On a Cauchy Problem in Banach Spaces

Abstract

In this paper, we consider solvability of the Cauchy initial problem $dx/{dt}=f(t,x),\; x(0)=x_0$ 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 $x\in C^1([0,a],X)$ for all $a>0$ under the assumption that $f:{\mathbb {R}}^+\oplus X\rightarrow X$ 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, $\ell _p$ and $L_p(\Omega ,\sum ,\mu )$ ($1<p<\infty$ and $(\Omega ,\sum ,\mu )$ is $\sigma$-finite), and for all $a>0$ there is a bounded nowhere locally Lipschitz function $f:{\mathbb {R}}^+\oplus X\rightarrow X$ which is weak-to-weak continuous on some bounded set so that the Cauchy initial problem has a solution $x\in C^1([0,a],X)$.