Some Applications of Connectedness

In some applications of connectedness, we shall define two fixed point theorems in connection with the application of connectedness. Fixed point theorems are useful in obtaining the (unique) solutions of differential and integral equations.

 

Fixed Point

Let f:X \to X be a self mapping. A point x \in X is called a fixed point of f, if f\left( x \right) = x. It may be noted that not every mapping has a fixed point.

Examples:
• Let f:\mathbb{R} \to \mathbb{R} be defined by f\left( x \right) = x. Then each point x \in \mathbb{R} is a fixed point.
• Let f:\mathbb{R} \to \mathbb{R} be defined by f\left( x \right) = x + 1. Then f has no fixed point.
• Let f:\mathbb{R} \to \mathbb{R} be defined by f\left( x \right) = - x. Then f has exactly one fixed point “0”.

Theorems
• Let f:\left[ {0,1} \right] \to \left[ {0,1} \right] be a continuous self mapping. Then there exists a point x \in \left[ {0,1} \right] such that f\left( x \right) = x.
• A contraction self mapping T defined on a complete metric space X has a unique fixed point.

 

Fixed Point Space

A space X is said to be a fixed point space if every continuous self mapping on X has a fixed point. For example, \left[ {0,1} \right] is a fixed point space, or in other words, we say that X has a fixed point property. Finally, we show that “a fixed point property” is a topological property.

Theorem

Let X and Y be homeomorphic spaces. Then each continuous mapping f:X \to X has a fixed point if and only if each continuous mapping g:Y \to Y has a fixed point.