Weak convergence of an iterative algorithm for accretive operators

Volume 10, Issue 8, pp 4099--4108

Publication Date: 2017-08-07



Hengjun Zhao - School of Science, Henan University of Engineering, Zhengzhou 451191, China.
Sun Young Cho - Center for General Education, China Medical University, Taichung, Taiwan.


In this paper, an iterative algorithm investigated for \(m\)-accretive and inverse-strongly accretive operators. Also, a weak convergence theorem for the sum of two accretive operators is established in a real uniformly convex and \(q\)-uniformly smooth Banach space.


Accretive operator, zero point, projection, splitting method, weak convergence.


[1] D. E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc., 82 (1981), 423–424.
[2] I. K. Argyros, S. George, On the convergence of inexact Gauss-Newton method for solving singular equations, J. Nonlinear Funct. Anal., 2016 (2016), 22 pages.
[3] B. A. Bin Dehaish, A. Latif, H. O. Bakodah, X.-L. Qin, A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 14 pages.
[4] B. A. Bin Dehaish, X.-L. Qin, A. Latif, H. O. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321–1336.
[5] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123–145.
[6] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041–1044.
[7] R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math., 32 (1979), 107–116.
[8] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103–120.
[9] S. Y. Cho, B. A. Bin Dehaish, X.-L. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427–438.
[10] S. Y. Cho, S. M. Kang, Approximation of common solutions of variational inequalities via strict pseudocontractions, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 1607–1618.
[11] S. Y. Cho, W.-L. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013), 14 pages.
[12] J. García Falset, W. Kaczor, T. Kuczumow, S. Reich, Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal., 43 (2001), 377–401.
[13] S.-P. Han, G. Lou, A parallel algorithm for a class of convex programs, SIAM J. Control Optim., 26 (1988), 345–355.
[14] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508–520.
[15] S.-T. Lv, Convergence analysis of a Halpern-type iterative algorithm for zero points of accretive operators, Commun. Optim. Theory, 2016 (2016), 9 pages.
[16] B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, (French) Rev. Franc¸aise Informat. Recherche Op´erationnelle, 4 (1970), 154–158.
[17] B. Martinet, Détermination approchée d’un point fixe d’une application pseudo-contractante, Cas de l’application prox, (French) C. R. Acad. Sci. Paris S´er. A-B, 274 (1972), 163–165.
[18] G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72 (1979), 383–390.
[19] X.-L. Qin, S.-S. Chang, Y. J. Cho, Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. Real World Appl., 11 (2010), 2963–2972.
[20] X.-L. Qin, S. Y. Cho, Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), 488–502.
[21] X.-L. Qin, J.-C. Yao, Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016), 9 pages.
[22] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97–116.
[23] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877–898.
[24] D. R. Sahu, J. C. Yao, A generalized hybrid steepest descent method and applications, J. Nonlinear Var. Anal., 1 (2017), 111–126.
[25] W. Takahashi, Weak and strong convergence theorems for families of nonlinear and nonself mappings in Hilbert spaces, J. Nonlinear Var. Anal., 1 (2017), 1–23.
[26] S. Takahashi, W. Takahashi, M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147 (2010), 27–41.
[27] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127–1138.
[28] S. Yang, Zero theorems of accretive operators in reflexive Banach spaces, J. Nonlinear Funct. Anal., 2013 (2013), 12 pages.


XML export