Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces. (English) Zbl 1114.49008

Summary: The approximate solvability of a generalized system for relaxed cocoercive nonlinear variational inequality in Hilbert spaces is studied, based on the convergence of projection methods. The results presented in this paper extend and improve the main results of R. U. Verma [J. Optimization Theory Appl. 121, No. 1, 203–210 (2004; Zbl 1056.49017); Adv. Nonlinear Var. Inequal. 7, No. 2, 155–164 (2004; Zbl 1079.49011); Appl. Math. Lett. 18, No. 11, 1286–1292 (2005; Zbl 1099.47054)], N. H. Xiu and J. Z. Zhang [J. Optimization Theory Appl. 115, No. 1, 211–230 (2002; Zbl 1091.49011)] and H. Nie, Z. Liu, K. H. Kim and S. M. Kang [Adv. Nonlinear Var. Inequal. 6, No. 2, 91–99 (2003; Zbl 1098.47055)].


49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
Full Text: DOI


[1] Verma, R. U., Generalized system for relaxed cocoercive variational inequalities and its projection methods, J. Optim. Theory Appl., 121, 1, 203-210 (2004) · Zbl 1056.49017
[2] Verma, R. U., Generalized class of partial relaxed monotonicity and its connections, Adv. Nonlinear Var. Inequal., 7, 2, 155-164 (2004) · Zbl 1079.49011
[3] Verma, R. U., General convergence analysis for two-step projection methods and applications to variational problems, Appl. Math. Lett., 18, 11, 1286-1292 (2005) · Zbl 1099.47054
[4] Xiu, N. H.; Zhang, J. Z., Local convergence analysis of projection type algorithms: Unified approach, J. Optim. Theory Appl., 115, 211-230 (2002) · Zbl 1091.49011
[5] Nie, H.; Liu, Z.; Kim, K. H.; Kang, S. M., A system of nonlinear variational inequalities involving strongly monotone and pseudocontractive mappings, Adv. Nonlinear Var. Inequal., 6, 2, 91-99 (2003) · Zbl 1098.47055
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