Resolvent Dynamical Systems Technique for Mixed General Variational Inequalities
Abstract
In this paper, we introduce new second order dynamical system approach for solving a class of mixed general variational inequalities. Using the forward finite difference schemes, we suggest some multi-step iterative methods for solving the mixed variational inequalities. Convergence analysis is investigated under certain mild conditions. Some special cases are discussed as applications of the results. It is an interesting problem to compare these methods with other technique for solving mixed variational inequalities and related optimizations.
References
Ames, W. F. (1992). Numerical methods for partial differential equations. Academic Press. https://doi.org/10.1142/9789814537735
Ashish, Rani, M., & Chugh, R. (2012). Dynamics of antifractals in Noor orbit. International Journal of Computer Applications, 57(4).
Ashish, Rani, M., & Chugh, R. (2014). Julia sets and Mandelbrot sets in Noor orbit. Applied Mathematics and Computation, 228(1), 615-631. https://doi.org/10.1016/j.amc.2013.11.077
Ashish, Chugh, R., & Rani, M. (2021). Fractals and chaos in Noor orbit: A four-step feedback approach. LAP Lambert Academic Publishing.
Ashish, Cao, J., & Noor, M. A. (2023). Stabilization of fixed points in chaotic maps using Noor orbit with applications in cardiac arrhythmia. Journal of Applied Analysis and Computation, 13(5), 2452-2470. https://doi.org/10.11948/20220350
Barbagallo, A., & Lo Bainco, S. G. (2024). A random elastic traffic equilibrium problem via stochastic quasi-variational inequalities. Communications in Nonlinear Science and Numerical Simulation, 131, 107798. https://doi.org/10.1016/j.cnsns.2023.107798
Bnouhachem, A. (2005). A self-adaptive method for solving general mixed variational inequalities. Journal of Mathematical Analysis and Applications, 309, 136-150. https://doi.org/10.1016/j.jmaa.2004.12.023
Bnouhachem, A., & Hamdi, A. (2014). Parallel LQP alternating direction method for solving variational inequality problems with separable structure. Journal of Inequalities and Applications, 2014, ID=392. https://doi.org/10.1186/1029-242X-2014-392
Bnouhachem, A., Noor, M. A., & Rassias, T. M. (2006). Three-step iterative algorithms for mixed variational inequalities. Applied Mathematics and Computation, 183, 436-446. https://doi.org/10.1016/j.amc.2006.05.086
Bnouhachem, A., Noor, M. A., Khalfaoui, M., & Zhaohan, S. (2011). A resolvent method for solving mixed variational inequalities. Journal of King Saud University-Science, 23(2), 235-240. https://doi.org/10.1016/j.jksus.2010.07.015
Brezis, H. (1973). Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert. North-Holland.
Chairatsiripong, C., & Thianwan, T. (2022). Novel Noor iterations technique for solving nonlinear equations. AIMS Mathematics, 7(6), 10958-10976. https://doi.org/10.3934/math.2022612
Cho, S. Y., Shahid, A. A., Nazeer, W., & Kang, S. M. (2006). Fixed point results for fractal generation in Noor orbit and $s$-convexity. SpringerPlus, 5, 1843.
Chugh, R., Rani, M., & Ashish, R. (2012). On the convergence of logistic map in Noor orbit. International Journal of Computer Applications, 43(18). https://doi.org/10.5120/6200-8739
Cottle, R. W., Pang, J.-S., & Stone, R. E. (2009). The linear complementarity problem. SIAM Publications. https://doi.org/10.1137/1.9780898719000
Cristescu, G., & Lupsa, L. (2002). Non-connected convexities and applications. Kluwer Academic Publishers. https://doi.org/10.1007/978-1-4615-0003-2
Dupuis, P., & Nagurney, A. (1993). Dynamical systems and variational inequalities. Annals of Operations Research, 44, 7-42. https://doi.org/10.1007/BF02073589
Glowinski, R., Lions, J. L., & Tremolieres, R. (1981). Numerical analysis of variational inequalities. North-Holland.
Glowinski, R., & Le Tallec, P. (1989). Augmented Lagrangian and operator splitting methods in nonlinear mechanics. SIAM. https://doi.org/10.1137/1.9781611970838
Hamdi, A. (2005). A Moreau-Yosida regularization of a difference of two convex functions. Applied Mathematics E-Notes, 5, 164-170.
Khan, A. G., Noor, M. A., & Noor, K. I. (2015). Dynamical systems for general quasi-variational inequalities. Annals of Functional Analysis, 6(1), 193-209. https://doi.org/10.15352/afa/06-1-14
Khan, A. G., Noor, M. A., & Noor, K. I. (2019). Dynamical systems for variational inclusions involving difference operators. Honam Mathematical Journal, 41(1), 207-225.
