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Approximation of common solutions of nonlinear problems involving various classes of mappings. (English) Zbl 07008522

Summary: A new iterative scheme is proposed which approximates a common solution of split equality fixed point problems involving \(\eta\)-demimetric mappings, finite family of \(\gamma \)-inverse strongly monotone mappings, finite family of relatively quasi-nonexpansive mappings and finite family of system of generalized mixed equilibrium problems in real Banach spaces which are 2-uniformly convex and uniformly smooth. Our theorems extend and complement several existing results in this area of research.

MSC:

47H06 Nonlinear accretive operators, dissipative operators, etc.
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05 Equations involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators
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[1] Alber, Y.I.: Metric and generalized projection operator in Banach space: properties and applications. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15-50. Dekker, New York (1996) · Zbl 0883.47083
[2] Alsulami, S.M., Takahashi, W.: The split common null point problem for maximal monotone mappings in Hilbert spaces and applications. J. Nonlinear Anal. 15, 793-808 (2014) · Zbl 1296.47044
[3] Beauzamy, B.: Introduction to Banach Spaces and Their Geometry, 2nd edn. Elsevier, North Holland (1995) · Zbl 0491.46014
[4] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 64, 123-145 (1994) · Zbl 0888.49007
[5] Bregman, L.M.: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR COmput. Math. Math. Phys. 7, 200-217 (1967)
[6] Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459-470 (1977) · Zbl 0383.47035
[7] Butnariu, D., Reich, S., Zalslavski, A.J.: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 7, 151-174 (2001) · Zbl 1010.47032
[8] Bryme, C., Censor, Y., Gibali, A., Reich, S.: The split common nullpoint problem. J. Nonlinear Convex Anal. 13, 759-775 (2012) · Zbl 1262.47073
[9] Bryme, C., Censor, Y., Gibali, A., Reich, S.: The split common nullpoint problem. Inverse Probl. 18, 441-453 (2012)
[10] Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer Algorithms 8(2-4), 221-239 (1994) · Zbl 0828.65065
[11] Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071-2084 (2005) · Zbl 1089.65046
[12] Censor, Y., Lent, A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321-353 (1981) · Zbl 0431.49042
[13] Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587600 (2009) · Zbl 1189.65111
[14] Chidume, C.E.: Geometric properties of Banach spaces and nonlinear iterations. Springer Verlag Series: Lecture Notes in Mathematics, vol. 1965, XVII, p. 326 (2009) (ISBN 978-1-84882-189-7) · Zbl 1167.47002
[15] Cioranescu, I.: Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer, Dordrecht, 1990, and its review by S. Reich. Bull. Am. Math. Soc. 26, 367-370 (1992)
[16] Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry and nonexpansive mappings. Marcel Dekker, New York (1984) · Zbl 0537.46001
[17] Hojo, M., Takahashi, W.: The split common fixed point problem and the hybrid method in Banach spaces. Linear and nonlinear Analysis 1(2), 305-315 (2015) · Zbl 1338.47093
[18] Kamimura, S., Takahashi, W.: Strong convergence of proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-945 (2002) · Zbl 1101.90083
[19] Kocourck, P., Takahashi, W., Yao, J.-C.: Fixed point theorems and weak convergence theorem for generalized hybrid mappings in Hilbert spaces. Taiwan. J. Math. 14, 2497-2511 (2010) · Zbl 1226.47053
[20] Kohasaka, F., Takahashi, W.: Proximal point algorithm with Bregman functions in Banach spaces. J. Nonlinear Convex Anal. 6, 505-523 (2005) · Zbl 1105.47059
[21] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979) · Zbl 0403.46022
[22] Mahdioui, H., Chadli, O.: On a system of generalized mixed equilibrium problems involving variational-like inequalities in Banach spaces: Existence and Algorithmic aspects. Adv. Oper. Res. 2012 (2012) (Article ID 843486) · Zbl 1246.49009
[23] Mainge, P.E.: Strong convergence of projected subgradient methods for nonsmooth and non-strictly convex minimization. Set-Valued Anal. 16, 899-912 (2008) · Zbl 1156.90426
[24] Martin-Marquez, V., Reich, S., Sabach, S.: Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discret. Contin. Dyn. Syst. Ser. 6, 1043-1063 (2013) · Zbl 1266.26023
[25] Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8(3), 367 (2007) · Zbl 1171.90009
[26] Moudafi, A.: The split common fixed point problem for demi-contractive mappings. Inverse Probl. 26, 055007 (2010) · Zbl 1219.90185
[27] Moudafi, A., Al-Shemas, E.: Simultaneous iterative methods for split equality problem. Trans. Math. Program. Appl. 1(2), 1-11 (2013)
