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

Fixed Point:
            Let  be a self mapping. A point  is called a fixed point of , if . It may be noted that not every mapping has a fixed point.

Examples:

• Let  be defined by . Then each point  is a fixed point.
• Let  be defined by . Then  has no fixed point.
• Let  be defined by . Then  has exactly one fixed point “0”.

Theorems:

• Let  be a continuous self mapping. Then there exist a point  such that .
• 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,  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  has a fixed point if and only if each continuous mapping  has a fixed point.