Abstract
In this paper, by using products of finitely many resolvents of monotone operators, we propose an iterative algorithm for finding a common zero of a finite family of monotone operators and a common fixed point of an infinitely countable family of nonexpansive mappings in Hadamard spaces. We derive the strong convergence of the proposed algorithm under appropriate conditions. A common fixed point of an infinitely countable family of quasi-nonexpansive mappings and a common zero of a finite family of monotone operators are also approximated in reflexive Hadamard spaces. In addition, we define a norm on X◇ := spanX∗ and give an application of this norm, where X is an Hadamard space with dual space X∗. A numerical example to solve a nonconvex optimization problem will be exhibited in an Hadamard space to support our main results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aoyama, K., Kamimura, Y., Takahashi, W., Toyoda, M.: Approximation of common fixed point of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 2350–2360 (2007)
Bačák, M.: Convex Analysis and Optimization in Hadamard Spaces, Volume 22 of De Gruyter Series in Nonlinear Analysis and Applications. De Gruyter, Berlin (2014)
Bačák, M., Reich, S.: The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces. J. Fixed Point Theory Appl. 16, 189–202 (2014)
Bauschke, H.H., Matoušková, E., Reich, S.: Projection and proximal point methods: Convergence results and counterexamples. Nonlinear Anal. 56, 715–738 (2004)
Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedi- cata 133, 195–218 (2008)
Bruck, R.E.: A strongly convergent iterative solution of 0 ∈ U(x) for a maximal monotone operator in Hilbert spaces. J. Math. Anal. Appl. 48, 114–126 (1974)
Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)
Bruck, R.E., Reich, S.: A general convergence principle in nonlinear functional analysis. Nonlinear Anal. 4, 939–950 (1980)
Chaipunya, P., Kumam, P.: On the proximal point method in Hadamard spaces. Optimization 66, 1647–1665 (2017)
Chaoha, P., Phon-on, A.: A note on fixed point sets in C A T(0) spaces. J. Math. Anal. Appl. 320, 983–987 (2006)
Cholamjiak, P.: The modified proximal point algorithm in C A T(0) spaces. J. Optim. Lett. 9, 1401–1410 (2014)
Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H. H., Burachik, R., Combettes, P. L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp 185–212. Springer, New York (2011)
Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Monotone point-to-set vector fields. Balkan J. Geom. Appl. 5, 69–79 (2000)
Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Contributions to the study of monotone vector fields. Acta Math. Hungar. 94, 307–320 (2002)
Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Prez, L.R., Németh, S.Z.: Convex- and monotone-transformable mathematical programming problems and a proximal-like point method. J. Global Optim. 35(1), 53–69 (2006)
Dehghan, H., Rooin, J.: A characterization of metric projection in C A T(0) spaces. In: International Conference on Functional Equation, Geometric Functions and Applications (ICFGA 2012) 11-12 th May 2012, pp. 41–43. Payame University, Tabriz (2012)
Dhompongsa, S., Kaewkhao, A., Panyanak, B.: On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on C A T(0) spaces. Nonlinear Anal. 75, 459–468 (2012)
Dhompongsa, S., Kirk, W.A., Sims, B.: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 65, 762–772 (2006)
Dhompongsa, S., Panyanak, B.: On Δ-convergence theorems in C A T(0) spaces. Comput. Math. Appl. 56, 2572–2579 (2008)
Eskandani, G.Z., Raeisi, M.: A new algorithm for finding fixed points of Bregman quasi-nonexpansive mappings and zeros of maximal monotone operators by using products of resolvents. Result Math. 71, 1307–1326 (2017)
Eskandani, G.Z., Azarmi, S., Eslamian, M.: Composition of resolvents and quasi-nonexpansive multivalued mappings in Hadamared spaces. Bult. Iran. Math. Soc. 6, 1939–1955 (2017)
