Consider the BVP:$$\frac{dy}{dt}=f(t,y)~,~y(0)=y_{0},$$ where $f(t,y)$ is a bounded function, continuous on the set $\Omega(t,y)=\{ |t| < a , |y-y_{0}| < b \}$ and satisfies a Lipschitz condition there with Lipschitz constant $K.$
Question. Find a condition which will ensure that a unique solution will exist inside $\Omega$ and that the condition has a maximal area on $\Omega.$
The solution of the IVP is given by $$y(t)=y_{0}+\int_{0}^{t} f(s,y(s))~ds.$$ I thought Lipshcitz being satisfy is enough to guarantee the uniqueness of the solution. Also how should I go by and comment about the area ? Any help in solving this is much appreciated.