Measures of noncompactness, Darbo maps and differential equations in abstract spaces (Q1912664)

Let \(B\) be a real Banach space. The existence theory for the initial value problem \(y'(t)= q(t) f(t, y(t))\), \(t\in (0, T]\), \(y(0)= a\in B\), \(q\in C (0, T]\), \(q>0\) and \(\int^T_0 q(s) ds< \infty\), and for the Dirichlet boundary value problem \(y''+ \beta y' - \varepsilon y= q(t) f(t, y, y')\), \(0< t< 1\), \(y(0)= a\in B\), \(y(1)= b\in B\); \(\beta, \varepsilon\in \mathbb{R}\), \(q\in C (0,1)\), \(q>0\) and \(\int^1_0 q(s)ds< \infty\) is considered in the case when \(f\) has a splitting of the form \(f= g+h\) with \(g\), \(h\) continuous and \(g\) satisfying some compactness condition. The existence theory is based on a Leray-Schauder type nonlinear alternative [see \textit{J. Dugundji} and \textit{A. Granas}, Fixed point theory. Warszawa (1982; Zbl 0483.47038)] and on the Kuratowski measure of noncompactness.