International Journal of Nonlinear Analysis and Applications

A new extragradient-viscosity algorithm for finite families of asymptotically nonexpansive mappings and variational inequality problems in Banach spaces

Document Type : Research Paper

Authors

Department of Mathematics, Micheal Okpara University of Agriculture, Umudike, Umuahia Abia State, Nigeria

Abstract

ln this paper, a new approach for finding common element of the set of solutions of the variational inequality problem for accretive mappings and the set of fixed points for asymptotically nonexpansive mappings is introduced and studied. Consequently, strong convergence results for finite families of asymptotically nonexpansive mappings and variational inequality problems are established in the setting of uniformly convex Banach space and 2-uniformly smooth Banach space. Furthermore, we prove that a slight modification of our novel scheme could be applied in finding common element of solution of variational inequality problems in Hilbert space. Our results improve, extend and generalize several recently announced results in literature.

Keywords

[1] I.K. Agwu and D.I. Igbokwe, Approximation of common fixed points of finite family of mixed-type total asymptotically quasi-pseudo contractive-type mappings in uniformly convex Banach spaces, Adv. Inequal. Appl. Appl. 2020 (2020), no. 4.
[2] K. Aoyama, H. Iiduka, W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed Point Theory and Appl. 2006 (2006), 13 pages.
[3] F.E. Browder, Convergence theorem for sequence of nonlinear operators in Banach space, Math. Z 100 (1967), no. 3, 201–225.
[4] F.E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660–665.
[5] F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228.
[6] C.E. Chidume, Geometric properties of Banach space and nonlinear iterations, Series: Lecture Notes in Mathematics, Springer Verlag, 2009.
[7] C.E. Chidume, J. Li and A. Udoemene, Convergence of path and approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 113 (2005), 473–480.
[8] I. Ciranescu, Geometry of Banach space, duality mapping and nonlinear problems, vol 62 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, Netherland, 1990.
[9] G. Cai, Y. Shehu and O.S. Iyiola, Iterative algorithm for solving variational inequalities and fixed point problems for asymptotically nonexpansive mappings in Banach spaces, Numer. Algor. 2016 (2016), 35 pages.
[10] G. Cai and S. Bu, Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces, Math. Comput. Model 55 (2012), 538–546.
[11] G. Cai and S. Bu, An iterative algorithm for a general system of variational inequality problem and fixed point problems in q-uniformly smooth Banach spaces, Optim. Lett. 7 (2013), 267–287.
[12] V. Censor, A.N. Iusem and S.A. Zenios, An interior point method with Bregman functions for the variational inequality problem with para monotone operators, Math. Program. 81 (1998), 373–400.
[13] Y.C. Cho, H. Zhou and G. Guo, Weak and strong convergence theorems for three-step iteration with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004), 707–717.
[14] S.S. Chang, H.W.J. Lee, C.K. Chan and J.K. Kim, Approximating solutions of variational inequalities for asymptotically nonexpansive mappings, Appl. Math. Comput. 212 (2009), 51–59.
[15] S.S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal. 30 (1997), 4197–4208.
[16] L.C. Ceng, C. Wang and J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008), 375–390.
[17] W.G. Dotson(Jr), Fixed points of quasi-nonexpansive mappings, Aust. Math. Soc. 13 (1992), 167–170.
[18] K. Goebel and W. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), no. 1, 171–174.
[19] K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings, Marcel Dekker, New York, 1984.
[20] B. Halpern, Fixed point of nonexpansive maps, Bull. Amer. Math. Soc. 3 (1967), 957–961.
[21] A.N. Iusem and A.R. De Pierro, On the convergence of Hans method for convex programming quadratic objective, Math. Program. Ser. B. 52 (1991), 265–284.
[22] D.I. Igbokwe and S.J. Uko, Weak and strong convergence theorems for approximating fixed points of nonexpansive mappings using composite hybrid iteration method, J. Nig. Math. Soc. 33 (2014), 129–144.
[23] D.I. Igbokwe and S.J. Uko, Weak and strong convergence of hybrid iteration methods for fixed points of asymptotically nonexpansive mappings, Adv. Fixed Point Theory 5 (2015), no. 1, 120–134.
