An extragradient iterative scheme for common fixed point problems and variational inequality problems with applications

Petruşel, Adrian; Sahu, D.R.; Sagar, Vidya

doi:10.1515/auom-2015-0017

Abstract

In this paper, by combining a modified extragradient scheme with the viscosity approximation technique, an iterative scheme is developed for computing the common element of the set of fixed points of a sequence of asymptotically nonexpansive mappings and the set of solutions of the variational inequality problem for an α-inverse strongly monotone mapping. We prove a strong convergence theorem for the sequences generated by this scheme and give some applications of our convergence theorem.

References

[1] R.P. Agarwal, Donal O’Regan and D.R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Series: Topological Fixed Point Theory and Its Applications Springer, New York, 2009.10.1155/2009/439176
Search in Google Scholar Back to article
[2] J. Chen, L. Zhang and T. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl. 334 (2007), 1450-1461.10.1016/j.jmaa.2006.12.088
Search in Google Scholar Back to article
[3] Y.J. Cho, H. Zhou and G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl., 47 (2004), 707-717.10.1016/S0898-1221(04)90058-2
Search in Google Scholar Back to article
[4] Q.-L. Dong and S. He, A viscosity projection method for class 𝒯 mappings, Analele Ştiinţifice ale Univ. Ovidius Constanţa 21 (2013), 95-109.10.2478/auom-2013-0025
Search in Google Scholar Back to article
[5] K. Goebel and W.A. Kirk, Topics on Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.10.1017/CBO9780511526152
Search in Google Scholar Back to article
[6] G.M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonomika i Matematicheskie Metody 12 (1976), 747-756.
Search in Google Scholar Back to article
[7] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.10.1007/s10957-005-7564-z
Search in Google Scholar Back to article
[8] M.A. Noor, Modified projection method for pseudomonotone variational inequalities, Appl. Math. Lett. 15 (2002), 315-320.10.1016/S0893-9659(01)00137-9
Search in Google Scholar Back to article
[9] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.10.1090/S0002-9904-1967-11761-0
Search in Google Scholar Back to article
[10] A. Petruşel and J.-C. Yao, An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems, Cent. Eur. J. Math. 7 (2009), 335-347.10.2478/s11533-009-0003-x
Search in Google Scholar Back to article
[11] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88.10.1090/S0002-9947-1970-0282272-5
Search in Google Scholar Back to article
[12] D.R. Sahu, Zequing Liu, and Shin Min Kang, Iterative approaches to common fixed points of asymptotically nonexpansive mappings, Rocky Mountain J. Math. 39 (2009), 281-304.10.1216/RMJ-2009-39-1-281
Search in Google Scholar Back to article
[13] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), 153-159.10.1017/S0004972700028884
Search in Google Scholar Back to article
[14] G. Stampacchia, Formes bilineaires coercivities sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.
Search in Google Scholar Back to article
[15] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
Search in Google Scholar Back to article
[16] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), 417-428.10.1023/A:1025407607560
Search in Google Scholar Back to article
[17] H.K. Xu and T.H. Kim, Convergence of hybrid sleepest-descend methods for variational inequalities, J. Optim. Theory Appl. 119 (2003), 185-201.10.1023/B:JOTA.0000005048.79379.b6
Search in Google Scholar Back to article
[18] L.C. Zeng and J.-C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwan. J. Math., 10 (2006), 1293-1303.10.11650/twjm/1500557303
Search in Google Scholar Back to article

An extragradient iterative scheme for common fixed point problems and variational inequality problems with applications

Abstract

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