Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2925–2933 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Resolvent dynamical systems and mixed variational inequalities Bandar Bin-Mohsin a , Muhammad Aslam Noor a,b,* , Khalida Inayat Noor b , Raﬁa Latif b a Department of Mathematics, King Saud University, Riyadh, Saudi Arabia. b Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan. Communicated by A. Atangana Abstract In this paper, we use the dynamical systems technique to suggest and investigate some inertial proximal methods for solving mixed variational inequalities and related optimization problems. It is proved that the convergence analysis of the proposed methods requires only the monotonicity. Some special cases are also considered. Our method of proof is very simple as compared with other techniques. Ideas and techniques of this paper may be extended for other classes of variational inequalities and equilibrium problems. c 2017 All rights reserved. Keywords: Variational inequalities, dynamical systems, inertial proximal methods, convergence. 2010 MSC: 26D15, 26D10, 90C23, 49J40. 1. Introduction and Preliminaries Variational inequalities, which were introduced and investigated by Stampacchia [30]. Variational inequalities can be viewed as signiﬁcant and natural generalization of the variational principles, the origin of which can be traced back to Euler, Lagrange, Newton and Bernoulli’s brothers. It is remarkable and amazing that a wide class of unrelated problems, which in pure, applied and engineering sciences can be studied in the general and uniﬁed framework of variational inequalities. The ideas and techniques of this theory are being applied in a variety of diverse areas of pure and applied sciences and proved to innovative, see [1–4, 6–11, 13, 15–17, 19, 20, 22, 24, 26–33]. Variational inequalities involving the nonlinear term is called the mixed variational inequality or vari- ational inequality of the second kind. Mixed variational inequalities have applications in elasticity, struc- tural engineering and electronic network, see [12] and the references therein. Due to the presence of the nonlinear term, the projection method and its variant form can be used to establish the equivalence be- tween the mixed variational inequalities and the ﬁxed point problem. However, if the nonlinear term is a proper, convex and lower-semi continuous, then one can show that the mixed variational inequalities are equivalent to the ﬁxed point problem using the resolvent operator technique. This alternative formulation * Corresponding author Email addresses: balmohsen@ksu.edu.s (Bandar Bin-Mohsin), noormaslam@gmail.com (Muhammad Aslam Noor), khalidanoor@hotmail.com (Khalida Inayat Noor), rafialatif818@gmail.com (Raﬁa Latif) doi:10.22436/jnsa.010.06.07 Received 2017-02-13