A Viscosity Nonlinear Algorithm for Split Generalized Equilibrium Problem

Yıl 2019, Cilt: 16 Sayı: 1, 35 - 53, 31.05.2019

Hamid Reza Sahebi , Mahdi Azhini

Öz

In this paper, we proposed a viscosity iterative algorithm to approximate a common solution of split generalized equilibrium problem and fixed point problem for a
nonexpansive semigroups in real Hilbert spaces. Under certain conditions control on
parameters, the iteration sequences generated by the proposed algorithms are proved
to be strongly convergent to a solution of split generalized equilibrium problem. Our
results can be viewed as a generalization and improvement of various existing results
in the current literature. Some numerical examples to guarantee the main result of
this paper.

Anahtar Kelimeler

Nonexpansive semigroup, split general equilibrium problems system, strongly positive linear bounded operator

Kaynakça

• Ya. I. Alber, C. E. Chidume and H. Zegeye, Approximating fixed points of total asymptotically nonexpansive mappings, Fixed Point Theory Appl., (2006), 2006:10673, 20 pp.
• E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63, (1994),123-145.
• F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.
• C. Byrne, Y. Censor, A. Gibali, S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13(4), (2012), 759-775.
• Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity modulated radiation theory, Phys. Med. Biol. 51, (2006), 2353-2365.
• Y. Censor, T. Elfving, A multi projection algorithm using Bregman projections in product space, Numer. Algorithm. 8, (1994), 221-239.
• Y. Censor, A. Gibali, S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithm. 59(2), (2012), 301-323.
• S. S. Chang, J. Lee, H. W. Chan, An new method for solving equilibrium problem, fixed point problem and variational inequality problem with application to optimization, Nonlinear Analysis. 70, (2009), 3307-3319.
• F. Cianciaruso, G. Marino, L. Muglia, Iterative methods for equilibrium and fixed point problems for non expansive semigroups in Hilbert space, J. Optim. Theory Appl. 146, (2010), 491-509.
• F. Cianciaruso, G. Marino, L. Muglia, Y. Yao, A hybrid projection algorithm for finding solution of mixed equilibrium problem and variational inequality problem, Fixed Point Theory Appl. 2010, (2010), 383-740.
• G. Crombez, A hicrarchical presentation of operators with fixed points on Hilbert spaces, Numer. Funct. Anal. Optim. 27, (2006), 259-277.
• J. Kang, Y. Su and X. Zhang, Genaral iterative algorithm for non expansive semigroups and variational inequalitis in Hilbert space, Journal of Inequalities and Applications, ( 2010) Article ID.264052, 10 pages.
• K.R. Kazmi, S.H. Rizvi, Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for non expansive semigroup, Math. Sci. 7, (2013). Art. 1.
• M. A. A. Khan, Convergence characteristics of a shrinking projection algorithm in the sense of Mosco for split equilibrium problem and fixed point problem in Hilbert spaces, Linear Nonlinear Anal., 3, (2017), 423-435.
• M. A. A. Khan, Y. Arafat and A.R. Butt, A shrinking projection approach to solve split equilibrium problems and fixed point problems in Hilbert spaces, (Accepted for Publication) University Politehnica of Bucharest, Scientific Bulletin, Series A, Applied Mathematics and Physics.
• Z. Ma and L. Wang, An algorithm with strong convergence for the split common fixed point problem of total asymptotically strict pseudo contraction mappings, J. Inequal. Appl.,(2015), 2015:40 DOI 10.1186/s13660-015- 0562-2.
• G. Marino, HK, Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, Math. Appl. 318, (2006), 43-52.
• A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241, (2000), 46-55.
• A. Moudafi, The split common fixed point problem for demicontractive mappings, Invers Probl. 26, (2010) ,055007.
• A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl. 150, (2011), 275-283.
• A. Moudafi, M. Thera, Proximal and Dynamical Approaches to Equilibrium Problems, in: Lecture Notes in Economics and Mathematical Systems, Springer, vol. 477, (1999), 187-201.
• N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mapping and monotone mapping, J. Optim. Theory Appl. 128, (2006), 191-201.
• Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc. 73(4), (1967), 595-597.
• S. Plubtieng, R. Punpaeng, Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Model. 48, (2008), 279-286.
• H. R. Sahebi, A. Razani, A solution of a general equilibrium problem, Acta Mathematica Scientia 33B(6), (2013), 1598-1614.
• H. R. Sahebi, A. Razani, An iterative algorithm for finding the solution of a general equilibrium problem system, Faculty of Sciences and Mathematics, University of Nis, Serbia, 7, (2014), 1393-1415.
• H. R. Sahebi, S. Ebrahimi, An explicit viscosity iterative algorithm for finding the solutions of a general equilibrium problem systems, Tamkang Journal Of Mathematics 46(3), (2015), 193-216.
• H. R. Sahebi, S. Ebrahimi, A Viscosity iterative algorithm for the optimization problem system, Faculty of Sciences and Mathematics, University of Nis, Serbia, 8, (2017), 2249-2266.
• T. Shimizu, W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211, (1997), 71-83.
• T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305, (2005), 227-239.
• M. Taherian, M. Azhini, Strong convergence theorems for fixed point problem of infinite family of non self mapping annd generalized equilibrium problems with perturbation in Hilbert spaces, Advances and Applications in Mathematical Sciences 15(2), (2016), 25-51.
• W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, 118, (2003), 417-428.
• S. Takahashi, W. Takahashi, Viscosity approximation method for equilibrium and fixed point problems in Hilbert space, J. Math. Anall. Appl. 331, (2007), 506-515.
• S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69, (2008), 1025-1033.
• H. H. Xu, T. H. Kim, Convergence of hybrid steepest-descent methods for variational inequalites, J. Optim. Theory Appl. 119, (2003), 185-201.
• Y. Yao, J. C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186, (2007), 1551-1558.
• L. C. Zeng, Y. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math. 10, (2006), 1293-1303.
• H. K. Xu, Viscosity approximation method for nonexpansive semigroups, J. Math. Anal. Appl. 298, (2004), 279- 291.
• Y. Zhang, Y. Gui, Strong convergence theorem for split equilibrium problem and fixed point problem in Hilbert spaces, Int. Math. 12(9), (2017), 413-427.