In this paper, we consider solvability of the Cauchy initial problem dx/dt=f(t,x),x(0)=x0\documentclass[12pt]{minimal}
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\begin{document}$$dx/{dt}=f(t,x),\; x(0)=x_0$$\end{document} in a Banach space X. As a result, we generalize Peano’s existence theorem in the following manner: For every Banach space X, the problem always has a solution x∈C1([0,a],X)\documentclass[12pt]{minimal}
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\begin{document}$$x\in C^1([0,a],X)$$\end{document} for all a>0\documentclass[12pt]{minimal}
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\begin{document}$$a>0$$\end{document} under the assumption that f:R+⊕X→X\documentclass[12pt]{minimal}
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\begin{document}$$f:{\mathbb {R}}^+\oplus X\rightarrow X$$\end{document} is weak-to-weak continuous on some bounded set with a relatively weakly compact range. We also show that for any infinite dimensional reflexive Banach space X with an unconditional basis, in particular, ℓp\documentclass[12pt]{minimal}
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\begin{document}$$\ell _p$$\end{document} and Lp(Ω,∑,μ)\documentclass[12pt]{minimal}
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\begin{document}$$L_p(\Omega ,\sum ,\mu )$$\end{document} (1<p<∞\documentclass[12pt]{minimal}
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\begin{document}$$1<p<\infty $$\end{document} and (Ω,∑,μ)\documentclass[12pt]{minimal}
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\begin{document}$$(\Omega ,\sum ,\mu )$$\end{document} is σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-finite), and for all a>0\documentclass[12pt]{minimal}
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\begin{document}$$a>0$$\end{document} there is a bounded nowhere locally Lipschitz function f:R+⊕X→X\documentclass[12pt]{minimal}
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\begin{document}$$f:{\mathbb {R}}^+\oplus X\rightarrow X$$\end{document} which is weak-to-weak continuous on some bounded set so that the Cauchy initial problem has a solution x∈C1([0,a],X)\documentclass[12pt]{minimal}
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\begin{document}$$x\in C^1([0,a],X)$$\end{document}.
