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05-02-2020 | Original Paper | Issue 3/2020

Hybrid inertial accelerated algorithms for split fixed point problems of demicontractive mappings and equilibrium problems

Journal:: Numerical Algorithms > Issue 3/2020

Authors:: Adisak Hanjing, Suthep Suantai

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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) MathSciNetCrossRef

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) MathSciNetCrossRef

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) MathSciNetCrossRef

Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, Berlin (2017) CrossRef

Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009) MathSciNetCrossRef

Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 1–23 (1993) MathSciNet

Byrne, C.: Iterative oblique projection onto convex subsets ant the split feasibility problem. Inverse Probl. 18, 441–453 (2002) CrossRef

Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012) MathSciNetMATH

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) MathSciNetCrossRef

10.

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) CrossRef

11.

Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms. 8, 221–239 (1994) MathSciNetCrossRef

12.

Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple set split feasibility problem and its applications. Inverse Problems. 21, 2071–2084 (2005) MathSciNetCrossRef

13.

Censor, Y., Gibali, A., Reich, S.: Algorithm for the split variational inequality problem. Numerical Algorithms 59, 301–323 (2012) MathSciNetCrossRef

14.

Censor, Y., Motova, A., Segal, A.: Perturbed projections and subgradient projections for the multiple-sets feasibility problem. J. Math. Anal. 327, 1244–1256 (2007) MathSciNetCrossRef

15.

Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009) MathSciNetMATH

16.

Chuang, C.S.: Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications. Optimization. 66, 777–792 (2017) MathSciNetCrossRef

17.

Combettes, P.L., Hirstoaga, A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005) MathSciNetMATH

18.

Cui, H., Wang, F.: Iterative methods for the split common fixed point problem in a Hilbert spaces. Fixed Point Theory Appl. 2014, 78 (2014) MathSciNetCrossRef

19.

Dafermos, S.: Exchange price equilibria and variational inequalities. Math. Progam. 46, 391–402 (1990) MathSciNetCrossRef

20.

Farajzadeh, A.P., Zafarani, J.: Equilibrium problem and variational inequalities in topological vector space. Optimization 59(4), 485–499 (2010) MathSciNetCrossRef

21.

Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York (1984)

22.

Iusem, A., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. 52, 621–635 (2003) MathSciNetCrossRef

23.

Konnov, I.: Combined Relaxation Methods for Variational Inequalities. Springer, New York (2001) CrossRef

24.

Mainge, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007) MathSciNetCrossRef

25.

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) MathSciNetMATH

26.

Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bulletin de la Societé Mathématique de France 93, 273–299 (1965) MathSciNetCrossRef

27.

Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26, 587–600 (2010) MathSciNetCrossRef

28.

Moudafi, A.: A note on the split common fixed point problem for quasinonexpansive operators. Nonlinear Anal. 74, 4083–4087 (2011) MathSciNetCrossRef

29.

Moudafi, A., Al-Shemas, E.: Simultaneous iterative methods for split equality problem. Trans. Math. Program. Appl. 1, 1–11 (2013)

30.

Nasri, M., Sosa, W.: Equilibrium problems and generalized Nash games. Optimization 60, 1161–1170 (2011) MathSciNetCrossRef

31.

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)

32.

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) MathSciNetCrossRef

33.

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) MathSciNetCrossRef

34.

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) MathSciNetMATH

35.

Tada, A., Takahashi, W. Takahashi, W, Tanaka, T (eds.): Strong Convergence Theorem for an Equilibrium Problem and a Nonexpansive Mapping. Yokohama Publishers, Yokohama (2005)

36.

Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009) MATH

37.

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) MathSciNetCrossRef

38.

Thung, K., Raveendran, P.: A Survey of Image Quality Measures. In: 2009 International Conference for Technical Postgraduates (TECHPOS), pp 1–4 (2009)

39.

Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002) MathSciNetCrossRef

Title: Hybrid inertial accelerated algorithms for split fixed point problems of demicontractive mappings and equilibrium problems
Authors:: Adisak Hanjing
Suthep Suantai
Publication date: 05-02-2020
DOI: https://doi.org/10.1007/s11075-019-00855-y
Publisher: Springer US
Journal: Numerical Algorithms
Issue 3/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265

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