On a weakly degenerate first-order linear differential equation in a Banach space (Q2480404)

Consider the equation \[ \varphi(t) x'(t)= Ax(t)+ f(t)\quad\text{for }t\in (0,\infty)\tag{\(*\)} \] in a Banach space \(E\) under the following assumptions (i) \(A: D(A)\subset E\to E\) is an unbounded linear operator and (ii) \(\varphi\in C((0,\infty), (0,\infty))\) satisfies \(\lim_{t\to +0}{\varphi(t)\over t^\alpha}= K\) with \(0< \alpha< 1\), \(0< K<\infty\). The author gives additional conditions on \(A\) and \(f\) such that \((*)\) has a solution bounded for \(t= 0\).