A system of variational inclusions with \(P\)-\(\eta \)-accretive operators. (English) Zbl 1350.49007
Summary: We introduce and study a system of variational inclusions with \(P-\eta \)-accretive operators in real \(q\)-uniformly smooth Banach spaces. By using the resolvent operator technique associated with \(P-\eta \)-accretive operators, we prove the existence and uniqueness of solutions for this system of variational inclusions and construct a Mann iterative algorithm to approximate the unique solution. The results in this paper extend and improve some known results in the literature.
MSC:
| 49J40 | Variational inequalities |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
Keywords:
\(P-\eta \)-accretive operators; system of variational inclusions; iterative algorithm; resolvent operator technique; \(q\)-uniformly smooth Banach spacesReferences:
| [1] | Agarwal, R. P.; Cho, Y. J.; Huang, N. J., Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett., 13, 6, 19-24 (2000) · Zbl 0960.47035 |
| [2] | Agawal, R. P.; Huang, N. J.; Tan, M. Y., Sensitivity Analysis for a new system of generalized Nonlinear mixed quasi-variational inclusions, Appl. Math. Lett., 17, 345-352 (2004) · Zbl 1056.49008 |
| [3] | Allevi, E.; Gnudi, A.; Konnov, I. V., Generalized vector variational inequalities over product sets, Nonlinear Anal., 47, 573-582 (2001) · Zbl 1042.49509 |
| [4] | Ansari, Q. H.; Shaible, S.; Yao, J. C., Systems of vector equilibrium problems and its applications, J. Optim. Theory Appl., 107, 547-557 (2000) · Zbl 0972.49009 |
| [5] | Ansari, Q. H.; Yao, J. C., A fixed point theorem and its applications to a system of variational inequalities, Bull. Austral. Math. Soc., 59, 433-442 (1999) · Zbl 0944.47037 |
| [6] | M. Bianchi, Pseudo \(P\)-monotone operators and variational inequalities, Report 6, Istituto di econometria e Matematica per le decisioni economiche, Universita Cattolica del Sacro Cuore, Milan, Italy, 1993.; M. Bianchi, Pseudo \(P\)-monotone operators and variational inequalities, Report 6, Istituto di econometria e Matematica per le decisioni economiche, Universita Cattolica del Sacro Cuore, Milan, Italy, 1993. |
| [7] | Chang, S. S., Some problems and results in the study of nonlinear analysis, Nonlinear Anal. Theory Methods Appl., 30, 4197-4208 (1997) · Zbl 0901.47036 |
| [8] | Chen, G. Y.; Craven, B. D., Approximate dual and approximate vector variational inequality for multiobjective optimization, J. Austral. Math. Soc. Ser. A, 47, 418-423 (1989) · Zbl 0693.90089 |
| [9] | Chidume, C. E.; Kazmi, K. R.; Zegeye, H., Iterative approximation of a solution of a general variational-like inclusion in Banach spaces, Internat. J. Math. Math. Sci., 22, 1159-1168 (2004) · Zbl 1081.47053 |
| [10] | Cho, Y. J.; Fang, Y. P.; Huang, N. J.; Hwang, H. J., Algorithms for systems of nonlinear variational inequalities, J. Korean Math. Soc., 41, 489-499 (2004) · Zbl 1057.49010 |
| [11] | Cohen, G.; Chaplais, F., Nested monotony for variational inequalities over a product of spaces and convergence of iterative algorithms, J. Optim. Theory Appl., 59, 360-390 (1988) · Zbl 0628.90069 |
| [12] | Ding, X. P.; Luo, C. L., Perturbed proximal point algorithms for generalized quasi-variational-like inclusions, J. Comput. Appl. Math., 113, 153-165 (2000) · Zbl 0939.49010 |
| [13] | Fang, Y. P.; Huang, N. J., \(H\)-Accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett., 17, 647-653 (2004) · Zbl 1056.49012 |
| [14] | Fang, Y. P.; Huang, N. J., \(H\)-monotone operators and system of variational inclusions, Commun. Appl. Nonlinear Anal., 11, 1, 93-101 (2004) · Zbl 1040.49007 |
| [15] | Fang, Y. P.; Huang, N. J., \(H\)-Monotone operator resolvent operator technique for quasi-variational inclusions, Appl. Math. Comput., 145, 795-803 (2003) · Zbl 1030.49008 |
