Fix $0 < \lambda < 1$. Choose $\epsilon > 0$ such that $\epsilon + \frac { \lambda ^ { 2 } } { 4 } \leq \frac { \lambda } { 2 }$. Consider the metric space
$$X : = \left\{ \psi \in \mathcal { C } ^ { 1 } ( [ - \epsilon , \epsilon ] ) : | y + \psi ( y ) | \leq \frac { \lambda } { 2 } \text { for all } y \in [ - \epsilon , \epsilon ] \right\}$$
with the induced supremum metric from $\mathcal { C } ^ { 1 } ( [ - \epsilon , \epsilon ] )$, which we denote by $d$. (Recall: $\mathcal { C } ^ { 1 } ( [ - \epsilon , \epsilon ] )$ is the set of real-valued differentiable functions on $[ - \epsilon , \epsilon ]$ whose derivative is continuous.)
(A) (1 mark) Show that there is a function $A : X \longrightarrow X$ given by
$$( A \psi ) ( y ) = - ( y + \psi ( y ) ) ^ { 2 } .$$
(B) (2 marks) Show that if $\psi \in X$ is such that $A \psi = \psi$, then the function $x = y + \psi ( y )$ is an inverse to the function $y = x + x ^ { 2 }$, locally near the origin. In the next few steps, we show that such a $\psi$ exists.
(C) (2 marks) Show that $d \left( A \psi _ { 1 } , A \psi _ { 2 } \right) \leq \lambda d \left( \psi _ { 1 } , \psi _ { 2 } \right)$.
(D) (4 marks) Let $\phi \in X$. Show that the sequence $A ^ { n } \phi , n \geq 1$ is a Cauchy sequence, and it has a limit. By $A ^ { n }$, we mean the $n$-fold composition $A \circ A \circ \cdots \circ A$. (You may use the fact that $X$ is complete with respect to d.)
(E) (1 mark) Show that there exists $\psi \in X$ such that $A \psi = \psi$.