Iterative approximation methods for mixed equilibrium problems for a countable family of quasi-\(\phi\)-asymptotically nonexpansive multivalued mappings in Banach spaces. (English) Zbl 1476.47079
Summary: In this paper, we prove the existence of a solution of the mixed equilibrium problem \(MEP(f ,\varphi, C)\) by using the KKM mapping in a Banach space setting. Then, by virtue of this result, we introduce a hybrid iterative scheme for finding a common element of the set of solutions of \(MEP(f ,\varphi, C)\) and the set of common fixed points of a countable family of quasi-\(\phi\)-asymptotically nonexpansive multivalued mappings. Furthermore, we prove that the sequences generated by the hybrid iterative scheme converge strongly to a common element of the set of solutions of \(MEP(f ,\varphi, C)\) and the set of common fixed points of a countable family of quasi-\(\phi\)-asymptotically nonexpansive multivalued mappings.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H04 | Set-valued operators |
Keywords:
mixed equilibrium problems; shrinking projection method; KKM mappings; multivalued mappings; strong convergence; Banach space; hybrid iterative scheme; quasi-\(\phi\)-asymptotically nonexpansive multivalued mappingsReferences:
| [1] | Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems.J. Comput. Appl. Math. 2008, 214:186-201. · Zbl 1143.65049 · doi:10.1016/j.cam.2007.02.022 |
| [2] | Cholamjiak P, Suantai S: Existence and iteration for a mixed equilibrium problem and a countable family of nonexpansive mappings in Banach spaces.Comput. Math. Appl. 2011, 61:2725-2733. · Zbl 1221.49003 · doi:10.1016/j.camwa.2011.03.029 |
| [3] | Fang YP, Huang NJ: Variational-like inequalities with generalized monotone mappings in Banach spaces.J. Optim. Theory Appl. 2003,118(2):327-338. · Zbl 1041.49006 · doi:10.1023/A:1025499305742 |
| [4] | Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.Math. Stud. 1994,63(1-4):123-145. · Zbl 0888.49007 |
| [5] | Chadli O, Schaible S, Yao JC: Regularized equilibrium problems with an application to noncoercive hemivariational inequalities.J. Optim. Theory Appl. 2004, 121:571-596. · Zbl 1107.91067 · doi:10.1023/B:JOTA.0000037604.96151.26 |
| [6] | Chadli O, Wong NC, Yao JC: Equilibrium problems with applications to eigenvalue problems.J. Optim. Theory Appl. 2003,117(2):245-266. · Zbl 1141.49306 · doi:10.1023/A:1023627606067 |
| [7] | Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces.J. Math. Anal. Appl. 1979, 67:274-276. · Zbl 0423.47026 · doi:10.1016/0022-247X(79)90024-6 |
| [8] | Matsushita S, Takahashi W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces.Fixed Point Theory Appl. 2004, 2004:37-47. · Zbl 1088.47054 · doi:10.1155/S1687182004310089 |
| [9] | Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space.J. Approx. Theory 2005, 134:257-266. · Zbl 1071.47063 · doi:10.1016/j.jat.2005.02.007 |
| [10] | Takahashi, W.; Zembayashi, K., Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, No. 2008 (2008) · Zbl 1187.47054 |
| [11] | Aleyner A, Reich S: Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach spaces.J. Math. Anal. Appl. 2008,343(1):427-435. · Zbl 1167.90010 · doi:10.1016/j.jmaa.2008.01.087 |
| [12] | Plubtieng, S.; Ungchittrakool, K., Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, No. 2008 (2008) · Zbl 1173.47051 |
| [13] | Plubtieng S, Ungchittrakool K: Strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces.J. Nonlinear Convex Anal. 2007,8(3):431-450. · Zbl 1142.47041 |
| [14] | Zegeye, H.; Shahzad, N., Convergence theorems for a common point of solutions of equilibrium and fixed point of relatively nonexpansive multivalued mapping problems, No. 2012 (2012) · Zbl 1256.47057 |
| [15] | Chang, SS; Kim, JK; Wang, XR, Modified block iterative algorithm for solving convex feasibility problems in Banach spaces, No. 2010 (2010) · Zbl 1187.47045 |
| [16] | Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces.J. Nonlinear Convex Anal. 2005,6(1):117-136. · Zbl 1109.90079 |
| [17] | Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces.Nonlinear Anal. 2009, 70:45-57. · Zbl 1170.47049 · doi:10.1016/j.na.2007.11.031 |
| [18] | Tada, A.; Takahashi, W.; Takahashi, W. (ed.); Tanaka, T. (ed.), Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, 609-617 (2007), Yokohama · Zbl 1122.47055 |
| [19] | Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem.J. Optim. Theory Appl. 2007,133(3):359-370. · Zbl 1147.47052 · doi:10.1007/s10957-007-9187-z |
| [20] | Homaeipour, S.; Razani, A., Weak and strong convergence theorems for relatively nonexpansive multi-valued mappings in Banach spaces, No. 2011 (2011) · Zbl 1315.47064 |
| [21] | Yi, L., Strong convergence theorems for modifying Halpern iterations for quasi-ϕ-asymptotically nonexpansive multivalued mapping in Banach spaces with applications, No. 2012 (2012) · Zbl 1263.47083 |
| [22] | Tan, JF; Chang, SS, Strong convergence theorems for a generalized mixed equilibrium problem and a family of total quasi-ϕ-asymptotically nonexpansive multivalued mappings in Banach spaces, No. 2012 (2012) |
| [23] | Cioranescu, I., Mathematics and Its Applications 62 (1990), Dordrecht · Zbl 0712.47043 · doi:10.1007/978-94-009-2121-4 |
| [24] | Alber, YI; Kartsatos, AG (ed.), Metric and generalized projection operators in Banach spaces: properties and applications, 15-50 (1996), New York · Zbl 0883.47083 |
| [25] | Fan K: A generalization of Tychonoffs fixed point theorem.Math. Ann. 1961, 142:305-310. · Zbl 0093.36701 · doi:10.1007/BF01353421 |
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