Strong convergence for a common solution of variational inequalities, fixed point problems and zeros of finite maximal monotone mappings


Authors

Yang-Qing Qiu - Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China.
Jin-Zuo Chen - Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China.
Lu-Chuan Ceng - Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China.


Abstract

In this paper, by the strongly positive linear bounded operator technique, a new generalized Mann-type hybrid composite extragradient CQ iterative algorithm is first constructed. Then by using the algorithm, we find a common element of the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping, the set of zeros of two families of finite maximal monotone mappings and the set of fixed points of an asymptotically \(\kappa\)-strict pseudocontractive mappings in the intermediate sense in a real Hilbert space. Finally, we prove the strong convergence of the iterative sequences, which extends and improves the corresponding previous works.


Keywords


References

[1] H. H. Bauschke, P. L. Combettes, A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. Oper. Res., 26 (2001), 248-264.
[2] L. C. Ceng, S. M. Guu, J. C. Yao, Hybrid viscosity CQ method for finding a common solution of a variational inequality, a general system of variational inequalities, and a fixed point problem, Fixed Point Theory Appl., 2013 (2013), 25 pages.
[3] L. C. Ceng, S. M. Guu, J. C. Yao, Hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense, J. Global Optim., 60 (2014), 617-634.
[4] L. C. Ceng, C. W. Liao, C. T. Pang, C. F. Wen, Multistep hybrid iterations for systems of generalized equilibria with constraints of several problems, Abstr. Appl. Anal., 2014 (2014), 27 pages.
[5] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, (1990).
[6] H. Iiduka, W. Takahashi, Strong convergence theorem by a hybrid method for nonlinear mappings of nonexpansive and monotone type and applications, Adv. Nonlinear Var. Inequal., 9 (2006), 1-10.
[7] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747-756.
[8] G. Marino, H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-52.
[9] N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16 (2006), 1230-1241.
[10] X. Qin, M. Shang, S. M. Kang, Strong convergence theorems of modified Mann iterative process for strict pseudo- contractions in Hilbert spaces, Nonlinear Anal., 70 (2009), 1257-1264.
[11] Y. Q. Qiu, L. C. Ceng, J. Z. Chen, H. Y. Hu, Hybrid iterative algorithms for two families of finite maximal monotone mappings, Fixed Point Theory Appl., 2015 (2015), 18 pages.
[12] J. Radon, Theorie und anwendungen der absolut additiven mengenfunktionen, Wien. Ber., 122 (1913), 1295-1438.
[13] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75-88.
[14] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877-898.
[15] D. R. Sahu, H. K. Xu, J. C. Yao, Asymptotically strict pseudocontractive mappings in the intermediate sense, Nonlinear Anal., 70 (2009), 3502-3511.
[16] L. Wei, R. Tan, Strong and weak convergence theorems for common zeros of finite accretive mappings, Fixed Point Theory Appl., 2014 (2014), 17 pages.
[17] Y. Yao, G. Marino, L. Muglia, A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality, Optimization, 63 (2014), 559-569.
[18] Y. Yao, G. Marino, H. K. Xu, Y. C. Liou, Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces, J. Inequal. Appl., 2014 (2014), 14 pages.
[19] Y. Yao, M. Postolache, S. M. Kang, Strong convergence of approximated iterations for asymptotically pseudocontractive mappings, Fixed Point Theory Appl., 2014 (2014), 13 pages.
[20] E. Zeidler, Nonlinear functional analysis and its applications, II/B: Nonlinear monotone Operators, Springer Verlag, Berlin, Germany, (1990).