# Importance of Fixed-point theorems [duplicate]

I have a more general question on the importance of fixed-point theorems. In mathematics youre being introduced to so many fixed-point theorems but i still could not figure out why they are so important. Why would be a simply looking statement as $$f(x)=x$$ be so important. I would appreciate any answer. Thanks in advance. On wikipedia it says nothing about the importance, contextualisation of theorems in mathematics is sometimes not given.

One important reason is that the existence of solutions to systems of equations are equivalent to fixed-points of appropriate functions. Suppose you want to show $$f(x)=0$$ for some $$x$$. This is equivalent to $$f(x)+x=x$$, which means that the function $$F$$ defined by $$F(x)=f(x)+x$$ has a fixed-point.