• Jittiporn Tangkhawiwetkul Mathematics, Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok.
  • Narin Petrot
Keywords: dynamical system, variational inclusion, resolvent operator, Gronwall's inequality


In this paper, we consider the system of variational inclusion and introduce the resolvent equation which is equivalent to the system of variational inclusion. The dynamical system associated with the system of ariational inclusion is presented. Furthermore, the solution of such dynamical system is proved. The results in this paper improve and extend the variational inclusion problems which have been appeared in literature.


Download data is not yet available.


Agarwal RP, Huang NJ, Tan MY. Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions, Applied Mathematics Letters. 2004; 17: 345-352.

Agarwal RP, Verma RU. General implicit variational inclusion problems based on A-maximal (m)-relaxed monotonicity (AMRM) frameworks, Applied Mathematics and Computation. 2009; 215(1): 367-379.

Ansari QH, Balooee J, Yao JC. Extended general nonlinear quasi variational inequalities and Projection dynamical systems, Taiwanese Journal of Mathematics. 2013; 17(4): 1321-1352.

Bahiana M, Oono Y. Cell dynamical system approach to block copolymers, Physical Review Journals. 1990; 41(12): 6763-6771.

Biswas SKr., Chakraborty S. Interacting dark energy in f(T) cosmology: A dynamical system analysis, International Journal of Modern Physics D. 2015; 24(07): 21 pages.

Bliemer M, Bovy P. Quasi variational inequality formulation of the multi-class dynamic traffic assignment problem, Transportation Reseach. 2003; 37: 501-519.

Dong J, Zhang D, Nagurney A. A projected dynamical systems model of general financial equilibrium with stability analysis, Mathematical and Computer Modelling. 1996; 24(2): 35-44.

Dupuis P, Nagurney A. Dynamical systems and variational inequalities, Annals of Operations Research. 1993; 44: 9-42.

Fang YP, Huang NJ. H-monotone operators and system of variational inclusions, Communications on Applied Nonlinear Analysis. 2004; 11(1): 93-101.

Ha Nguyen TT, Strodiot Jean J, Vuong Phan T. On the global exponential stability of a projected dynamical system for strongly pseudomonotone variational inequalities, Optimization Letters . 2018; 12: 1-14.

Isac G, Cojocaru MG. The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems, 2002.

Lin LJ. Systems of variational inclusion problems and differential inclusion problems with applications, Journal of Global Optimization. 2009; 44DOI 10.1007/s10898-008-9359-x: 579–591.

Nagurney A, Zhang AD. Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic: Boston; 1996.

Noor MA. Equivalence of Variational Inclusions with Resolvent Equations, Nonlinear Analysis. 2000; 14: 963-970.

Noor MA. Resolvent dynamical systems for mixed variational inequalities, The Korean Journal of Computational & Applied Mathematics. 2002a; 9(1): 15-26.

Noor MA. Implicit resolvent dynamical systems for quasi variational inclusion, Journal of Mathematical Analysis and Applications. 2002b; 269: 216-226.

Scrimali L. The financial equilibrium problem with implicit budget constraints, Central European Journal of Operations Research. 2008; 16: 191-203.

Suantai S, Petrot N. Existence and Stability of Iterative Algorithms for the System of Nonlinear Quasi-Mixed Equilibrium Problems, Applied Mathematics Letters. 2011; 24: 308-313.

Suwannawit J, Petrot N. Existence and Stability of Iterative Algorithm for a System of Random Set- Valued Variational Inclusion Problems Involving (A, m, η)-Generalized Monotone Operators, Journal of Applied Mathematics. 2012; ID 590676: 21 pages.

Verma RU. A-monotonicity and applications to nonlinear variational inclusion problems, Journal of Applied Mathematics and Stochastic Analysis. 2004; 2: 193-195.

Yan WY, Fang YP, Huang NJ. A new system of set-valued variational inclusions with H-monotone operators, Mathematical Inequalities & Applications. 2005; 8(3): 537-546.
How to Cite
Tangkhawiwetkul, J., & Petrot, N. (2019). DYNAMICAL SYSTEM FOR THE SYSTEM OF VARIATIONAL INCLUSION PROBLEM. Life Sciences and Environment Journal, 20(2), 236-247. Retrieved from https://ph01.tci-thaijo.org/index.php/psru/article/view/172028
Research Articles