Vtyjkyuk

Let $F : [0,a] times [b-r,b+r] rightarrow mathbbR$, $a, b, r$ fixed real numbers, satisfying

(1) for each $k in [b-r,b+r]$, $g_k : [0,a] rightarrow mathbbR$ defined by $$g_k(t) = F(t,k)$$ is continuous,

(2) for each $m in [0,a],$ $$h_m(t) = F(m,t)$$ is continuous

(3) there exists $M > 0$ such that $$|F(s,t)| leq M$$ for all $s,t.$

If $aM leq r$, then the initial value problem $$y' = F(u,y(u))$$ with $u in [0,a]$ and $y(0) = y_0$ has a solution.

$textbfHint :$ Let $S = f(t) - y_0.$ Define $T : S rightarrow S$ by $$T(f)(x) = y_0 + int_0^x F(s,f(s)) ds.$$

$textbfAttemp$ Follow that hint, I defined that $T$. I saw one similar problem, and usual trick is to use Banach Fixed point theorem. So that is my plan.

Step 1 : $X$ is complete.

Since $X = overlineB(y_0,r)$ a closed ball in $C[0,a]$, it is complete.

Step 2 : $T$ is well-defined ($T(f) in X$)
Seem like the boundedness of $F$ and $aM leq r$ makes $T$ well-defined (I can already show it.)

So what left is to show that $T$ is a contraction (i.e. there is $C < 1$ such that $||T(f) - T(g)||_infty leq C ||f-g||_infty.$)

Ususally, $T$ has to be $textitLipschitz$ in the second variable,but not here. I try to use the continuity condition on $F$ too, but not work.

Any help please ?