Approximation of solutions of quasi-variational inclusions and fixed points of nonexpansive mappings

Volume 9, Issue 5, pp 3002--3009 http://dx.doi.org/10.22436/jnsa.009.05.92

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1389 Downloads
• 2555 Views

Authors

Dongfeng Li - School of Information Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450011, China. Juan Zhao - School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power University, Zhengzhou 450011, China.

Abstract

The purpose of this article is to study common solution problems of quasi-variational inclusion problems and nonlinear operator equations involving nonexpansive mappings. Strong convergence theorems are obtained without any compactness assumptions imposed on the operators and the spaces.

Share and Cite

Keywords

• Hilbert space
• convergent theorem
• fixed point
• contractive mapping
• monotone operator.

MSC

•  47H05
•  65J15.

References

• [1] V. Barbu, Nonlinear semigroups and differential equations in Banach space , Noordhoff International Publishing, Bucharest (1976)


• [2] B. A. Bin Dehaish, A. Latif, H. O. Bakodah, X. Qin, A viscosity splitting algorithm for solving inclusion and equilibrium problems, J. Inequal. Appl. , 2015 (2015), 14 pages.


• [3] B. A. Bin Dehaish, X. Qin, A. Latif, H. O. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321-1336.


• [4] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.


• [5] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci., 54 (1965), 1041-1044.


• [6] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.


• [7] S. S. Chang, Existence and approximation of solutions of set-valued variational inclusions in Banach spaces, Nonlinear Anal., 47 (2001), 583-594.


• [8] S. Y. Cho, W. Li, S. M. Kang , Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013), 14 pages.


• [9] S. Y. Cho, X. Qin , On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems, Appl. Math. Comput., 235 (2014), 430-438.


• [10] S. Y. Cho, X. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 15 pages.


• [11] P. L. Combettes, V. R. Wajs , Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200.


• [12] V. F. Demyanov, G. E. Stavroulakis, L. N. Polyakova, P. D. Panagiotopoulos, Quasi differentiability and nonsmooth modeling in mechanics, engineering and economics, Kluwer Academic, Dordrecht (1996)


• [13] K. Fan, A minimax inequality and applications, Academic Press, New york (1972)


• [14] P. Hartman, G. Stampacchia, On some nonlinear elliptic differential equations, Acta Math., 115 (1966), 271-310.


• [15] J. K. Kim, P. N. Anh, Y. M. Nam, Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems, J. Korean Math. Soc., 49 (2012), 187-200.


• [16] J. K. Kim, S. Y. Cho, X. Qin , Some results on generalized equilibrium problems involving strictly pseudocontractive mappings, Acta Math. Sci. Ser., 31 (2011), 2041-2057.


• [17] Y. Kimura, Shrinking projection methods for a family of maximal operators, Nonlinear Funct. Anal. Appl., 16 (2011), 481-489.


• [18] L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114-125.


• [19] M. A. Noor, K. I. Noor, Sensitivity analysis of quasi variational inclusions, J. Math. Anal. Appl., 236 (1999), 290-299.


• [20] X. Qin, S. Y. Cho, S. M. Kang , An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings, J. Global Optim., 49 (2011), 679-693.


• [21] X. Qin, S. Y. Cho, L. Wang, Iterative algorithms with errors for zero points of m-accretive operators, Fixed Point Theory Appl., 2013 (2013), 17 pages.


• [22] X. Qin, S. Y. Cho, L.Wang, A regularization method for treating zero points of the sum of two monotone operators, Fixed Point Theory Appl., 2014 (2014), 10 pages.


• [23] S. Takahashi, W. Takahashi, M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147 (2010), 27-41.


• [24] Z.-M. Wang, W. Lou, A new iterative algorithm of common solutions to quasi-variational inclusion and fixed point problems , J. Math. Comput. Sci., 3 (2013), 57-72.


• [25] Z.-M. Wang, X. Zhang, Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal., 2014 (2014), 25 pages.


• [26] C. Wu , Strong convergence theorems for common solutions of variational inequality and fixed point problems, Adv. Fixed Point Theory, 4 (2014), 229-244.


• [27] H. Zegeye, N. Shahzad , A hybrid approximation method for equilibrium, variational inequality and fixed point problems , Nonlinear Anal. Hybrid Syst., 4 (2010), 619-630.


• [28] H. Zegeye, N. Shahzad, Strong convergence theorem for a common point of solution of variational inequality and fixed point problem, Adv. Fixed Point Theory, 2 (2012), 374-397.


• [29] L. Zhang, H. Tong, An iterative method for nonexpansive semigroups, variational inclusions and generalized equilibrium problems, Adv. Fixed Point Theory, 4 (2014), 325-343.


• [30] J. Zhao, Strong convergence theorems for equilibrium problems, xed point problems of asymptotically nonexpan- sive mappings and a general system of variational inequalities, Nonliear Funct. Anal. Appl., 16 (2011), 447-464.