Strong convergence of the shrinking projection method for the split equilibrium problem and an infinite family of relatively nonexpansive mappings in Banach spaces. (English) Zbl 1463.47189
Summary: In this paper, we use the shrinking projection method to prove a strong convergence theorem for finding a common solution of the split equilibrium problem and fixed point problem of a relatively quasi-nonexpansive mapping. Consequently, our main theorem can apply to find a common solution of the split equilibrium problem and common fixed point problem for an infinite family of relatively nonexpansive mappings in Banach spaces.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H05 | Monotone operators and generalizations |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
split equilibrium problem; equilibrium problem; relatively quasi-nonexpansive; relatively nonexpansive; common fixed point; shrinking projection method; Banach space; strong convergenceReferences:
| [1] | Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algor. 8 (1994) 221-239. · Zbl 0828.65065 |
| [2] | S.M. Alsulami, W. Takahashi, The split common null point problem for maximal monotone mappings in Hilbert spaces and applications, J. Nonlinear Convex Anal. 15 (2014) 793-808. · Zbl 1296.47044 |
| [3] | C. Byrne, Y. Censor, A. Gibali, S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13 (2012) 759-775. · Zbl 1262.47073 |
| [4] | W. Takahashi, H.-K. Xu, J. -C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. 23 (2015) 205-221. · Zbl 1326.47099 |
| [5] | H.-K. Xu, A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems. 22 (2006) 2021-2034. · Zbl 1126.47057 |
| [6] | W. Takahashi, The split feasibility problem in Banach spaces, J. Nonlinear Convex Anal. 15 (2014) 1349-1355. · Zbl 1319.47063 |
| [7] | W. Takahashi, The split common null point problem in two Banach spaces, Arc. Math. 104 (2015) 357-365. · Zbl 1458.47034 |
| [8] | S. Suantai, Y. Shehu, P. Cholamjiak, O.-S. Iyiola, Strong convergence of a selfadaptive method for the split feasibility problem in Banach spaces, J. Fixed Point Theory Appl. (2018) 20:68. · Zbl 1518.47107 |
| [9] | W. Takahashi, The split common null point problem in two Banach spaces, J. Nonlinear Convex Anal. 16 (2015) 2343-2350. · Zbl 1336.47059 |
| [10] | W. Takahashi, The split common null point problem for generalized resolvents in two Banach spaces, Numer. Algorithms. 75 (2017) 1065-1078. · Zbl 06787454 |
| [11] | S. Takahashi, W. Takahashi, The split common null point problem and the shrinking projection method in two Banach spaces, Linear Nonlinear Anal. 1 (2015) 297-304. · Zbl 1338.47109 |
| [12] | Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal. 16 (2009) 587-600. · Zbl 1189.65111 |
| [13] | A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems 26 (2010) 6 pages. · Zbl 1219.90185 |
| [14] | W. Takahashi, The split common fixed point problem and strong convergence theorems by hybrid methods in two Banach spaces, J. Nonlinear Convex Anal. 17 (2016) 1051-1067. · Zbl 1448.47075 |
| [15] | W. Takahashi, The split common fixed point problem and the hybrid method for demigeneralized mappings in two Banach spaces and applications, J. Nonlinear Convex Anal. 18 (2017) 29-45. · Zbl 1477.47074 |
| [16] | W. Takahashi, C.-F. Wen, J. -C. Yao, The split common fixed point problem with families of mappings and hybrid methods in two Banach spaces, J. Nonlinear Convex Anal. 17 (2016) 1643-1658. · Zbl 06661908 |
| [17] | H. Nikaidˆo, K. Isoda, Note on noncooperative convex games. Pac. J. Math. 5 (1955) 807-815. · Zbl 0171.40903 |
| [18] | K. Fan, A minimax inequality and applications, in Inequalities, III, O. Shisha, Ed., Academic Press, New York, NY, USA, (1972) 103-113. · Zbl 0302.49019 |
| [19] | E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student 63 (1994) 123-145. · Zbl 0888.49007 |
| [20] | P. L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 117-136. · Zbl 1109.90079 |
