Approximation of the Jungck Multistep Iteration Process Generated by a Generalised Contractive-like Operator in Cone Banach Spaces
Keywords:
cone Banach space, contractive-like operators, convergence, weakly compatible maps
Abstract
In this paper, we investigate an approximation theorem on the common fixed point of the Jungck multistep iteration process in a cone Banach space. The results obtained in this paper are improvements and generalisations of several results on common fixed points in the Literature.
References
Abbas M., Jungck G. (2008) Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematics Analysis and Applications, 341(1): 416420.
Arshad M., Azam A., Vetro P. (2009) Some common fixed point results in cone metric spaces. Fixed Point Theory and Applications, Vol 2009, Article ID 493965, 11 pages.
Berinde V. (2004) On the convergence of the Ishikawa iteration in the class of quasi-contractive operators, Acta Mathematica Universitatis Comenianae, 73(1): 119126.
Chatterjea S. K. (1972) Fixed point theorems, Comptes Rendus de V Academic Bulgare des Sciences, 25(6): 727730.
Djoudi A., Aliouche A. (2007) Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type, Journal of Mathematical Analysis and Applications, 329(1): 3145.
Ishikawa S. (1974) Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44(1): 147150.
Jungck G. (1976) Commuting mappings and fixed points. The American Mathematical Monthly, 83(4): 261263.
Jungck G., Radenovic S., Rakocevic S., Rakocevic V. (2009) Common fixed point theorems of weakly compatible pairs on cone metric spaces, Fixed Point Theory and Applications, Vol 59, Article ID 643840, 13 pages.
Huang G., Zhang X. (2007) Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332: 14681476.
Kannan R. (1968) Some results on fixed points, Bulletin Calcutta Mathematical Society, 10: 7176.
Mann W. R. (1953) Mean Value methods in iterations, Proceedings of the American Mathematical Society. 44: 506510.
Karapinar E. ( 2009) Fixed point theorems in cone Banach spaces, Fixed Point Theory and Applications, Vol 2009, Article ID 609281, 9 pages.
Karapinar E., Turkoglu D. (2010) Best Approximations Theroem for a Couple in Cone Banach Spaces, Fixed Point Theory and Applications, Vol 2010, Article ID 784578, 9 pages.
Noor M. A. (2000) New approximation schemes for general variational inequalities. Journal of Mathematical Analysis and Appliacations, 251: 217229.
Olaleru J. O. (2011) Common fixed points of three self-mappings in cone metric space. Applied Mathematics E-notes, 11: 4149.
Olaleru J. O., Akewe H. (2010) On multistep iterative scheme for approximating the common fixed points of generalized contractive-like operations, International Journal of Mathematics and Mathematical Sciences, Vol 2010, Article ID 530964, 11 pages.
Olatinwo M. O., Imoru C. O. (2008) Some convergence results of the Jungck-Mann and Jungck-Ishikawa iterations processes in the class of generalized Zamfirescu operators, Acta Mathematica Universitatis Comenianae, 77(2): 229304.
Olatinwo M. O. (2008) A generalization of some convergence results using the Jungck-Noor three step iteration process in arbitrary Banach space, Fasciculli Mathematici, 40: 3743.
Osilike M. O. (1999) Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings, Indian Journal of Pure and Applied Mathematics, 30(12): 12291234.
Sabetghadam F., Masiha H. P., Sanatpour A. H. (2009) Some coupled fixed point theorems in cone metric spaces, Fixed Point Theory and Applications, Volume 2009, Article ID 125426, 8 pages.
Rhoades B. E. (1977) A comparison of various definitions of contractive mappings, Transactions of the American Mathematical Society, 226: 225290.
Singh S. L., Bhatnagar C., Mishra S. N. (2005) Stability of Jungck-type iteration procedures, International Journal of Mathematics and Mathematical Sciences, 19: 30353043.
Vetro P. (2007) Common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo, 56(3): 416420.
