Algorithms of common solutions for a fixed point of hemicontractive-type mapping and a generalized equilibrium problem
DOI:
https://doi.org/10.14419/ijams.v5i1.7270Keywords:
Fixed Points of Mappings, Generalized Equilibrium Problem, Hemicontractive-type Multi-valued Mapping, Iterative Algorithm, Strong Convergence.Abstract
In this paper, we introduce and study an iterative algorithm for finding a common element of the set of fixed points of a Lipschitz hemicontractive-type multi-valued mapping and the set of solutions of a generalized equilibrium problem in the framework of Hilbert spaces. Our results improve and extend most of the results that have been proved previously by many authors in this research area.References
[1] R. P. Agrawal, D. O’Regan, D. R. Sahu, “Fixed Point Theory for Lipschitzian-type Mappings with Applications,†Springer, New York, 2009.
[2] V. Berinde, M. Pcurar, “The role of Pompeiu-Hausdorff metric in ï¬xed point theory,†Creat. Math. Inform., 22 (2) (2013) 143–150.
[3] E. Blum, W. Oettli, “From optimization and variational inequalities to equilibrium problems,†Math. Stud., 63 (1994) 123-145.
[4] L. Ceng, A. Petrusel, M. Wong, “Strong convergence theorem for a generalized equilibrium problem and a pseudocontractive mapping in a Hilbert space,†Taiwan J. Math., 14 (2010) 1881-1901.
[5] C. E. Chidume, C.O. Chidume, N. Djitte, M.S. Minjibir, “Convergence theorems for ï¬xed points of multi-valued strictly pseudocontractive mapping in Hilbert spaces,†Abstr. Appl. Anal., 2013, Article ID 629468, 10 pages. http://dx.doi.org/10.1155/2013/629468.
[6] P. L. Combettes, S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,†J. Nonlinear Convex Anal., 6 (2005) 117-136.
[7] H. Iiduka, W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,’’ Nonlinear Analysis, 61 (2005) 341-350.
[8] F. O. Isiogugu, M.O. Osilike, “Convergence theorems for new classes of multi-valued hemicontractive-type mappings, ’’ Fixed point theory Appl., 2014 (2014):93, 12 pages.
[9] P. E. Mainge, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,’’ Set-Valued Anal., 16 (2008) 899-912.
[10]T. H. Meche, M. G. Sangago, H. Zegeye, “Iterative methods for a fixed point of hemicontractive-type mapping and a solution of a variational inequality problem,’’ Creat. Math. Inform., 25 (2) (2016) 183-196.
[11]Moudafi, “Weak convergence theorems for nonexpansive mapping and equilibrium,’’ Nonlinear Convex Anal., 9(1) (2008), 37-43.
[12]S. B. Nadler, Jr., “Multi-valued contraction mappings,’’ Pacific J. Math., 30 (1969) 475-487.
[13]Rafiq, “On Mann iteration in Hilbert spaces,’’ Nonlinear Analysis, 66 (2007) 2230-2236.
[14]K.P.R. Sastry, G.V.R. Babu, “Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point,’’ Czechoslovak Math. J., 55 (130) (2005) 817-826.
[15]N. Shahzad, H. Zegeye, “On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces,’’ Nonlinear Analysis: Theory, Method and Application, 71 (2009) 838-844.
[16]S. Takahashi, W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,’’ J. Math. Anal. Appl., 331 (2007) 506-515.
[17]S. Takahashi, W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and nonexpansive mapping in a Hilbert space,’’ Nonlinear Anal., 69 (2008) 1025-1033.
[18]S. Wang, H. Zhou, J. Song, “Viscosity approximation methods for equilibrium problems and fixed point problems of nonexpansive mappings and inverse-strongly monotone mappings,’’ Method Appl. Anal., 14 (4) (2007) 405-420.
[19]S. T. Woldeamanuel, M. G. Sangago, H. Zegeye, “Strong convergence theorems for a common fixed point of a finite family of Lipschitz hemicontractive-type multi-valued mappings,’’ Adv. Fixed Point Theory, 5 (2) (2015) 228-253.
[20]H. K. Xu, “Another control condition in an iterative method for nonexpansive mappings,’’ Bull. Aust. Math. Soc., 65 (2002) 109-113.
[21]Y. Yu, Z. Wu, P. Yang, “An iterative algorithm for hemi-contractive mappings in Banach spaces,’’ Abstr. Appl. Anal., 2012, Article ID 264103, 11 pages. doi:10.1155/2012/264103.
[22]H. Zegeye, N. Shahzad, “Convergence of Manns type iteration method for generalized asymptotically nonexpansive mappings,’’ Comput. Math. Appl., 62 (2011) 4007-4014.