Existence of weak solutions for an infinite system of second order differential equations (Q1714445)

The authors deal with a boundary value problem for an infinite system of second-order differential equations which is turned into the following system of perturbed quadratic integral equations: \[ u_n(t)=g_n(t,u(t))+h_n(t,u(t))\int_0^tG(t,s)f_n(s,u(s))\,ds,\quad n\in\mathbb N, \] where \(t\in I=[0,1]\), \(G\) is a suitable Green's function, and the functions \(f_n, g_n: I\times \mathbb R^{\infty}\to \mathbb R\) and \(h_n: I\times \mathbb R^{\infty}\to \mathbb R \setminus\{0\}\) (\(n\in \mathbb N\)) satisfy some additional assumptions. The main result of this paper concerns the existence of a weak solution to the problem mentioned above. Its proof is based on a certain version of the Krasnosel'skii fixed point theorem and the technique characteristic to the measures of weak non-compactness. The authors illustrate their considerations giving a suitable example.