Stability results for neutral differential equations by Krasnoselskii fixed point theorem (Q2020165)

In this paper, the neutral differential equations with variable delays \[ x'(t) = -a(t)x(t-\tau_1(t)) + c(t)x'(t-\tau_1(t)) + b(t)[x(t-\tau_2(t))]^\sigma, \quad t \ge t_0 \] is considered, where \(\sigma\in(0,1)\) is a quotient with odd positive integer denominator, \(a\), \(b\), \(c\), \(\tau_1\), \(\tau_2 \in C[t_0,\infty)\), \(\tau_i(t)\ge 0\) for \(t\ge t_0\), and \(t-\tau_i(t)\to\infty\) as \(t\to\infty\) for \(i=1\), \(2\). Sufficient conditions for the boundedness of solutions and necessary and sufficient conditions ensuring the asymptotic stability of solutions are derived.