Kikuchi, N., & Oden, J. T. (1988). Contact problems in elasticity. SIAM Publishing. https://doi.org/10.1137/1.9781611970845
Kinderlehrer, D., & Stampacchia, G. (2000). An introduction to variational inequalities and their applications. SIAM. https://doi.org/10.1137/1.9780898719451
Korpelevich, G. M. (1976). The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody, 12, 747-756.
Kwuni, Y. C., Shahid, A. A., Nazeer, W., Butt, S. I., Abbas, M., & Kang, S. M. (2019). Tricorns and multicorns in Noor orbit with s-convexity. IEEE Access, 7. https://doi.org/10.1109/ACCESS.2019.2928796
Lions, J., & Stampacchia, G. (1967). Variational inequalities. Communications on Pure and Applied Mathematics, 20, 493-519. https://doi.org/10.1002/cpa.3160200302
Nagurney, A., & Zhang, D. (1996). Projected dynamical systems and variational inequalities with applications. Kluwer Academic Publishers. https://doi.org/10.1007/978-1-4615-2301-7
Nammanee, K., Noor, M. A., & Suantai, S. (2006). Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings. Journal of Mathematical Analysis and Applications, 314, 320-334. https://doi.org/10.1016/j.jmaa.2005.03.094
Niculescu, C. P., & Persson, L. E. (2018). Convex functions and their applications. Springer-Verlag. https://doi.org/10.1007/978-3-319-78337-6_1
Noor, M. A. (1975). On variational inequalities (PhD thesis). Brunel University, London, UK.
Noor, M. A. (1988). General variational inequalities. Applied Mathematics Letters, 1, 119-121. https://doi.org/10.1016/0893-9659(88)90054-7
Noor, M. A. (1988). Quasi variational inequalities. Applied Mathematics Letters, 1(4), 367-370. https://doi.org/10.1016/0893-9659(88)90152-8
Noor, M. A. (1988). Fixed point approach for complementarity problems. Journal of Mathematical Analysis and Applications, 33, 437-448. https://doi.org/10.1016/0022-247X(88)90413-1
Noor, M. A. (1997). Some recent advances in variational inequalities: Part II. Other concepts. New Zealand Journal of Mathematics, 26, 229-255.
Noor, M. A. (1977). Generalized mixed variational inequalities and resolvent equations. Positivity, 1, 145-154.
Noor, M. A. (2000). Variational inequalities for fuzzy mappings (III). Fuzzy Sets and Systems, 110(1), 101-108. https://doi.org/10.1016/S0165-0114(98)00131-6
Noor, M. A. (2000). New approximation schemes for general variational inequalities. Journal of Mathematical Analysis and Applications, 251, 217-230. https://doi.org/10.1006/jmaa.2000.7042
Noor, M. A. (2001). Three-step iterative algorithms for multivalued quasi variational inclusions. Journal of Mathematical Analysis and Applications, 255(2), 589-604. https://doi.org/10.1006/jmaa.2000.7298
Noor, M. A. (2002). Resolvent dynamical systems for mixed variational inequalities. Korean Journal of Computational and Applied Mathematics, 9(1), 15-26. https://doi.org/10.1007/BF03012337
Noor, M. A. (2004). Some developments in general variational inequalities. Applied Mathematics and Computation, 152, 199-277. https://doi.org/10.1016/S0096-3003(03)00558-7
Noor, M. A. (2001). Three-step iterative algorithms for multivalued quasi variational inclusions. Journal of Mathematical Analysis, Applications, 255(2), 589-604. https://doi.org/10.1006/jmaa.2000.7298
Noor, M. A. (2003). Mixed quasi variational inequalities. Applied Mathematics and Computation, 146, 553-578. https://doi.org/10.1016/S0096-3003(02)00605-7
Noor, M. A. (2009). Extended general variational inequalities. Applied Mathematics Letters, 22, 182-185. https://doi.org/10.1016/j.aml.2008.03.007
Noor, M. A. (2008). Differentiable non-convex functions and general variational inequalities. Applied Mathematics and Computation, 199(2), 623-630. https://doi.org/10.1016/j.amc.2007.10.023
Noor, M. A., & Al-Said, E. A. (1999). Change of variable method for generalized complementarity problems. Journal of Optimization Theory and Applications, 100(2), 389-395. https://doi.org/10.1023/A:1021790404792
Noor, M. A. (2010). Some iterative schemes for general mixed variational inequalities. Journal of Applied Mathematics and Computation, 34(1), 57-70. https://doi.org/10.1007/s12190-009-0306-x
Noor, M. A., & Noor, K. I. (2008). Proximal-resolvent methods for mixed variational inequalities. CUBO Mathematics Journal, 10(03), 1-12.
