On general implicit hybrid iteration method for triple hierarchical variational inequalities with hierarchical variational inequality constraints

References

• Anh PN, Kim JK, Muu LD. An extragradient algorithm for solving bilevel pseudomonotone variational inequalities. J Glob Optim. 2012;52:627–639. doi: 10.1007/s10898-012-9870-y
   Web of Science ®Google Scholar
• Iiduka H. Iterative algorithm for solving triple-hierarchical constrained optimization problem. J Optim Theory Appl. 2011;148:580–592. doi: 10.1007/s10957-010-9769-z
   Web of Science ®Google Scholar
• Iiduka H. Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem. Nonlinear Anal. 2009;71:e1292–e1297. doi: 10.1016/j.na.2009.01.133
   Web of Science ®Google Scholar
• Ceng LC, Wong NC, Yao JC. Regularized hybrid iterative algorithms for triple hierarchical variational inequalities. J Inequal Appl. 2014;2014. Artical ID 490.
   Web of Science ®Google Scholar
• Cho SY, Qin XL, Yao JC, et al. Viscosity approximation splitting methods for monotone and nonexpansive operators in Hilbert spaces. J Nonlinear Convex Anal. 2018;19:251–264.
   Web of Science ®Google Scholar
• Ceng LC, Petrusel A, Yao JC. Multi-step hybrid iterative method for triple hierarchical variational inequality problem with equilibrium problem constraint. J Nonlinear Convex Anal. 2012;13:475–502.
   Web of Science ®Google Scholar
• Ceng LC. Approximation of common solutions of a split inclusion problem and a fixed-point problem. J Appl Numer Optim. 2019;1:1–12.
   Google Scholar
• Chang SS, Wen CF, Yao JC. Zero point problem of accretive operators in Banach spaces. Bull Malays Math Sci Soc. 2019;42:105–118. doi: 10.1007/s40840-017-0470-3
   Web of Science ®Google Scholar
• Ceng LC, Ansari QH, Yao JC. Relaxed hybrid steepest-descent methods with variable parameters for triple-hierarchical variational inequalities. Appl Anal. 2012;91:1793–1810. doi: 10.1080/00036811.2011.614602
   Web of Science ®Google Scholar
• Zeng LC, Wong MM, Yao JC. Strong convergence of relaxed hybrid steepest-descent methods for triple hierarchical constrained optimization. Fixed Point Theory Appl. 2012;2012. Article ID 29. doi: 10.1186/1687-1812-2012-29
   Google Scholar
• Qin XL, Cho SY, Wang L. Strong convergence of an iterative algorithm involving nonlinear mappings of nonexpansive and accretive type. Optimization. 2018;67:1377–1388. doi: 10.1080/02331934.2018.1491973
   Web of Science ®Google Scholar
• Cho SY, Bin Dehaish BA, Qin XL. Weak convergence of a splitting algorithm in Hilbert spaces. J Appl Anal Comput. 2017;7:427–438.
   Google Scholar
• Zhao XP, Ng KF, Li C, et al. Linear regularity and linear convergence of projection-based methods for solving convex feasibility problems. Appl Math Optim. 2018;78:613–641. doi: 10.1007/s00245-017-9417-1
   Web of Science ®Google Scholar
• Zhao XP, Köbis MA. On the convergence of general projection methods for solving convex feasibility problems with applications to the inverse problem of image recovery. Optimization. 2018;67:1409–1427. doi: 10.1080/02331934.2018.1474355
   Web of Science ®Google Scholar
• Zhao XP, Fang DH. Strong convergence of general projection methods for solving convex feasibility problem. J Nonlinear Convex Anal. 2018;19:515–524.
   Web of Science ®Google Scholar
• Zhao XP, Sahu DR, Wen CF. Iterative methods for system of variational inclusions involving accretive operators and applications. Fixed Point Theory. 2018;19:801–822. doi: 10.24193/fpt-ro.2018.2.59
   Web of Science ®Google Scholar
• Chadli O, Saidi A, Zhao XP. A vector auxiliary principle technique approach to systems of mixed vector equilibrium problems. Optimization. 2018. doi:10.1080/02331934.2018.1553973.
   Web of Science ®Google Scholar
• Zhao XP, Naraghirad E. Strong convergence of Halpern iteration for nonsmooth variational inequality in Hilbert space. J Nonlinear Convex Anal. 2017;18:993–999.
   Web of Science ®Google Scholar
• Sahu DR, Singh KK, Zhao XP. On the convergence analysis of a Newton-like method under weak smoothness assumptions. J Nonlinear Convex Anal. 2015;16:1425–1437.
