Verma, R. U. Generalized system for relaxed cocoercive variational inequalities and projection methods. (English) Zbl 1056.49017 J. Optimization Theory Appl. 121, No. 1, 203-210 (2004). Summary: Let \(K\) be a nonempty closed convex subset of a real Hilbert space \(H\). The approximate solvability of a system of nonlinear variational inequality problems, based on the convergence of projection methods, is discussed as follows: find an element \((x^*,y^*)\in K\times K\) such that \[ \bigl\langle\rho T(y^*,x^*)+x^*-y^*,x-x^*\bigr \rangle\geq 0,\quad \forall x\in K\text{ and }\rho>0, \]\[ \bigl\langle\eta T(x^*,y^*)+y^*-x^*,x-y^*\bigr\rangle\geq 0,\quad \forall x\in K\text{ and }\eta>0, \] where \(T:K\times K\to H\) is a nonlinear mapping on \(K\times K\). Cited in 11 ReviewsCited in 106 Documents MSC: 49J40 Variational inequalities 47J20 Variational and other types of inequalities involving nonlinear operators (general) Keywords:relaxed cocoercive nonlinear variational inequalities; projection methods; relaxed cocoercive mappings; convergence of projection methods PDFBibTeX XMLCite \textit{R. U. Verma}, J. Optim. Theory Appl. 121, No. 1, 203--210 (2004; Zbl 1056.49017) Full Text: DOI References: [1] VERMA, R. U., Projection Methods, Algorithms, and a New System of Nonlinear Variational Inequalities, Computers and Mathematics with Applications, Vol.41, 1025–1031, 2001. · Zbl 0995.47042 · doi:10.1016/S0898-1221(00)00336-9 [2] NIE, H., LIU, Z., KIM, K. H., and KANG, S. M., A System of Nonlinear Variational Inequalities Involving Strongly Monotone and Pseudocontractive Mappings, Advances in Nonlinear Variational Inequalities, Vol.6, 91–99, 2003. · Zbl 1098.47055 [3] VERMA, R. U., Projection Methods and a New System of Cocoercive Variational Inequality Problems, International Journal of Differential Equations and Applications, Vol.6, 359–367, 2002. · Zbl 1052.49014 [4] DUNN, J. C., Convexity, Monotonicity, and Gradient Processes in Hilbert Spaces, Journal of Mathematical Analysis and Applications, Vol.53, 145–158, 1976. · Zbl 0321.49025 · doi:10.1016/0022-247X(76)90152-9 [5] HE, B. S., A New Method for a Class of Linear Variational Inequalities, Mathematical Programming, Vol.66, 137–144, 1994. · Zbl 0813.49009 · doi:10.1007/BF01581141 [6] KINDERLEHRER, D., and STAMPACCHIA, G., An Introduction to Variational Inequalities, Academic Press, New York, NY, 1980. · Zbl 0457.35001 [7] VERMA, R. U., A Class of Quasivariational Inequalities Involving Cocoercive Mappings, Advances in Nonlinear Variational Inequalities, Vol.2, 1–12, 1999. · Zbl 1007.49512 [8] VERMA, R. U., An Extension of a Class of Nonlinear Quasivariational Inequality Problems based on a Projection Method, Mathematical Sciences Research Hotline, Vol.3, 1–10, 1999. · Zbl 0954.49008 [9] VERMA, R. U., Variational Inequalities in Locally Convex Hausdorff Topological Vector Spaces, Archiv der Mathematik, Vol.71, 246–248, 1998. · Zbl 0908.49010 · doi:10.1007/s000130050260 [10] VERMA, R. U., Nonlinear Variational and Constrained Hemivariational Inequalities Involving Relaxed Operators, Zeitschrift fur Angewandte Mathematik und Mechanik, Vol.77, 387–391, 1997. · Zbl 0886.49006 · doi:10.1002/zamm.19970770517 [11] WITTMANN, R., Approximation of Fixed Points of Nonexpansive Mappings, Archiv der Mathematik, Vol.58, 486–491, 1992. · Zbl 0797.47036 · doi:10.1007/BF01190119 [12] ZEIDLER, E., Nonlinear Functional Analysis and Its Applications, Vol. II/B, Springer Verlag, New York, NY, 1990. · Zbl 0684.47029 [13] XIU, N. H., and ZHANG, J. Z., Local Convergence Analysis of Projection Type Algorithms: Unified Approach, Journal of Optimization Theory and Applications, Vol.115, 211–230, 2002. · Zbl 1091.49011 · doi:10.1023/A:1019637315803 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.