The generalized viscosity implicit rule of nonexpansive semigroup in Banach spaces

Jaiboon, Chaichana ¹ ; Plubtieng, Somyot ² ; Katchang, Phayap ³

¹ Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin, Nakhon Pathom 73170, Thailand
² Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
³ Rajamangala University of Technology Lanna Tak, Division of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand, Tak 63000, Thailand

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 6, p. 746-761.

Résumé

In this research, we focus on a common fixed point problem of a nonexpansive semigroup with the generalized viscosity methods for implicit iterative algorithms. Our main objective is to construct the new strong convergence theorems under certain appropriate conditions in uniformly convex and uniformly smooth Banach spaces. Specifically, the main results make a contribution to the implicit midpoint theorems. The findings for theorems in Hilbert spaces and the other forms of a nonexpansive semigroup can be used in several practical purposes. Finally, a numerical example in 3 dimensions is provided to support our main results.

Détail

DOI : 10.22436/jnsa.011.06.02

Classification : 47H10, 47H20, 47J20
Keywords: Nonexpansive semigroup, fixed point, generalized viscosity, implicit, Banach space

Bibliographie

[1] Atsushiba, S.; W. Takahashi Strong convergence of Mann’s-type iterations for nonexpansive semigroups in general Banach spaces, Nonlinear Anal., Volume 61 (2005), pp. 881-899 | DOI | Zbl

[2] Gossez, J.-P.; Dozo, E. Lami Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math., Volume 40 (1972), pp. 565-573

[3] W.-B. Guan An iterative method for variational inequality problems, J. Inequal. Appl., Volume 2013 (2013), pp. 1-10 | DOI

[4] Ke, Y.; C. Ma The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., Volume 2015 (2015), pp. 1-21 | DOI | Zbl

[5] Lau, A. T.-M. Amenability and fixed point property for semigroup of nonexpansive mapping, Fixed point theory Appl. (Marseille, (1989)), Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow, 1991

[6] Lau, A. T.-M.; Y. Zhang Fixed point properties of semigroups of non-expansive mappings, J. Funct. Anal., Volume 254 (2008), pp. 2534-2554 | DOI

[7] L.-C. Lim On characterizations of Meir-Keeler contractive maps , Nonlinear Anal., Volume 46 (2001), pp. 113-120 | DOI | Zbl

[8] Sunthrayuth, P.; Kumam, P. Viscosity approximation methods based on generalized contraction mappings for a countable family of strict pseudo-contractions, a general system of variational inequalities and a generalized mixed equilibrium problem in Banach spaces, Math. Comput. Modelling, Volume 58 (2013), pp. 1814-1828 | Zbl | DOI

[9] T. Suzuki Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., Volume 305 (2005), pp. 227-239 | DOI | Zbl

[10] Suzuki, T. Moudafi’s viscosity approximations with MeirKeeler contractions, J. Math. Anal. Appl., Volume 325 (2007), pp. 342-352 | DOI

[11] Wang, Y.; Xu, H.-K. Hybrid method for a class of accretive variational inequalities involving nonexpansive mappings, J. Inequal. Appl., Volume 2014 (2014), pp. 1-9 | DOI | Zbl

[12] Xu, H.-K. Iterative algorithms for nonlinear operators, J. London Math. Soc., Volume 66 (2002), pp. 240-256 | DOI

[13] Xu, H.-K.; Alghamdi, M. A.; Shahzad, N. The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., Volume 2015 (2015), pp. 1-12 | Zbl | DOI

[14] Yan, Q.; Cai, G.; P. Luo Strong convergence theorems for the generalized viscosity implicit rules of nonexpansive mappings in uniformly smooth Banach spaces, J. Nonlinear Sci. Appl., Volume 9 (2016), pp. 4039-4051 | Zbl