# Article

Full entry | PDF   (0.3 MB)
Keywords:
Ordinary differential equations; integral equations; periodic boundary value problems
Summary:
We investigate the problem with perturbed periodic boundary values $\left\rbrace \begin{array}{ll}y^{\prime \prime \prime }(x) + a_2(x) y^{\prime \prime }(x) + a_1(x) y^{\prime }(x) + a_0(x) y(x) = f(x) , y^{(i)}(T) = c y^{(i)}(0), \ i = 0, 1, 2; \ 0 < c < 1 \end{array}\right.$ with $a_2, a_1, a_0 \in C[0,T]$ 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 $a_2$, $a_1$ and $a_0$ 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.
References:
[1] H. Brezis: Analyse Fonctionnelle, Théorie et Applications. Masson, Paris, 1983. MR 0697382 | Zbl 0511.46001
[2] R. Brown: A Topological Introduction to Nonlinear Analysis. Birkhäuser, Boston, 1993. MR 1232418 | Zbl 0794.47034
[3] J. A. Cochran: Analysis of Linear Integral Equations. McGraw Hill, New York, 1972. MR 0447991 | Zbl 0233.45002
[4] R. Kress: Linear Integral Equations. Springer-Verlag, New York, 1989. MR 1007594 | Zbl 0671.45001
[5] M. Reed, and B. Simon: Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. Academic Press, Orlando, Florida, 1980. MR 0751959

Partner of