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On the strong convergence of a general-type Krasnosel’skii-Mann’s algorithm depending on the coefficients. (English) Zbl 1333.47051

Summary: Let \(H\) be a Hilbert space, \((W_n)_{n \in \mathbb N}\) a suitable family of mappings, \(S\) a nonexpansive mapping and \(D\) a strongly monotone operator. We are interested in the strong convergence of the general scheme \[ x_{n+1} =\gamma x_n+(1-\gamma) W_n \left(\alpha_n Sx_n +(1-\alpha)(I-\mu_n D)x_n\right),\quad \gamma\in [0,1), \] in dependence of the coefficients \((\alpha_{n})_{n \in \mathbb N}\) and \((\mu_n)_{n \in \mathbb N}\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] Atsushiba S., Takahashi W.: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian J. Math. 41, 435-453 (1999) · Zbl 1055.47514
[2] Baillon J.-B., Haddad G.: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Israel J. Math. 26, 137-150 (1977) · Zbl 0352.47023
[3] F. Cianciaruso, G. Marino, L. Muglia and Y. Yao, On a two-step algorithm for hierarchical fixed point problems and variational inequalities. J. Inequal. Appl. 2009 (2009), Article ID 208692, 13 pages. · Zbl 1180.47040
[4] K. Deimling, Nonlinear Functional Analysis. Dover Publications, 2010 (originally publlished by Springer, Berlin, 1985). · Zbl 0559.47040
[5] Halpern B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957-961 (1967) · Zbl 0177.19101
[6] Hundal H. S.: An alternating projection that does not converge in norm. Nonlinear Anal. 57, 35-61 (2004) · Zbl 1070.46013
[7] N. Hussain, G. Marino and A. A. N. Abdou, On Mann’s method with viscosity for nonexpansive and nonspreading mappings in Hilbert spaces. Abstr. Appl. Anal. 2014 (2014), Article ID 152530, 11 pages. · Zbl 1428.47025
[8] N. Hussain, G. Marino, L. Muglia and B. Alamri, On some Mann’s type iterative algorithms. Fixed Point Theory Appl. 2015 (2015), doi:10.1186/s13663-015-0267-8, 16 pages. · Zbl 1321.47141
[9] Hussain N., Takahashi W.: Weak and strong convergence theorems for semigroups of mappings without continuity in Hilbert spaces. J. Nonlinear Convex Anal. 14, 769-783 (2013) · Zbl 1418.47004
[10] Ishikawa S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Amer. Math. Soc. 59, 65-71 (1976) · Zbl 0352.47024
[11] Maingé P.-E., Moudafi A.: Strong convergence of an iterative method for hierarchical fixed-points problems. Pac. J. Optim. 3, 529-538 (2007) · Zbl 1158.47057
[12] Mann W. R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506-510 (1953) · Zbl 0050.11603
[13] G. Marino, F. Cianciaruso and N. Hussain, Ergodic approximations via matrix regularization approach. Fixed Point Theory Appl. 2015 (2015), doi:10.1186/s13663-015-0321-6, 11 pages. · Zbl 1346.47051
[14] Marino G., Muglia L.: On the auxiliary mappings generated by a family of mappings and solutions of variational inequalities problems. Optim. Lett. 9, 263-282 (2015) · Zbl 1310.47095
[15] Marino G., Muglia L., Yao Y.: The uniform asymptotical regularity of families of mappings and solutions of variational inequality problems. J. Nonlinaer Convex Anal. 15, 477-492 (2014) · Zbl 1338.47102
[16] Marino G., Xu H.-K.: Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 149, 61-78 (2011) · Zbl 1221.49012
[17] Moudafi A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46-55 (2000) · Zbl 0957.47039
[18] A. Moudafi and P.-E. Maingé, Towards viscosity approximations of hierarchical fixed-points problems. Fixed Point Theory Appl. 2006 (2006), Art. ID 95453, 10 pages. · Zbl 1143.47305
[19] Nakajo K., Takahashi W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372-379 (2003) · Zbl 1035.47048
[20] Reich S., Xu H.-K.: An iterative approach to a constrained least squares problem. Abstr. Appl. Anal. 8, 503-512 (2003) · Zbl 1053.65041
[21] Shimoji K., Takahashi W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 5, 387-404 (2001) · Zbl 0993.47037
[22] Takahashi W., Toyoda M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417-428 (2003) · Zbl 1055.47052
[23] Xu H.-K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. (2) 66, 240-256 (2002) · Zbl 1013.47032
[24] Xu H.-K.: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360-378 (2011) · Zbl 1233.90280
[25] Xu H.-K., Kim T. H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185-201 (2003) · Zbl 1045.49018
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