| [21] | A. Moudafi, M. Th´era, Proximal and dynamical approaches to equilibrium problems, in: Lecture Notes in Economics and Mathematical Systems, Springer 477 (1999) 187-201. · Zbl 0944.65080 |
| [22] | S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506-515. · Zbl 1122.47056 |
| [23] | S. Dhompongsa, W. Inthakon, W. Takahashi, Strong and weak convergence theorems for equilibrium problems and generalized nonexpansive mappings in Banach spaces, in Fixed Point Theory and its Applications (L. J. Lin , A. Petru¸sel and H. K. Xu, eds.), Yokohama Publishers, Yokohama (2009) 39-54. · Zbl 1221.47114 |
| [24] | W. Inthakon, Strong convergence theorems for generalized nonexpansive mappings with the system of equilibrium problems in Banach spaces, J. Nonlinear Convex Anal. 15 (2014) 753-763. · Zbl 1297.47080 |
| [25] | P. Kumam, P. Katchang, A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings, Nonlinear Anal. Hybrid Syst. 3 (2009) 475- 486. · Zbl 1221.49011 |
| [26] | A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. Nonlinear Convex Anal. 9 (2008) 37-43. · Zbl 1167.47049 |
| [27] | S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007) 455-469. · Zbl 1127.47053 |
| [28] | S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008) 1025-1033. · Zbl 1142.47350 |
| [29] | W. Takahashi, J.-C. Yao, Nonlinear operators of monotone type and convergence theorems with equilibrium problems in Banach spaces, Taiwanese J. Math. 1 (2011) 787-818. · Zbl 1248.47050 |
| [30] | W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009) 45-57. · Zbl 1170.47049 |
| [31] | Z. He, The split equilibrium problem and its convergence algorithms, J. Ineq. Appl. (2012) 2012:162. · Zbl 1291.47054 |
| [32] | A. Bnouhachem, Strong convergence algorithm for split equilibrium problems and hierarchical fixed point problems, Scientific World J. (2014) Article ID 390956. · Zbl 1314.49002 |
| [33] | K. R. Kazmi, S. H. Rizvi, Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, J. Egyptian Math. Soc. 21 (2013) 44-51. · Zbl 1277.49009 |
| [34] | S.-H. Wang, X.-Y. Gong, A.A. Abdou, Y.J. Cho, Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications, Fixed Point Theory Appl. (2016) 2016:4. · Zbl 1346.47068 |
| [35] | B.H. Guo, P. Ping, H. Q. Zhao, Y.J. Cho, Strong and weak convergence theorems for spilt equilibrium problems and fixed point problem in Banach spaces, J. Nonlinear Sci. 10 (2017) 2886-2901. · Zbl 1412.47033 |
| [36] | W. Inthakon, N. Niyamosot, The split equilibrium problem and common fixed points of two relatively quasi-nonexpansive mappings in Banach spaces, J. Nonlinear Convex Anal. 20 (2019), 685-702. · Zbl 1478.47079 |
| [37] | W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 341 (2008) 276-286. · Zbl 1134.47052 |
| [38] | S. Suantai, W. Phuengrattana, A hybrid shrinking projection method for common fixed points of a finite family of demicontractive mappings with variational inequality problems, Banach J. Math. Anal. 11 (2017) 661-675. · Zbl 06754307 |
| [39] | W. Takahashi, K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl. (2008) Article ID 528476. · Zbl 1187.47054 |
| [40] | Y.I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., Dekker, New York 178 (1996) 15-50. · Zbl 0883.47083 |
| [41] | S. Reich, Constructive techniques for accretive and monotone operators, Applied Nonlinear Analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978), Academic Press, New York, (1979) 335-345. · Zbl 0444.47042 |
| [42] | D. Butnariu, S. Reich, A.J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7 (2001) 151-174. · Zbl 1010.47032 |
| [43] | S.-Y. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2) (2005) 257-266. · Zbl 1071.47063 |
| [44] | F. Kohsaka, W. Takahashi, The set of common fixed points of an infinite family of relatively nonexpansive mappings, In: Banach and Function Spaces II, Yokohama Publ., Yokohama (2008) 361-373. · Zbl 1157.47036 |
| [45] | W. |
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