Imoru, M. O. Olatinwo, “On the stability of Picard and Mann iteration processes”, Carpathian J. Math. 19 (2) (2003) 155-160.

      [8] C.O. Imoru, M. O. Olatinwo, “Some stability theorems for some iteration processes”, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 45 (2006) 81-88.

      [9] S. Ishikawa, “Fixed points by a new iteration method “, Proc. Amer Math. Soc., 44 (1) (1974), 147-150.

      [10] W. A. Kirk, “On successive approximations for nonexpansive mappings in Banach spaces”, Glasgow Mathematical Journal, 12 (1971) 6-9.

      [11] W. R. Mann, “Mean value methods in iteration”, Proc., Amer. Math. Soc., 44 (1953), 506-510.

      [12] M.A. Noor, “New approximation schemes for general variational inequalities”, J. Math. Anal. Appl., 251 (2000), 217-229.

      [13] J.O. Olaleru, H. Akewe, “The convergence of Jungck-type iterative schemes for generalized contractive-like operators”, Fasc. Math., 45 (2010) 87-98.

      [14] M.O. Olatinwo, “A generalization of some convergence results using the Jungck-Noor three-step iteration process in arbitrary Banach space”, Fasc. Math., 40 (2008) 37-43.

      [15] M. O. Olatinwo, “On some stability results for fixed point iteration procedures”, Journal of Mathematics and statistics 2 (1) (2006) 339-342.

      [16] M. O. Olatinwo, “Some stability results for nonexpansive and quasi-nonexpansive operators in uniformly convex Banach space using two new iterative processes of Kirk-type”, Fasc. Math., 43 (2010) 101-114.

      [17] M. O. Olatinwo, “Some stability results for two hybrid fixed point iterative algorithms in normed linear space”, Math. Vesnik, 61 (4) (2009) 247-256.

      [18] M. O. Olatinwo, “Some stability results in complete metric space”, Acta Univ. Palacki. Olomuc., Fac. rer. nat. Mathematica, 48 (2009) 83-92.

      [19] M. O. Osilike, “Some stability results for fixed point iteration procedures”, J. Nigerian Math. Soc., 14/15 (1995) 17-29.

      [20] M. O. Osilike, “Stability of the Mann and Ishikawa iteration procedures for $phi$-strong pseudocontractions and nonlinear equations of the $phi$-strongly accretive type”, J. Math. Anal. Appl. 227 (2) (1998) 319-334.

      [21] A. M. Ostrowski, “The round-off stability of iterations”, Z. Angew. Math. Mech., 47 (1967) 77-81.

      [22] B. E. Rhoades, S. M. Soltuz, “The equivalence between Mann-Ishikawa iteration and multistep iteration”, Nonlinear Anal., 58 (2004) 219-228.

      [23] S. L. Singh, C. Bhatnagar and S. N. Mishra, “Stability of Jungck-Type iterative procedures”, International Journal of Mathematics and Mathematical Sciences, 19 (2005) 3035-3043.

      [24] T. Zamfirescu, “Fix point in metric spaces”, Archiv der Mathematik, 23 (1)(1972) 292-298.


 

View

Download

Article ID: 3892
 
DOI: 10.14419/ijamr.v4i1.3892




Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.