Document Type : Research Paper
Authors
- Mohsen Tahernia ^{1}
- Sirous Moradi ^{} ^{1}^{, 2}
- Somaye Jafari ^{1}
^{1} Department of Mathematics, Faculty of Science, Arak University, 38156-8-8349, Arak, Iran.
^{2} Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Iran.
Abstract
In this paper, we consider a proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in real Hilbert spaces. Also, we give a necessary and sufficient condition for the common zero set of finite operators to be nonempty, and by showing that in this case, this iterative sequence converges strongly to the metric projection of some point onto the set of common zeros of operators.
Keywords
[1] H.H. Bauschke, P.L. Combettes, and S. Reich, The asymptotic behavior of the composition of two resolvents, Nonlinear Anal. (TMA), 60 (2005), pp. 283-301.
[2] H.H. Bauschke, E. Matouskova, and S. Reich, Projection and proximal point methods: convergence results and counterexamples, Nonlinear Anal. (TMA), 56 (2004), pp. 715-738.
[3] O.A. Boikanyo and G. Morosanu, A contraction proximal point algorithm with two monotone operators, Nonlinear Anal. (TMA), 75 (2012), pp. 5686-5692.
[4] O.A. Boikanyo and G. Morosanu, On the method of alternating resolvents, Nonlinear Anal. (TMA), 74 (2011), pp. 5147-5160.
[5] O.A. Boikanyo and G. Morosanu, Strong convergence of the method of alternating resolvents, J. Nonlinear Convex Anal., 14 (2013), pp. 221-229.
[6] O.A. Boikanyo and G. Morosanu, The method of alternating resolvents revisited, Numer. Funct. Anal. Optim., 33 (2012), pp. 1280-1287.
[7] L.M. Bregman, The method of successive projection for finding a common point of convex sets, Sov. Math. Dokl., 6 (1965), pp. 688-692.
[8] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.
[9] H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. (TMA), 57 (2004), pp. 35-61.
[10] E. Kopecka and S. Reich, A note on the von Neumann alternating projections algorithm, J. Nonlinear Convex Anal., 5 (2004), pp. 379-386.
[11] P.E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), pp. 899-912.
[12] E. Matouskova and S. Reich, The Hundal example revisited, J. Nonlinear Convex Anal., 4 (2003), pp. 411-427.
[13] G. Morosanu, Nonlinear Evolution Equations and Applications, Reidel, Dordrecht, 1988.
[14] L. Nasiri and A. Sameripour, The spectral properties of differential operators with matrix coefficients on elliptic systems with boundary conditions, Sahand Commun. Math. Anal., 10 (2018), pp. 37-46.
[15] N. Nimit, A.P. Farajzadeh, and N. Petrot, Adaptive subgradient method for the split quasi-convex feasibility problems, Optimization, 65 (2016), pp. 1885-1898.
[16] H.K. Xu, A regularization method for the proximal point algorithm, J. Glob. Optim., 36 (2006), pp. 115-125.
[17] P. Yatakoat, A new approximation method for common fixed points of a finite family of nonexpansive non-self mappings in Banach spaces, Int. J. Nonlinear Anal. Appl., 9 (2018), pp. 223-234.