Abstract
In this paper, we propose a hybrid iterative method to approximate a common solution of split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. We prove that sequences generated by the proposed hybrid iterative method converge strongly to a common solution of these problems. Further, we discuss some applications of the main result. We also discuss a numerical example to demonstrate the applicability of the iterative method. The method and results presented in this paper extend and unify the corresponding known results in this area.
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Aoyama, K., Kimura, Y., Takahashi, W.: Maximal monotone operators and maximal monotone functions for equilibrium problems. J. Convex Anal. 15(2), 395–409 (2008)
Atsushiba, S., Takahashi, W.: Strong Convergence Theorems for a Finite Family of Nonexpansive Mappings and Applications, in: B.N. Prasad birth centenary commemoration volume. Indian J. Math. 41(3), 435–453 (1999)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Bnouhachem, A.: Strong convergence algorithm for split equilibrium problem and hierarchical fixed point problems. Sci. World J. 2014, Article ID 390956, 12 (2014)
Bnouhachem, A., Ansari, Q.H., Yao, J.C.: An iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings. Fixed Point Theory Appl. 2015:111, 13 (2015)
Bnouhachem, A., Ansari, Q.H., Yao, J.C.: Stromg convergence algorithm for hierarchical fixed point problems of a finite family of nonexpansive mappings. Fixed Point Theory 17(1), 47–62 (2016)
Bre~zis, H.: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Mathematical Studies(Amsterdam: North-Holand) 5, 759–775 (1973)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13(4), 759–775 (2012)
Cabot, A.: Proximal point algorithm controlled by a slowly vanishing term: application to hierarchical minimization. SIAM J. Optim. 15, 555–572 (2005)
Ceng, L.C., Petruşel, A.: Krasnoselski-mann iterations for hierarchical fixed point problems for a finite family of nonself mappings in Banach spaces. J. Optim. Theory Appl. 146, 617–639 (2010)
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59(2), 301–323 (2012)
Colao, V., Marino, G., Xu, H.K.: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)
Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron Physics 95, 155–453 (1996)
Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Geobel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)
Kazmi, K.R., Ali, R., Furkan, M.: Krasnoselski-mann type iterative method for hierarchical fixed point problem and split mixed equilibrium problem. Numerical Algorithms (2017), https://doi.org/10.1007/s11075-017-0316-y
Kazmi, K.R., Rizvi, S.H.: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J. Egy. Math. Soc. 44–51, 21 (2013)
Kazmi, K.R., Rizvi, S.H.: An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping. Optim. Letters 8, 1113–1124 (2014)
Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)
Marino, G., Colao, V., Muglia, L., Yao, Y.: Krasnoselski-mann iteration for hierarchical fixed-points and equilibrium problem. Bull. Aust. Math. Soc. 79, 187–200 (2009)
Moudafi, A.: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 23, 1635–1640 (2007)
Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)
Moudafi, A., Mainge, P.-E.: Towards viscosity approximations of hierarchical fixed-point problems. Fixed Point Theory Appl. 2006, Article ID 95453 (2006)
Moudafi, A., Mainge, P.-E.: Strong convergence of an iterative method for hierarchical fixed-point problems. Pacific J. Optim. 3, 529–538 (2007)
Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroup. J. Math. Anal. Appl. 279, 372–379 (2003)
Shehu, Y., Ogbuisi, F.U.: An iterative method for solving split monotone variational inclusion and fixed point problems. RACSAM 110(2), 503–518 (2016)
Takahashi, W.: Weak and strong convergence theorems for families of nonexpansive mappings and their applications. Ann. Univ. Mariae Curie-Sklodowska. 51, 277–292 (1997)
Takahashi, W., Shimoji, K.: Convergence theorems for nonexpansive mappings and feasibility problems. Math. Comput. Modelling. 32, 1463–1471 (2000)
Yamada, I., Ogura, N.: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004)
Yang, Q., Zhao, J.: Generalized KM theorem and their applications. Inverse Probl. 22, 833–844 (2006)
Yao, Y.: A general iterative method for a finite family of nonexpansive mappings. Nonlinear Anal. 66, 2676–2687 (2007)
Yao, Y., Liou, Y.C.: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 24, 501–508 (2008)
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The authors are grateful to the anonymous referees for their constructive and helpful comments and suggestions towards the improvement of the original manuscript.
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Kazmi, K.R., Ali, R. & Furkan, M. Hybrid iterative method for split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings. Numer Algor 79, 499–527 (2018). https://doi.org/10.1007/s11075-017-0448-0
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DOI: https://doi.org/10.1007/s11075-017-0448-0
Keywords
- Split monotone variational inclusion problem
- Hierarchical fixed point problem
- Hybrid iterative method
- Maximal monotone operators
- Nonexpansive mapping
- Strong convergence