New Modified Implicit Iterative Algorithm for Finite Families of Two Total Asymptotically Pseudocntractive Mappings

Document Type : Original Article

Authors
Austine Ofem 1 ΒΆ
DONATUS IKECHI IGBOKWE 2
1 Department of Mathematics, University of Uyo, Uyo Nigeria.
2 Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria
10.22034/maco.2.2.4
Abstract
In this article, we proposed a modified implicit iterative algorithm for approximation of common fixed point of finite families of two uniformly L-Lipschitzian total asymptotically pseudocontractive mappings in Banach spaces. Our new iterative algorithm contains some well known iterative algorithm which has been used by several authors for approximating fixed
points of different classes of mappings. We prove some convergence theorems of our new iterative method and validate our main result with a numerical ex-
ample. Our result is an improvement and generalization of the results of some well known authors in the existing literature.

Keywords

[1] G. L. Acedo and H.K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 67, 2258–2271, 2007.
[2] C. E. Chidume and N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal., TMA, 62(6), 1149–1156, 2005.
[3] E. E. Chima and M. O. Osilike, Split Common Fixed Point Problem for a Class of Total Asymptotic Pseudocontractions, Journal of Applied Mathematics Volume 2016, Article ID 3435078, 10 pages.
[4] C. Ding and J. Quan, A Strong Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces, Abstract and Applied Analysis, Vol. 2012, Article ID 127851, 8 pages, 2012.
[5] G. Marino and H. K. Xu, Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces, J. Math. Anal. Appl., 329, 336–346, 2007.
[6] A. E. Ofem, Strong convergence of modified implicit hybrid S-iteration scheme for finite family of nonexpansive and asymptotically generalized Φ-hemicontractive mappings, Malaya Journal of Matematik, 8(4), 1643–1649, 2020
[7] A. E. Ofem, Strong convergence of a multi-step implicit iterative scheme with errors for common fixed points of uniformly L-Lipschitzian total asymptotically strict pseudocontractive mappings, Results in Nonlinear Analysis, 3(2), 100–116, 2020.
[8] A. E. Ofem, D. I. Igbokwe, X. A. Udo-utun, Implicit iteration process for Lipschitzian α-hemicontraction semigroup, MathLAB Journal, 7, 43–52, 2020.
[9] M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl., 294(1), 73–81, 2004.
[10] M.O. Osilike, B.G Akuchu, Common fixed points of finite family of asymptotically pseudocontractive maps, Fixed Point Theory Appl., 2004, 81–88, 2004.
[11] R.D. Chen, Y.S. Song, H. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl., 314, 701–706, 2006.
[12] X. Qin, S. Y. Cho, and J. K. Kim, Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense, Fixed Point Theory and Applications, vol. 2010, Article ID 186874, 14 pages, 2010.
[13] X. Qin, S. Y. Cho, and S. M. Kang, Weak Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces, Fixed Point Theory and Applications, Vol. 2011, Article ID 859795, 11 pages, 2011.
[14] G. S. Saluja, Convergence of the explicit iteration method for strictly asymptotically pseudocontractive mappings in the intermediate sense, Novi Sad J. Math., 44(1), 75-90, 2014.
[15] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, Journal of Mathematical Analysis and Applications, 158(2), 407–413, 1991.
[16] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bulletin of the Australian Mathematical Society, vol. 43(1), 153–159, 1991.
[17] Z. H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasinonexpansive mappings, J. Math. Anal. Appl., 286, 351–358, 2003.
[18] B. S. Thakur, R. Dewangan and M. Postolache, general composite implicit iteration process for a finite family of asymptotically pseudo-contractive mappings, Thakur et al. Fixed Point Theory and Applications 2014, 2014:90.
[19] B. E. Rhoades, Comments on two fixed point iteration methods, Journal of Mathematical Analysis and Applications, 56(3), 741–750, 1976.
[20] Y. Wang and C. Wang, Convergence of a new modified Ishikawa type iteration for common fixed Points of total asymptotically strict pseudocontractive semigroups, Volume 2013, Article ID 319241, 7 pages.
[21] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16, 1127–1138, 1991.
[22] H.K. Xu and R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22, 767–773, 2001.
[23] Y. Zhou and S. S. Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim., 23, 911–921, 2002.
Mathematical Analysis and Convex Optimization
Volume 2, Issue 2
Autumn 2021
Pages 31-44