Existence results for trifunction equilibrium problems and fixed point problems. (English) Zbl 07074226
Summary: In this paper, we establish the existence and uniqueness solutions of trifunction equilibrium problems using the generalized relaxed \(\alpha \)-monotonicity in Banach spaces. By using the generalized \(f\)-projection operator, a hybrid iteration scheme is presented to find a common element of the solutions of a system of trifunction equilibrium problems and the set of fixed points of an infinite family of quasi-\(\phi \)-nonexpansive mappings. Moreover, the strong convergence of our new proposed iterative method under generalized relaxed \(\alpha \)-monotonicity is considered.
MSC:
| 47J05 | Equations involving nonlinear operators (general) |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 49J25 | Optimal control problems with equations with ret. arguments (exist.) (MSC2000) |
Keywords:
quasi-\(\phi \)-nonexpansive mapping; normalized duality mapping; general metric projection; general \(f\)-projection operator; generalized relaxed \(\alpha \)-monotonicityReferences:
| [1] | Preda, V., Beldiman, M., Bătătorescu, A.: On variational-like inequalities with generalized monotone mappings. In: Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol. 583, pp. 415-431. Springer, Berlin (2007) · Zbl 1278.47062 |
| [2] | Fang, Y.P., Huang, N.J.: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Optim. Theory Appl. 118(2), 327-338 (2003) · Zbl 1041.49006 · doi:10.1023/A:1025499305742 |
| [3] | Bai, M.R., Zhou, S.Z., Ni, G.Y.: Variational-like inequalities with relaxed \[\eta -\alpha\] η-α pseudomonotone mappings in Banach spaces. Appl. Math. Lett. 19(6), 547-554 (2006) · Zbl 1144.47047 · doi:10.1016/j.aml.2005.07.010 |
| [4] | Onjai-uea, N., Kumam, P.: Existence and convergence theorems for the new system of generalized mixed variational inequalities in banach spaces. J. Inequal. Appl. 2012(1), 9 (2012) · Zbl 1486.47112 · doi:10.1186/1029-242X-2012-9 |
| [5] | Petrot, N., Wattanawitoon, K., Kumam, P.: A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces. Nonlinear Anal. Hybrid Syst. 4(4), 631-643 (2010) · Zbl 1292.47051 · doi:10.1016/j.nahs.2010.03.008 |
| [6] | Saewan, S., Kumam, P.: A modified Mann iterative scheme by generalized f-projection for a countable family of relatively quasi-nonexpansive mappings and a system of generalized mixed equilibrium problems. Fixed Point Theory Appl. 2011(1), 1-21 (2011) · Zbl 1311.47092 · doi:10.1186/1687-1812-2011-1 |
| [7] | Saewan, S., Kumam, P.: Existence and algorithm for solving the system of mixed variational inequalities in Banach spaces. J. Appl. Math. 2012 (2012). doi:10.1155/2012/413468 · Zbl 1245.49016 |
| [8] | Saewan, S., Kumam, P.: A strong convergence theorem concerning a hybrid projection method for finding common fixed points of a countable family of relatively quasi-nonexpansive mappings. J. Nonlinear Convex Anal. 13(2), 313-330 (2012) · Zbl 1248.47058 |
| [9] | 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(1), 20-30 (2009) · Zbl 1165.65027 · doi:10.1016/j.cam.2008.06.011 |
| [10] | Qin, X., Shang, M., Su, Y.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal Theory Methods Appl 69(11), 3897-3909 (2008) · Zbl 1170.47044 · doi:10.1016/j.na.2007.10.025 |
| [11] | Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331(1), 506-515 (2007) · Zbl 1122.47056 · doi:10.1016/j.jmaa.2006.08.036 |
| [12] | Takahashi, W., Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. Theory Methods Appl. 70(1), 45-57 (2009) · Zbl 1170.47049 · doi:10.1016/j.na.2007.11.031 |
| [13] | Chang, S., Lee, H.W.J., Chan, C.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. Theory Methods Appl. 70(9), 3307-3319 (2009) · Zbl 1198.47082 · doi:10.1016/j.na.2008.04.035 |
| [14] | Plubtieng, S., Ungchittrakool, K.: Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space. J. Approx. Theory 149(2), 103-115 (2007) · Zbl 1137.47056 · doi:10.1016/j.jat.2007.04.014 |
| [15] | Zhang, S.: Generalized mixed equilibrium problem in Banach spaces. Appl. Math. Mech. 30(9), 1105-1112 (2009) · Zbl 1178.47051 · doi:10.1007/s10483-009-0904-6 |
| [16] | Saewan, S., Kumam, P.: A generalized f-projection method for countable families of weak relatively nonexpansive mappings and the system of generalized Ky fan inequalities. J. Glob. Optim. 56, 1-23 (2012) · Zbl 1300.90065 |
| [17] | Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44(1), 147-150 (1974) · Zbl 0286.47036 · doi:10.1090/S0002-9939-1974-0336469-5 |
| [18] | Pettis, B.J.: A proof that every uniformly convex space is reflexive. Duke Math. J. 5(2), 249-253 (1939) · JFM 65.1312.04 · doi:10.1215/S0012-7094-39-00522-3 |
| [19] | Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62. Springer, Dordrecht (1990) · Zbl 0712.47043 · doi:10.1007/978-94-009-2121-4 |
| [20] | Cioranescu, I.: Book review. Bull. Am. Math. Soc 26, 367-370 (1992) · doi:10.1090/S0273-0979-1992-00287-2 |
| [21] | Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. Lecture Notes, Pure and Applied Mathematics, pp. 15-50. (1996) · Zbl 0883.47083 |
| [22] | Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000) · Zbl 0997.47002 |
| [23] | Wu, K., Huang, N.: The generalised f-projection operator with an application. Bull. Aust. Math. Soc. 73(2), 307-318 (2006) · Zbl 1104.47053 · doi:10.1017/S0004972700038892 |
| [24] | Fan, J., Liu, X., Li, J.: Iterative schemes for approximating solutions of generalized variational inequalities in banach spaces. Nonlinear Anal. Theory Methods Appl. 70(11), 3997-4007 (2009) · Zbl 1219.47110 · doi:10.1016/j.na.2008.08.008 |
| [25] | Li, X., Huang, N., O’Regan, D.: Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications. Comput. Math. Appl. 60(5), 1322-1331 (2010) · Zbl 1201.65091 · doi:10.1016/j.camwa.2010.06.013 |
| [26] | Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985) · Zbl 0559.47040 · doi:10.1007/978-3-662-00547-7 |
| [27] | Yang, F., Zhao, L., Kim, J.K.: Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinite family of asymptotically quasi-\[ \phi\] ϕ-nonexpansive mappings in banach spaces. Appl. Math. Comput. 218(10), 6072-6082 (2012) · Zbl 1246.65086 |
| [28] | Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13(3), 938-945 (2002) · Zbl 1101.90083 · doi:10.1137/S105262340139611X |
| [29] | Mahato, N.K., Nahak, C.: Hybrid projection methods for the general variational-like inequality problems. J. Adv. Math. Stud. 6(1), 143-158 (2013) · Zbl 1278.47076 |
| [30] | Fan, K.: A minimax inequality and applications. Inequalities 3, 103-113 (1972) · Zbl 0302.49019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