[24] P. Kumama, N. Petrot and R. Wangkeeree, A hybrid iterative scheme for equilibrium problems and fixed point problems for asymptotically k-strictly pseudocontractions, J. Comput. Appl. Math. 233 (2010), 2013–2026.
[25] P. Kumama and K. Wattanawitoon, A general composite explicit iterative scheme fixed point problems of variational inequalities for nonexpansive semigroup, Math. Comput. Model. 53 (2011), 998–1006.
[26] S. Kamimura and W. Takahashi, Strong convergence of proximal-type algorithm in Banach space, SIAM J. Optim. 13 (2002), 938–945.
[27] E. Kopecka and S. Reich, Nonexpansive retracts in Banach spaces, Banach Center Pub. 77 (2007), 161–174.
[28] S. Kitahara and W. Takahashi, Image recovery by convex combination of sunny nonexpansive retractions, Topo. Methods. Nonlinear Anal. 2 (1993), no. 2, 333–342.
[29] T.H. Kim and D.H. Kim, Demiclosedness principle for continuous TAN mappings, RIMS Koykuroku 1821 (2013), 90–106.
[30] J. Lou, L. Zhang and Z. He, Viscosity approximation method for asymptotically nonexpansive mappings, Appl. Math. Comput. 203 (2008), 171–177.
[31] T.C. Lim and H.K. Xu, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal. 22 (1994), 1345–1355.
[32] G. Marino and H.K. Xu, Weak and strong convergence theorems for strict pseudocontractive in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), 336–346.
[33] A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl. 241 (2000), 46–55.
[34] M.A. Noor, Some development in general variational inequalities, Appl. Math. Comput. 152 (2004), 199–277.
[35] X. Qin and L. Wang, On asymptotic quasi-ϕ-nonexpansive mappings in the intermediate sense, Abstr. Appl. Anal. 2012 (2012), 13 pages.
[36] X. Qin and Y.C. Cho, Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2963–2972.
[37] S. Reich, Asymptotic behavour of contractions in Banach space, J. Math. Anal. Appl. 44 (1973), no. 1, 57–70.
[38] S. Reich, Product formulas, accretive operators and nonlinear mappings, J. Funct. Anal. 36 (1980), 147–168.
[39] S. Reich, Constructive techniques for accretive and monotone operators in applied nonlinear analysis, Academic Press, New York, 1979.
[40] S. Reich, Extension problems for accretive sets in Banach spaces, J. Funct. Anal. 26 (1977), 1378–395.
[41] G.S. Saluja, Convergence to common fixed point of two asymptotically quasi-nonexpansive mappings in the intermediate sense in Banach spaces, Math. Morvica 19 (2015), 33–48.
[42] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroup without Bochner integrals, J. Math. Anal. Appl. 305 (2005), 227–239.
[43] Y.L. Song and C. Ceng, A general iterative scheme for variational inequality problems and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces, J. Glob. Optim. 57 (2013), 1325–1348.
[44] T.M.M. Sow, M. Sene, M. Ndiaye, Y.B. El Yekheir and N. Djitte, Algorithms for a system of variational inequality problems and fixed point problems with demicontractive mappings, J. Nig. Math. Soc. 38 (2019), no. 3, 341–361.
[45] K. Tae-hwa, Review on some examples of nonlinear mappings, RIMS Kokyuroku 2114 (2019), 48–72.
[46] M.O. Osilike, A. Udoemene, D.I. Igbokwe and B.G. Akuchu, Demiclosedness principle and convergence theorems for k-strictly pseudocontractive maps, J. Math. Anal. Appl. 326 (2007), no. 3, 1334–1345.
[47] K. Sitthithakerngkiet, P. Sunthrayuth and P. Kumam, Some iterative methods for finding a common zero of finite family of accretive operators in Banach spaces, Bull. Iran. Math. Soc. 43 (2017), no. 1, 239–258.
[48] H.K. Xu and T.H. Kim, Convergence of Hybrid steepest-descent methods for variational inequalites, J. Optim. Theory Appl. 119 (2003), 185–201.
[49] H.K. Xu, Iterative algorithm for nonlinear operators, J. London. Math. Soc. 66 (2002), no. 2, 240–256.
[50] H.K. Xu, Viscosity approximation method for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), 279–291.
[51] Y. Yao, M. Aslam Noor, K. Inayat Noor, Y.C. Liou and H. Yaqoob, Modified extragradient methods for a system of variational inequalities in Banach spaces, Acta Appl. Math. 110 (2010), no. 3, 1211–1224.
[52] Y. Yao and S. Maruster, Strong convergence of an iterative scheme for variational inequalities in Banach spaces, Math. Comput. Model. 54 (2011), 325–329.
[53] Y. Yao, C. Lou, S.M. Kang and Y. Yu, Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces, Nonlinear Anal. 74 (2011), 624–634.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 409-433
  • Receive Date: 20 January 2021
  • Revise Date: 13 February 2021
  • Accept Date: 09 March 2021