On a small stabilizing perturbation of a singular differential equation with a constant operator and a degenerate coefficient of the general form (Q1778287)

Let \(E\) be a Banach space, let \(A\) be the infinitesimal generator of a \(C_0\)-semigroup \(\{U(t)\}\). Let \(f \in C([0, \infty); E)\) be such that \(f(t)\in D(A)\) for all \(t\geq 0\) and \(Af \in C([0, \infty); E)\) and let \(\varphi \in C((0, \infty); (0, \infty))\) be such that \(\varphi(+0)=0\) and \(\lim_{t \to +0}\varphi(t)/t^\alpha = K\), with \(\alpha \geq 1\) and \(K >0\). Setting \(J_\varepsilon(s, t) = \int_s^t \frac{d\tau}{\varphi(\tau + \varepsilon)}\), where \(\varepsilon\) is a small parameter, the author proves that the problem \[ \varphi(t+\varepsilon)x^\prime_\varepsilon(t) = Ax_\varepsilon(t) + f(t),\;t \geq 0, x_\varepsilon(0)=x_{\varepsilon, 0}, \;x_{\varepsilon, 0} \in D(A), \] has the solution \[ J_\varepsilon(t) = U(J_\varepsilon(0, t))x_{\varepsilon, 0}+ \int_0^t U(J_{\varepsilon}(s, t))\frac{f(s)}{\varphi(s+\varepsilon)}\;ds \] and deduces various properties of it.