Reproducing kernel method of solving singular integral equation with cosecant kernel.

*(English)*Zbl 1152.45007This paper deals with equations containing integral operators with the cosecant kernel. The authors use a standard transformation to extract the singular term with successive application of the method mentioned in the title.

It should be noted that the motivation in considering such equations is not entirely clear. Moreover, such equations, can easily be reduced to equations with cotangent kernels, the theory of which is well developed and very well presented in the literature. In addition, this paper contains numerous errors and inaccuracies. For example, in Section 3 operators \(A\) and \(B\) are mentioned three times but the reader cannot find any definition of the operator \(B\). Further, the authors believe that equation (3.3) has a unique solution (see the end of the proof of Theorem 3.1). It is obviously not true. Thus if one takes the coefficient \(b=b(x)\) to be equal to \(0\) on \([0,2\pi]\), then one obtains a Fredholm equation which can either be not solvable or, if it is, it also might have infinitely many solutions. However, Theorem 3.2 presents the unique solution without any assumptions about the equation whatsoever.

There are also other inaccuracies, so the reader should be very careful with this paper.

It should be noted that the motivation in considering such equations is not entirely clear. Moreover, such equations, can easily be reduced to equations with cotangent kernels, the theory of which is well developed and very well presented in the literature. In addition, this paper contains numerous errors and inaccuracies. For example, in Section 3 operators \(A\) and \(B\) are mentioned three times but the reader cannot find any definition of the operator \(B\). Further, the authors believe that equation (3.3) has a unique solution (see the end of the proof of Theorem 3.1). It is obviously not true. Thus if one takes the coefficient \(b=b(x)\) to be equal to \(0\) on \([0,2\pi]\), then one obtains a Fredholm equation which can either be not solvable or, if it is, it also might have infinitely many solutions. However, Theorem 3.2 presents the unique solution without any assumptions about the equation whatsoever.

There are also other inaccuracies, so the reader should be very careful with this paper.

Reviewer: Victor Didenko (Brunei)

##### MSC:

45H05 | Integral equations with miscellaneous special kernels |

65R20 | Numerical methods for integral equations |

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\textit{H. Du} and \textit{J. Shen}, J. Math. Anal. Appl. 348, No. 1, 308--314 (2008; Zbl 1152.45007)

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