A global positive solution of a delay differential equation with indefinite coefficients (Q441917)

Consider the delay-differential equation NEWLINE\[NEWLINEx'(t)+ a_1(t) x(t)+ a_2(t) x(t- h(t))= 0,\quad t\geq t_0> 0.\tag{\(*\)}NEWLINE\]NEWLINE Let \(h: (t_0,\infty)\to \mathbb{R}^+\) be bounded such that \(h'\) exists and is bounded on \((t_0,\infty)\), \(T_0:= \text{inf}_{t\geq t_0} (t- h(t))\).NEWLINENEWLINE Associate with \((*)\) the initial condition NEWLINE\[NEWLINEx(t)= \zeta(t)\quad\text{for }t\in[T_0, t_0]\quad\text{with }\zeta\in C([T_0, t_0],\mathbb{R}).\tag{\(**\)}NEWLINE\]NEWLINE The author proves that the initial value problem \((*)\), \((**)\) has a global positive solution satisfying NEWLINE\[NEWLINEx(t)\to 0\qquad\text{as }t\to\inftyNEWLINE\]NEWLINE under the hypothesesNEWLINENEWLINE (H1) \(a_1'\) exists and is bounded on \([T_0,\infty)\) with \(a_1(t)\to 0\) as \(t\to\infty\).NEWLINENEWLINE (H2) \(a_2'\) exists and is bounded on \([T_0,\infty)\), \(a_1\) is continuous on \([T_0,\infty)\) with \(a_2(t)\to 0\) as \(t\to\infty\).NEWLINENEWLINE The proof is based on Schauder's fixed point theorem.