Abstract
The aim of this work is twofold; first, to give an extension to split mixed equilibrium problem. Secondly, we suggest and analyze an iterative method based on Mann iterative method and Halpern iterative method to find a common solution of split mixed equilibrium problem and fixed point problem for a nonexpansive mapping in the framework of real Hilbert spaces. Further, under some mild conditions, strong convergence theorem is obtained by the sequences generated by the proposed iterative method, which solves the variational inequality problem. Furthermore, we derived some consequences from our main result and provide some numerical experiments. Our results can be viewed as significant extension and generalization of the previously known results for solving equilibrium problems.
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References
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Daniele, P., Giannessi, F., Mougeri, A. (eds.): Equilibrium Problems and Variational Models, Nonconvex Optimization and its Application, vol. 68. Kluwer, Norwell (2003)
Giannessi, F., (Eds): Vector Variational Inequalities and Vector Equilibria Mathematical Theories, Nonconvex Optimization and its Applications, vol. 38. Kluwer, Dordrecht (2000)
Moudafi, A.: Second order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4(1), Art. 18 (2003)
Huang, N.J., Deng, C.X.: Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. J. Math. Anal. Appl. 256, 345–359 (2001)
Kazmi, K.R., Khan, F.A.: Existence and iterative approximation of solutions of generalized mixed equilibrium problems. Comput. Math. Appl. 56(5), 1314–1321 (2008)
Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004)
Ceng, L.C., Yao, J.C.: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186–201 (2008)
Chang, S.S., Lee, H.W., Chan, C.K.: A new method for solving equilibrium problem, fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009)
Cianciaruso, F., Marino, G., Muglia, L., Yao, Y.: A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl., Article ID 383740 (2010)
Flam, S.D., Antipin, A.S.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997)
Kazmi, K.R., Rizvi, S.H.: A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem. Appl. Math. Comput. 218, 5439–5452 (2012)
Tada, A., Takahashi, W.: Strong convergence theorem for an equilibrium problem and and a nonexpansive mapping. In: Takahashi, W., Tanaka, T. (eds.) Nonlinear Analysis and Convex Analysis, pp. 609–617, Yokohama (2006)
Kazmi, K.R., Rizvi, S.H.: Implicit iterative method for approximating a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup. Arab J. Math. Sci. 20(1), 57–75 (2014)
Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 21(4), 1319–1344 (2011)
Reich, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp. Math. 568, 225–240 (2012)
Mann, W.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)
Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 591–597 (1967)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59(2), 301–323 (2012)
Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)
Kazmi, K.R., Rizvi, S.H.: 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(1), 44–51 (2013)
Moudafi, A., Théra, M.: Proximal and Dynamical Approaches to Equilibrium Problems. Lecture Notes in Economics and Mathematical Systems, vol. 477. Springer, New York, pp. 187–201 (1999)
Moudafi, A.: Mixed equilibrium problems sensitivity analysis and algorithmic aspect. Comput. Math. Appl. 44, 1099–1108 (2002)
Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equation. Acta Math. 115, 271–310 (1966)
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in product space. Numer. Algorithms 8, 221–239 (1994)
Masad, E., Reich, S.: A note on the multiple split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8(3), 367–371 (2007)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Byrne, C.: A unified treatment for some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron. Phys. 95, 155–453 (1996)
Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118(2), 417–428 (2003)
Moudafi, A.: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9(1), 37–43 (2008)
Takahashi, S., Takahashi, W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)
Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133(3), 359–370 (2007)
Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)
Bnouhachem, A.: Strong convergence algorithm for split equilibrium problems and hierarchical fixed point problems. Sci. World J., Article ID 390956 (2014)
Kazmi, K.R., Rizvi, S.H.: Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup. Math. Sci. 7(1), 10 (2013)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984)
Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73(4), 595–597 (1967)
Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005)
Xu, H.K.: Viscosity approximation method for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)
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Rizvi, S.H. A strong convergence theorem for split mixed equilibrium and fixed point problems for nonexpansive mappings. J. Fixed Point Theory Appl. 20, 8 (2018). https://doi.org/10.1007/s11784-018-0487-8
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DOI: https://doi.org/10.1007/s11784-018-0487-8