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A new system of variational inclusions with \((H,\eta)\)-monotone operators in Hilbert spaces. (English) Zbl 1068.49003

Summary: We introduce and study a new system of variational inclusions involving \((H,\eta)\)-monotone operators in Hilbert spaces. Using the resolvent operator associated with the \((H,\eta)\)-monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm.

MSC:

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H10 Fixed-point theorems
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[1] Adly, S., Perturbed algorithm and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl., 201, 609-630 (1996) · Zbl 0856.65077
[2] Ahmad, R.; Ansari, Q. H., An iterative algorithm for generalized nonlinear variational inclusions, Appl. Math. Lett., 13, 5, 23-26 (2000) · Zbl 0954.49006
[3] Ansari, Q. H.; Yao, J. C., A fixed point theorem and its applications to a system of variational inequalities, Bull. Austral. Math. Soc., 59, 3, 433-442 (1999) · Zbl 0944.47037
[4] Ding, X. P.; Luo, C. L., Perturbed proximal point algorithms for generalized quasi-variational-like inclusions, J. Comput. Appl. Math., 210, 153-165 (2000) · Zbl 0939.49010
[5] Fang, Y. P.; Huang, N. J., \(H\)-Monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput., 145, 795-803 (2003) · Zbl 1030.49008
[6] Y.P. Fang and N.J. Huang, Mann iterative algorithm for a system of operator inclusions, Publ. Math. Debrecen; Y.P. Fang and N.J. Huang, Mann iterative algorithm for a system of operator inclusions, Publ. Math. Debrecen · Zbl 1082.47059
[7] Fang, Y. P.; Huang, N. J., Research Report (2003), Sichuan University: Sichuan University New York
[8] Giannessi, F.; Maugeri, A., Variational Inequalities and Network Equilibrium Problems (1995), Plenum · Zbl 0834.00044
[9] Hassouni, A.; Moudafi, A., A perturbed algorithm for variational inequalities, J. Math. Anal. Appl., 185, 706-712 (1994) · Zbl 0809.49008
[10] Huang, N. J., Generalized nonlinear variational inclusions with noncompact valued mappings, Appl. Math. Lett., 9, 3, 25-29 (1996) · Zbl 0851.49009
[11] Huang, N. J., Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasivariational inclusions, Computers Math. Applic., 35, 10, 1-7 (1998) · Zbl 0999.47057
[12] Huang, N. J.; Fang, Y. P., A new class of general variational inclusions involving maximal \(η\)-monotone mappings, Publ. Math. Debrecen, 62, 1-2, 83-98 (2003) · Zbl 1017.49011
[13] Huang, N. J.; Fang, Y. P., Fixed point theorems and a new system of multivalued generalized order complementarity problems, Positivity, 7, 257-265 (2003) · Zbl 1042.90047
[14] Kassay, G.; Kolumbán, J., System of multi-valued variational inequalities, Publ. Math. Debrecen, 56, 185-195 (2000) · Zbl 0989.49010
[15] Kassay, G.; Kolumbán, J.; Páles, Z., Factorization of Minty and Stampacchia variational inequality system, European J. Oper. Res., 143, 2, 377-389 (2002) · Zbl 1059.49015
[16] Kazmi, K. R., Mann and Ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl., 209, 572-584 (1997) · Zbl 0898.49007
[17] Lee, C. H.; Ansari, Q. H.; Yao, J. C., A perturbed algorithm for strongly nonlinear variational-like inclusions, Bull. Austral. Math. Soc., 62, 417-426 (2000) · Zbl 0979.49008
[18] Liu, L. W.; Li, Y. Q., On generalized set-valued variational inclusions, J. Math. Anal. Appl., 261, 1, 231-240 (2001) · Zbl 1070.49006
[19] Siddiqi, A. H.; Ansari, Q. H., General strongly nonlinear variational inequalities, J. Math. Anal. Appl., 116, 386-392 (1992) · Zbl 0770.49006
[20] Siddiqi, A. H.; Ahmad, R.; Husain, S., A perturbed algorithm for generalized nonlinear quasi-variational inclusions, Math. Comput. Appl., 3, 3, 177-184 (1998) · Zbl 0931.49004
[21] Verma, R. U., A system of generalized auxiliary problems principle and a system of variational inequalities, Math. Ineq. Appl., 4, 443-453 (2001) · Zbl 0999.49007
[22] Verma, R. U., Projection methods, algorithms, and a new system of nonlinear variational inequalities, Comput. Math. Appl., 41, 1025-1031 (2001) · Zbl 0995.47042
[23] Verma, R. U., Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Theory Appl., 121, 1, 203-210 (2004) · Zbl 1056.49017
[24] Yuan, G. X.Z., KKM Theory and Applications in Nonlinear Analysis (1999), Marcel Dekker: Marcel Dekker New York · Zbl 0936.47034
[25] Zeidler, E., Nonlinear Functional Analysis and its Applications II: Monotone Operators (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0583.47051
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