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Consider the following problem $$x'(t)=f(t,x(t)),\qquad x(t_0)=x_0.$$ If $Q_b=\{(t,x)\in\mathbb{R}\times X\,|\,|t-t_0|\le a\, \|x-x_0\|\le b\}$ and function $f\colon Q_b\to X$ is continuous and Lipschitz, i.e., $$\forall_{(t,x_1),(t,x_2)\in Q_{b}}\,\|f(t,x_1)-f(t,x_2)\| \le L \|x_1-x_2\|,$$ $\|f(t,x)\|\le K$ for all $(t,x)\in Q_{b}$, then using fixed points arguments, we can prove that diff. equation has a unique solution $x\in C([t_0-c,t_0+c], X)$, where $c=\min (a, b/K)$.

I wonder what happens if we forget about Lipschitz condition. Then, we can solve the so called Peano's problem. As a result we obtain a solution but not unique, I guess.

Anyway, do you know any differential equations in Banach spaces (of infinite dimension) which do not have any solution? I heard that there are some when $X=\mathbb{c}_0$ but unfortunatelly I cannot figure them out on my own.

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