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doi: 10.3934/naco.2021038
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## Triple-hierarchical problems with variational inequality

 1 Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna, 99 Chiangrai 57120, Thailand 2 Department of Mathematics, Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan 3 Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, 126 Bangkok 10140, Thailand

* Corresponding author: Thanyarat Jitpeera

Received  September 2020 Revised  August 2021 Early access September 2021

Fund Project: The first author is supported by RMUTL

In this paper, we suggest and analyze an iterative scheme for finding the triple-hierarchical problem in a real Hilbert space. We also consider the strong convergence for the proposed method under some assumptions. Our results extend ones of Ceng et. al (2011) [2], Yao et. al (2011) [24].

Citation: Thanyarat JItpeera, Tamaki Tanaka, Poom Kumam. Triple-hierarchical problems with variational inequality. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021038
##### References:
 [1] A. Bnouhachem, S. A. Homidan and Q. H. Ansari, An iterative method for common solutions of equilibrium problems and hierarchical fixed point problems, Fixed Point Theory Appl., (2014), Article number: 194. doi: 10.1186/1687-1812-2014-194.  Google Scholar [2] L. C. Ceng, Q. H. Ansari and J. C Yao, Iterative methods for triple hierarchical variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 151 (2011), 489-512.  doi: 10.1007/s10957-011-9882-7.  Google Scholar [3] L. C. Ceng, A. Latif, Q. H. Ansari and J. C Yao, Hybrid extragradientmethod for hierarchical variational inequalities, Fixed Point Theory Appl., (2014), Article number: 222. doi: 10.1186/1687-1812-2014-222.  Google Scholar [4] P. L. Combettes, A block-itrative surrogate constraint splitting method for quadratic signal recovery, IEEE Trans. Signal Process, 51 (2003, ) 1771–1782. doi: 10.1109/TSP.2003.812846.  Google Scholar [5] W. Q. Deng, New viscosity method for hierarchical fixed point approach tovariational inequalities, Fixed Point Theory Appl., (2013), Article number: 219. doi: 10.1186/1687-1812-2013-219.  Google Scholar [6] P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math., 115 (1966), 271-310.  doi: 10.1007/BF02392210.  Google Scholar [7] S. A. Hirstoaga, Iterative selection method for common fixed point problems, J. Math. Anal. Appl., 324 (2006), 1020-1035.  doi: 10.1016/j.jmaa.2005.12.064.  Google Scholar [8] H. Iiduka, Decentralized hierarchical constrained convex optimization, Optim. Engineer, 21 (2020), 181-213.  doi: 10.1007/s11081-019-09440-7.  Google Scholar [9] H. Iiduka, Fixed point optimization algorithm and its application to power control in CDMA data networks, Math. Program., Ser. A, 133 (2012), 227-242.  doi: 10.1007/s10107-010-0427-x.  Google Scholar [10] H. Iiduka, Iterative algorithm for solving triple-hierarchical constrained optimization problem, J. Optim. Theory Appl., 148 (2011), 580-592.  doi: 10.1007/s10957-010-9769-z.  Google Scholar [11] H. Iiduka, Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem, Nonlinear Anal., 71 (2009), e1292–e1297. doi: 10.1016/j.na.2009.01.133.  Google Scholar [12] H. Iiduka, W. Takahashi and M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, Pananmer. Math. J., 14 (2004), 49-61.   Google Scholar [13] T. Jitpeera and P. Kumam, Algorithms for solving the variational inequality problem over the triple hierarchical problem, Abstract Appl. Anal., (2012), Article ID 827156. doi: 10.1155/2012/827156.  Google Scholar [14] T. Jitpeera and P. Kumam, A new explicit triple hierarchical problem over the set of fixed point and generalized mixed equilibrium problem, J. Ineq. Appl., (2012), Article number: 82. doi: 10.1186/1029-242X-2012-82.  Google Scholar [15] W. A. Kirk, Fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly, 72 (1965), 1004-1006.  doi: 10.2307/2313345.  Google Scholar [16] P. E. Maing$\acute{e}$ and A. Moudafi, Strong convergence of an iterative method for hierarchical fixed-point problems, Pacific. J. Optim., 3 (2007), 529-538.   Google Scholar [17] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 595-597.  doi: 10.1090/S0002-9904-1967-11761-0.  Google Scholar [18] K. Slavakis and I. Yamada, Robust wideband beamforming by the hybrid steepest descent method, IEEE Trans. Signal Process, 55 (2007), 4511-4522.  doi: 10.1109/TSP.2007.896252.  Google Scholar [19] K. Slavakis, I. Yamada and K. Sakaniwa, Computation of symmetric positive definite Toeplitz matrices by the hybrid steepest descent method, Signal Process, 83 (2003), 1135-1140.   Google Scholar [20] N. Wairojjana and P. Kumam, General iterative algorithms for hierarchical fixed points approach to variational inequalities, J. Appl. Math., (2012), Article ID 174318. doi: 10.1155/2012/174318.  Google Scholar [21] I. Yamada, The hybrid steepest descent method for the variational inequality problems over the intersection of fixed point sets of nonexpansive mappings, In Inherently Paralle Algorithms for Feasibllity and Optimization and Their Applications (eds. D. Butnariu, Y. Censor and S. Reich), Elsevier, Amsterdam, (2001), 473–504. doi: 10.1016/S1570-579X(01)80028-8.  Google Scholar [22] I. Yamada and N. Ogura, Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mapping, Numer. Funct. Anal. Optim., 25 (2004), 619-655.  doi: 10.1081/NFA-200045815.  Google Scholar [23] I. Yamada, N. Ogura and N. Shirakawa, A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems, in: Inverse Problems, Image Analysis, and Medical Imaging, in: Contemp. Math. (eds. Z. Nashed, O. Scherzer), Amer. Math. Soc., 313 (2002), 269–305. doi: 10.1090/conm/313/05379.  Google Scholar [24] Y. Yao, Y. C. Liou and S. M. Kang, Algorithms construction for variational inequaliies, Fixed Point Theory Appl., (2011), Article number: 794203. doi: 10.1155/2011/794203.  Google Scholar [25] H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar

