Geodesic
Unique solvability of a linear problem with perturbed periodic boundary values
Czechoslovak Mathematical Journal, Tome 49 (1999) no. 2, pp. 351-362 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 34B05, 34B15, 34C10, 45B05
Keywords: Ordinary differential equations; integral equations; periodic boundary value problems 
@article{CMJ_1999_49_2_a11,
     author = {Mehri, Bahman and Nojumi, Mohammad H.},
     title = {Unique solvability of a linear problem with perturbed periodic boundary values},
     journal = {Czechoslovak Mathematical Journal},
     pages = {351--362},
     year = {1999},
     volume = {49},
     number = {2},
     mrnumber = {1692528},
     zbl = {0955.34007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_1999_49_2_a11/}
}
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%A Nojumi, Mohammad H.
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Mehri, Bahman; Nojumi, Mohammad H. Unique solvability of a linear problem with perturbed periodic boundary values. Czechoslovak Mathematical Journal, Tome 49 (1999) no. 2, pp. 351-362. http://geodesic.mathdoc.fr/item/CMJ_1999_49_2_a11/

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