Asymptotic behavior of mild solutions to fractional Cauchy problems in Banach spaces (Q1985409)

The author studies in a Banach space \(X\) the asymptotic behaviour of the solutions to the fractional Cauchy problems \[ \partial^{\alpha}_{t}u(t)=Au(t)+f(t),\ t\in [0,+\infty),\ u(0)=x,\ u'(0)=y, \] and \[ \partial^{\alpha}u(t)=Au(t)+f(t),\ t\in [0,+\infty),\ (g_{2-\alpha}*u)(0)=x,\ (g_{2-\alpha}*u)'(0)=y, \] where \(f\) is a suitable function, \(A:D(A)\subset X\to X\) is a closed and linear operator, the generator of an \((\alpha,\beta)\)-resolvent family, \(x, y \in X\) for \(1 < \alpha < 2\), \(\partial^{\alpha}_{t}\) and \(\partial^{\alpha}\) denote, respectively, the Caputo and Riemann-Liouville fractional derivatives, and for \(\mu > 0\), \(g_{\mu}(t) := t^{\mu-1}/\Gamma(\mu)\) (here, \(\Gamma(\mu)\) is the Gamma function) and \(*\) denotes the usual finite convolution.