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We introduce the Jungck-multistep iteration and show that it converges strongly to the unique common fixed point of a pair of weakly compatible generalized contractive-like opera- tors defined on a Banach space. As corollaries, the results show that the Jungck-Mann, Jungck-Ishikawa and Jungck-Noor itera- tions can also be used to approximate the common fixed points of such maps. The results are improvements, generalizations and extensions of the work of Olatinwo and Imoru [13], Olatinwo [14-15]. Consequently, several results in literature are generalized.
Czasopismo
Rocznik
Tom
Strony
87--98
Opis fizyczny
Bibliogr. 24 poz.
Twórcy
Bibliografia
- [1] Abbas M., Jungck G., Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341(2008), 416-420.
- [2] Berinde V., On the convergence of Ishikawa iteration in the class of quasicontractive operators, Acta Math. Univ. Comenianae, LXXIII(1)(2004), 119-126.
- [3] Chatterjea S.K., Fixed point theorems, Comptes rendus de l'Academic bulgare des Sciences, 25(6)(1972), 727-730.
- [4] Das K.M., Naik K.V., Common fixed point theorems for commuting maps on metric spaces, Proc. Amer. Math. Soc., 77(1979), 369-373.
- [5] Ishikawa S., Fixed points by a new iteration method, Proc. Amer. Math. Soc., 149(1974), 147-150.
- [6] Jungck G., Commuting mappings and fixed points, Amer. Math. Monthly, 83(1976), 261-263.
- [7] Kannan R., Some results on fixed points II, Amer. Math. Monthly, 76(1969), 405-408.
- [8] Mann W.R., Mean value methods in iteration, Proc. Amer. Math. Soc., 4(1953), 506-510.
- [9] Noor M.A., New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and Applications, 251(1)(2004), 217-229.
- [10] Olaleru J.O., On the convergence of the Mann iteration in locally convex spaces, Carpathian Journal of Mathematics, 22(1-2)(2006), 115-120.
- [11] Olaleru J.O., On the equivalence of Picard, Mann and Ishikawa iterations for a class of quasi-contractive operators, J. Nig. Assoc. Math. Phys., 11(2007), 51-56.
- [12] Olaleru J.O., A new approximation method for the common fixed points of two weakly compatible mappings, (submitted).
- [13] Olatinwo M.O., Imoru C.O., Some convergence results for the Jungck Mann and Jungck-Ishikawa iteration process in the class of generalized Zamfirescu operators, Acta Math. Univ. Comenianae, 77(2)(2008), 299-304.
- [14] Olatinwo M.O., Some stability and strong convergence results for the Jungck-Ishikawa iteration process, Creative Math. and Info., 17(2008), 33-42.
- [15] Olatinwo M.O., A generalization of some convergence results using the Jungck-Noor three step iteration process in arbitrary Banach space, Fasc. Math., 40(2008), 37-43.
- [16] Osilike M.O., Stability results for Ishikawa fixed point iteration procedure, Indian J. Pure Appl. Math., 26(10)(1995), 937-941.
- [17] Rafiq A., On the equivalence of Mann and Ishikawa iteration methods with errors, Math. Comm., 11(2006), 143-152.
- [18] Rhoades B.E., Comments on two fixed points iteration methods, J. Math. Anal.Appl., 56(2)(1976), 741-750.
- [19] Rhoades B.E., Soltuz S.M., The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Anal., 58(2004), 219-228.
- [20] Popescu O., Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Math. Comm., 12(2007), 195-202.
- [21] Singh S.L., On Common fixed points of commuting maps, Math. Sem. Notes Kobe Univ., 5(1977), 131-134.
- [22] Singh S.L., Bhatnagar C., Mishra S.N., Stability of Jungck-type iterative procedures, International J. Math. and Math. Sc., 19(2005), 3035-3043.
- [23] Zhiqun X., Remarks of equivalence among Picard, Mann and Ishikawa iterations in normed space, Fixed Point Theory and Applications, Vol. 2007, Article ID 61434, 5 pp.
- [24] Zamfirescu T., Fixed point theorems in metric spaces, Arch. Math. (Basel), 23(1972), 292-298.
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Bibliografia
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bwmeta1.element.baztech-article-BPP3-0003-0077