Eskandani, G.Z., Azarmi, S., Raeisi, M.: Products of resolvents and multivalued hybrid mappings in C A T(0) spaces. Acta Math. Sci. 38, 791–804 (2018)
Ferreira, O.P., Lucambio Pérez, L.R., Németh, S.Z.: Singularities of monotone vector fields and an extragradient-type algorithm. J. Global Optim. 31, 133–151 (2005)
Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002)
Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Volume 83 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York (1984)
Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)
Huang, S., Kimura, Y.: A projection method for approximating fixed points of quasi-nonexpansive mappings in Hadamard spaces. Fixed Point Theory Appl. 36, 1–13 (2016)
Kakavandi, B.A.: Weak topologies in complete C A T(0) metric spaces. Proc. Amer. Math. Soc. 141, 1029–1039 (2013)
Kakavandi, B.A., Amini, M.: Duality and subdifferential for convex functions on complete C A T(0) metric spaces. Nonlinear Anal. 73, 3450–3455 (2010)
Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 13, 226–240 (2000)
Khatibzadeh, H., Ranjbar, S.: Monotone operators and the proximal point algorithm in complete C A T(0) metric spaces. J. Aust. Math. Soc. 103, 70–90 (2017)
Kirk, W.A., Panyanak, B.: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008)
Kopecká, E., Reich, S.: Hyperbolic monotonicity in the Hilbert ball. Fixed Point Theory Appl. Art. ID 78104, 15 pages (2006)
Kopecká, E., Reich, S.: Asymptotic behavior of resolvents of coaccretive operators in the Hilbert ball. Nonlinear Anal. 70, 3187–3194 (2009)
Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009)
Maingé, P.E.: Strong convergence of projected pubgradient methods for nonsmooth and nonstrictly convex minimization. Set-valued Anal. 16, 899–912 (2008)
Martinet, B.: Régularisation dinéquations variationelles par approximations successives. Revue Fr. Inform. Rech. Oper. 4, 154–159 (1970)
Nevanlinna, O., Reich, S.: Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces. Isr J. Math. 32, 44–58 (1979)
Németh, S.Z.: Monotone vector fields. Publ. Math. Debrecen. 54, 437–449 (1999)
Raeisi, M., Eskandani, G.Z., Eslamian, M.: A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings. Optimization 68, 309–327 (2018)
Reich, S.: Constructive techniques for accretive and monotone operators. In: Applied Nonlinear Analysis, pp 335–345. Academic Press, New York (1979)
Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)
Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. In: Theory and Applications of Nonlinear Operators, pp 313–318. Marcel Dekker, New York (1996)
Reich, S., Salinas, Z.: Infinite products of discontinuous operators in Banach and metric spaces. Linear Nonlinear Anal. 1, 169–200 (2015)
Reich, S., Salinas, Z.: Weak convergence of infinite products of operators in Hadamard spaces. Rend. Circ. Mat. Palermo 65, 55–71 (2016)
Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537–558 (1990)
Reich, S., Shemen, L.: A note on Halpern’s algorithm in the Hilbert ball. J. Nonlinear Convex Anal. 14, 853–862 (2013)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Saejung, S.: Halpern’s iteration in C A T(0) spaces. Fixed Point Theory Appl. 2010, 471781 (2010)
Shioji, N., Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Amer. Math. Soc. 125, 3641–3645 (1997)
Suparatulatorn, R., Cholamjiak, P., Suantai, S.: On solving the minimization problem and the fixed-point problem for nonexpansive mapping in C A T(0) spaces. Optim. Meth. Soft., 182–192 (2017)
Wangkeeree, R., Preechasilp, P.: Viscosity approximation methods for nonexpansive mappings in C A T(0) spaces. J. Inequal. Appl. 2013, 93 (2013)
Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Austral. Math. Soc. 65, 109–113 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Eskandani, G.Z., Raeisi, M. On the zero point problem of monotone operators in Hadamard spaces. Numer Algor 80, 1155–1179 (2019). https://doi.org/10.1007/s11075-018-0521-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-018-0521-3