Local and global solutions of well-posed integrated Cauchy problems

We study the local well-posed integrated Cauchy problem v'(t) = Av(t) + t(alpha) /Gamma(alpha+1) x, v(0) = 0 t epsilon [0, k) with k > 0 alpha >= 0, and x epsilon X, where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in alpha. The global case (k = infinity) is also treated in detail. Growth of solutions is given in both cases.