9 - Fixed Point Theorems and Their Applications

Summary

Let X be a set, and f : X → X be a mapping. A point x in X is said to be a fixed point of f if f(x) = x. There are a number of problems in mathematics whose solutions are fixed points of certain mappings. Sometimes it is not possible to find a fixed point of a mapping exactly. Approximation methods are applied to get a value which is near to the required fixed point. Newton's method in numerical analysis can be considered such an approximation method. Some of these problems will be discussed in this chapter.

Fixed Point Theorems

Let X be a set, and f : X → X be a mapping. Then f need not have a fixed point. However, if we impose a certain mathematical structure on X and f, then f may have a fixed point. Our first attempt in this direction is the following.

Theorem 9.1.1 Every continuous function f : [a, b] → [a, b] has atleast one fixed point.

Proof. If a or b is a fixed point of f, then there is nothing to prove. So we assume that f(a) ≠ a and f(b) ≠ b.

Let us define a function g : [a, b] → [a, b] by setting g(x) = f(x) - x. Then g is obviously a continuous function. Since f(a) is in [a, b], f(a) > a. Similarly, f(b) < b. This means that g(a) > 0, and g(b) < 0.