In this paper, we are going to estimate the domain of attraction of tumor-free equilibrium points in a perturbed cancer tumor model describing the tumor-immune system competition dynamics. The proposed method is based on an optimization problem solution for a chosen Lyapunov function that can be casted in terms of Linear Matrix Inequalities constraint and Taylor expansion of nonlinear terms. We find a specific Lyapunov function in order to vanish maximum perturbation of modeling error, aging or uncertainties which exist in this system. Using this method and appropriate Lyapunov function, we demonstrate that there is an invariant polytope that for the set of perturbed initial conditions belonging to such region, the convergence to the healthy state is guaranteed.

The purpose of this research is to develop the outsourcing model of the bank's electronic banking services using game theory to identify the equilibrium and win-win strategies between the bank and the applicant companies. For this purpose, an interview was held with bank and company experts. After qualitative data analysis,16 strategies were identified for the parties. According to the compatibility of the identified strategies with the marketing mix, the results were grouped into 4 product groups, price, distribution and promotion. Since the identified strategies did not have the same value for the parties, to identify the weight of the strategies. A pairwise comparison questionnaire was used and the opinions of experts from each side were collected separately. The analytical Hierarchy process was used to obtain the weight of each index and the Inconsistency Ratio index was used to check the reliability of the obtained results. After identifying the weight of the strategies, the game environment was implemented and according to the 16 identified strategies, 65536 scenarios were implemented for each side. After performing calculations, 8 scenarios with the same outcomes were identified as equilibrium scenarios or Nash Equilibrium games. The scenario that had the most consequences for the parties was introduced as the result of the game.

Based on the extragradient-like method combined with shrinking projection, we propose two algorithms, the first algorithm is obtained using sequential computation of extragradient-like method and the second algorithm is obtained using parallel computation of extragradient-like method, to find a common point of the set of fixed points of a nonexpansive mapping and the solution set of the equilibrium problem of a bifunction given as a sum of the finite number of H ̈, older continuous bifunctions. The convergence theorems for iterative sequences generated by the algorithms are established under widely used assumptions for the bifunction and its summands.

Unlike perfect competitive markets, in oligopoly electricity markets due to strategic producers and transmission constraints GenCos may increase their own profit through strategic biddings. This paper investigates the problem of developing optimal bidding strategies of GenCos considering participants' market power and transmission constraints. The problem is modeled as a bi-Ievel optimization that at the first level each GenCo maximizes its payoff through strategic bidding and at the second level, in order to consider transmission constraints a system dispatch is accomplished through an OPF problem. The AC power flow model is used for proposed OPF. Here it is assumed that each GenCo uses linear supply function model for its bidding and has information about initial bidding of other-competitors. The impact of optimal biddings on market characteristics as well as GenCos' payoffs are investigated and compared with perfect competitive markets where all the participants bid with their marginal costs. Furthermore, effects of exercising market power due to transmission constraints as well as different biddings of strategic generators on GenCos' optimal bidding strategies are presented. Finally IEEE-30 bus test system is used for case study to demonstrate simulation results.