International Journal of Nonlinear Analysis and Applications
Hybrid iterative algorithms for finding common solutions of a system of generalized mixed quasi-equilibrium problems and fixed point problems of nonexpansive semigroups
Document Type : Research Paper
Authors
Department of Mathematics, Jazan University, Jazan-45142, KSA
Abstract
In this paper, we introduced a hybrid iterative method for finding the set of common solutions for a system of generalized mixed quasi-equilibrium problems, the set of common fixed points for nonexpansive semigroup and the set of solutions of quasi-variational inclusion problems with multi-valued maximal monotone mappings and inverse strongly monotone mappings in Hilbert spaces. Under suitable assumptions, we prove some strong convergence theorems for the iteration.
Keywords
[1] M.K. Ahmad and Salahuddin, Perturbed three step approximation process with errors for a general implicit nonlinear variational inequalities, Int. J. Math. Math. Sci. 2006 (2006), Article ID 43818, 1–14.
[2] M.K. Ahmad and Salahuddin, A stable perturbed algorithms for a new class of generalized nonlinear implicit quasi variational inclusions in Banach spaces, Adv. Pure Math. 2 (2012), no. 2, 139–148.
[3] S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal. 30 (1997), 4197–4208.
[4] L.C. Ceng and J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), 186–201.
[5] R. Chen and H. He, Viscosity approximation of common fixed points of nonexpansive semigroups in Banach spaces, Appl. Math. Lett. 20 (2007), 751–757.
[6] M.M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–514.
[7] J.K. Kim and S.S. Chang, Generalized mixed equilibrium problems for an infinite family of quasi-ϕ-nonexpansive mappings in Banach spaces, Nonlinear Anal. Convex Anal. RIMS Kokyuroku, Kyoto Univ. 1923 (2014), 28–41.
[8] J.K. Kim, S.Y. Cho and X.L. Qin, Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings, J. Inequal. Appl. 2010 (2010), Article ID 312062, 17 pages.
[9] J.K. Kim, Salahuddin and W.H. Lim, An iterative algorithm for generalized mixed equilibrium problems and fixed points of nonexpansive semigroups, J. Appl. Math. Phys. 5 (2017), 276–293.
[2] M.K. Ahmad and Salahuddin, A stable perturbed algorithms for a new class of generalized nonlinear implicit quasi variational inclusions in Banach spaces, Adv. Pure Math. 2 (2012), no. 2, 139–148.
[3] S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal. 30 (1997), 4197–4208.
[4] L.C. Ceng and J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), 186–201.
[5] R. Chen and H. He, Viscosity approximation of common fixed points of nonexpansive semigroups in Banach spaces, Appl. Math. Lett. 20 (2007), 751–757.
[6] M.M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–514.
[7] J.K. Kim and S.S. Chang, Generalized mixed equilibrium problems for an infinite family of quasi-ϕ-nonexpansive mappings in Banach spaces, Nonlinear Anal. Convex Anal. RIMS Kokyuroku, Kyoto Univ. 1923 (2014), 28–41.
[8] J.K. Kim, S.Y. Cho and X.L. Qin, Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings, J. Inequal. Appl. 2010 (2010), Article ID 312062, 17 pages.
[9] J.K. Kim, Salahuddin and W.H. Lim, An iterative algorithm for generalized mixed equilibrium problems and fixed points of nonexpansive semigroups, J. Appl. Math. Phys. 5 (2017), 276–293.
[10] J.N. Ezeora, Approximating solution of generalized mixed equilibrium problem and fixed point of multi valued strongly pseudocontractive mappings, Nigerian Math. Soc. 38 (2019), no. 3, 515–532.
[11] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006), 191–201.
[12] A.T.M. Lau and W. Takahashi, Invariant means and fixed point properties for nonexpansive representations of topological semigroups, Topological Meth. Nonlinear Anal. 5 (1995), 39–57.
[13] A.T.M. Lau, H. Miyake and W. Takahashi, Approximation of fixed points for semigroups of nonexpansive mappings in Banach spaces, Nonlinear Anal. TMA, 67 (2007), 1211–1225.
