In this paper we establish convergence and stability results using general contractive condition, quasi-nonexpansive mapping and mean non expansive mapping for K-iteration process. We shall also generalize the K-iteration process for a pair of distinct mappings and with the help of example we claim that the generalized iteration process has better convergence rate than the K-iteration process for single mapping and some of the existing iteration processes. Suitable examples are given in the support of main results.
References
[1]
Mann, W.R. (1953) Mean Value Methods in Iteration. Proceedings of the American Mathematical Society, 4, 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
[2]
Khan, S.H. (2013) A Picard-Mann Hybrid Iterative Process. Fixed Point Theory and Applications, 2013, 69. https://doi.org/10.1186/1687-1812-2013-69
[3]
Ishikawa, S. (1974) Fixed Points by a New Iteration Method. Proceedings of the American Mathematical Society, 44, 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
[4]
Gursoy, F. and Karakaya, V. (2014) A Picard-S Hybrid Type Iteration Method for Solving a Differential Equation with Retared Arguments. 1-16.
[5]
Karakaya, V., Bouzara, N.E.H., Dogan, K. and Atalan, Y. (2015) On Different Results for a New Two Step Iteration Method under Weak Contraction Mapping in Banach Spaces. 1-10. arXiv:1507.00200v1
[6]
Thakur, B.S., Thakur, D. and Postolache, M. (2016) A New Iterative Scheme for Numerical Reckoning Fixed Points of Suzuki’s Generalized Non-Expansive Mappings. Applied Mathematics and Computation, 275, 147-155. https://doi.org/10.1016/j.amc.2015.11.065
[7]
Opial, Z. (1967) Weak Convergence of the Sequence of Successive Approximations for Non-Expansive Mappings. Bulletin of the American Mathematical Society, 73, 595-597. https://doi.org/10.1090/S0002-9904-1967-11761-0
[8]
Hussain, N., Ullah, K. and Arshad, M. (2018) Fixed Point Approximation of Suzuki Generalized Non-Expansive Mapping via New Faster Iterative Process. arxiv 1802.09888v
[9]
Zhang, S.S. (1975) About Fixed Point Theory for Mean Non Expansive Mappings in Banach Spaces. Journal of Sichuan University, 2, 67-78.
[10]
Harder, A.M. (1987) Fixed Point Theory and Stability Results for Fixed Point Iteration Procedure. University of Missouri-Rolla, Missouri.
[11]
Weng, X. (1991) Fixed Point Iteration for Local Strictly Pseudo-Contractive Mapping. Proceedings of the American Mathematical Society, 113, 727-731. https://doi.org/10.1090/S0002-9939-1991-1086345-8
[12]
Berinde, V. (2002) On the Stability of Some Fixed Point Procedures. Buletinul ？tiin？ific al Univer-sitatii Baia Mare, Seria B, Fascicola matematic？-informatic？, 18, 7-14.
[13]
Sahu, J. (1991) Weak and Strong Convergence to Fixed Points of Asymptotically Non-Expansive Mappings. Bulletin of the Australian Mathematical Society, 43, 153-159. https://doi.org/10.1017/S0004972700028884
[14]
Akewe, H. and Okeke, G.A. (2015) Convergence and Stability Theorems for the Picard-Mann Hybrid Iterative Scheme for a General Class of Contractive-Like Operators. Fixed Point Theory and Applications, 2015, 66.
[15]
Chidume, C.E. (2014) Strong Convergence and Stability of Picard Iteration Sequence for General Class of Contractive-Type Mappings. Fixed Point Theory and Applications, 2014, 233. https://doi.org/10.1186/1687-1812-2014-233
[16]
Bosede, A.O. and Rhoades, B.E. (2010) Stability of Picard and Mann Iteration for a General Class of Functions. Journal of Advanced Mathematical Studies, 3, 23.
