Unique solvability of a linear problem with perturbed periodic boundary values

We investigate the problem with perturbed periodic boundary values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ \begin{gathered} y'''(x) + a_2 (x)y''(x) + a_1 (x)y'(x) + a_0 (x)y(x) = f(x), \hfill \\ y^{(i)} (T) = cy^{(i)} (0), i = 0,1,2;\;\;0 < c < 1 \hfill \\ \end{gathered} \right.$$ \end{document} with a2, a1, a0 ∈ C[0, T] for some arbitrary positive real number T, by transforming the problem into an integral equation with the aid of a piecewise polynomial and utilizing the Fredholm alternative theorem to obtain a condition on the uniform norms of the coefficients a2, a1 and a0 which guarantees unique solvability of the problem. Besides having theoretical value, this problem has also important applications since decay is a phenomenon that all physical signals and quantities (amplitude, velocity, acceleration, curvature, etc.) experience.