In this paper, the operator approach based on the fixed point principles of Banach and Schaefer is used to establish the existence of solutions to stationary Kirchhoff equations with reaction terms. Next, for a coupled system of Kirchhoff equations, it is proved that under suitable assumptions, there exists a unique solution which is a Nash equilibrium with respect to the energy functionals associated to the equations of the system. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls are established. The solution is obtained by using an iterative process based on Ekeland's variational principle and whose development simulates a noncooperative game.

The paper deals with some control problems related to the Kolmogorov system for two interacting populations. For the first problem, the control acts in time over the per capita growth rates of the two populations in order for the ratio between their sizes to follow a prescribed evolution. For the second problem, the control is a constant which adjusts the per capita growth rate of a single population so that it reaches the desired size at a certain time. For the third problem the control acts on the growth rate of one of the populations in order that the total population to reach a prescribed level. The solution of the three problems is done within an abstract scheme, by using operator-based techniques. Some examples come to illustrate the results obtained. One refers to a system that models leukemia, and another to the SIR model with vaccination.