I have some questions regarding the following theorems:
Theorem Let $A$ be an open subset of a Banach space $W$, let $I$ be an open interval in $\mathbb{R}$, and let $F$ be a continuous mapping from $I \times A$ to $W$ which is locally uniformly Lipschitz in its second variable. Then for any point $
the solution function is proved to be the fixed point of the mapping $K: f\rightarrow g$, where $f:J \rightarrow A$ is any continuous mapping and $g:J \rightarrow W$ is defined by $ g(t)=\alpha_0+\int^t_{t_0}F(s,f(s))ds$
the proof starts by choosing a neighborhood $L \times U$ of $
- Question 1: why not choose $I \times A$ instead of the neighborhood $L \times U$. Is the purpose to make $F$ bounded?
A lemma to the above theorem: Let $g_1$ and $g_2$ be any two solutions of $d\alpha/dt=F(t,\alpha)$ through $
Proof by contradiction. Otherwise there is a point $s$ in $J$ such that $g_1(s) \neq g_2(s)$. Suppose that $s>t_0$, and set $C=\{t:t>t_0 \mbox{and} g_1(t)\neq g_2(t)\}$ and x = glbC. The set $C$ is open, and therefore $x$ is not in $C$. That is $g_1(x)=g_2(x)$
- Question 2: why is the set $C$ open?
Call this common value $\alpha$ and apply the theorem to $
- Question 3: why $B_r(\alpha)$ has to be a subset of $C$? Why is this possible?
The above lemma allows us to remove the restriction on the range of $f$
- Question 4: Can you elaborate this? and why removing the restriction useful?
Thanks
Edit (T.B.): This is taken from Section 6 of Lynn H. Loomis, Shlomo Sternberg, Advanced Calculus, Jones and Bartlett Publishers, 1990.