Local and global solutions of well-posed integrated Cauchy problems
Volume 187 / 2008
Studia Mathematica 187 (2008), 219-232
MSC: 47D62, 26A33.
DOI: 10.4064/sm187-3-2
Abstract
We study the local well-posed integrated Cauchy problem $$ v'(t)=Av(t)+{t^{\alpha }\over {\mit\Gamma} (\alpha+1 )} \, x,\ \quad v(0)=0, \ \quad t\in [0, \kappa), $$ with $\kappa>0$, $\alpha \ge 0$, and $x\in X$, where $X$ is a Banach space and $A$ a closed operator on $X$. We extend solutions increasing the regularity in $\alpha $. The global case $(\kappa=\infty)$ is also treated in detail. Growth of solutions is given in both cases.
