Document Type : Research Paper

Authors

1 Department of Mathematics, University of Uyo, Uyo, Nigeria

2 Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria

Abstract

It is our purpose in this paper to introduce the concept of $\alpha$-demicontractive semigroup. Also, we construct a new implicit iterative scheme for approximating the common fixed points of $\alpha$-demicontractive semigroup. We prove strong convergence of our new iterative scheme to the common fixed points of $\alpha$-demicontractive semigroup in Banach spaces. Our result is an improvement and generalization of several well known results in the existing literature.

Keywords

Main Subjects

###### ##### References
[1] A. Aleyner, S. Reich, An explicit construction of sunny nonexpansive retractions in Banach spaces, Fixed Point Theory Appl. 3 (2005), 295-305.
[2] S. S. Chang, C. K. Chan, H. W. Joseph Lee, L. Yang, A system of mixed equilibrium problems, fixed point problems of strictly pseudocontractive mappings and nonexpansive semigroups, Appl. Math. Comput. 216 (1) (2010), 51-60.
[3] S. S. Chang, Y. J. Cho, H. W. Joseph Lee, C. K. Chan, Strong convergence theorems for Lipschitzian
demicontraction semigroups in Banach spaces, Fixed Point Theory Appl. 2011, 2011:583423.
[4] G. E. Kim, Approximating common fixed points of Lipschitzian pseudocontraction semigroups, J. Fixed Point Theory Appl. 18 (2016), 927-934.
[5] S. Li, L. H. Li, F. Su, General iteration methods for a one-parameter nonexpansive semigroups in Hilbert spaces, Nonlinear Anal. 70 (9) (2009), 3065-3071.
[6] L. Maruster, S. Maruster, Strong convergence of the Mann iteration for α-demicontractive mappings, Math. Comput. Modelling. 54 (2011), 2486-2492.
[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 Anal. 3 (2) (2020), 100-116.
[8] A. E. Ofem, Strong convergence of modified implicit hybrid S-iteration scheme for finite family of nonexpansive and asymptotically generalized Φ-hemicontractive mappings, Malaya J. Matematik. 8 (4) (2020), 1643-1649.
[9] A. E. Ofem, D. I. Igbokwe, X. A. Udo-utun, Implicit iteration process for Lipschitzian α-hemicontraction semigroups, MathLAB J. 7 (2020), 43-52.
[10] A. E. Ofem, U. E. Udofia, Iterative solutions for common fixed points of nonexpansive mappings and strogly mappings and strongly pesudocontractive mappings with applications, Canad. J. Appl. Math. 3 (1) (2021), 18-36.
[11] M. O. Osilike, A. C. Onah, Strong convergence of the Ishikawa Iteration for Lipschitz Hemicontractive
mappings, Annals of West University of Timisoara (Seria Mathematica Informatica). LIII 1 (2015), 151-161.
[12] J. Quan, S. Chang, M. Liu, Strong and weak convergence of an implicit iterative process for pseudocontractive semigroups in Banach space, Fixed Point Theory Appl. 2012, 2012:16.
[13] N. Shioji, W. Takahashi, Strong convergence theorems for asymptotically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 34 (1) (1998), 87-99.
[14] T. Suzuki, On strong convergence to a common fixed point of nonexpansive semigroups in Hilbert spaces, Proc. Am. Math. Soc. 131 (7) (2003), 2133-2136.
[15] T. Suzuki, Fixed point property for nonexpansive mappings versus that for nonexpansive semigroups, Nonlinear Anal. 70 (2009), 3358-3361.
[16] D. V. Thong, An implicit iteration process for nonexpansive semigroups, Nonlinear Anal. 74 (2011), 6116-6120.
[17] D. V. Thong, On Mann type implicit iteration process for strictly pseudocontraction semigroups, Annals of the University of Craiova, Math. Comput Sci. Ser. 38 (3) (2011), 101-108.
[18] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (12) (1991), 1127-1138.
[19] H. K. Xu, Strong convergence theorem for contraction semigroups in Banach spaces, Bull. Aust. Math. Soc. 72 (3) (2005), 371-379.
[20] S. S. Zhang, Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semi-groups in Banach spaces, Appl. Math. Mech. 30 (2009), 145-152.
[21] S. S. Zhang, Weak convergence theorem for Lipschitzian pseudocontraction semigroups in Banach spaces, Acta. Math. Sinica. 26 (2010), 337-344.
[22] S. S. Zhang, L. Yang, J. A. Liu, Strong convergence theorem for nonexpansive semigroups in Banach spaces, Appl. Math. Mech. 28 (10) (2007), 1287-1297.