Abstract
In this research, we modified iterative scheme for finding common element of the set of fixed point of total quasi-\(\phi \)-asymptotically nonexpansive multivalued mappings, the set of solution of an equilibrium problem and the set of fixed point of relatively nonexpansive mappings in Banach spaces. In addition, the strong convergence for approximating common solution of our mentioned problems is proved under some mild conditions. Our results extend and improve some recent results announced by some authors. We divide our research details into three main sections including Introduction, Preliminaries, Main Results. First, we introduce the backgrounds and motivations of this research and follow with the second section, Preliminaries, which mention about the tools that will be needed to prove our main results. In the last section, Main Results, we propose the theorem and corollary which is the most important part in our research.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Reich S (1996) A weak convergence theorem for the alternating method with Breman distance. In: Kaetsatos AG (ed) Theory and applications of nonlinear operators of accretive and monotone type. Marcel Dekker, New York, pp 313–318
Nilsrakoo W, Saejung S (2008) Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings. Fixed Point Theor Appl 2008(312454):19
Su Y, Wang D (2008) Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theor Appl 2008(284613):8
Zegeye H, Shahzad N (2009) Strong convergence for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal 70:2707–2716
Butnariu D, Reich S, Zaslavski AJ (2001) Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J Math Anal Appl 7(2):151–174
Cens Y, Reich S (1996) Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37:323–339
Takahashi W, Takeuchi Y, Kubota R (2008) Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J Math Anal Appl 341:276–286
Matsushita S, Takahashi W (2004) Weakly and strong convergence theorems for relatively nonexpansive mappings in a Banach space. Fixed Point Theor Appl 2004:37–47
Matsushita S, Takahashi W (2005) A Strong convergence theorem for relatively nonexpansive mappings in a Banach space. J Approximation Theor 134(2):257–266
Chang SS, Lee HWJ, Chan CK, Zhang WB (2012) A modified Halpern-type iteration algorithm for totally quasi-\(\phi \)-asymptotically nonexpansive mappings with applications. Appl Math Comput 218:6489–6497
Tang J, Chang SS (2012) Strong convergence theorems for total quasi-\(\phi \)-asymptotically nonexpansive multi-value mappings in Banach spaces. Fixed Point Theor Appl 2012:63. doi: 10.1186/1687-1812-2012-63
Wattanawitoon K, Witthayarat U, Kumam P (2013) Strong convergence theorems of multivalued nonexpansive mappings and maximal monotone operators in banach spaces. Lecture Notes in Engineering and Computer Science: Proceedings of The International MultiConference of Engineers and Computer Scientists 2013, Hong Kong, 13–15 Mar 2013, pp 1194–1199
Cioranescu I (1990) Geometry of banach spaces, duality mappings and nonlinear problems of mathematics and its applications. Kluwer Academic Publishers, Dordrecht, The Netherlands
Takahashi W (2000) Nonlinear functional analysis, fixed point theory and its applications. Yokohama Publishers, Yokohama, Japan
Alber YI (1996) Metric and generalized projection operators in Banach spaces: properties and applications, theory and applications of nonlinear operators of accretive and monotone type. In: Kartsatos AG (ed) Marcel Dekker, New York, vol 178, pp 15–50
Kamimura S, Takahashi W (2002) Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim 13(3):938–945
Blum E, Oettli W (1994) From optimization and variational inequalities to equilibrium problems. Math Stud 63:123–145
Combettes PL, Hirstoaga SA (2005) Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 6:117–136
Takahashi W, Zembayashi K (2008) Strong convergence theorems by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theor Appl 2008(528476):11
Acknowledgments
The authors would like to thank the referees for the valuable suggestions which helped to improve this manuscript. K. Wattanawitoon gratefully acknowledges support provided by the King Mongkut’s University of Technology Thonburi (KMUTT) during the second author’s stay at the King Mongkut’s University of Technology Thonburi (KMUTT) as a post doctoral fellow (KMUTT-Post-doctoral Fellowship).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Witthayarat, U., Wattanawitoon, K., Kumam, P. (2014). Modified Iterative Scheme for Multivalued Nonexpansive Mappings, Equilibrium Problems and Fixed Point Problems in Banach Spaces. In: Yang, GC., Ao, SI., Huang, X., Castillo, O. (eds) Transactions on Engineering Technologies. Lecture Notes in Electrical Engineering, vol 275. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7684-5_20
Download citation
DOI: https://doi.org/10.1007/978-94-007-7684-5_20
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-7683-8
Online ISBN: 978-94-007-7684-5
eBook Packages: EngineeringEngineering (R0)