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Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces. (English) Zbl 1219.47110
Authors’ abstract: We introduce two iterative schemes for approximating solutions of generalized variational inequalities in the setting of Banach spaces. The existence of solutions of this general problem and the convergence of the proposed iterative schemes to a solution are established.

MSC:
47J25 Iterative procedures involving nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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[1] Alber, Ya., Metric and generalized projection operators in Banach spaces: properties and applications, (), 15-50 · Zbl 0883.47083
[2] Alber, Ya., Proximal projection method for variational inequalities and cesro averaged approximations, Comput. math. appl., 43, 1107-1124, (2002) · Zbl 1050.65065
[3] Alber, Ya.; Guerre-Delabriere, S, On the projection methods for fixed point problems, Analysis, 21, 17-39, (2001) · Zbl 0985.47044
[4] Chang, S.S., On chidume’s open questions and approximate solutions of multivalued strongly accretive mapping in Banach spaces, J. math. anal. appl., 216, 94-111, (1997) · Zbl 0909.47049
[5] Chidume, C.E., Iterative solutions of nonlinear equations in smooth Banach spaces, Nonlinear anal., 26, 1823-1834, (1996) · Zbl 0868.47039
[6] Chidume, C.E.; Li, J., Projection methods for approximating fixed points of Lipschitz suppressive operators, Panamer. math. J., 15, 29-40, (2005) · Zbl 1086.47018
[7] Fan, J.H., A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces, J. math. anal. appl., 337, 1041-1047, (2008) · Zbl 1140.49011
[8] Huang, N.J., Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions, Comput. math. appl., 35, 10, 1-7, (1998) · Zbl 0999.47057
[9] Li, J., On the existence of solutions of variational inequalities in Banach spaces, J. math. anal. appl., 295, 115-126, (2004) · Zbl 1045.49008
[10] Li, J., The generalized projection operator on reflexive Banach spaces and its application, J. math. anal. appl., 306, 55-71, (2005) · Zbl 1129.47043
[11] Li, J.; Rhoades, B.E., An approximation of solutions of variational inequalities in Banach spaces, Fixed point theory appl., 3, 377-388, (2005) · Zbl 1165.47310
[12] Kazmi, K.R., Mann and Ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions, J. math. anal. appl., 209, 572-584, (1997) · Zbl 0898.49007
[13] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. optim., 13, 938-945, (2002) · Zbl 1101.90083
[14] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 50-65, (1953)
[15] Rhoades, B.E.; Soltuz, S.M., The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map, J. math. anal. appl., 283, 681-688, (2003) · Zbl 1045.47057
[16] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Publishers
[17] Wu, K.Q.; Huang, N.J., The generalized f-projection operator with an application, Bull. austral. math. soc., 73, 307-317, (2006) · Zbl 1104.47053
[18] Wu, K.Q.; Huang, N.J., Properties of the generalized f-projection operator and its applications in Banach spaces, Comput. math. appl., 54, 399-406, (2007) · Zbl 1151.47057
[19] Xu, H.K., Inequalities in Banach spaces with applications, Nonlinear anal., 16, 1127-1138, (1991) · Zbl 0757.46033
[20] Zeng, L.C.; Yao, J.C., Existence theorems for variational inequalities in Banach spaces, J. optim. theory appl., 132, 321-337, (2007) · Zbl 1149.49015
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