In the present paper, we introduce and study a fuzzy vector equilibrium problem and prove some existence results with and without convexity assumptions by using some particular forms of results of Kim and Lee [W.K. Kim and K.H. Lee, Generalized fuzzy games and fuzzy equilibria, Fuzzy Sets and Systems, 122 (2001), 293-301] and Tarafdar [E. Tarafdar, Fixed point theorems in H-spaces and equilibrium points of abstract economies, J. Aust. Math. Soc. (Series A), 53 (1992), 252-260]. An example is also constructed in support of fuzzy vector equilibrium problem.

In this paper, first a short history of the notion of equilibrium problem in Economics and Nash game theory is stated. Also the relationship between equilibrium problem and important mathematical problems like optimization problem, nonlinear programming, variational inequality problem, fixed point problem and complementarity problem are given. The concept of generalized pseudomonotonicity for vector valued bifunctions is introduced and by using it some existence results for the vector equilibrium problem, in the setting of topological vector spaces, are presented. Some examples in order to illustrate the main results and compare them with the corresponding published results are furnished. Further, the compactness of the solution set of vector equilibrium problem is investigated.

In this paper, first, the traffic problem is studied from the Finsler geometric point of view and it is shown that the related metric is a Finsler metric of Randers type. Therefore, the time optimal paths of traffic problem are geodesics of this Randers metric. A real example is given by using the Maple program. Then a nonlinear complementary problem for traffic equilibrium problem is studied and it is shown that a Randers metric can be considered as a gap function in the nonlinear complementarity problem. Next, we extend the nonlinear complementarity problem of traffic equilibrium for a dynamical network. These results may be used in the different equilibrium problems.

Non-cooperative intelligent control agents (ICAs) with dedicated cost functions, can lead the system to poor performance and in some cases, closed-loop instability. A robust solution to this challenge is to place the ICAs at the feedback Nash equilibrium point (FNEP) of the differential game between them. This paper introduces the designation of a robust decentralized infinite horizon LQR control system based on the FNEP for a linear time-invariant system. For this purpose, two control strategies are defined. The first one is a centralized infinite horizon LQR (CIHLQR) problem (i.e. a supervisory problem), and the second one is a decentralized control problem (i.e. an infinite horizon linear-quadratic differential game). Then, while examining the optimal solution of each of the above strategies on the performance of the other, the necessary and sufficient conditions for the equivalence of the two problems are presented. In the absence of the conditions, by using the least-squares error criterion, an approximated CIHLQR controller is presented. It is shown that the theorems could be extended from a two-agent control system to a multi-agent system. Finally, the results are evaluated using the simulation results of a Two-Area non-reheat power system.

The problem of personal identity among others may stem from the following question—what does be the person that you are, from one day to the next, necessarily consist of? The diachronic problem of personal identity raises question on the necessary and sufficient conditions for the identity of the person over time. The synchronic problem is grounded in the question of what features or traits characterize a given person at one time. To answer these questions, John Locke discarded the soul and the body as necessary and sufficient substances for personal identity over time. He accepted consciousness as the only criterion for personal identity; the only thing capable of remaining the same and preserving personal identity through change. Though Locke’s argument is somewhat clear and coherent but what remains vague and incoherent is embedded in the question—what exactly is consciousness? How and why should it be the basis or criterion for the determination of personal identity? Using the method of critical analysis, I argue that Locke’s choice of consciousness as the determinant of personal identity, though quite novel, is incoherent and vague. Secondly, Locke had already presumed and anticipated clearly though fallaciously the very thing he wishes to substantiate. I therefore conclude that Locke’s argument is just another way of trying to escape but inadvertently prolonging the difficulty of apparently articulating a distinction between the psychological approach and physiological approach to the problem of personal identity. However, in my submission, I propose the concept of the “other” as alternative approach— a sort of an extrinsic-intrinsic approach to the problem of personal identity.

In this paper, using the Bergman distance, we introduce a new projection-type algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points. Then, the strong convergence of the sequence generated by the algorithm under suitable conditions is proved. In fact, we prove that the sequence generated by the algorithm converges to the projection of an element in the intersection of the fixed points set and the solutions set of an equilibrium problem. For this purpose, we introduce a Bergman Lipschitz-type condition for a pseudomonotone bifunction. Then, we see an application for a variational inequality problem and we apply our result for finding a common element of the solution set of a variational inequality problem and the set of fixed points of a nonexpansive mapping. Finally, using MATLAB software, we provide a numerical example to illustrate the convergence performance of the main algorithm.