[14] A.T.M. Lau, N. Shioji and W. Takahashi, Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorem in Banach spaces, J. Funct. Anal. 161 (1999), 62–75.
[15] S. Li, L. Li, L. Cao, X. He and X. Yue, Hybrid extragradient method gor generalized mixed equilibrium problems and fixed point problems in Hilbert spaces, Fixed point Theo. Appl. 2013 (2013), 240.
[16] J. L. Lions and G. Stampacchia, Variational Inequalities, Comm. Pure Appl. Math. 20 (1967), 493–517.
[17] D. Pascali, Nonlinear mappings of monotone type, Sijthoff and Noordhoff International Publishers, The Netherlands 1978.
[18] I. Inchan, Hybrid extragradient metheod for general equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Anal.: Hybrid Syst. 5 (2011), 467–478.
[19] G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006), 43–52.
[20] S. Saeidi, Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of family and semigroups of nonexpansive mappings, Nonlinear Anal. 70 (2009), 4195– 4209.
[21] S. Saeidi, Existence of ergodic retractions for semigroups in Banach spaces, Nonlinear Anal. 69 (2008), 3417–3422.
[22] T. Suzuki, Strong convergence of Krasnoselski and Mann type sequence for one parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005), 227–239.
[23] T. Shimizu and W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997), 71–83.
[24] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), 506–515.
[25] H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theo. Appl. 116(3) (2003), 659–678.
[26] L. Yang, F. Zhao and J.K. Kim, Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinity family of asymptotically quasi ϕ-nonexpansive mappings in Banach spaces, Appl. Math. Comput. 218 (2012), 6072–6082.
[27] S. Zhang, J.H. Joseph Lee and C.K. Chan, Algorithm of common solutions for quasi variational inclusions and fixed point problems, Appl. Math. Mech. 29 (2008), 1–11.
[11] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006), 191–201.
[12] A.T.M. Lau and W. Takahashi, Invariant means and fixed point properties for nonexpansive representations of topological semigroups, Topological Meth. Nonlinear Anal. 5 (1995), 39–57.
[13] A.T.M. Lau, H. Miyake and W. Takahashi, Approximation of fixed points for semigroups of nonexpansive mappings in Banach spaces, Nonlinear Anal. TMA, 67 (2007), 1211–1225.
[14] A.T.M. Lau, N. Shioji and W. Takahashi, Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorem in Banach spaces, J. Funct. Anal. 161 (1999), 62–75.
[15] S. Li, L. Li, L. Cao, X. He and X. Yue, Hybrid extragradient method gor generalized mixed equilibrium problems and fixed point problems in Hilbert spaces, Fixed point Theo. Appl. 2013 (2013), 240.
[16] J. L. Lions and G. Stampacchia, Variational Inequalities, Comm. Pure Appl. Math. 20 (1967), 493–517.
[17] D. Pascali, Nonlinear mappings of monotone type, Sijthoff and Noordhoff International Publishers, The Netherlands 1978.
[18] I. Inchan, Hybrid extragradient metheod for general equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Anal.: Hybrid Syst. 5 (2011), 467–478.
[19] G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006), 43–52.
[20] S. Saeidi, Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of family and semigroups of nonexpansive mappings, Nonlinear Anal. 70 (2009), 4195– 4209.
[21] S. Saeidi, Existence of ergodic retractions for semigroups in Banach spaces, Nonlinear Anal. 69 (2008), 3417–3422.
[22] T. Suzuki, Strong convergence of Krasnoselski and Mann type sequence for one parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005), 227–239.
[23] T. Shimizu and W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997), 71–83.
[24] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), 506–515.
[25] H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theo. Appl. 116(3) (2003), 659–678.
[26] L. Yang, F. Zhao and J.K. Kim, Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinity family of asymptotically quasi ϕ-nonexpansive mappings in Banach spaces, Appl. Math. Comput. 218 (2012), 6072–6082.
[27] S. Zhang, J.H. Joseph Lee and C.K. Chan, Algorithm of common solutions for quasi variational inclusions and fixed point problems, Appl. Math. Mech. 29 (2008), 1–11.
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Volume 14, Issue 1
January 2023Pages 2771-2786
- Receive Date: 01 December 2022
- Accept Date: 02 January 2023