On some applications of the controllability principle for fixed point equations

Abstract

The aim of this paper is to draw attention to a general principle for solving control problems for operator equations with the help of fixed point techniques. Three distinct applications are presented: a control problem related to the Lotka–Volterra system, the nonlinear Stokes system, and radial solutions of the Neumann problem for ϕ-Laplace equations. Only for the first application, the control is an explicit external one, while for the next two applications, the control is dependent on the models and arises from the necessity to conform to the actual modeled process or to a certain boundary condition. From the perspective of those interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method thus paving the way for new applications. From a theoretical perspective, the method leading to fixed point equations with composed operators is suitable to be related to advanced research in fixed point theory for single-valued and multi-valued operators.

Keywords

• Control problem
• Fixed point
• Boundary value problem
• Stokes system
• Radial solution
• Lotka–Volterra system