×

Algorithms for general system of generalized resolvent equations with corresponding system of variational inclusions. (English) Zbl 1245.49011

Summary: Very recently, Ahmad and Yao (2009) introduced and considered a system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. In this paper, we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between general systems of generalized resolvent equations and general system of variational inclusions. The iterative algorithms for finding the approximate solutions of general systems of generalized resolvent equations are proposed. The convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. Our results represent the generalization, improvement, supplement, and development the results of Ahmad and Yao.

MSC:

49J40 Variational inequalities
47J25 Iterative procedures involving nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Problems, John Wiley & Sons, New York, NY, USA, 1984. · Zbl 0551.49007
[2] G. Isac, V. A. Bulavsky, and V. V. Kalashnikov, Complementarity, Equilibrium, Efficiency and Economics, vol. 63 of Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002. · Zbl 1081.90001
[3] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 88 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1980. · Zbl 0457.35001
[4] I. Konnov, Combined Relaxation Methods for Variational Inequalities, vol. 495 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 2001. · Zbl 0982.49009
[5] A. Nagurney, Network Economics: A Variational Inequality Approach, vol. 1 of Advances in Computational Economics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. · Zbl 0873.90015
[6] P. D. Panagiotopoulos and G. E. Stavroulakis, “New types of variational principles based on the notion of quasidifferentiability,” Acta Mechanica, vol. 94, no. 3-4, pp. 171-194, 1992. · Zbl 0756.73096 · doi:10.1007/BF01176649
[7] M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, vol. 23 of Applied Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. · Zbl 0913.65058
[8] A. H. Siddiqi and Q. H. Ansari, “An algorithm for a class of quasivariational inequalities,” Journal of Mathematical Analysis and Applications, vol. 145, no. 2, pp. 413-418, 1990. · Zbl 0697.49005 · doi:10.1016/0022-247X(90)90409-9
[9] A. Hassouni and A. Moudafi, “A perturbed algorithm for variational inclusions,” Journal of Mathematical Analysis and Applications, vol. 185, no. 3, pp. 706-712, 1994. · Zbl 0809.49008 · doi:10.1006/jmaa.1994.1277
[10] M. A. Noor and K. I. Noor, “Multivalued variational inequalities and resolvent equations,” Mathematical and Computer Modelling, vol. 26, no. 7, pp. 109-121, 1997. · Zbl 0893.49005 · doi:10.1016/S0895-7177(97)00189-1
[11] R. U. Verma, “Projection methods, algorithms, and a new system of nonlinear variational inequalities,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 1025-1031, 2001. · Zbl 0995.47042 · doi:10.1016/S0898-1221(00)00336-9
[12] J.-S. Pang, “Asymmetric variational inequality problems over product sets: applications and iterative methods,” Mathematical Programming, vol. 31, no. 2, pp. 206-219, 1985. · Zbl 0578.49006 · doi:10.1007/BF02591749
[13] G. Cohen and F. Chaplais, “Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms,” Journal of Optimization Theory and Applications, vol. 59, no. 3, pp. 369-390, 1988. · Zbl 0628.90069 · doi:10.1007/BF00940305
[14] M. Binachi, “Pseudo p-monotone operators and variational inequalities,” Tech. Rep. 6, Instituto di Econometria e Matematica per Decisioni Economiche, Universita Cattolica del Sacro Cuore, Milan, Italy, 1993.
[15] Q. H. Ansari and J.-C. Yao, “A fixed point theorem and its applications to a system of variational inequalities,” Bulletin of the Australian Mathematical Society, vol. 59, no. 3, pp. 433-442, 1999. · Zbl 0944.47037 · doi:10.1017/S0004972700033116
[16] R. P. Agarwal, N.-J. Huang, and M.-Y. Tan, “Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions,” Applied Mathematics Letters, vol. 17, no. 3, pp. 345-352, 2004. · Zbl 1056.49008 · doi:10.1016/S0893-9659(04)90073-0
[17] J. Peng and D. Zhu, “A new system of generalized mixed quasi-variational inclusions with (H,\eta )-monotone operators,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 175-187, 2007. · Zbl 1104.49012 · doi:10.1016/j.jmaa.2006.04.015
