The map g(x)=sqrt(x**2+1) has no fixed point. However |g(x)-g(y)| < |x-y| for all reals x, y.
counterexample-banach-fixed-point-theorem-image
The map g(x)=sqrt(x**2+1) has no fixed point. However |g(x)-g(y)| < |x-y| for all reals x, y.