Let $f(x,y)$ be continuously differentiable on $[0,1]\times \mathbb{R}$, and satisfy $$0<m<\frac{\partial f}{\partial y}(x,y)\leq M.$$
Show that there is a unique continuous function, $\Phi(x)$ such that $$f(x,\Phi(x))=0, \forall x\in(0,1).$$
We're given a hint to define $(T\Phi)(x)=\Phi(x)-\lambda f(x,\Phi(x))$, for $\Phi\in C([0,1])$, and then find a $\lambda$ that would make $T$ into a contraction.
I've actually got the rest of the problem figured out. I'm just struggling to find the appropriate $\lambda$.
Using the max-difference metric on $C([0,1])$, I know that
$$d(T\Phi,T\hat{\Phi})=\max_{x\in[0,1]}\{|T\Phi(x)-T\hat{\Phi}(x)|\}=$$ $$\max_{x\in[0,1]}\{|\Phi(x)-\lambda f(x,\Phi(x)) - \hat{\Phi}(x)+\lambda f(x,\hat{\Phi}(x))|\} = $$
$$\max_{x\in[0,1]}\{|\Phi(x)-\hat{\Phi}(x)-\lambda (f(x,\Phi(x)) - f(x,\hat{\Phi}(x)))|\}.$$
Moreover, since $f$ has a bounded derivative in $y$, I know that it's lipschitz in y. Which means I should be able to bound the $f(x,\Phi(x)) - f(x,\hat{\Phi}(x))$ expression in my last term.
But it's not clear to me how to pick a value of $\lambda$ that would work for all functions $\Phi\in C([0,1])$. i.e. How to find a $\lambda$ so that the last expression is less than $$\rho\max_{x\in[0,1]}\{|\Phi(x)-\hat{\Phi}(x)|\},$$ for some $\rho<1$
All I really need at this point is just some direction on finding such a $\lambda$.
Thanks in advance.
From $$0<m<\frac{\partial f}{\partial y}(x,y)\leq M$$ we know that $$m(\Phi(x)-\hat\Phi(x))\leq f(x,\Phi(x)) - f(x,\hat{\Phi}(x)) \leq M(\Phi(x)-\hat\Phi(x)),~\text{if }\Phi(x)\geq\hat\Phi(x)$$
$$M(\Phi(x)-\hat\Phi(x))\leq f(x,\Phi(x)) - f(x,\hat{\Phi}(x)) \leq m(\Phi(x)-\hat\Phi(x)),~\text{if }\Phi(x)<\hat\Phi(x)$$
So (considering $\lambda=1/M$) $$0\leq\Phi(x)-\hat{\Phi}(x)-\lambda (f(x,\Phi(x)) - f(x,\hat{\Phi}(x)))\leq (1-m/M)(\Phi(x)-\hat\Phi(x)),~\text{if }\Phi(x)\geq\hat\Phi(x)$$
$$(1-m/M)(\Phi(x)-\hat\Phi(x))\leq\Phi(x)-\hat{\Phi}(x)-\lambda (f(x,\Phi(x)) - f(x,\hat{\Phi}(x)))\leq 0,~\text{if }\Phi(x)<\hat\Phi(x)$$
The rest is straightforward.
This proof does not use the fact that the derivative of $f(x,y)$ is continuous, I don't know if it is really necessary.