It has been proved recently that the two-direction refinement equation of the form $f\left(x\right)={\sum }_{n\in }{c}_{n,1}f(kx-n)+{\sum }_{n\in \mathbb{Z}}{c}_{n,-1}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f\left(x\right)={\sum }_{n\in \mathbb{Z}}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f\left(x\right)={\int }_{ℝ}c\left(y\right)f(kx-y)dy$ has also various interesting applications....

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.