Obtaining New Fixed Point Theorems Using Generalized Banach-Contraction Principle

Yıl 2020, Cilt: 35 Sayı: 3, 34 - 45, 04.01.2020

Yunus Atalan

Öz

In this study, a new three step iterative algorithm was introduced with the help of Jungck-contraction principle which is one of the remerkable generalizations of Banach-contraction principle. Also, the convergence and stability results were obtained for the pair of nonself mappings which satisfy a certain contractive condition by using this iterative algorithm in any Banach space. In addition, it was shown that the new iterative algorithm has a better convergence speed when compared the other Jungck-type iterative algorithms in the current literature, and to support this result, numerical examples were given.

Anahtar Kelimeler

Jungck-contraction principle, New iterative algorithm, Convergence, Stability

Kaynakça

• [1] Banach, S. 1922. Sur les operations dans les ensembles abstraites et leurs applications aux equations integrales. Fund. Math., 3, 133-181.
• [2] Ciric, L. J. 1974. A generalization of Banach’s contraction principle. Proceedings of American Mathematical Society, 45, 267-273.
• [3] Edelstein, M. 1961. An extension of Banach contraction principle. Proceedings of American Mathematical Society, 1, 7-10.
• [4] Jungck, G. 1976. Commuting mappings and fixed points. American Mathematical Monthly, 83(4), 261-263.
• [5] Presic, S. B. 1965. Sur Une Classe D’inequations Aux Differences Finite Et. Sur La Convergence De Certaines Suites. Publications de l'Institut Mathématique, 5(25), 75-78.
• [6] Mann, W.R. 1953. Mean value methods in iteration. Proceedings of the American Mathematical Society, 4, 506–510.
• [7] Ishikawa, S. 1974. Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44(1), 147–150.
• [8] Noor, M.A. 2000. New approximation schemes for general variational inequalities. Journal of Mathematical Analysis and Applications, 251(1), 217–229.
• [9] Atalan, Y. 2019. On Numerical Approach to The Rate of Convergence and Data Dependence Results for a New Iterative Scheme. Konuralp Journal of Mathematics, 7(1), 97-106.
• [10] Khan, A. R., Gürsoy, F., and Karakaya, V., 2016. Jungck-Khan iterative scheme and higher convergence rate. International Journal of Computer Mathematics, 93(12), 1029-2105.
• [11] Singh, S.L., Bhatnagar, C. Mishra, S.N. 2005. Stability of Jungck-type iterative procedures. International Journal of Mathematics and Mathematical Sciences, 19, 3035–3043.
• [12] Olatinwo, M.O. 2008. Some stability and strong convergence results for the Jungck-Ishikawa iteration process. Creative Mathematics and Informatics, 17, 33–42.
• [13] Olatinwo, M.O. 2008. A generalization of some convergence results using a Jungck-Noor three-step iteration process in arbitrary Banach space. Polytechnica Posnaniensis, 40, 37–43.
• [14] Chugh, R., Kumar, V. 2011. Strong Convergence and Stability results for Jungck-SP iterative scheme. International Journal of Computer Applications, 36(12), 40-46.
• [15] Hussain, N., Kumar, V., Kutbi, M. A. 2013. On rate of convergence of Jungck-type iterative schemes, Abstract and Applied Analysis, 2013, 1-15.
• [16] Khan, A. R., Kumar, V., Hussain, N. 2014. Analytical and numerical treatment of Jungck-type iterative schemes. Applied Mathematics and Computation, 231, 521-535.
• [17] Phuengrattana W., Suantai, S. 2011. On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. Journal of Computational and Applied Mathematics, 235(9), 3006–3014.
• [18] Chugh, R., Kumar, V., Kumar, S. 2012. Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces. American Journal of Computational Mathematics, 2, 345-357.
• [19] Chugh, R., Kumar, S. 2013. On the Stability and Strong Convergence for Jungck-Agarwal et al. Iteration Procedure International Journal of Computer Applications, 64(7), 39-44.
• [20] Jungck, G., Hussain, N. 2007. Compatible maps and invariant approximations. J. Math. Anal. Appl., 325 (2), 1003-1012.
• [21] Berinde, V. 2007. Iterative Approximation of Fixed Points, Springer, Berlin.
• [22] Berinde, V. 2004. Picard iteration converges faster than the Mann iteration in the class of quasi-contractive operators. Fixed Point Theory Appl., 2, 97-105.