Zamfirescu T. (1972) Fixed point theorem in metric spaces, Archiv der Mathenatik (Basel), 23: 292298.
Arshad M., Azam A., Vetro P. (2009) Some common fixed point results in cone metric spaces. Fixed Point Theory and Applications, Vol 2009, Article ID 493965, 11 pages.
Berinde V. (2004) On the convergence of the Ishikawa iteration in the class of quasi-contractive operators, Acta Mathematica Universitatis Comenianae, 73(1): 119126.
Chatterjea S. K. (1972) Fixed point theorems, Comptes Rendus de V Academic Bulgare des Sciences, 25(6): 727730.
Djoudi A., Aliouche A. (2007) Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type, Journal of Mathematical Analysis and Applications, 329(1): 3145.
Ishikawa S. (1974) Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44(1): 147150.
Jungck G. (1976) Commuting mappings and fixed points. The American Mathematical Monthly, 83(4): 261263.
Jungck G., Radenovic S., Rakocevic S., Rakocevic V. (2009) Common fixed point theorems of weakly compatible pairs on cone metric spaces, Fixed Point Theory and Applications, Vol 59, Article ID 643840, 13 pages.
Huang G., Zhang X. (2007) Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332: 14681476.
Kannan R. (1968) Some results on fixed points, Bulletin Calcutta Mathematical Society, 10: 7176.
Mann W. R. (1953) Mean Value methods in iterations, Proceedings of the American Mathematical Society. 44: 506510.
Karapinar E. ( 2009) Fixed point theorems in cone Banach spaces, Fixed Point Theory and Applications, Vol 2009, Article ID 609281, 9 pages.
Karapinar E., Turkoglu D. (2010) Best Approximations Theroem for a Couple in Cone Banach Spaces, Fixed Point Theory and Applications, Vol 2010, Article ID 784578, 9 pages.
Noor M. A. (2000) New approximation schemes for general variational inequalities. Journal of Mathematical Analysis and Appliacations, 251: 217229.
Olaleru J. O. (2011) Common fixed points of three self-mappings in cone metric space. Applied Mathematics E-notes, 11: 4149.
Olaleru J. O., Akewe H. (2010) On multistep iterative scheme for approximating the common fixed points of generalized contractive-like operations, International Journal of Mathematics and Mathematical Sciences, Vol 2010, Article ID 530964, 11 pages.
Olatinwo M. O., Imoru C. O. (2008) Some convergence results of the Jungck-Mann and Jungck-Ishikawa iterations processes in the class of generalized Zamfirescu operators, Acta Mathematica Universitatis Comenianae, 77(2): 229304.
Olatinwo M. O. (2008) A generalization of some convergence results using the Jungck-Noor three step iteration process in arbitrary Banach space, Fasciculli Mathematici, 40: 3743.
Osilike M. O. (1999) Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings, Indian Journal of Pure and Applied Mathematics, 30(12): 12291234.
Sabetghadam F., Masiha H. P., Sanatpour A. H. (2009) Some coupled fixed point theorems in cone metric spaces, Fixed Point Theory and Applications, Volume 2009, Article ID 125426, 8 pages.
Rhoades B. E. (1977) A comparison of various definitions of contractive mappings, Transactions of the American Mathematical Society, 226: 225290.
Singh S. L., Bhatnagar C., Mishra S. N. (2005) Stability of Jungck-type iteration procedures, International Journal of Mathematics and Mathematical Sciences, 19: 30353043.
Vetro P. (2007) Common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo, 56(3): 416420.
Zamfirescu T. (1972) Fixed point theorem in metric spaces, Archiv der Mathenatik (Basel), 23: 292298.
Published
2018-07-29
How to Cite
Akewe, H., & Olaluwa, H. (2018). Approximation of the Jungck Multistep Iteration Process Generated by a Generalised Contractive-like Operator in Cone Banach Spaces. Journal of Scientific Research and Development, 17(1), 73-79. Retrieved from http://jsrd.unilag.edu.ng/article/view/45
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