On a Cauchy Problem in Banach Spaces

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 epsilon C-1 ([0, a], X) for all a > 0 under the assumption that f : R+circle plus X -> 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, l(p) and L-p(Omega, Sigma, mu) (1 < p < infinity and (Omega, Sigma, mu) is sigma-finite), and for all a > 0 there is a bounded nowhere locally Lipschitz function f : R+ circle plus X -> X which is weak-to-weak continuous on some bounded set so that the Cauchy initial problem has a solution x epsilon C-1 ([0, a], X).