Abstract
Our contribution in this paper, we introduce and analyze two new hybrid algorithms by combining Mann iteration and inertial method for solving split fixed point problems of demicontractive mappings and equilibrium problems in a real Hilbert space. By using a new technique of choosing step size, our algorithms do not need any prior information on the operator norm. In fact, an inertial type algorithm was proposed in order to accelerate its convergence rate. We then prove weak and strong convergence of proposed methods under some control conditions. Moreover, some numerical experiments for image restoration problems and oligopolistic market equilibrium problems are also provided for supporting our main results.
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Alvarez, F., Attouch, H.: An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)
Anh, P.N., Le Thi, H.A.: An Armijo-type method for pseudomonotone equilibrium problems and its applications. J. Glob. Optim. 57, 803–820 (2013)
Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities. J. Optim Theory Appl. 124, 285���306 (2005)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, Berlin (2017)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 1–23 (1993)
Byrne, C.: Iterative oblique projection onto convex subsets ant the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)
Ceng, L.C., Yao, J.C.: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186–201 (2008)
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., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms. 8, 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple set split feasibility problem and its applications. Inverse Problems. 21, 2071–2084 (2005)
Censor, Y., Gibali, A., Reich, S.: Algorithm for the split variational inequality problem. Numerical Algorithms 59, 301–323 (2012)
Censor, Y., Motova, A., Segal, A.: Perturbed projections and subgradient projections for the multiple-sets feasibility problem. J. Math. Anal. 327, 1244–1256 (2007)
Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)
Chuang, C.S.: Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications. Optimization. 66, 777–792 (2017)
Combettes, P.L., Hirstoaga, A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Cui, H., Wang, F.: Iterative methods for the split common fixed point problem in a Hilbert spaces. Fixed Point Theory Appl. 2014, 78 (2014)
Dafermos, S.: Exchange price equilibria and variational inequalities. Math. Progam. 46, 391–402 (1990)
Farajzadeh, A.P., Zafarani, J.: Equilibrium problem and variational inequalities in topological vector space. Optimization 59(4), 485–499 (2010)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York (1984)
Iusem, A., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. 52, 621–635 (2003)
Konnov, I.: Combined Relaxation Methods for Variational Inequalities. Springer, New York (2001)
Mainge, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007)
Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)
Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bulletin de la Societé Mathématique de France 93, 273–299 (1965)
Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26, 587–600 (2010)
Moudafi, A.: A note on the split common fixed point problem for quasinonexpansive operators. Nonlinear Anal. 74, 4083–4087 (2011)
Moudafi, A., Al-Shemas, E.: Simultaneous iterative methods for split equality problem. Trans. Math. Program. Appl. 1, 1–11 (2013)
Nasri, M., Sosa, W.: Equilibrium problems and generalized Nash games. Optimization 60, 1161–1170 (2011)
Peng, J.W., Liou, Y.C., Yao, J.C.: An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions. Fixed Point Theory Appl. 2009 Article ID 794178 (2009)
Reich, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemporary Math. 568, 225–240 (2012)
Shehu, Y., Mewomo, O.T., Ogbuisi, F.U.: Further investigation into approximation of a common solution of fixed point problems and split feasibility problems. Acta Math Sci. 36B, 913–930 (2016)
Shehu, Y., Ogbuisi, F.U.: An iterative algorithm for approximating a solution of split common fixed point problem for demi-contractive maps. Dynam. Cont. Dis. Ser. B. 23, 205–216 (2016)
Tada, A., Takahashi, W. Takahashi, W, Tanaka, T (eds.): Strong Convergence Theorem for an Equilibrium Problem and a Nonexpansive Mapping. Yokohama Publishers, Yokohama (2005)
Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009)
Tan, K.-K, Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178(2), 301–308 (1993)
Thung, K., Raveendran, P.: A Survey of Image Quality Measures. In: 2009 International Conference for Technical Postgraduates (TECHPOS), pp 1–4 (2009)
Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002)
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This study was financially supported by Chiang Mai University.
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Hanjing, A., Suantai, S. Hybrid inertial accelerated algorithms for split fixed point problems of demicontractive mappings and equilibrium problems. Numer Algor 85, 1051–1073 (2020). https://doi.org/10.1007/s11075-019-00855-y
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DOI: https://doi.org/10.1007/s11075-019-00855-y