[28] Moudafi, A., Thera, M.: Proximal and dynamical approaches to equilibrium problems. In: Lecture notes in Economics and Mathematics Systems, vol. 477, pp. 187-201. Springer, Berlin (1999) · Zbl 0944.65080
[29] Nilsrakoo, W., Saejung, S.: Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings. Fixed Points Theory Appl. 2008, 312454 (2008). https://doi.org/10.1155/2008/312454 · Zbl 1203.47061
[30] Ofoedu, E.U.: A general approximation scheme for solutions of various problems in fixed point theory. Int. J. Anal. 2013, 18 (2013). https://doi.org/10.1155/2013/762831. (Article ID 76281) · Zbl 1268.47081 · doi:10.1155/2013/762831
[31] Ofoedu, E.U., Shehu, Y.: Convergence analysis for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems. Appl. Math. Comp. 217, 9142-9150 (2011) · Zbl 1308.47074
[32] Ofoedu, E.U., Malonza, D.M.: Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type. Appl. Math. Comp. 217(13), 6019-6030 (2011) · Zbl 1215.65105
[33] Phelps, R.P.: Convex functions, Monotone operators and differentiablity, 2nd edn. In: Lecture Notes in Mathematics, vol. 1364. Springer, Berlin (1993)
[34] 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, 20-30 (2009) · Zbl 1165.65027
[35] Reich, S.: A weak convergence theorem for the alternating method with Bregmann distance. In: ’Theory and applications of nonlinear operators’, pp. 313-318. Marcel Dekker, New York (1996) · Zbl 0943.47040
[36] Sunthrayuth, P., Kuman, P.: A system of generalized mixed equilibrium problems, maximal monotone operators, and fixed point problems with application to optimization problems. Abstract and Applied Analysis (2012) (Article ID 316276) · Zbl 1232.49037
[37] Takahashi, W.: Nonlinear Functional Analysis—Fixed Point Theory and Applications. Yokohama Publishers Inc., Yokohama (2000). (In Japanese) · Zbl 0997.47002
[38] Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers Inc., Yokohama (2000) · Zbl 0997.47002
[39] Takahashi, W.: Convex analysis and approximation of fixed points, pp. iv+276. Yokohama Publishers, Yokohama (2000) (4-946552-04-9: Japanese)
[40] Takahashi, W.: The split common null point problem in two Banach spaces. J. Nonlinear Convex Anal. 16(12), 2343-2350 (2015) · Zbl 1336.47059
[41] Takahashi, W.: The split common fixed point problem and the shrinking projection method in Banach spaces. J. Convex Anal. (to appear) · Zbl 1503.47097
[42] Takahashi, W.: Iterative methods for split feasibility problems and split common null point problems in Banach spaces. In: The 9th International Conference on Nonlinear Analysis and Convex Analysis, Chiang Rai, Thailand, 21-25 Jan (2015)
[43] Takahashi, W., Wen, C.-F., Yao, J.-C.: Strong convergence theorems by shrinking projection method for new nonlinear mappings in Banach spaces and applications. Optimization 66(4), 609-621 (2017) · Zbl 1373.49005
[44] Takahashi, W., Yao, J.-C.: Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces. Fixed Point Theory Appl. 2015, 87 (2015) · Zbl 1344.47049
[45] Tahahashi, W., Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45-57 (2009) · Zbl 1170.47049
[46] Wang, Y., Kim, T.-H.: Simultaneous iterative algorithm for the split equality fixed-point problem of demicontractive mappings. J. Nonlinear Sci. Appl. 10, 154165 (2017)
[47] Wattanawitoon, K., Kuman, P.: Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 3, 11-20 (2009) · Zbl 1166.47060
[48] Xu, H.K.: Inequalities of Banach spaces with applications. Nonlinear Anal. 16(2), 1127-1138 (1991) · Zbl 0757.46033
[49] Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109-113 (2002) · Zbl 1030.47036
[50] Zegeye, H.: Strong convergence theorems for split equality fixed point problems of \[\eta\] η-demimetric mappings in Banach spaces. (to appear) · Zbl 1522.47102
[51] Zhang, J., Su, Y., Cheng, Q.: The approximation of common element for maximal monotone operator, generalized mixed equilibrium problem and fixed point problem. J. Egypt. Math. Soc. 23, 326-333 (2015) · Zbl 1326.47103
[52] Zhang, S.: Generalized mixed equilibrium problem in Banach spaces. Appl. Math. Engl. Ed. 30, 1105 (2009) · Zbl 1178.47051
[53] Zhao, J.: Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operator norms. Optimization 64(12), 2619-2630 (2015) · Zbl 1326.47104
[54] Zhao, J., Yang, Q.: Several solution methods for the split feasibility problem. Inverse Probl. 21(5), 1791-1799 (2005) · Zbl 1080.65035
[55] Zhu, J., Chang, S-S., Liu, M.: Generalized mixed equilibrium problems and fixed point problem for a countable family of total quasi-\[ \phi\] ϕ-asymtotically nonexpansive mappings in Banach spaces. J. Appl. Math. 2012, 961560 (2012). https://doi.org/10.1155/2012/961560
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