Strong convergence results for random Jungck­Ishikawa and Jungck­Noor iterative schemes in Banach spaces

Document Type: Original Article

Authors

1 Department of Mathematics, Faculty of Science, Assuit University, Assuit 71516, Egypt.

2 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt.

Abstract

In this paper, we introduce a new random Jungck-Ishikawa and Jungck-Noor iterative schemes and
discuss the strong convergence of them to a unique common random fixed point for two nonself random
mappings under a general contractive condition in separable Banach spaces. Our results generalize and
extend many results in this direction.

Keywords


[1] A. Alotaibi, V. Kumar and N. Hussain, Convergence comparison and stability of Jungck-Kirk-type algorithms for
common fixed point problems, Fixed Point Theory Appl., 173 (2013), 1-30.

[2] A. T. Bharucha-Reid, Random Integral Equations, Academic Press, New York, (1972).

[3] A. T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc., 82 (1976), 641-657.

[4] A. O. Bosede, On the stability of Jungck-Mann, Jungck-Krasnoselskij and Jungck iteration processes in arbitrary
Banach spaces, Acta Univ. Palacki. Olomuc. Fac. Rer. Nat. Mathematica, 50 (2011), 17-22.

[5] R. S. Chandelkar, A. Pariya, V. H. Badshah and V. K. Gupta, A Common random fixed point theorem using
random Jungck Mann type iteration on Banach spaces, Int. J. Advanced Computer Math. Sci., 6 (2015), 1-5.

[6] B. S. Choudhury, Convergence of a random iteration scheme to a random fixed point, J. Appl. Math. Stochastic
Anal., 8 (1995), 139-142.

[7] B. S. Choudhury, Random Mann iteration scheme, Appl. Math. Lett., 16 (2003), 93-96.

[8] B. S. Choudhury, A random fixed point iteration for three random operators on uniformly convex Banach spaces,
Anal. Theory Appl., 19 (2003), 99-107.

[9] B. S. Choudhury, An iteration for fi nding a common random fixed point, J. Appl. Math. Stochastic Anal., 4
(2004), 385-394.

[10] B. S. Choudhury and M. Ray, Convergence of an iteration leading to a solution of a random operator equation,
J. Appl. Math. Stoch. Anal., 12 (1999), 161-168.

[11] B. S. Choudhury and A. Upadhyay, An iteration leading to random solutions and fixed points of operators, Soochow
J. Math., 25 (1999), 395-400.

[12] C. J. Himmelberg, Measurable relations, Fund Math., 87 (1975), 53-71.

[13] T. Ioana, Stability of Jungck-type iterative procedure for some contractive type mappings via implicit relations,
Miskolc Math. Notes, 13 (2012), 555-567.

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

[15] P. K. Jhade and A. S. Saluja, Strong convergence theorems by modifi ed four step iterative scheme with errors for
three nonexpansive mappings, Kyungpook Math. J., 55 (2015), 667-678.

[16] S. H. Khan, Common fixed points by a generalized iteration scheme with errors, Surveys Math. Appl., 6 (2011),
117-126.

[17] T. H. Kim and H. K. Xu, Strong convergence of modi fied Mann iterations, Nonlinear Anal., 61 (2005), 51-60.

[18] T. C. Lin, Random approximation and random fixed point theorems for non-self mappings, Proc. Amer. Math.
Soc., 103 (1988), 1129-1135.

[19] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.

[20] M. A. Noor, Three-step iterative algorithms for multivalued quasi-variational inclusion, J. Math. Anal. Appl.,
255 (2001), 589-604.

[21] M. O. Olatinwo, A Generalization of some convergence results using a Jungck-Noor three-step iteration process
in arbitrary Banach space, Fasciculi Math., 40 (2008), 37-43.

[22] M. O. Olatinwo, Convergence results for Jungck-type iterative process in convex metric spaces, Acta Univ. Palacki
Olomuc, Fac. rer. nat. Math., 51 (2012), 79-87. 

[23] M. O. Olatinwo and C. O. Imoru, Some convergence results for the Jungck-Mann and the Jungck-Ishikawa
iteration processes in the class of generalized Zam rescu operators, Acta Math. Univ. Comenianae, 77 (2008),
299-304.

[24] R. A. Rashwan and H. A. Hammad, Convergence and almost sure (S,T)-stability for random iterative schemes,
Int. J. Advances Math., 2016, (2016), 1-16. 

[25] R. A. Rashwan and H. A. Hammad, Convergence and stability of modifi ed random SP-iteration for a generalized asymptotically quasi-nonexpansive mapping, Math. Interdisciplinary Research, 2 (2017), 9-21.

[26] R. A. Rashwan, A common fixed point theorem of two random operators using random Ishikawa iteration scheme,
Bull. Int. Math. Virtual Inst., 1 (2011), 45-51.

[27] B. E. Rhoades, Iteration to obtain random solutions and fixed points of operators in uniformly convex Banach
spaces, Soochow J. Math., 27 (2001), 401-404.

[28] S. L. Singh, C. Bhatnagar, and S. N. Mishra, Stability of Jungck-type iterative procedures, Int. J. Math. Math.
Sci., 19 (2005), 3035-3043.