Document Type: Research Paper
Authors
- Tesfalem Hadush Meche ^{1}
- Habtu Zegeye ^{} ^{} ^{2}
^{1} Department of Mathematics, College of Natural and Computational Sciences, Aksum University, P.O.Box 1020, Aksum, Ethiopia.
^{2} Department of Mathematics and Statistical Sciences, Faculty of Sciences, Botswana International University of Science and Technology, Private Mail Bag 16, Palapye, Botswana.
Abstract
In this article, we introduced an iterative scheme for finding a common element of the set of fixed points of a multi-valued hemicontractive-type mapping, the set of common solutions of a finite family of split equilibrium problems and the set of common solutions of a finite family of variational inequality problems in real Hilbert spaces. Moreover, the sequence generated by the proposed algorithm is proved to be strongly convergent to a common solution of these three problems under mild conditions on parameters. Our results improve and generalize many well-known recent results existing in the literature in this field of research.
Keywords
[1] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), pp. 123-145.
[2] Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. algorithms, 59 (2012), pp. 301-323.
[3] C.E. Chidume, C.O. Chidume, N. Djitte and M.S. Minjibir, Convergence theorems for fixed points of multi-valued strictly pseudocontractive mapping in Hilbert spaces, Abstr. Appl. Anal., 2013, Article ID 629468, 10 pages.
[4] S.Y. Cho, Approximation of solutions of a generalized variational inequality problem based on iterative methods, Commun. Korean. Math. Soc., 25 (2010), pp. 207-214.
[5] P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), pp. 117-136.
[6] M. Eslamian, General algorithms for split common fixed point problem of demicontractive mappings, Optimization, 65 (2016), pp. 443-465.
[7] Z. He, The split equilibrium problem and its convergence algorithms, J. inequal. Appl., 2012, 2012: 162.
[8] J.U. Jeong, Nonlinear algorithms for a common solution of a system of variational inequalities, a split equilibrium problem and fixed point problems, Korean J. Math. 24 (2016), pp. 495-524.
[9] S.B. Jeong, A. Raﬁq and S.M. Kang, On implicit mann type iteration process for strictly hemicontractive mappings in real smooth Banach spaces, Int. J. Pure and Applied Math., 89 (2013), pp. 95-103.
[10] K.R. Kazmi and S.H. Rizvi, Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, J. Egypt. Math. Soc., 21 (2013), pp. 44-51.
[11] J.K. Kim and N. Buong, An iterative method for common solution of a system of equilibrium problems in Hilbert spaces, Fixed Point Theory Appl., 2011 (2011), 15 pages.
[12] R. Kraikaew and S. Saejung, On a hybrid extragradient-viscosity method for monotone operator and fixed point problems, Numerical Fun. Anal. Optim., 35 (2014), pp. 32-49.
[13] P.E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), pp. 899-912.
[14] T.H. Meche, M.G. Sangago and H. Zegeye, Iterative methods for a fixed point of hemicontractive-type mapping and a solution of a variational inequality problem, Creat. Math. Inform., 25 (2016), pp. 183-196.
[15] T.H. Meche, M.G. Sangago and H. Zegeye, Approximating a common solution of a finite family of generalized equilibrium and fixed point problems, SINET: Ethiop. J. Sci., 38(2015), pp. 17-28.
[16] T.H. Meche, M.G. Sangago and H. Zegeye, Iterative methods for common solution of split equilibrium, variational inequality and fixed point problems of multi-valued nonexpansive mapping, (2017) (in press).
[17] C. Mongkolkeha, Y.J. Cho and P. Kumam, Convergence theorems for $k-$demicontractive mapping in Hibert spaces, Math. inequal. App., 16 (2013), pp. 1065-1082.
[18] S.B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math., 30 (1969), pp. 475-487.
[19] C.C. Okeke and O.T. Mewomo, On split equilibrium problem, variational inequality problem and fixed point problem for multi-valued mappings, Ann. Acad. Rom. Sci. Ser. Math. Appl., 9 (2017), pp. 223-248.
[20] S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75 (2012), pp. 742-750.
[21] K.P.R. Sastry and G.V.R. Babu, Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point, Czechoslovak Math. J., 55 (2005), pp. 817-826.
[22] N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal-Theory, 71 (2009), pp. 838-844.
[23] Y. Shehu and O. Iyiola, Nonlinear iteration method for proximal split feasibility problems, Wiley, (2016).
[24] G. Stampacchia, Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Sciences, Paris, 258 (1964), pp. 4413-4416.
[25] W. Takahashi, Nonlinear functional analysis, Yokohama Publishere, Yokohama, Japan, 2000.
[26] S. Takahashi and W. Takahashi, The split common null point problem and the shrinking projection method in Banach spaces, Optimization, 65 (2016), pp. 281-287.
[27] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), pp. 417-428.
[28] G.C. Ugwunnadi and B. Ali, Approximation methods for solutions of system of split equilibrium problems, Adv. Oper. Theory, 1 (2016), pp. 164-183.
[29] J. Vahid, A. Latif and M. Eslamian, New iterative scheme with strict pseudo-contractions and multi-valued nonexpansive mappings for fixed point problems and variational inequality problems, Fixed Point Theory Appl., (2013) 2013:213.
[30] S.H. Wang and M.J. Chen, Iterative algorithm for a split equilibrium problem and fixed point problem for finite family of asymptotically nonexpansive mappings in Hilbert space, Filomat, 31:5 (2017), pp. 1423-1434.
[31] S. Wang, X. Gong, A.A. Abdou and Y.J. Cho, Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications, Fixed Point Theory Appl., 2016:4 (2016).
[32] F. Wang and H.K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), pp. 4105-4111.
[33] S. Wang and C. Zhou, New iterative schemes for finite families of equilibrium, variational inequality and fixed point problems in Banach spaces, Fixed point Theory Appl., Vol. 2011, Article ID 372975, 18 pages.
[34] S. Wang, H. Zhou and J. Song, Viscosity approximation methods for equilibrium problems and fixed point problems of nonexpansive mappings and inverse-strongly monotone mappings, Method Appl. Anal., 14 (2007), pp. 405-420.
[35] S.T. Woldeamanuel, M.G. Sangago and H. Zegeye, Strong convergence theorems for a common fixed point of a finite family of Lipchitz hemicontractive-type multi-valued mappings, Adv. Fixed Point Theory, 5 (2015), pp. 228-253.
[36] H.K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109-113.
[37] H. Zegeye, An iterative approximation method for a common fixed point of two pseudocontractive mappings, ISRN Math. Anal., 2011 (2011), 14 pages.
[38] H. Zegeye, T.H. Meche and M.G. Sangago, Algorithms of common solutions for a fixed point of hemicontractive-type mapping and a generalized equilibrium problem, Inter. J. Adv. Math. Sci., 5 (2017), pp. 20-26.
[39] H. Zegeye and N. Shahzad, Convergence of Manns type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), pp. 4007-4014.
[40] H. Zegeye and N. Shahzad, A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems, Nonlinear Anal., 74 (2011), pp. 263-272.
[41] X. Zheng, Y. Yao, Y.C. Liou and L. Leng, Fixed point algorithms for split problem of demicontractive operators, J. Nonlinear Sci. Appl., 10 (2017), pp. 1263-1269.