Abstract
In this paper, we introduce Halpern-type proximal point algorithm for approximating a common solution of monotone inclusion problem and fixed point problem. We obtain a strong convergence of the proposed algorithm to a common solution of finite family of monotone inclusion problem and fixed point problem for nonexpansive mappings in complete CAT(0) spaces. Nontrivial application and numerical example were given. Our results complement and extend some recent results in literature.
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Ahmadi Kakavandi, B., Amini, M.: Duality and subdifferential for convex functions on complete CAT(0) metric spaces. Nonlinear Anal. 73, 3450–3455 (2010)
Bačák, M.: The proximal point algorithm in metric spaces. Israel J. Math. 194, 689–701 (2013)
Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedicata 133, 195–218 (2008)
Bridson, M.R., Haefliger, A.: Metric Spaces of Non-Positive Curvature, Fundamental Principle of Mathematical Sciences, vol. 319. Springer, Berlin, Germany (1999)
Bruhat, F., Tits, J.: Groupes Réductifs sur un Corp Local, I. Donneés Radicielles Valuées, 41 Institut des Hautes Études Scientifiques, (1972)
Burago, D., Burago, Y., Ivanov, S.: A Course in metric geometry, graduate studies in mathematics, 33 American Mathematical Society, Providence, (2001)
Chaoha, P., Phon-on, A.: A note on fixed point sets in CAT(0) spaces. J. Math. Anal. Appl. 320(2), 983–987 (2006)
Dehghan, H., Rooin, J.: Metric projection and convergence theorems for nonexpansive mapping in Hadamard spaces, 5 Oct. (2014) arXiv:1410.1137VI [math.FA]
Dhompongsa, S., Panyanak, B.: On \(\Delta \)-convergence theorems in CAT(0) spaces. Comput. Math. Appl 56, 2572–2579 (2008)
Dhompongsa, S., Kirk, W.A., Panyanak, B.: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear and Convex Anal. 8, 35–45 (2007)
Dhompongsa, S., Kirk, W.A., Sims, B.: Fixed points of unifromly Lipschitzian mappings. Nonlinear Amalysis 64(4), 762–772 (2006)
Espínola, R., Fernández-León, A.: CAT(k)-spaces, weak convergence and fixed points. J. Math. Anal. Appl. 353, 410–427 (2009)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Marcel Dekker, New York (1984)
Gromov, M., Bates, S.M.: Metric Structures for Riemannian and Non-Riemannian Spaces, with Appendices by M. Katz, P. Pansu and S. Semmes. In: Lafontaine, S.M., Pansu, P. (eds.) Progr. Math., vol. 152. BirkhNauser, Boston (1999)
Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)
Jost, J.: Nonpositive Curvature: Geometric and Analytic Aspects, Lectures Math. ETH ZNurich. BirkhNauser, Basel (1997)
Kakavandi, B.A., Amini, M.: Duality and subdifferential for convex functions on complete CAT(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 106, 226–240 (2000)
Khatibzadeh, H., Ranjbar, S.: Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces. J. Aust. Math Soc. (2017). https://doi.org/10.1017/S1446788716000446
Kirk, W.A.: Geodesic geometry and fixed point theory. II, International Conference on Fixed Point Theory and Applications, pp. 113–142. Yokohoma Publisher, Yokohoma (2004)
Kirk, W.A., Panyanak, B.: A concept of convergence in geodesic spaces. Nonlinear Analysis: Theory, Methods & Applications 56, 3689–3696 (2008)
Leustean, L.: Nonexpansive iterations uniformly cover W-hyperbolic spaces. Nonlinear Analysis and Optimization 1: Nonlinear Analysis. Contemporary Math. Am. Math. Soc., Providence 513, 193–209 (2010)
Lim, T.C.: Remarks on some fixed point theorems. Proc. Amer. Math. Soc. 60, 179–182 (1976)
Martinet, B.: R\(\acute{e}\)gularisation d’In\(\acute{e}\)quations Variationnelles par Approximations Successives. Rev.Fran\(\acute{c}\)aise d’Inform. et de Rech. Op\(\acute{e}\)rationnelle 3, 154–158 (1970)
Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)
Ogbuisi, F.U., Mewomo, O.T.: Iterative solution of split variational inclusion problem in real Banach space. Afr. Mat. 28, 295–309 (2017)
Ranjbar, S., Khatibzadeh, H.: Strong and \(\Delta \)-convergence to a zero of a monotone operator in CAT(0) spaces, Mediterr. J. Math., 14 (2), Art. 56, 15 pp. (2017)
Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537–558 (1990)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Ugwunnadi, G.C., Ali, B.: Approximation of Common Fixed Points of Total Asymptotically Nonexpansive mapping in CAT(0) spaces. Advances in Nonlinear Variational Inequalities 19(1), 36–47 (2016)
Xu, H.K.: Iterative algorithms for nonlinear operators. J. London. Math. Soc. 2, 240–256 (2002)
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Ugwunnadi, G.C., Izuchukwu, C. & Mewomo, O.T. Strong convergence theorem for monotone inclusion problem in CAT(0) spaces. Afr. Mat. 30, 151–169 (2019). https://doi.org/10.1007/s13370-018-0633-x
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DOI: https://doi.org/10.1007/s13370-018-0633-x