We prove the generalized Hyers-Ulam stability of the 2nd-order linear differential equation of the form , with condition that there exists a nonzero in such that and is an open interval. As a consequence of our main theorem, we prove the generalized Hyers-Ulam stability of several important well-known differential equations.

In each community, there is a lot of information about the disadvantages and risks of drug use and its negative effects on health, work, honor, and other living funds of people. A group of individuals, who can be called responsive individuals, will be safe from the risk of drug abuse, by receiving and understanding such information. In this paper, by proposing mathematical models, we investigate the effect of the distribution of this kind of information on the transformation of susceptible individuals into responsive individuals as well as their effect in preventing the occurrence of substance abuse epidemics. In these models, we take into account the fact that the spirit of responsiveness of these individuals can be decayed with time, and these people can become susceptible people, and eventually to addicts. We analyze the dynamical properties of the models, such as local and global stability of equilibrium points and the occurrence of backward bifurcation. The results of this study show that the higher the rate of conversion of susceptible individuals to those responsive, the prevention of drug epidemy is easier.

In this paper, we present a model of Partnership Game with respect to the important role of partnership and cooperation in nowdays life. Since such interactions are repeated frequently, we study this model as a Stage Game in the structure of infinitely repeated games with a discount factor δ and Trigger strategy. We calculate and compare the payoffs of cooperation and violation and as an important result of this study, we show that each partner will adhere to the cooperation.

In this study, considering the importance of treatment of addiction as an illness, we have analyzed a model of harvesting of facilities for the rehabilitation and treatment as a new application of game theory. According to the results of this research, it is possible to avoid a waste of money, energy, and facilities with a better management of allocating facilities and we can treat more addicts.