# On solutions of integral equations with analytic kernels and rotations

Nguyen Van Mau; Nguyen Minh Tuan

Annales Polonici Mathematici (1996)

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

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topNguyen 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

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

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