Iteration Methods for a General Variational Inequality System and Common Fixed Point Problems of Nonexpansive Mappings in q-Uniformly Smooth Banach Spaces

Article Sidebar

PDF
Published: Mar 26, 2022
DOI: https://doi.org/10.14456/past.2022.1
Keywords:
Banach space Fixed point Inverse-strongly accretive mapping q-uniformly smooth General variational inequality system

Main Article Content

Uamporn Witthayarat
Department of Mathematics, School of Science, University of Phayao
Kriengsak Wattanawitoon
Division of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak
Phayap Katchang
Division of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak

Abstract

This research proposed the iteration method for finding a common fixed point of an infinite family of nonexpansive mappings and two inverse strongly accretive mappings in -uniformly smooth Banach spaces. Furthermore, our method can also solve a new general variational inequality system and its strong convergence theorem is proved under some appropriate conditions. Our result improves and extends the previous outcomes in the literature.

Article Details

How to Cite
1.
Witthayarat U, Wattanawitoon K, Katchang P. Iteration Methods for a General Variational Inequality System and Common Fixed Point Problems of Nonexpansive Mappings in q-Uniformly Smooth Banach Spaces. Prog Appl Sci Tech. [Internet]. 2022 Mar. 26 [cited 2024 May 8];12(1):1-10. Available from: https://ph02.tci-thaijo.org/index.php/past/article/view/245557
Issue
Vol. 12 No. 1 (2022): January - April 2022
Section
Mathematics and Applied Statistics
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Crossref
Scopus
Google Scholar
Europe PMC

References

Cioranescu I. Geometry of Banach spaces, duality mappings and nonlinear problemsz: Kluwer, Dordrecht; 1990.

Reich S. Review of Geometry of Banach spaces, Duality Mappings and Nonlinear Problems by Ioana Cioranescu, Kluwer Academic Publishers, Dordrecht, 1990. Bulletin of the American Mathematical Society. 1992; 26, 367-370.

Aoyama K, Kimura Y, Takahashi W, Toyoda M. Approximation of common fixed point of a countable family of nonexpansive mapping in a Banach space. Nonlinear Anal. Theory Methods Appl. 2007; 67, 2350-2360.

Reich S. Asymptotic behavior of contractions in Banach spaces. Journal of mathematical Analysis and Appications. 1973; 44, 57-70.

Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings: Marcel Dekker, New York; 1984.

Kopecka E, Reich S. Nonexpansive retracts in Banach spaces. Banach Center Publications. 2007; 77, 161-174.

Chang SS, Lee HWJ, Chan CK. Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces. Appl. Math. Letters. 2007; 20, 329-334.

Cho YJ, Yao Y, Zhou H. Strong convergence of an iterative algorithm for accretive operators in Banach spaces. J. Com. Anal Appl. 2008; 10, 113-125.

Qin X, Kang AM, Shang M. Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces. Applicable Analisis. 2008; 87, 421-430.

Moudafi A. Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000; 241, 46-55.

Ceng LC, Yao JC, Muglia L. An extragradient-like approximation method for variational inequality problems and fixed point problems. Appl. Math. Comput. 2007; 190, 205-215.

Imnang S. Viscosity iterative method for a new general system of variational inequalities in Banach spaces. J. Inequalities Appl. 2013; 249.

Qin X, Cho SY, Kang SM. Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math. 2009; 233, 231-240.

Sen MDL, Muglia L. Fixed and best proximity points of cyclic jointly accretive and contractive self-mappings. J. Appl. Math. 2012; 419-429.

Pan C, Wang Y. Generalized viscosity implicit iterative process for asymptotically non-expansive mappings in Banach spaces. Mathematics. 2019; 7, 379.

Iiduka H, Takahashi W, Toyoda M. Approximation of solutions of variational inequalities for monotone mappings. Pan. Math. J. 2004; 14, 49-61.

Aoyama K, Liduka H, Takahashi W. Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory and Appliications. 2006; Article ID 35390, 13 pages

Yao Y, Noor MA, Noor KI, Liou YC. Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl. Math. 2010; 110, 1211-1224.

Song Y, Ceng L. A general iteration scheme for variational inequality problem and common _xed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces. J. Glob. Optim. 2013; 57:1327-1348.

Wang Y, Pan C. Viscosity approximation methods for a general variational inequality system and fixed point problems in Banach spaces. symmetry. 2020; 12, 36.

Mitrinovic DS. Analytic inequalities: Springer, New York; 1970.

Suntrayuth P, Kumam P. Iterative methods for variational inequality problems and fixed point problems of a countable family of strict pseudo-contractions in a q-uniformly smooth Banach space. Fixed Point Theory Appl. 2012; 65.

Bruck RE. Properties of fixed point sets of nonexpansivemappings in Banach spaces. Trans. Am. Math. Soc. 1973; 179, 251-262.

Kitahara S, Takahashi W. Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization. 2002; 13, 938-945.

Xu HK. Inequalities in Banach spaces with applications. Nonlinear Analysis. 1991; 16, 11127-1138.

Xu HK. Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 2002; 2, 1-17.

Song Y, Ceng L. A General iteration scheme for variational in equality problem and common fixed point problems of nonexpansive mapping in q-unformly smooth Banach spaces. Journal of Globle Optimization. 2013; 57, 1327-1348.

Xu HK. Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004; 298, 279-291.