- Volume 51 Issue 3
In this paper, we introduce a new approximating method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solution of system variational inequalities for finite family of inverse strongly monotone mappings and strictly pseudo-contractive of Browder-Petryshyn type mappings. We show that the sequence converges strongly to a common element the above two sets under some parameter controling conditions. Our results improve and extend the results announced by many others.
- H. Piri, Strong convergence for a minimization problem on solutions of systems of equilibrium problems and common fixed points of an infinite family and semigroup of nonexpansive mappings, Comput. Math. Appl. 61 (2011), no. 9, 2562-2577. https://doi.org/10.1016/j.camwa.2011.02.049
- H. Piri and A. H. Badali, Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities, Fixed Point Theory Appl. 2011 (2011), doi:10.1186/1687-1812-2011-55. https://doi.org/10.1186/1687-1812-2011-55
- S. Plubtieng and R. Wangkeeree, Strong convergence theorems for three-step iterations with errors for non-lipschitzian nonself-mappings in Banach spaces, Comput. Math. Appl. 51 (2006), no. 6-7, 1093-1102. https://doi.org/10.1016/j.camwa.2005.08.035
- R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88. https://doi.org/10.1090/S0002-9947-1970-0282272-5
- N. Xiu, Y. Wang, and X. Zhang, Modified fixed-point equations and related iterative methods for variational inequalities, Comput. Math. Appl. 47 (2004), no. 6-7, 913-920. https://doi.org/10.1016/S0898-1221(04)90075-2
- H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003), no. 3, 659-678. https://doi.org/10.1023/A:1023073621589
- Y. Yao and J. C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2007), no. 2, 1551-1558. https://doi.org/10.1016/j.amc.2006.08.062
- T. Jitpeera and P. Kumam, A general iterative algorithm for generalized mixed equilibrium problems and variational inclusions approach to variational inequalities, Int. J. Math. Math. Sci. 2011 (2011), Article ID 619813, 25 pages.
- L. C. Ceng, S. M. Gu, and J. C. Yao, A general iterative method with strongly positive operators for general variational inequalities, Comput. Math. Appl. 59 (2010), no. 4, 14411452. https://doi.org/10.1016/j.camwa.2009.11.007
- J. Chen, L. Zhang, and T. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl. 334 (2007), no. 2, 1450-1461. https://doi.org/10.1016/j.jmaa.2006.12.088
- T. Jitpeera and P. Kumam, An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed point of finitely many nonexpansive mappings, J. Nonlinear Anal. Optim. 1 (2010), no. 1, 71-91.
- T. Jitpeera and P. Kumam, A composite iterative method for generalized mixed equilibrium problems and variational inequality problems, J. Comput. Anal. Appl. 13 (2011), no. 2, 345-361.
- J. S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005), no. 2, 509-520. https://doi.org/10.1016/j.jmaa.2004.08.022
- P. Katchang and P. Kumam, An iterative algorithm for finding a common solution of fixed points and a general system of variational inequalities for two inverse strongly accretive operators, Positivity 15 (2011), no. 2, 281-295. https://doi.org/10.1007/s11117-010-0074-8
- G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006), no. 1, 43-52. https://doi.org/10.1016/j.jmaa.2005.05.028
- M. O. Osilike and D. I. Igbokwe, Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. Math. Appl. 40 (2000), no. 4-5, 559-567. https://doi.org/10.1016/S0898-1221(00)00179-6
- H. Piri, A general iterative method for finding common solutions of system of equilibrium problems, system of variational inequalities and fixed point problems, Math. Comput. Modell. 55 (2012), 1622-1638. https://doi.org/10.1016/j.mcm.2011.10.069
- H. Piri, Solutions of variational inequalities on fixed points of nonexpansive mappings, Bull. Iranian Math. Soc. 39 (2013), no. 4, 743-764.
- H. Piri, Hybrid pseudo-viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non-expansive map- pings, Nonlinear Anal. 74 (2011), no. 17, 6788-6804. https://doi.org/10.1016/j.na.2011.06.056
- R. P. Agarwal, D. O. Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian Type Mappings with Applications, Topological Fixed Point Theory and Its Applications, 6, Springer, New York, 2009.
- F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-222. https://doi.org/10.1016/0022-247X(67)90085-6