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Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type. (English) Zbl 1215.65105
Authors’ abstract: We study a hybrid iterative scheme for finding a common element of a set of solutions of generalized mixed equilibrium problem, a set of common fixed points of a finite family of weak relatively nonexpansive mappings and null spaces of a finite family of $\gamma$-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. Our results extend, improve and generalize the results of several authors which are announced recently. Application of our theorem to solution of equations of Hammerstein-type is of independent interest.
##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H30 Particular nonlinear operators 47H09 Mappings defined by “shrinking” properties
##### References:
 [1] Alber, Y. I.: Metric and generalized projection operators in Banach spaces: properties and applications, Lecture notes in pure and appl. Math 178, 15-50 (1996) · Zbl 0883.47083 [2] Beauzamy, B.: Introduction to Banach spaces and their geometry, (1995) [3] Blum, E.; Oettli, W.: From optimization and variational inequalities to equilibrium problems, Math. student 63, 123-145 (1994) · Zbl 0888.49007 [4] Breźis, H.; Browder, F. E.: Some new results about Hammerstein equations, Bull. amer. Math. soc. 80, 567-572 (1974) · Zbl 0286.45007 · doi:10.1090/S0002-9904-1974-13500-7 [5] Browder, F. E.; De Figueiredo, D. G.; Gupta, P.: Maximal monotone operators and nonlinear integral equations of Hammerstein type, Bull. amer. Math. soc. 76, 700-705 (1970) · Zbl 0197.41101 · doi:10.1090/S0002-9904-1970-12511-3 [6] Browder, F. E.; Gupta, P.: Monotone operators and nonlinear integral equations of Hammerstein type, Bull. amer. Math. soc. 75, 1347-1353 (1969) · Zbl 0193.11204 · doi:10.1090/S0002-9904-1969-12420-1 [7] Chang, S. S.: On chidume’s open questions and approximation solutions of multi-valued strongly accretive mapping equations in Banach spaces, J. math. Anal. appl. 216, 94-111 (1997) · Zbl 0909.47049 · doi:10.1006/jmaa.1997.5661 [8] Chang, S. S.; Lee, H. W. Joseph; Chan, C. K.: A new hybrid method for solving generalized equilibrium problems, variational inequality and common fixed point in Banach spaces with applications, Nonlinear anal. (2010) [9] Chepanovich, R. Sh.: Nonlinear Hammerstein equations and fixed points, (1984) [10] Chidume, C. E.: Fixed point iterations for nonlinear Hammerstein equations involving nonexpansive and accretive mappings, Indian J. Pure appl. Math. 120, 129-135 (1989) · Zbl 0672.47047 [11] De Figueiredo, D. G.; Gupta, C. P.: On the variational method for the existence of solutions to nonlinear equations of Hammerstein type, Proc. amer. Math. soc. 40, 470-476 (1973) · Zbl 0269.47030 · doi:10.2307/2039394 [12] Dolezale, V.: Monotone operators and its applications in automation and network theory, Studies in automation and control 3 (1979) [13] Figiel, T.: On the moduli of convexity and smoothness, Studia math. 56, 21-155 (1976) · Zbl 0344.46052 [14] Hammerstein, A.: Nichtlineare integragleichungen nebst anwendungen, Acta math. Soc. 54, 117-176 (1930) · Zbl 56.0343.03 · doi:10.1007/BF02547519 [15] Kamimura, S.; Takahashi, W.: Strong convergence of proximal-type algorithm in a Banach space, SIAM J. Optim. 13, 938-945 (2002) · Zbl 1101.90083 · doi:10.1137/S105262340139611X [16] Kang, J.; Su, Y.; Zhang, X.: Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications, Nonlinear anal. Hybrid systems (2010) [17] Kohasaka, F.; Takahashi, W.: Strong convergence of an iterative sequence for maximal monotone operators in Banach spaces, Abstr. appl. Anal. 70, No. 7, 2707-2716 (2009) [18] Kohasaka, F.; Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19, 824-835 (2008) · Zbl 1168.47047 · doi:10.1137/070688717 [19] Matsushita, S.; Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. approx. Theory 134, 257-266 (2005) · Zbl 1071.47063 · doi:10.1016/j.jat.2005.02.007 [20] Matsushita, S.; Nakajo, K.; Takahashi, W.: Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces, Nonlinear anal. 73, No. 6, 1466-1480 (2010) · Zbl 1201.49033 · doi:10.1016/j.na.2010.04.007 [21] Moudafi, A.; Thera, M.: Proximal and dynamical approaches to equilibrium problems, Lecture notes in economics and mathematics systems 477, 187-201 (1999) · Zbl 0944.65080 [22] D. Pascali and Sburlan, Nonlinaer mappings of monotone type, editura academiae, Bucaresti, Romania, 1978. [23] Reich, S.: A weak convergence theorem for the alternating method with Bergman distance, Lecture notes in pure and appl. Math 178, 313-318 (1996) · Zbl 0943.47040 [24] Takahashi, W.; Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach soaces, Nonlinear anal. 70, 45-57 (2009) · Zbl 1170.47049 · doi:10.1016/j.na.2007.11.031 [25] Takahashi, W.: Nonlinear functional analysis, (2000) · Zbl 0997.47002 [26] Xu, H. K.: Inqualities in Banach spaces with applications, Nonlinear anal. 16, 1127-1138 (1991) [27] Xu, Z. B.; Roach, G. F.: Characteristic inequalities for uniformly convex and uniformly smooth Banach spaces, J. math. Anal. appl. 157, 189-210 (1991) · Zbl 0757.46034 · doi:10.1016/0022-247X(91)90144-O [28] Su, Yong; Gao, Junyu; Zhou, Haiyun: Monotone CQ algorithm of fixed points for weak relatively nonexpansive mappings and applications, J. math. Res. exposition 28, No. 4, 957-967 (2008) · Zbl 1199.47295 · doi:10.3770/j.issn:1000-341X.2008.04.028 [29] Zhang, S. S.: On the generalized mixed equilibrium problem in Banach space, Appl. math. Mech. 30, No. 9, 1105-1112 (2009) · Zbl 1178.47051 · doi:10.1007/s10483-009-0904-6 [30] Zegeye, H.; Ofoedu, E. U.; Shahzad, N.: Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings, Appl. math. Comput. 216, No. 12, 3439-3449 (2010) · Zbl 1198.65100 · doi:10.1016/j.amc.2010.02.054 [31] Zegeye, H.; Shahzad, N.: Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings, Nonlinear anal. 70, No. 7, 2707-2716 (2009) · Zbl 1223.47108 · doi:10.1016/j.na.2008.03.058 [32] Zegeye, H.; Shahzad, N.: A hybrid approximation method for equilibrium, variational inequality and fixed point problems, Nonlinear anal. Hybrid syst. 4, No. 4, 619-630 (2010) [33] Zegeye, H.; Shahzad, N.: A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems, Nonlinear anal. 74, 263-272 (2011) [34] Zegeye, H.; Shahzad, N.: Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces, Optim lett. (2010)