Convergence results for solutions of a first-order differential equation

Abstract

We consider the first order differential problem: \[ (P_n) \begin{cases} u'(t) = f_n(t, u(t)),\,\,\,\,\, \texttt{for almost every} \quad t \in [0, 1],\\ u(0) = 0. \end{cases} \] Under certain conditions on the functions \(f_n\), the problem \((P_n)\) admits a unique solution \(u_n \in W^{1;1}([0; 1];E)\). In this paper, we propose to study the limit behavior of sequences \((u_n)_{n\in \mathbb{N}}\) and \((u'_n)_{n\in \mathbb{N}}\), when the sequence \((f_n)_{n\in \mathbb{N}}\) is subject to different growing conditions.