91Ó°ÊÓ

Give further examples of mappings in elementary geometry which have ( \(a\) ) a single fixed point, \((b)\) infinitely many fixed points.

Show that \(f\) defined by \(f(t, x)=|\sin x|+t\) satisfies a Lipschitz condition on the whole \(t x\)-plane with respect to its second argument, but \(\partial f / \partial x\) does not exist when \(x=0\). What fact does this illustrate?

Solve the following integral equation ( \(a\) ) by a Neumann series, \((b)\) by a direct approach. $$ x(t)-\mu \int_{0}^{1} x(\tau) d \tau=1 $$