| [16] | Fang, Y. P.; Huang, N. J., Iterative algorithm for a system of variational inclusions involving \(H\)-accretive operators in Banach spaces, Acta Math. Hungar., 108, 3, 183-195 (2005) · Zbl 1094.49007 |
| [17] | Fang, Y. P.; Huang, N. J.; Thompson, H. B., A new system of variational inclusions with \((H, \eta)\)-monotone operators in Hilbert spaces, Comput. Math. Appl., 49, 365-374 (2005) · Zbl 1068.49003 |
| [18] | Giannessi, F.; Marafdar, E., Variational Inequalities and Network Equilibrium Problems (1995), Plenum Press: Plenum Press New York |
| [19] | Harker, P. T.; Pang, J. S., Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming, 48, 161-220 (1990) · Zbl 0734.90098 |
| [20] | Kassay, G.; Kolumbán, J., System of multi-valued variational inequalities, Publ. Math. Debrecen, 54, 267-279 (1999) · Zbl 0932.49009 |
| [21] | Kassay, G.; Kolumbán, J.; Páles, Z., Factorization of Minty and Stampacchia variational inequality system, European J. Oper. Res., 143, 377-389 (2002) · Zbl 1059.49015 |
| [22] | Kazmi, K. R.; Bhat, M. I., Iterative algorithm for a system of nonlinear variational-like inclusions, Comput. Math. Appl., 48, 1929-1935 (2004) · Zbl 1059.49016 |
| [23] | K.R. Kazmi, F.A. Khan, Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: a \(P-\operatorname{Η;} \)-proximal-point mapping approach, J. Math. Anal. Appl. 325 (2007) 665-674.; K.R. Kazmi, F.A. Khan, Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: a \(P-\operatorname{Η;} \)-proximal-point mapping approach, J. Math. Anal. Appl. 325 (2007) 665-674. · Zbl 1125.47045 |
| [24] | Kim, J. K.; Kim, D. S., A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Convex Anal., 11, 1, 235-243 (2004) · Zbl 1061.49010 |
| [25] | Marcotte, P.; Zhu, D. L., Weak sharp solutions and the finite convergence of algorithms for solving variational inequalities, SIAM J. Optim., 9, 179-189 (1999) · Zbl 1032.90050 |
| [26] | Noor, M. A., Modified resolvent splitting algorithms for general mixed variational inequalities, J. Comput. Appl. Math., 135, 111-124 (2001) · Zbl 0997.65091 |
| [27] | Pang, J. S., Asymmetric variational inequality problems over product sets: applications and iterative methods, Math. Programming, 31, 206-219 (1985) · Zbl 0578.49006 |
| [28] | Peng, J. W., System of generalized set-valued quasi-variational-like inequalities, Bull. Austral. Math. Soc., 68, 501-515 (2003) · Zbl 1063.47060 |
| [29] | Peng, J. W., Set-valued variational inclusions with T-accretive, Appl. Math. Lett., 19, 273-282 (2006) · Zbl 1102.47050 |
| [30] | Peng, J. W.; Yang, X. M., On existence of a solution for the system of generalized vector quasi-equilibrium problems with upper semicontinuous set-valued maps, Internat. J. Math. Math. Sci., 15, 2409-2420 (2005) · Zbl 1106.49031 |
| [31] | Verma, R. U., On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line, 3, 8, 65-68 (1999) · Zbl 0970.49011 |
| [32] | Verma, R. U., Projection methods, algorithms and a new system of nonlinear variational inequalities, Comput. Math. Appl., 41, 1025-1031 (2001) · Zbl 0995.47042 |
| [33] | Verma, R. U., Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Variance Inequality, 4, 1, 117-124 (2001) · Zbl 1014.47050 |
| [34] | Verma, R. U., Generalized system for relaxed cocoercive variational inequalities and problems and projection methods, J. Optim. Theory Appl., 121, 203-210 (2004) · Zbl 1056.49017 |
| [35] | Verma, R. U., General convergence analysis for two-step projection methods and application to variational problems, Appl. Math. Lett., 18, 1286-1292 (2005) · Zbl 1099.47054 |
| [36] | Xu, Z. B., Inequalities in smooth Banach spaces with applications, Nonlinear Anal. Theory Methods Appl., 16, 1127-1138 (1991) · Zbl 0757.46033 |
| [37] | Xu, Z. B.; Roach, G. F., Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl., 157, 189-210 (1991) · Zbl 0757.46034 |
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.
