Abstract
In this paper, we construct a forward–backward splitting algorithm for approximating a zero of the sum of an \(\alpha \)-inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a nonexpansive mapping in the algorithm and prove that the generated sequence converges strongly to a common element of a fixed points set of a nonexpansive mapping and zero points set of the sum of monotone operators. We apply our main result both to equilibrium problems and convex programming.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alber, Y.; Yao, J.C.: Another version of the proximal point algorithm in a Banach space. Nonlinear Anal. 70, 3159–3171 (2009)
Aoyama, K.; Kimura, Y.; Takahashi, W.; Toyoda, M.: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal. 8, 471–489 (2007)
Boikanyo, O.A.; Morosanu, G.: A proximal point algorithm converging strongly for general errors. Optim. Lett. 4, 635–641 (2010)
Boikanyo, O.A.: The viscosity approximation forward-backward splitting method for zeros of the sum of monotone operators. Abstr. Appl. Anal. 2016, 2371857 (2016). https://doi.org/10.1155/2016/2371857
Bruck, R.E.; Passty, G.B.: Almost convergence of the infinite product of resolvents in Banach spaces. Nonlinear Anal. 3, 279–282 (1979)
Burachik, R.S.; Scheimberg, S.: A proximal point method for the variational inequality problem in Banach spaces. SIAM J. Control Optim. 39, 1633–1649 (2000)
Chang, S.S.; Lee, H.J.; Chan, C.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. Theory Methods Appl. 70(9), 3307–3319 (2009)
Cho, S.Y.; Qin, X.; Kang, S.M.: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 57, 1429–1446 (2013)
Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62. Mathematics and Its ApplicationsKluwer Academic Publishers, Dordrecht (1990)
Dadashi, V.: Shrinking projection algorithms for the split common null point problem. Bull. Aust. Math. Soc. 96, 299–306 (2017)
Dadashi, V.: On a hybrid proximal point algorithm in Banach spaces. Univ. Politeh. Buchar. Ser. A 80(3), 45–54 (2018)
Dadashi, V.; Khatibzadeh, H.: On the weak and strong convergence of the proximal point algorithm in reflexive Banach spaces. Optimization 66(9), 1487–1494 (2017)
Dadashi, V.; Postolache, M.: Hybrid proximal point algorithm and applications to equilibrium problems and convex programming. J. Optim. Theory Appl. 174(2), 518–529 (2017)
Goebel, K.; Kirk, W.A.: Topics in Metric Fixed Point Theory, vol. 28. Cambridge Studies in Advanced MathematicsCambridge University Press, Cambridge (1990)
Hadjisavvas, N.I.; Khatibzadeh, H.: Maximal monotonicity of bifunctions. Optimization 59(2), 147–160 (2010)
Kamimura, S.; Takahashi, W.: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set Valued Anal. 8, 361–374 (2000)
Kang, S.; Cho, S.; Liu, Z.: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010(1), 827082 (2010)
Khatibzadeh, H.: Some remarks on the proximal point algorithm. J. Optim. Theory Appl. 153, 769–778 (2012)
Kinderlehrer, D.; Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
Li, L.; Song, W.: Modified proximal point algorithm for maximal monotone operators in Banach spaces. J. Optim. Theory Appl. 138, 45–64 (2008)
Liu, L.S.: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 194(1), 114–125 (1995)
Lv, S.: Generalized systems of variational inclusions involving (A, \(\eta \))-monotone mappings. Adv. Fixed Point Theory 1(1), 15 (2011)
Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Française Informat. Recherche Opérationnelle 3, 154–158 (1970)
Matsushita, S.; Xu, L.: On convergence of the proximal point algorithm in Banach spaces. Proc. Am. Math. Soc. 139, 4087–4095 (2011)
Moudafi, A.: On the regularization of the sum of two maximal monotone operators. Nonlinear Anal. Theory Methods Appl. 42(7), 1203–1208 (2000)
Moudafi, A.: On the convergence of the forward-backward algorithm for null-point problems. J. Nonlinear Var. Anal. 2, 263–268 (2018)
Moudafi, A.; Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155(2), 447–454 (2003)
Moudafi, A.; Thera, M.: Finding a zero of the sum of two maximal monotone operators. J. Optim. Theory Appl. 94(2), 425–448 (1997)
Nadezhkina, N.; Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006)
Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72(2), 383–390 (1979)
Qin, X.; Su, Y.: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 329, 415–424 (2007)
Qin, X.; Cho, Y.J.; Kang, S.M.: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 225(1), 20–30 (2009)
Qin, X.; Cho, S.Y.; Wang, L.: A regularization method for treating zero points of the sum of two monotone operators. Fixed Point Theory Appl. 2014, 75 (2014). https://doi.org/10.1186/1687-1812-2014-75
Rockafellar, R.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33(1), 209–216 (1970)
Rockafellar, R.T.: Maximal monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Rouhani, B.D.; Khatibzadeh, H.: On the proximal point algorithm. J. Optim. Theory Appl. 137, 411–417 (2008)
Schu, J.: Weak and strong convergence to fixed points of a asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43, 153–159 (1991)
Suzuki, T.: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 103–123 (2005)
Takahashi, W.: Approximating solutions of accretive operators by viscosity approximation methods in Banach spaces. In: Applied Functional Analysis, pp. 225–243, Yokohama Publishers, Yokohama (2007)
Takahashi, S.; Takahashi, W.; Toyoda, M.: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147(1), 2741 (2010)
Tian, C.; Wang, F.: The contraction-proximal point algorithm with square-summable errors. Fixed Point Theory Appl. 2013, 93 (2013). https://doi.org/10.1186/1687-1812-2013-93
Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)
Wang, F.; Cui, H.: On the contraction-proximal point algorithms with multi-parameters. J. Glob. Optim. 54(3), 485–491 (2012)
Wang, F.; Cui, H.: Convergence of the generalized contraction-proximal point algorithm in a Hilbert space. Optimization 64(4), 709–715 (2015)
Yao, Y.; Noor, M.A.: On convergence criteria of generalized proximal point algorithms. J. Comput. Appl. Math. 217(1), 46–55 (2008)
Yao, Y.; Shahzad, N.: Strong convergence of a proximal point algorithm with general errors. Optim. Lett. 6(4), 621–628 (2012)
Yuan, H.: A splitting algorithm in a uniformly convex and 2-uniformly smooth Banach space. J. Nonlinear Funct. Anal. 2018, 26 (2018). https://doi.org/10.23952/jnfa.2018.26
Zhang, S.; Lee, J.H.W.; Chan, C.K.: Algorithms of common solutions to quasi variational inclusion and fixed point problems. Appl. Math. Mech. Engl. Ed. 29(5), 571581 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Dadashi, V., Postolache, M. Forward–backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators. Arab. J. Math. 9, 89–99 (2020). https://doi.org/10.1007/s40065-018-0236-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-018-0236-2