On the asymptotic properties of the solutions of some functional equations (Q1707066)

The authors study the equation \[ x(t)=a_1x(t-r_1)+\cdots +a_{n_0}x(t-r_{n_0})+b_1x(q_1t)+\cdots +b_{n_1}x(q_{n_1}t), \] where \(\{a_k, b_k\}\subset \mathbb{R}, r_k>0\), and \(0<q_k<1\). In the first theorem, the authors obtain sufficient conditions such that the solution \(x(t)\) can be estimated as \[ \left|x^{(m)}(t)\right|\leq K_m(t_0, \nu)t^\nu \max \left\{\sup_{s\in[r(t_0), t_0]}\left|x^{(m)}(s)\right|, \ldots, \sup_{s\in[r(t_0), t_0]}\left|x^{(j)}(s)\right|\right\}, \; t\geq r(t_0). \] Then they discuss the special case \[ x(t)=a_1x(t-r_1)+\cdots +a_{n_0}x(t-r_{n_0})+bx(qt). \] Concretely, Theorem 2 shows that, under a certain set of conditions, the \(j\) times continuously differentiable solutions \(x(t)\) of such a special case can be estimated as \[ \left|x(t)\right|\leq K(t_0)t^{\nu_*} \max \left\{\sup_{s\in[r(t_0), t_0]}\left|x(s)\right|, \ldots, \sup_{s\in[r(t_0), t_0]}\left|x^{(j)}(s)\right|\right\}, \; t\geq r(t_0). \] Theorem 3 provides the asymptotic behavior of the above equation. Theorem 4 contains an estimate for the solution of the inhomogeneous equation \[ x(t)=a_1x(t-r_1)+\cdots +a_{n_0}x(t-r_{n_0})+b_1x(q_1t)+\cdots +b_{n_1}x(q_{n_1}t)+f_0, \] where \(f_0=f_0(x(t-r_1),\ldots,x(t-r_{n_0}),x(q_1t),\ldots,x(q_{n_1}t))\).