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Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. (English) Zbl 1198.65100

Let $E$ be a real Banach space and $C\subset E$ closed, convex and nonempty. A mapping $A:D\left(A\right)\subset E\to {E}^{\text{'}}$ is said to be $\gamma$-inverse strongly monotone if there exists $\gamma >0$ such that

$\left(Ax-Ay,x-y\right)\ge \gamma \parallel Ax-Ay{\parallel }^{2},\phantom{\rule{1.em}{0ex}}\forall x,y\in D\left(A\right)·$

The authors introduce an iterative process of the type:

${x}_{n-1}={{\Pi }}_{{C}_{n-1}}\left({x}_{0}\right);\phantom{\rule{1.em}{0ex}}{x}_{0}\in {C}_{0}\left(\equiv C\right)\phantom{\rule{1.em}{0ex}}\text{arbitrary}\phantom{\rule{1.em}{0ex}}n\ge 0,$

converging strongly to a common element of the set of common fixed points of the countably infinite family of closed relatively quasi-nonexpensive mappings, the solution set of a generalized equilibrium problem and the solution set of a variational inequality problem for a $\gamma$-inverse strongly monotone mapping in Banach spaces. The theorems of the paper improve, generalize, unify and extend several known results.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J25 Iterative procedures (nonlinear operator equations) 47J40 Equations with hysteresis operators 65K15 Numerical methods for variational inequalities and related problems 47H09 Mappings defined by “shrinking” properties
##### References:
 [1] Alber, Ya.: 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] Blum, E.; Oettli, W.: From optimization and variational inequalities to equilibrium problems, Math. student 63, 123-145 (1994) · Zbl 0888.49007 [3] Butanriu, D.; Reich, S.; Zaslavski, A. J.: Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. appl. Anal. 7, 151-174 (2001) · Zbl 1010.47032 · doi:10.1515/JAA.2001.151 [4] Genel, A.; Lindenstress, J.: An example concerning fixed points, Israel J. Math. 22, 81-86 (1975) · Zbl 0314.47031 · doi:10.1007/BF02757276 [5] 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 [6] Kohasaka, F.; Takahashi, W.: Strong convergence of an iterative sequence for maximal monotone operators in Banach spaces, Abstr. appl. Anal. 3, 239-249 (2004) · Zbl 1064.47068 · doi:10.1155/S1085337504309036 [7] Kumam, P.: A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive, Nonlinear anal., hybrid syst. (2008) [8] Mann, M. R.: Mean value methods in iteration, Proc. am. Math. soc. 4, 503-510 (1953) · Zbl 0050.11603 · doi:10.2307/2032162 [9] Nakajo, K.; Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semi-groups, J. math. Anal. appl. 279, 372-379 (2003) · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4 [10] 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. (2008) [11] Reich, S.: Weak convergence theorem for nonexpansive mappings, J. math. Anal. appl. 67, 274-276 (1979) · Zbl 0423.47026 · doi:10.1016/0022-247X(79)90024-6 [12] 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 [13] Rockafellar, R. T.: Monotone operators and the proximal point algorithm, SIAM J. Control optim. 14, 877-898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056 [14] Su, Y.; Wang, D.; Shang, M.: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings, Fixed point theory appl. (2008) · Zbl 1203.47078 · doi:10.1155/2008/284613 [15] Tada, A.; Takahashi, W.: Weak and strong convergence theorems for nonexpansive mappings and equilibrium problems, J. optim. Theory appl. 133, 359-370 (2007) · Zbl 1147.47052 · doi:10.1007/s10957-007-9187-z [16] Takahashi, W.: Nonlinear functional analysis, (1988) [17] Takahashi, 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 · doi:10.1016/j.na.2007.11.031 [18] Xu, H. K.: Inequalities in Banach spaces with applications, Nonlinear anal. 16, 1127-1138 (1991) · Zbl 0757.46033 · doi:10.1016/0362-546X(91)90200-K [19] Zegeye, H.; Shahzad, N.: Strong convergence for monotone mappings and relatively weak nonexpansive mappings, Nonlinear anal. (2008) [20] Zegeye, H.; Problems, A. Hybrid Iteration Scheme For Equilibrium: Variational inequality problems and common fixed point problems in Banach spaces, Nonlinear anal. 72, 2136-2146 (2010) · Zbl 1225.47121 · doi:10.1016/j.na.2009.10.014