General Iterative Scheme for Split Mixed Equilibrium Problems, Variational Inequality Problems and Fixed Point Problems in Hilbert Spaces

Jitsupa Deepho, Poom Kumam

Abstract


The purpose in this paper is to study the strong convergence of general iterative scheme to find a common element of the set of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping and the set of solutions of split mixed equilibrium problem. Our results extend recent results announced by many others.


Keywords


Split mixed equilibrium problem; Fixed point problem; Hilbert spaces; Relaxed cocoercive mapping; Finite family of nonexpansive mappings

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References


S. Atsushiba and W. Takahashi, Strong convergence theorems for a finite family of nonexpansive mappings and applications, Indian Journal of Mathematics 41 (1999), 435 – 453.

H. H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, Journal of Mathematical Analysis and Applications 202 (1996), 150 – 159, DOI: 10.1006/jmaa.1996.0308.

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review 38 (1996), 367 – 426, DOI: 10.1137/S0036144593251710.

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Mathematics Students 63 (1994), 123 – 145.

A. Bnouhachem, Algorithms of common solutions for a variational inequality, a split equilibrium problem and a hierarchical fixed point problem, Fixed Point Theory and Applications 2013 (2013), Article number 278, DOI: 10.1186/1687-1812-2013-278.

A. Bnouhachem, Strong convergence algorithm for split equilibrium problems and hierarchical fixed point problems, The Scientific World Journal 2014 (2014), Article ID 390956, DOI: 10.1155/2014/390956.

L. C. Ceng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, Journal of Computational and Applied Mathematics 214 (2008), 186 – 201, DOI: 10.1016/j.cam.2007.02.022.

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms 8 (1994), 221 – 239, DOI: 10.1007/BF02142692.

Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Physics in Medicine & Biology 51 (2006), 2353 – 2365, DOI: 10.1088/0031-9155/51/10/001.

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems 21 (2005), 2071 – 2084, DOI: 10.1088/0266-5611/21/6/017.

Y. Censor, A. Motova and A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, Journal of Mathematical Analysis and Applications 327 (2007), 1244 – 1256, DOI: 10.1016/j.jmaa.2006.05.010.

T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia (2005), DOI: 10.1137/1.9780898717877.

J. M. Chen, L. J. Zhang and T. G. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, Journal of Mathematical Analysis and Applications 334 (2007), 1450 – 1461, DOI: 10.1016/j.jmaa.2006.12.088.

P. L. Combettes, Constrained image recovery in a product space, in Proceedings of the IEEE International Conference on Image Processing, Washington, DC, 1995, IEEE Computer Society Press, California, pp. 2025 – 2028 (1995), DOI: 10.1109/ICIP.1995.537406.

P. I. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis 6 (2005), 117 – 136, http://www.ybook.co.jp/online2/jncav6.html.

J. Deepho, W. Kumam and P. Kumam, A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems, Journal of Mathematical Modelling and Algorithms in Operations Research 13(4) (2014), 405 – 423, DOI: 10.1007/s10852-014-9261-0.

J. Deepho, J. Martinez-Moreno and P. Kumam, A viscosity of Cesaro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems, Journal of Nonlinear Science and Applications 9 (2016), 1475 – 1496, DOI: 10.22436/jnsa.009.04.07.

F. Deutsch and H. Hundal, The rate of convergence of Dykstras cyclic projections algorithm: The polyhedral case, Numerical Functional Analysis and Optimization 15 (1994), 537 – 565, DOI: 10.1080/01630569408816580.

F. Deutsch and I. Yamada, Minimizing certain convex functions over the intersection of the fixed point set of nonexpansive mappings, Numerical Functional Analysis and Optimization 19 (1998), 33 – 56, DOI: 10.1080/01630569808816813.

S. D. Flåm and A. S. Antipin, Equilibrium programming using proximal-like algorithm, Mathematical Programming 78 (1997), 29 – 41, DOI: 10.1007/BF02614504.

H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inversestrongly monotone mappings, Nonlinear Analysis: Theory, Methods & Applications 61 (2005), 341 – 350, DOI: 10.1016/j.na.2003.07.023.

A. N. Iusem and A. R. De Pierro, On the convergence of Hans method for convex programming with quadratic objective, Mathematical Programming 52 (1991), 265 – 284, DOI: 10.1007/BF01582891.

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, Journal of the Egyptian Mathematical Society 21 (2013), 44 – 51, DOI: 10.1016/j.joems.2012.10.009.

W. Kumam, J. Deepho and P. Kumam, Hybrid extragradient method for finding a common solution of the split feasibility and systemof equilibrium problems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 21(6) (2014), 367 – 388.

Z. Ma, L. Wang, S. S. Chang and W. Duan, Convergence theorems for split equality mixed equilibrium problems with applications, Fixed Point Theory and Applications 2015 (2015), Article number 31, DOI: 10.1186/s13663-015-0281-x.

G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications 318 (2006), 43 – 52, DOI: 10.1016/j.jmaa.2005.05.028.

A. Moudafi, Viscosity approximation methods for fixed points problems, Journal of Mathematical Analysis and Applications 241 (2000), 46 – 55, https://core.ac.uk/download/pdf/82181463.pdf.

Z. Opial,Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bulletin of American Mathematical Society 73 (1967), 591 – 597, DOI: 10.1090/S0002-9904-1967-11761-0.

R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Transactions of the American Mathematical Society 149 (1970), 75 – 88, DOI: 10.1090/S0002-9947-1970-0282272-5.

S. Suantai, P. Cholamjiak, Y. J. Cho and W. Cholamjiak, On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces, Fixed Point Theory and Applications 2016 (2016), Article number 35, DOI: 10.1186/s13663-016-0509-4.

W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications 118 (2003), 417 – 428, DOI: 10.1023/A:1025407607560.

R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and its projection methods, Journal of Optimization Theory and Applications 121(1) (2004), 203 – 210, DOI: 10.1023/B:JOTA.0000026271.19947.05.

R. U. Verma, General convergence analysis for two-step projection methods and application to variational problems, Applied Mathematics Letters 18(11) (2005), 1286 – 1292, DOI: 10.1016/j.aml.2005.02.026.

H. K. Xu, Iterative algorithms for nonlinear operators, Journal of London Mathematical Society 66 (2002), 240 – 256, DOI: 10.1112/S0024610702003332.

H. K. Xu, An iterative approach to quadratic optimization, Journal of Optimization Theory and Applications 116 (2003), 659 – 678, DOI: 10.1023/A:1023073621589.

I. Yamada, The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings, in Inherently Parallel Algorithm for Feasibility and Optimization, D. Butnariu, Y. Censor and S. Reich (editors), Elsevier, pp. 473 – 504 (2001).

D. C. Youla, Mathematical theory of image restoration by the method of convex projections, in Image Recovery: Theory and Applications, H. Stark (editor), Academic Press, Florida, pp. 29 – 77 (1987), https://books.google.co.in/books?hl=en&lr=&id=xs7d049z-6sC&oi=fnd&pg=PA29&ots=kaEdd9mBqV&sig=0b573ujfdmxDVBwKE2-pTRJcy8w&redir_esc=y#v=onepage&q&f=false.




DOI: http://dx.doi.org/10.26713%2Fcma.v11i1.1298

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