show all references

##### References:
 [1] A. Bnouhachem, S. A. Homidan and Q. H. Ansari, An iterative method for common solutions of equilibrium problems and hierarchical fixed point problems, Fixed Point Theory Appl., (2014), Article number: 194. doi: 10.1186/1687-1812-2014-194.  Google Scholar [2] L. C. Ceng, Q. H. Ansari and J. C Yao, Iterative methods for triple hierarchical variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 151 (2011), 489-512.  doi: 10.1007/s10957-011-9882-7.  Google Scholar [3] L. C. Ceng, A. Latif, Q. H. Ansari and J. C Yao, Hybrid extragradientmethod for hierarchical variational inequalities, Fixed Point Theory Appl., (2014), Article number: 222. doi: 10.1186/1687-1812-2014-222.  Google Scholar [4] P. L. Combettes, A block-itrative surrogate constraint splitting method for quadratic signal recovery, IEEE Trans. Signal Process, 51 (2003, ) 1771–1782. doi: 10.1109/TSP.2003.812846.  Google Scholar [5] W. Q. Deng, New viscosity method for hierarchical fixed point approach tovariational inequalities, Fixed Point Theory Appl., (2013), Article number: 219. doi: 10.1186/1687-1812-2013-219.  Google Scholar [6] P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math., 115 (1966), 271-310.  doi: 10.1007/BF02392210.  Google Scholar [7] S. A. Hirstoaga, Iterative selection method for common fixed point problems, J. Math. Anal. Appl., 324 (2006), 1020-1035.  doi: 10.1016/j.jmaa.2005.12.064.  Google Scholar [8] H. Iiduka, Decentralized hierarchical constrained convex optimization, Optim. Engineer, 21 (2020), 181-213.  doi: 10.1007/s11081-019-09440-7.  Google Scholar [9] H. Iiduka, Fixed point optimization algorithm and its application to power control in CDMA data networks, Math. Program., Ser. A, 133 (2012), 227-242.  doi: 10.1007/s10107-010-0427-x.  Google Scholar [10] H. Iiduka, Iterative algorithm for solving triple-hierarchical constrained optimization problem, J. Optim. Theory Appl., 148 (2011), 580-592.  doi: 10.1007/s10957-010-9769-z.  Google Scholar [11] H. Iiduka, Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem, Nonlinear Anal., 71 (2009), e1292–e1297. doi: 10.1016/j.na.2009.01.133.  Google Scholar [12] H. Iiduka, W. Takahashi and M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, Pananmer. Math. J., 14 (2004), 49-61.   Google Scholar [13] T. Jitpeera and P. Kumam, Algorithms for solving the variational inequality problem over the triple hierarchical problem, Abstract Appl. Anal., (2012), Article ID 827156. doi: 10.1155/2012/827156.  Google Scholar [14] T. Jitpeera and P. Kumam, A new explicit triple hierarchical problem over the set of fixed point and generalized mixed equilibrium problem, J. Ineq. Appl., (2012), Article number: 82. doi: 10.1186/1029-242X-2012-82.  Google Scholar [15] W. A. Kirk, Fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly, 72 (1965), 1004-1006.  doi: 10.2307/2313345.  Google Scholar [16] P. E. Maing$\acute{e}$ and A. Moudafi, Strong convergence of an iterative method for hierarchical fixed-point problems, Pacific. J. Optim., 3 (2007), 529-538.   Google Scholar [17] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 595-597.  doi: 10.1090/S0002-9904-1967-11761-0.  Google Scholar [18] K. Slavakis and I. Yamada, Robust wideband beamforming by the hybrid steepest descent method, IEEE Trans. Signal Process, 55 (2007), 4511-4522.  doi: 10.1109/TSP.2007.896252.  Google Scholar [19] K. Slavakis, I. Yamada and K. Sakaniwa, Computation of symmetric positive definite Toeplitz matrices by the hybrid steepest descent method, Signal Process, 83 (2003), 1135-1140.   Google Scholar [20] N. Wairojjana and P. Kumam, General iterative algorithms for hierarchical fixed points approach to variational inequalities, J. Appl. Math., (2012), Article ID 174318. doi: 10.1155/2012/174318.  Google Scholar [21] I. Yamada, The hybrid steepest descent method for the variational inequality problems over the intersection of fixed point sets of nonexpansive mappings, In Inherently Paralle Algorithms for Feasibllity and Optimization and Their Applications (eds. D. Butnariu, Y. Censor and S. Reich), Elsevier, Amsterdam, (2001), 473–504. doi: 10.1016/S1570-579X(01)80028-8.  Google Scholar [22] I. Yamada and N. Ogura, Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mapping, Numer. Funct. Anal. Optim., 25 (2004), 619-655.  doi: 10.1081/NFA-200045815.  Google Scholar [23] I. Yamada, N. Ogura and N. Shirakawa, A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems, in: Inverse Problems, Image Analysis, and Medical Imaging, in: Contemp. Math. (eds. Z. Nashed, O. Scherzer), Amer. Math. Soc., 313 (2002), 269–305. doi: 10.1090/conm/313/05379.  Google Scholar [24] Y. Yao, Y. C. Liou and S. M. Kang, Algorithms construction for variational inequaliies, Fixed Point Theory Appl., (2011), Article number: 794203. doi: 10.1155/2011/794203.  Google Scholar [25] H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar
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