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

## Main Article Content

## 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.

### Downloads

## Article Details

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

## 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.