# On solutions of integral equations with analytic kernels and rotations

• Volume: 63, Issue: 3, page 293-300
• ISSN: 0066-2216

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## Abstract

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We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*)     x(t) + a(t)(Tx)(t) = b(t), where $T={M}_{n₁,k₁}...{M}_{{n}_{m},{k}_{m}}$ and ${M}_{{n}_{j},{k}_{j}}$ are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial ${P}_{T}\left(t\right)=t³-t$. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation (*) in closed form.

## How to cite

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Nguyen Van Mau, and Nguyen Minh Tuan. "On solutions of integral equations with analytic kernels and rotations." Annales Polonici Mathematici 63.3 (1996): 293-300. <http://eudml.org/doc/262785>.

@article{NguyenVanMau1996,
abstract = {We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*)     x(t) + a(t)(Tx)(t) = b(t), where $T = M_\{n₁,k₁\} ... M_\{n_m,k_m\}$ and $M_\{n_j,k_j\}$ are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial $P_T(t) = t³ - t$. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation (*) in closed form.},
author = {Nguyen Van Mau, Nguyen Minh Tuan},
journal = {Annales Polonici Mathematici},
keywords = {integral operators; singular integral equations; algebraic operators; Riemann boundary value problems; integral equations on the unit circle; analytic continuation; Fredholm equation},
language = {eng},
number = {3},
pages = {293-300},
title = {On solutions of integral equations with analytic kernels and rotations},
url = {http://eudml.org/doc/262785},
volume = {63},
year = {1996},
}

TY - JOUR
AU - Nguyen Van Mau
AU - Nguyen Minh Tuan
TI - On solutions of integral equations with analytic kernels and rotations
JO - Annales Polonici Mathematici
PY - 1996
VL - 63
IS - 3
SP - 293
EP - 300
AB - We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*)     x(t) + a(t)(Tx)(t) = b(t), where $T = M_{n₁,k₁} ... M_{n_m,k_m}$ and $M_{n_j,k_j}$ are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial $P_T(t) = t³ - t$. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation (*) in closed form.
LA - eng
KW - integral operators; singular integral equations; algebraic operators; Riemann boundary value problems; integral equations on the unit circle; analytic continuation; Fredholm equation
UR - http://eudml.org/doc/262785
ER -

## References

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1. [1] F. D. Gakhov, Boundary Value Problems, Oxford, 1966 (3rd Russian complemented and corrected edition, Moscow, 1977). Zbl0141.08001
2. [2] Nguyen Van Mau, Generalized algebraic elements and linear singular integral equations with transformed argument, Wydawnictwa Politechniki Warszawskiej, Warszawa, 1989.
3. [3] Nguyen Van Mau, Boundary value problems and controllability of linear systems with right invertible operators, Dissertationes Math. 316 (1992). Zbl0749.34034
4. [4] D. Przeworska-Rolewicz and S. Rolewicz, Equations in Linear Spaces, Monografie Mat. 47, PWN, Warszawa, 1968. Zbl0181.40501

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