[18] H.-Y. Lan, J. H. Kim, and Y. J. Cho, “On a new system of nonlinear A-monotone multivalued variational inclusions,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 481-493, 2007. · Zbl 1118.49013 · doi:10.1016/j.jmaa.2005.11.067
[19] R. Ahmad, Q. H. Ansari, and S. S. Irfan, “Generalized variational inclusions and generalized resolvent equations in Banach spaces,” Computers & Mathematics with Applications, vol. 49, no. 11-12, pp. 1825-1835, 2005. · Zbl 1081.49004 · doi:10.1016/j.camwa.2004.10.044
[20] S. S. Chang, Y. J. Cho, B. S. Lee, and I. H. Jung, “Generalized set-valued variational inclusions in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 409-422, 2000. · Zbl 1031.49017 · doi:10.1006/jmaa.2000.6795
[21] N.-J. Huang, “A new class of generalized set-valued implicit variational inclusions in Banach spaces with an application,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 937-943, 2001. · Zbl 0998.47044 · doi:10.1016/S0898-1221(00)00331-X
[22] L. C. Ceng, S. Schaible, and J. C. Yao, “On the characterization of strong convergence of an iterative algorithm for a class of multi-valued variational inclusions,” Mathematical Methods of Operations Research, vol. 70, no. 1, pp. 1-12, 2009. · Zbl 1182.47051 · doi:10.1007/s00186-008-0227-8
[23] L.-C. Zeng, S.-M. Guu, and J.-C. Yao, “Characterization of H-monotone operators with applications to variational inclusions,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 329-337, 2005. · Zbl 1080.49012 · doi:10.1016/j.camwa.2005.06.001
[24] X. P. Ding, J.-C. Yao, and L.-C. Zeng, “Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces,” Computers & Mathematics with Applications, vol. 55, no. 4, pp. 669-679, 2008. · Zbl 1291.49004 · doi:10.1016/j.camwa.2007.06.004
[25] L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Iterative approximation of solutions for a class of completely generalized set-valued quasi-variational inclusions,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 978-987, 2008. · Zbl 1155.65355 · doi:10.1016/j.camwa.2008.01.026
[26] L.-C. Zeng, S.-M. Guu, and J.-C. Yao, “Three-step iterative algorithms for solving the system of generalized mixed quasi-variational-like inclusions,” Computers & Mathematics with Applications, vol. 53, no. 10, pp. 1572-1581, 2007. · Zbl 1152.47305 · doi:10.1016/j.camwa.2006.05.024
[27] L.-C. Zeng, S.-M. Guu, and J.-C. Yao, “Iterative algorithm for completely generalized set-valued strongly nonlinear mixed variational-like inequalities,” Computers & Mathematics with Applications, vol. 50, no. 5-6, pp. 935-945, 2005. · Zbl 1081.49011 · doi:10.1016/j.camwa.2004.12.017
[28] L.-C. Zeng, S.-M. Guu, and J.-C. Yao, “An iterative method for generalized nonlinear set-valued mixed quasi-variational inequalities with H-monotone mappings,” Computers & Mathematics with Applications, vol. 54, no. 4, pp. 476-483, 2007. · Zbl 1131.49012 · doi:10.1016/j.camwa.2007.01.025
[29] L.-C. Zeng, Q. H. Ansari, and J.-C. Yao, “General iterative algorithms for solving mixed quasi-variational-like inclusions,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2455-2467, 2008. · Zbl 1165.65355 · doi:10.1016/j.camwa.2008.05.016
[30] L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Iterative algorithm for finding approximate solutions of mixed quasi-variational-like inclusions,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 942-952, 2008. · Zbl 1155.49300 · doi:10.1016/j.camwa.2008.01.024
[31] R. Ahmad and J. C. Yao, “System of generalized resolvent equations with corresponding system of variational inclusions,” Journal of Global Optimization, vol. 44, no. 2, pp. 297-309, 2009. · Zbl 1178.47039 · doi:10.1007/s10898-008-9327-5
[32] Ya. Alber and J.-C. Yao, “Algorithm for generalized multi-valued co-variational inequalities in Banach spaces,” Functional Differential Equations, vol. 7, no. 1-2, pp. 5-13, 2000. · Zbl 1057.47514
[33] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15-50, Dekker, New York, NY, USA, 1996. · Zbl 0883.47083
[34] S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475-488, 1969. · Zbl 0187.45002 · doi:10.2140/pjm.1969.30.475
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.