   Web of Science ®Google Scholar
• Chang SS, Wen CF, Yao JC. Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces. Optimization. 2018;67:1183–1196. doi: 10.1080/02331934.2018.1470176
   Web of Science ®Google Scholar
• Qin X, Yao JC. Projection splitting algorithms for nonself operators. J Nonlinear Convex Anal. 2017;18:925–935.
   Web of Science ®Google Scholar
• Moudafi A, Xu HK. A DC regularization of split minimization problems. Appl Anal Optim. 2018;2:285–297.
   Google Scholar
• Cholamjiak W, Suantai S, Suparatulatorn R, et al. Viscosity approximation methods for fixed point problems in Hilbert spaces endowed with graphs. J Appl Numer Optim. 2019;1:25–38.
   Google Scholar
• Ceng LC. Convergence analysis of a Mann-like iterative algorithm in reflexive Banach spaces. Appl Set-Valued Anal Optim. 2019;1:1–18.
   Google Scholar
• Qin XL, Yao JC. Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators. J Inequal Appl. 2016;2016, Article ID 232. doi: 10.1186/s13660-016-1163-4
   Web of Science ®Google Scholar
• Yamada I. The hybrid steepest-descent method for the variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings. In: Batnariu D, Censor Y, Reich S, editors. Inherently parallel algorithms in feasibility and optimization and their applications. Amsterdam: North-Holland; 2001. p. 473–504.
   Google Scholar
• Ceng LC, Ansari QH, Yao JC. Iterative methods for triple hierarchical variational inequalities in Hilbert spaces. J Optim Theory Appl. 2011;151:489–512. doi: 10.1007/s10957-011-9882-7
   Web of Science ®Google Scholar
• Ceng LC, Ansari QH, Wong MM, et al. Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems. Fixed Point Theory. 2012;13:403–422.
   Web of Science ®Google Scholar
• Ceng LC, Kong ZY, Wen CF. On general systems of variational inequalities. Comput Math Appl. 2013;66:1514–1532. doi: 10.1016/j.camwa.2013.08.016
   Web of Science ®Google Scholar
• Ceng LC, Latif A, Ansari QH, et al. Hybrid extragradient method for hierarchical variational inequalities. Fixed Point Theory Appl. 2014;2014. Article ID 222.
   Google Scholar
• Ceng LC, Yao JC. Relaxed and hybrid viscosity methods for general system of variational inequalities with split feasibility problem constraint. Fixed Point Theory Appl. 2013;2013. Article ID 43.
   Google Scholar
• Cai G, Shehu Y, Iyiola OS. Strong convergence results for variational inequalities and fixed point problems using modified viscosity implicit rules. Numer Algorithms. 2018;77:535–558. doi: 10.1007/s11075-017-0327-8
   Web of Science ®Google Scholar
• Korpelevich GM. The extragradient method for finding saddle points and other problems. Matecon. 1976;12:747–756.
   Google Scholar
• Zeng LC, Yao JC. Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings. Nonlinear Anal. 2006;64:2507–2515. doi: 10.1016/j.na.2005.08.028
   Web of Science ®Google Scholar
• Ceng LC, Petrusel A, Yao JC, et al. Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces. Fixed Point Theory. 2018;19:487–502. doi: 10.24193/fpt-ro.2018.2.39
   Web of Science ®Google Scholar
• Ceng LC, Liou YC, Wen CF, et al. Hybrid implicit steepest-descent methods for triple hierarchical variational inequalities with hierarchical variational inequality constraints. J Nonlinear Sci Appl. 2017;10:3963–3987. doi: 10.22436/jnsa.010.07.47
   Google Scholar
• Goebel K, Kirk WA. Topics on metric fixed-point theory. Cambridge: Cambridge University Press; 1990.
   Google Scholar
• Yao YY, Liou YC, Kang SM. Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. Comput Math Appl. 2010;59:3472–3480. doi: 10.1016/j.camwa.2010.03.036
   Web of Science ®Google Scholar
• Xu HK, Kim TH. Convergence of hybrid steepest-descent methods for variational inequalities. J Optim Theory Appl. 2003;119:185–201. doi: 10.1023/B:JOTA.0000005048.79379.b6
   Web of Science ®Google Scholar
• Xue ZQ, Zhou HY, Cho YJ. Iterative solutions of nonlinear equations for m-accretive operators in Banach spaces. J Nonlineaer Convex Anal. 2000;1:313–320.
   Google Scholar
• Suzuki T. Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005;2005. Article ID 685918. doi: 10.1155/FPTA.2005.103
   Google Scholar
• Yao YY, Liou YC, Yao JC. An extragradient method for fixed point problems and variational inequality problems. J Inequal Appl. 2007;2007, Article ID 38752. doi: 10.1155/2007/38752
   Web of Science ®Google Scholar