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Unique solvability of a linear problem with perturbed periodic boundary values

Authors:

Bahman Mehri Mohammad H. Nojumi

Affiliation:

(1) Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran

Abstract:

We investigate the problem with perturbed periodic boundary values

$left{ begin{gathered} y'(x) + a_2 (x)y'(x) + a_1 (x)y'(x) + a_0 (x)y(x) = f(x), hfill y^{(i)} (T) = cy^{(i)} (0), i = 0,1,2;;;0 < c < 1 hfill end{gathered} right.$

with a₂, a₁, a₀ isin

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.

Keywords:

Ordinary differential equations integral equations periodic boundary value problems

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