The asymptotic properties of solutions of the equation $u^{\text{'}\text{'}\text{'}}\left(t\right)=p_1\left(t\right)u\left({\tau }_1\left(t\right)\right)+p_2\left(t\right)u^{\text{'}}\left({\tau }_2\left(t\right)\right)$, are investigated where $p_i:[a,+\infty [\to R(i=1,2)$ are locally summable functions, ${\tau }_i:[a,+\infty [\to R(i=1,2)$ measurable ones and ${\tau }_i\left(t\right)\ge t(i=1,2)$. In particular, it is proved that if $p_1\left(t\right)\le 0$, $p_2^2\left(t\right)\le \alpha \left(t\right)\left|p_1\left(t\right)\right|$, $${\int }_a^{+\infty }{[{\tau }_1\left(t\right)-t]}^2{p}_1\left(t\right)dt<+\infty \text{and}{\int }_a^{+\infty }\alpha \left(t\right)dt<+\infty ,$$ then each solution with the first derivative vanishing at infinity is of the Kneser type and a set of all such solutions forms a one-dimensional linear space.

In this paper we study a linear integral equation $x\left(t\right)=a\left(t\right)-{\int }_0^{t}C(t,s)x\left(s\right)\mathrm{d}s$, its resolvent equation $R(t,s)=C(t,s)-{\int }_s^{t}C(t,u)R(u,s)\mathrm{d}u$, the variation of parameters formula $x\left(t\right)=a\left(t\right)-{\int }_0^{t}R(t,s)a\left(s\right)\mathrm{d}s$, and a perturbed equation. The kernel, $C(t,s)$, satisfies classical smoothness and sign conditions assumed in many real-world problems. We study the effects of perturbations of $C$ and also the limit sets of the resolvent. These results lead us to the study of nonlinear perturbations.