A Simple proof for the algorithms of relaxed $(u, v)$-cocoercive mappings and $\alpha$-inverse strongly monotone mappings

Document Type : Research Paper

Authors

1 Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran

2 Department of Mathematics Texas and University-Kingsville 700 University Blvd., MSC 172 Kingsville, Texas 78363-8202, USA

3 School of Mathematics, Statistics, National University of Ireland, Galway, Ireland

Abstract

In this paper, a simple proof is presented for the convergence of the algorithms for the class of relaxed $(u, v)$-cocoercive mappings and $\alpha$-inverse strongly monotone mappings. Based on $\alpha$-expansive maps, for example, a simple proof of the convergence of the recent iterative algorithms by relaxed $(u, v)$-cocoercive mappings due to Kumam-Jaiboon is provided. Also a simple proof for the convergence of the iterative algorithms by inverse-strongly monotone mappings due to Iiduka-Takahashi in a special case is provided. These results are an improvement as well as a refinement of previously known results.

Keywords

[1] F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (2005) 197–228.
[2] G. Cai and S. Bu, Strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in Banach spaces, Comput. Math. with Appl. 62 (2011) 2567–2579.
[3] W. Chantarangs, C. Jaiboon and P. Kumam, A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces, Abstr. Appl. Anal. (2010) Art. ID 390972:39.
[4] J. Chen, L. Zhang and T. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl. 334 (2007) 1450–1461.
[5] H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings Nonlinear Anal. 61 (2005) 341–350.
[6] C. Jaiboon and P. Kumam, Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities, J. Inequal. Appl. (2010) Art. ID 728028:43.
[7] T. Jitpeera and P. Kumam, A new hybrid algorithm for a system of mixed equilibrium problems, fixed point problems for nonexpansive semigroup, and variational inclusion problem, Fixed Point Theory Appl. (2011) Art. ID 217407:27.
[8] T. Jitpeera and P. Kumam, Hybrid algorithms for minimization problems over the solutions of generalized mixed equilibrium and variational inclusion problems, Math. Probl. Eng. (2011) Art. ID 648617:25.
[9] T. Jitpeera and P. Kumam, A new hybrid algorithm for a system of equilibrium problems and variational inclusion, Ann. Univ. Ferrara Sez. VII Sci. Mat. 57 (2011) 89–108.
[10] T. Jitpeera and P. Kumam, The shrinking projection method for common solutions of generalized mixed equilibrium problems and fixed point problems for strictly pseudocontractive mappings, J. Inequal. Appl. (2011) Art. ID 840319:25.
[11] J.S. Jung, A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems, J. Inequal. Appl. (2011) 51:23.
[12] A. Kangtunyakarn, Iterative methods for finding common solution of generalized equilibrium problems and variational inequality problems and fixed point problems of a finite family of nonexpansive mappings, Fixed Point Theory Appl. (2010) Art. ID 836714:29.
[13] K.R. Kazmi, Rehan Ali and Mohd Furkan, Hybrid iterative method for split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings, Numer. Algorithms79 (2018) 499–527.
[14] K.R. Kazmi and S.H. Rizvi, A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem, Appl. Math. Comput. 218 (2012) 5439–5452.
[15] P. Kumam and C. Jaiboon, A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems, Nonlinear Anal. Hybrid Syst. 3 (2009) 510–530.
[16] M. Lashkarizadeh Bami and E. Soori, Strong convergence of a general implicit algorithm for variational inequality problems and equilibrium problems and a continuous representation of nonexpansive mappings, Bull. Iranian Math. Soc. 40 (2014) 977–1001.
[17] N. Onjae-uea, C. Jaiboon and P. Kumam, Convergence of iterative sequences for fixed points of an infinite family of nonexpansive mappings based on a hybrid steepest descent methods, J. Inequal. Appl. (2012) 101:22.
[18] S. Peathanom and W. Phuengrattana, A hybrid method for generalized equilibrium, variational inequality and fixed point problems of finite family of nonexpansive mappings, Thai J. Math. 9 (2011) 95–119.
[19] H. Piri, A general iterative method for finding common solutions of system of equilibrium problems, system of variational inequalities and fixed point problems, Math. Comput. Modelling 55 (2012) 1622–1638.
[20] X. Qin, M. Shang and Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Anal. 69 (2008) 3897–3909.
[21] X. Qin, M. Shang and Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Modelling 48 (2008) 1033–1046.
[22] X. Qin, M. Shang and H. Zhou, Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces, Appl. Math. Comput. 200 (2008) 242–253.
[23] S. Saeidi, Comments on relaxed (γ, r)-cocoercive mappings, Int. J. Nonlinear Anal. Appl. 1 (2010) 54–57.
[24] S. Saewan and P. Kumam, The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-φ-nonexpansive mappings, Fixed Point Theory Appl. (2011) 9:25.
[25] S. Shan and N. Huang, An iterative method for generelized mixed vector equlibrium problems and fixed point of nonexpansive mappings and variathional inequalities, Taiwan. J. Math. 16 (2012)1681–1705.
[26] L. Sun, Hybrid methods for common solutions in Hilbert spaces with applications, (english summary), J. Inequal. Appl. 183(MR3239824)(2014):16.
[27] S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008) 1025–1033.
[28] W. Takahashi, Strong convergence theorems for maximal and inverse-strongly monotone mappings in Hilbert spaces and applications, J. Optim. Theory Appl. 157 (2013) 781–802.
[29] Y. Yao, Y. Liou and S. Kang, Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method , Comput. Math. Appl. 59 (2010) 3472–3480.
[30] M. Zhang, Iterative algorithms for common elements in fixed point sets and zero point sets with applications, Fixed Point Theory Appl. (2012) 2012:21.
Volume 12, Issue 2
November 2021
Pages 327-333
  • Receive Date: 22 July 2019
  • Accept Date: 17 October 2019