Problem: Give mapping $\phi: [a,b]\times\mathbb{R} \to \mathbb{R}, (t,x) \mapsto \phi(t,x)$ is continuous, $\phi'_x$ is continuous and exist $m,M$ that $m\le \phi'_x(t,x) \le M$. Prove that exist unique $x_* \in C[a,b]$ that $\phi[t,x_*(t)]=0,\forall t \in [a,b]$.
My direction:
Lemma: Suppose that $f:[0,a] \times [x_0-r,x_0+r] \to \mathbb{R}$ is continuous and satisfy Lipschitz condition follow the second variable, it mean exists $k\in (0,1)$ that $$\vert f(t,x)-f(t,y) \vert \le k\vert x-y \vert, \forall t \in [0,a], \forall x,y \in [x_0-r,x_0+r].$$ Set $S=\sup\left\{\vert f(t,x) \vert: x \in [x_0-r,x_0+r]\right\}$. Call $b$ is a number satisfy $0\le b \le a$, $bS \le r$, $bk<1$. Then equations $$x'(t)=f(t,x(t)), \forall t \in [0,b] \text{ and } x_0=x(0)$$ have unique solution in $[0,b]$.
Back to problem: I set $f(x)(t)=x(t)-\dfrac{2}{m+M}\phi[t,x(t)]$ and try to use the lemma but i still not prove the problem.