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Keywords:
nonexpansive mapping; fixed point problems; Variational inequality; relaxed extragradient approximation method; maximal monotone
nonexpansive mapping; fixed point problems; Variational inequality; relaxed extragradient approximation method; maximal monotone
Summary:
We introduce an iterative sequence for finding the common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for tree inverse-strongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to finding solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of paper we utilize our results to study the zeros of the maximal monotone and some convergence problem for strictly pseudocontractive mappings. Our results include the previous results as special cases extend and improve the results of Ceng et al., [Math. Meth. Oper. Res., 67:375–390, 2008] and many others.
References:
[1] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63 (1994), 123–145. MR 1292380 | Zbl 0888.49007
[2] Browder, F. E., Petryshyn, W. V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20 (1967), 197–228. DOI 10.1016/0022-247X(67)90085-6 | MR 0217658
[3] Ceng, L.–C., Wang, C.–Y., Yao, J.–C.: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67 (2008), 375–390. DOI 10.1007/s00186-007-0207-4 | MR 2403714 | Zbl 1147.49007
[4] Combettes, P. L., Hirstoaga, S. A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6 (1) (2005), 117–136. MR 2138105 | Zbl 1109.90079
[5] Goebel, K., Kirk, W. A.: Topics on Metric Fixed–Point Theory. Cambridge University Press, 1990. MR 1074005
[6] Haugazeau, Y.: Surles in équations variationnelles et la minimisation de fonctionnelles convexes, Thèse. Master's thesis, Université de Paris, 1968.
[7] Korpelevich, G. M.: An extragradient method for finding saddle points and for other problems. Ékonom. i Mat. Metody 12 (940) (1976), 747–756, Russian. MR 0451121
[8] Kumam, P.: Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space. Turkish J. Math. 33 (1) (2009), 85–98. MR 2524118 | Zbl 1223.47083
[9] Liu, F., Nashed, M. Z., Takahashi, W.: Regularization of nonlinear ill–posed variational inequalities and convergence rates. Set–Valued Anal. 6 (1998), 313–344. MR 1690160
[10] Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128 (2006), 191–201. DOI 10.1007/s10957-005-7564-z | MR 2201895 | Zbl 1130.90055
[11] Osilike, M. O., Igbokwe, D. I.: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput. Math. Appl. 40 (2000), 559–567. DOI 10.1016/S0898-1221(00)00179-6 | MR 1772655 | Zbl 0958.47030
[12] Plubtieng, S., Punpaeng, R.: A new iterative method for equilibrium problems 3 and fixed point problems of nonexpansive mappings and 4 monotone mappings. Appl. Math. Comput. (2007). DOI 10.1016/j.amc.2007.07.075
[13] Su, Y. et al.,: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. (2007). DOI 10.1016/j.na.2007.08.045
[14] Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one–parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305 (2005), 227–239. DOI 10.1016/j.jmaa.2004.11.017 | MR 2128124 | Zbl 1068.47085
[15] Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331 (1) (2007), 506–515. DOI 10.1016/j.jmaa.2006.08.036 | MR 2306020 | Zbl 1122.47056
[16] Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118 (2003), 417–428. DOI 10.1023/A:1025407607560 | MR 2006529 | Zbl 1055.47052
[17] Verma, R. U.: On a new system of nonlinear variational inequalities and associated iterative algorithms. Math. Sci. Res. Hot–Line 3 (8) (1999), 65–68. MR 1717779 | Zbl 0970.49011
[18] Verma, R. U.: Iterative algorithms and a new system of nonlinear quasivariational inequalities. Adv. Nonlinear Var. Inequal. 4 (1) (2001), 117–127. MR 1801652 | Zbl 1014.47050
[19] Xu, H. K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298 (2004), 279–291. DOI 10.1016/j.jmaa.2004.04.059 | MR 2086546 | Zbl 1061.47060
[20] Yao, J–C., Chadli, O.: Handbook of Generalized Convexity and Monotonicity. ch. Pseudomonotone complementarity problems and variational inequalities, pp. 501–558, Springer, Netherlands, 2005. MR 2098908
[21] Yao, Y., C., Liou Y., Yao, J.–C.: An extragradient method for fixed point problems and variational inequality problems. Journal of Inequalities and Applications 2007 (2007), 12, article ID 38752. DOI 10.1155/2007/38752 | MR 2291644 | Zbl 1137.47057
[22] Yao, Y., Yao, J.–C.: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 186 (2007), 1551–1558. DOI 10.1016/j.amc.2006.08.062 | MR 2316950 | Zbl 1121.65064
[23] Zeng, L. C., Wong, N. C., Yao, J.–C.: Strong convergence theorems for strictly pseudocontractive mapping of Browder–Petryshyn type. Taiwanese J. Math. 10 (4) (2006), 837–849. MR 2229625
[24] Zeng, L. C., Yao, J.–C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese J. Math. 10 (2006), 1293–1303. MR 2253379 | Zbl 1110.49013
[25] Zhang, S., Lee, J., Chan, C.: Algorithms of common solutions to quasi variational inclusion and fixed point problems. Appl. Math. Mech. (English Ed.) 29 (5) (2008), 571–581. DOI 10.1007/s10483-008-0502-y | MR 2414681 | Zbl 1196.47047
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