Noor, M. A., & Noor, K. I. (2022). Dynamical system technique for solving quasi variational inequalities. U.P.B. Science Bulletin, Series A, 84(4), 55-66.
Noor, M. A., & Noor, K. I. (2022). Some new trends in mixed variational inequalities. Journal of Advanced Mathematics Studies, 15(2), 105-140.
Noor, M. A., & Noor, K. I. (2023). Iterative schemes for solving general variational inequalities. Differential Equations and Applications, 15(2), 113-134. https://doi.org/10.7153/dea-2023-15-07
Noor, M. A., & Noor, K. I. (2024). Some novel aspects and applications of Noor iterations and Noor orbits. Journal of Advanced Mathematics Studies, 17(3), 274-284.
Noor, K. I., Noor, M. A., & Kankum, K. (2025). On a new generalization of the Lax-Milgram Lemma. Earthline Journal of Mathematical Sciences, 15(1), 23-34. https://doi.org/10.34198/ejms.15125.023034
Noor, M. A., Noor, K. I., & Khan, A. G. (2015). Dynamical systems for quasi variational inequalities. Annals of Functional Analysis, 6(1), 193-209. https://doi.org/10.15352/afa/06-1-14
Noor, M. A., Noor, K. I., & Bnouhachem, A. (2021). Some new iterative methods for variational inequalities. Canadian Journal of Applied Mathematics, 3(1), 1-17.
Noor, M. A., Noor, K. I., & Rassias, M. T. (2020). New trends in general variational inequalities. Acta Applicandae Mathematicae, 170(1), 981-1046. https://doi.org/10.1007/s10440-020-00366-2
Noor, M. A., Noor, K. I., & Rassias, T. M. (1993). Some aspects of variational inequalities. Journal of Computational and Applied Mathematics, 47, 285-312. https://doi.org/10.1016/0377-0427(93)90058-J
Noor, M. A., Noor, K. I., & Yaqoob, H. (2010). On general mixed variational inequalities. Acta Applicandae Mathematicae, 110, 227-246. https://doi.org/10.1007/s10440-008-9402-4
Noor, M. A., & Oettli, W. (1994). On general nonlinear complementarity problems and quasi equilibria. Le Mathematiche, 49, 31-31.
Paimsang, S., Yambangwai, D., & Thainwan, T. (2024). A novel Noor iterative method of operators with property $(E)$ as concerns convex programming applicable in signal recovery and polynomiography. Mathematical Methods in the Applied Sciences, 1-18. https://doi.org/10.1002/mma.10083
Patriksson, M. (1998). Nonlinear programming and variational inequalities: A unified approach. Kluwer Academic Publishers. https://doi.org/10.1007/978-1-4757-2991-7
Phuengrattana, W., & Suantai, S. (2011). On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. Journal of Computational and Applied Mathematics, 235(9), 3006-3014. https://doi.org/10.1016/j.cam.2010.12.022
Rattanaseeha, K., Imnang, S., Inkrong, P., & Thianwan, T. (2023). Novel Noor iterative methods for mixed-type asymptotically nonexpansive mappings from the perspective of convex programming in hyperbolic spaces. International Journal of Innovative Computing Information and Control, 19(6), 1717-1734.
Stampacchia, G. (1964). Formes bilineaires coercitives sur les ensembles convexes. Comptes Rendus de l’Académie des Sciences de Paris, 258, 4413-4416.
Suantai, S., Noor, M. A., Kankum, K., & Cholamjiak, P. (2021). Novel forward-backward algorithms for optimization and applications to compressive sensing and image inpainting. Advances in Difference Equations, 2021(265). https://doi.org/10.1186/s13662-021-03422-9
Ullah, S., & Noor, M. A. (2020). An efficient method for solving new general mixed variational inequality. Journal of Inequalities and Special Functions, 11(3), 1-9.
Tseng, P. (2000). A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization, 38, 431-446. https://doi.org/10.1137/S0363012998338806
Xia, Y. S., & Wang, J. (2000). A recurrent neural network for solving linear projection equations. Neural Networks, 13, 337-350. https://doi.org/10.1016/S0893-6080(00)00019-8
Xia, Y. S., & Wang, J. (2000). On the stability of globally projected dynamical systems. Journal of Optimization Theory and Applications, 106, 129-150. https://doi.org/10.1023/A:1004611224835
Zeng-Bao, W., & Yun-Zhi, Z. (2014). Global fractional-order projective dynamical systems. Communications in Nonlinear Science and Numerical Simulation, 19, 2811-2819. https://doi.org/10.1016/j.cnsns.2014.01.007
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