U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 2, 2018 ISSN 1223-7027 FRACTIONAL PROJECTED DYNAMICAL SYSTEM FOR QUASI VARIATIONAL INEQUALITIES Awais Gul Khan 1 , Muhammad Aslam Noor 2 , Khalida Inayat Noor 2 , Amjad Pervez 3 In this paper, we introduce a dynamical system associated with quasi vari- ational inequalities using projection operator technique. This dynamical system is called implicit fractional projected dynamical system. We show that the implicit fractional projected dynamical system is exponentially stable and converges to its unique equilib- rium point under some suitable conditions. Some special cases are discussed, which can be obtained from our results. Results obtained in this paper continue to hold for these problems. Keywords: Dynamical systems; Fractional derivative; Convergence; Quasi-variational inequalities. 1. Introduction Quasi variational inequalities were introduced and studied by Bensoussan and Lions [2, 3] in impulse control system. It is well known that the set involved in the quasi varia- tional inequalities depends upon the solution explicitly or implicitly. We remark that if the involved set does not depend upon the solution then quasi variational inequality reduces to the variational inequality, the origin of which can be traced back to Stampacchia [39]. Vari- ational inequalities and quasi variational inequalities provide us a unifying and an efficient framework to study various related and unrelated problems which arise in different branches of pure and applied sciences, see [1-46] and references therein. Dynamics is a concise term referring to the study of time evolving processes, and the corresponding system of equations, which describes this evolution, is called a dynamical system. Nonlinear systems are widely used as models to describe complex physical phenom- ena in various field of sciences, such as fluid dynamics, solid state physics, plasma physics, mathematical biology and chemical kinetics, vibrations, heat transfer and so on. It is well known that these problems can be studied via the quasi variational inequalities. Dynamical systems can be solved by using some analytical techniques such as Homotopy Perturbation Method, Variational Iteration Method, Neural Network techniques and their variant forms, see [19, 20, 24, 46] and references therein. For recent developments in nonlinear dynamical systems see [5, 6, 12, 21, 22]. In recent years, several dynamical systems associated with variational inequalities are being investigated using the projection operator methods and Wiener-Hopf equations. This can be traced back Dupuis and Nagurney [8], Friesz et al [10] and Noor [29]. The dynamical systems method is more attractive due to its wide applicability, flexibility and numerical efficiency. In this method, variational inequality problem is reformulated as an initial value 1 corresponding author, Department of Mathematics, Government College University, Allama Iqbal Road, Faisalabad, Pakistan, E-mail: awaisgulkhan@gmail.com 2 Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan. 3 Department of Mathematics, Government College University, Allama Iqbal Road, Faisalabad, Pakistan. 99