Implicit Iteration Process for Lipschitzian α-Hemicontraction Semigroups

Authors

  • Dr. A. E. Ofem Department of Mathematics, University of Uyo, U yo, Nigeria. https://orcid.org/0000-0001-8064-2326
  • Prof D. I. Igbokwe Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria.
  • Dr. X. A. Udo-Utun

Keywords:

Banach space, Fixed point, Implicit iteration process, Strong convergence, Weak convergence, $\alpha$-demicontraction semigroup, $\alpha$-hemicontraction semigroup, Normal duality mapping

Abstract

In the paper, the concept of α-demicontractive semigroup and α-hemicontractive semigroup is introduced. We study the strong and weak convergence of an implicit iterative scheme to the common fixed points of Lipschitzian α-hemicontractive semigroup. The result presented in this paper extend, generalize, improve and unify the results of several well known authors.
in the literature.

 

Downloads

Download data is not yet available.

Author Biography

Dr. A. E. Ofem, Department of Mathematics, University of Uyo, U yo, Nigeria.

Department of Mathematics, University of U yo, Uyo, Nigeria.

Researcher.

References

A. Aleyner and S. Reich, An explicit construction of sunny nonexpansive retractions in Banach spaces, Fixed Point Theory Appl., 3 (2005), 295-305.

S. S. Chang, C. K. Chan, H. W. Joseph Lee and L. Yang, A system of mixed equilibrium problems, fi xed point problems of strictly pseudocontractive mappings and nonexpansive semigroups. Appl Math Comput. 216(1)(2010),51{60. doi:10.1016/j.amc.2009.12.060.

S. S. Chang, Y. J. Cho, H. W. Joseph Lee and C. K. Chan, Strong convergence theorems

for Lipschitzian demicontraction semigroups in Banach spaces. Fixed Point Theory Appl

(2011). doi:10.1155/2011/583423.

S. Li, L. H. Li and F. Su, General iteration methods for a one-parameter nonexpansive semigroups in Hilbert spaces. NonlinearAnal. 70(9)(2009), 3065-3071. doi:10.1016/j.na.2008.04.007.

G. E. Kim, Approximating common fixed points of Lipschitzian pseudocontraction semigroups, J. Fixed Point Theory Appl., DOI 10.1007/s11784-016-0299-7.

L. Maruster and S. Maruster, Strong convergence of the Mann iteration for α-demicontractive mappings, Mathematical and Computer Modelling 54 (2011), 2486-2492.

M. O. Osilike and A. C. Onah, Strong convergence of the Ishikawa Iteration for Lipschitz

Hemicontractive mappings, Seria Mathematica Informatica, 1 (2015) LIII.

J. Quan, S. Chang and M. Liu, Strong and weak convergence of an implicit iterative process

for pseudocontractive semigroups in Banach space, Fixed Point Theory and Applications

, 2012:16 http://www. xedpointtheoryandapplications.com/content/2012.

N. Shioji and W. Takahashi, Strong convergence theorems for asymptotically nonexpan-

sive mappings in Hilbert spaces, Nonlinear Anal., 34(1)(1998),87{99. doi:10.1016/S0362-546X(97)00682-2.

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. doi:10.1090/S0002-9939-

-06844-2.

T. Suzuki, Fixed point property for nonexpansive mappings versus that for nonexpansive

semigroups. Nonlinear Anal., 70(2009), 3358-3361. doi:10.1016/j.na.2008.05.003.

D. V. Thong, An implicit iteration process for nonexpansive semigroups, Nonlinear Analysis, 74 (2011) 6116-6120.

D. V. Thong, On Mann type implicit iteration process for strictly pseudocontraction semi-

groups, Annals of the University of Craiova, Mathematics and Computer Science Series, 38(3)(2011), 101-108.

H. K. Xu, Strong convergence theorem for contraction semigroups in Banach spaces. Bull.

Aust. Math. Soc. 72(3)(2005), 371{379. doi:10.1017/S000497270003519X.

Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12)(1991),

-1138. doi:10.1016/0362{546X(91)90200-K.

Yang and Zhao Fixed Point Theory and Applications 2012, 2012:24

http://www. xedpointtheoryandapplications.com/content/2012/1/24 Page 9 of 10.

H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16(1991), 1127-

S. S. Zhang, L. Yang, H. W. Joseph Lee, C. K. Chan, Strong convergence theorem for

nonexpansive semigroups in Hilbert spaces. Acta Math Sinica. 52(2)(2009), 337-342.

S. S. Zhang, L. Yang and J. A. Liu, Strong convergence theorem for nonexpansive semi-

groups in Banach spaces. Appl Math. Mech. 28(10)(2007), 1287{1297. doi:10.1007/s10483-007-1002-x.

Zhang, SS: Convergence theorem of common fixed points for Lipshitzian pseudocontraction

semigroups in Banach spaces. Appl Math Mech. 30(2009), 145-152.

S. S. Zhang, Convergence theorem of common FIxed points for Lipschitzian pseudocontraction semi-groups in Banach spaces, Appl. Math. Mech. -Engl. Ed. 30 (2009), 145-152.

S. S. Zhang, Weak convergence theorem for Lipschitzian pseudocontraction semigroups in Banach spaces, Acta Mathematica Sinica, English Series 26 (2010), 337-344.

Downloads

Published

2020-12-31

How to Cite

Efut, A., Igbokwe, D., & Udo-Utun, X. (2020). Implicit Iteration Process for Lipschitzian α-Hemicontraction Semigroups. MathLAB Journal, 7, 43-52. Retrieved from https://purkh.com/index.php/mathlab/article/view/918

Issue

Section

Research Articles