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.
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.
• 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 “”.
• Let be a continuous self mapping. Then there exists a point such that .
• A contraction self mapping defined on a complete metric space has a unique fixed point.
Fixed Point Space
A space is said to be a fixed point space if every continuous self mapping on has a fixed point. For example, is a fixed point space, or in other words, we say that has a fixed point property. Finally, we show that “a fixed point property” is a topological property.
Let and be homeomorphic spaces. Then each continuous mapping has a fixed point if and only if each continuous mapping has a fixed point.