Vector-valued measures of noncompactness and the Cauchy problem with delay in a scale of Banach spaces (Q2307315)

In this paper, the authors investigate the following Cauchy problem with delay in a scale of Banach spaces: \[ \dfrac{\mathrm{d} u}{\mathrm{d}t} = f(t, u(t), u(h(t))), \quad t\in (0, T),\quad u(0) = u_0, \] where \(h:[0, T)\to [0,\infty)\) is continuous satisfying \(f (t) <t^{1/p}\) on \((0, T)\) for some \(p\in (0, 1)\) and \(f\) is an operator that satisfies \[ \alpha_r(f(t, \Omega_1, \Omega_2))\le L\left(\alpha_r(\Omega_1)+\frac{\alpha_s^p(\Omega_2)}{(s-r)^\gamma}\right), \quad r< s. \] Here, \(\alpha_s\) is the Kuratowski measure of noncompactness in the Banach space \(X_s\), where \((X_s, |\cdot|_s), s\in [a, b]\) is a scale of Banach spaces \(X_s\) with norms \( |\cdot|_s\), and \(\Omega_1\) and \(\Omega_2\) are bounded subsets of relevant Banach spaces \(X_s\). The numbers \(L\) and \(\gamma\) are positive. Under some additional conditions on \(f\), the authors prove that there exists a number \(\lambda>0\) such that the Cauchy problem has a solution \(u\) such that \(u(t) \in X_s\) for all \(t\in [0, (b-s)/\lambda),\; s\in [a, b).\) The methodology involves the Darbo-Sadovskii fixed point theorem for condensing operators on Fréchet spaces and techniques of vector-valued measure of noncompactness.