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Preface: Special Issue in Honor of the 60th Birthday of Sylvain Sorin
July  2014, 1(3): 331-346. doi: 10.3934/jdg.2014.1.331

## Asymptotic behavior of compositions of under-relaxed nonexpansive operators

 1 Université Paris 1 Panthéon-Sorbonne, SAMM – EA 4543, 75013 Paris, France 2 Sorbonne Universités – UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France 3 Universidad de Chile, Departamento de Ingeniería Industrial, Santiago, Chile

Received  April 2013 Revised  October 2013 Published  July 2014

In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.
Citation: Jean-Bernard Baillon, Patrick L. Combettes, Roberto Cominetti. Asymptotic behavior of compositions of under-relaxed nonexpansive operators. Journal of Dynamics and Games, 2014, 1 (3) : 331-346. doi: 10.3934/jdg.2014.1.331
##### References:
 [1] H. Attouch, L. M. Briceño-Arias and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim., 48 (2010), 3246-3270. doi: 10.1137/090754297. [2] J.-B. Baillon, P. L. Combettes and R. Cominetti, There is no variational characterization of the cycles in the method of periodic projections, J. Funct. Anal., 262 (2012), 400-408. doi: 10.1016/j.jfa.2011.09.002. [3] H. H. Bauschke and J. M. Borwein, On the convergence of von Neumann's alternating projection algorithm for two sets, Set-Valued Anal., 1 (1993), 185-212. doi: 10.1007/BF01027691. [4] H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke and H. Wolkowicz, eds., Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-9569-8. [5] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7. [6] H. H. Bauschke and M. R. Edwards, A conjecture by De Pierro is true for translates of regular subspaces, J. Nonlinear Convex Anal., 6 (2005), 93-116. [7] H. H. Bauschke, X. Wang and C. J. S. Wylie, Fixed points of averages of resolvents: Geometry and algorithms, SIAM J. Optim., 22 (2012), 24-40. doi: 10.1137/110823778. [8] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland/Elsevier, New York, 1973. [9] R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal., 18 (1975), 15-26. doi: 10.1016/0022-1236(75)90027-0. [10] C. L. Byrne, Applied Iterative Methods, A. K. Peters, Wellesley, MA, 2008. [11] A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, 2057, Springer, Heidelberg, 2012. [12] Y. Censor, P. P. B. Eggermont and D. Gordon, Strong under-relaxation in Kaczmarz's method for inconsistent systems, Numer. Math., 41 (1983), 83-92. doi: 10.1007/BF01396307. [13] P. L. Combettes, Inconsistent signal feasibility problems: Least-squares solutions in a product space, IEEE Trans. Signal Process., 42 (1994), 2955-2966. doi: 10.1109/78.330356. [14] A. R. De Pierro, From parallel to sequential projection methods and vice versa in convex feasibility: Results and conjectures, in Inherently Parallel Algorithms for Feasibility and Optimization, Elsevier, New York, 2001, 187-201. doi: 10.1016/S1570-579X(01)80012-4. [15] A. R. De Pierro and A. N. Iusem, A parallel projection method for finding a common point of a family of convex sets, Pesquisa Operacional, 5 (1985), 1-20. [16] L. G. Gubin, B. T. Polyak and E. V. Raik, The method of projections for finding the common point of convex sets, Comput. Math. Math. Phys., 7 (1967), 1-24. doi: 10.1016/0041-5553(67)90113-9. [17] W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl., 14 (1966), 276-284. doi: 10.1016/0022-247X(66)90027-8. [18] X. Wang and H. H. Bauschke, Compositions and averages of two resolvents: Relative geometry of fixed points sets and a partial answer to a question by C. Byrne, Nonlinear Anal., 74 (2011), 4550-4572. doi: 10.1016/j.na.2011.04.024. [19] E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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##### References:
 [1] H. Attouch, L. M. Briceño-Arias and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim., 48 (2010), 3246-3270. doi: 10.1137/090754297. [2] J.-B. Baillon, P. L. Combettes and R. Cominetti, There is no variational characterization of the cycles in the method of periodic projections, J. Funct. Anal., 262 (2012), 400-408. doi: 10.1016/j.jfa.2011.09.002. [3] H. H. Bauschke and J. M. Borwein, On the convergence of von Neumann's alternating projection algorithm for two sets, Set-Valued Anal., 1 (1993), 185-212. doi: 10.1007/BF01027691. [4] H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke and H. Wolkowicz, eds., Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-9569-8. [5] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7. [6] H. H. Bauschke and M. R. Edwards, A conjecture by De Pierro is true for translates of regular subspaces, J. Nonlinear Convex Anal., 6 (2005), 93-116. [7] H. H. Bauschke, X. Wang and C. J. S. Wylie, Fixed points of averages of resolvents: Geometry and algorithms, SIAM J. Optim., 22 (2012), 24-40. doi: 10.1137/110823778. [8] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland/Elsevier, New York, 1973. [9] R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal., 18 (1975), 15-26. doi: 10.1016/0022-1236(75)90027-0. [10] C. L. Byrne, Applied Iterative Methods, A. K. Peters, Wellesley, MA, 2008. [11] A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, 2057, Springer, Heidelberg, 2012. [12] Y. Censor, P. P. B. Eggermont and D. Gordon, Strong under-relaxation in Kaczmarz's method for inconsistent systems, Numer. Math., 41 (1983), 83-92. doi: 10.1007/BF01396307. [13] P. L. Combettes, Inconsistent signal feasibility problems: Least-squares solutions in a product space, IEEE Trans. Signal Process., 42 (1994), 2955-2966. doi: 10.1109/78.330356. [14] A. R. De Pierro, From parallel to sequential projection methods and vice versa in convex feasibility: Results and conjectures, in Inherently Parallel Algorithms for Feasibility and Optimization, Elsevier, New York, 2001, 187-201. doi: 10.1016/S1570-579X(01)80012-4. [15] A. R. De Pierro and A. N. Iusem, A parallel projection method for finding a common point of a family of convex sets, Pesquisa Operacional, 5 (1985), 1-20. [16] L. G. Gubin, B. T. Polyak and E. V. Raik, The method of projections for finding the common point of convex sets, Comput. Math. Math. Phys., 7 (1967), 1-24. doi: 10.1016/0041-5553(67)90113-9. [17] W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl., 14 (1966), 276-284. doi: 10.1016/0022-247X(66)90027-8. [18] X. Wang and H. H. Bauschke, Compositions and averages of two resolvents: Relative geometry of fixed points sets and a partial answer to a question by C. Byrne, Nonlinear Anal., 74 (2011), 4550-4572. doi: 10.1016/j.na.2011.04.024. [19] E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
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