In dynamic economics we extensively use the fixed-point theorems. From the close end, suffice it to mention that a difference equation is stable if it has a fixed-point. From the far end of applications, when iterating on the Bellman operator we want to find the unique value function satisfying the Bellman equation as a fixed-point of the operator. No matter where we go or what we do in economics, we encounter applications of the fixed-point theorems coming in various forms. In this lecture, we study the general form of mappings having properties that enable them to have fixed points. We start out with the general Lipschitz condition that we constrain to a form where a mapping has a unique fixed-point. Our most important result here is Banach’s fixed-point theorem. In the applications, most attention is paid to the integral operator as this is the operator that we most extensively use in stochastic dynamic programming. As a by-product, we also study the practice of turning a mapping having specific properties into an operator. As a final step, we also examine alternative strategies to have fixed points. Instead of trying to place restrictions on the mappings, we can add additional topological structure to the underlying spaces that ensure the existence and the uniqueness of fixed points.

Title page:(00:00)
Introduction:(00:10)
The rth power of T and its fixed point:(07:44)
The Lipschitz condition:(11:43)
Additional results:(22:46)
Operator norms:(28:30)
Meet the integral operator:(34:45)
An ODE and the integral operator:(49:34)
Another application of the integral operator:(65:58)
Constraining the spaces to have fixed points:(75:14)