Constructive method for the solution of singular integral equations with Hilbert nucleus in Hölder spaces (Q2923002)

The author provides a constructive method for solutions of a class of linear singular integral equations with Hilbert nucleus in Hölder spaces of the form NEWLINE\[NEWLINE(R\varphi)(t)=f(t),\tag{*}NEWLINE\]NEWLINE where, for \( \varphi\in H_{\alpha}\) \((0<\alpha\leq 1) \), the singular integral operator is given by NEWLINE\[NEWLINE(R\varphi)(t)=a(t)\varphi(t)+b(t)(S\varphi)(t)+(\chi \varphi)(t),NEWLINE\]NEWLINE where NEWLINE\[NEWLINE(S\varphi)(t)=\frac{1}{2\pi}\int_{0}^{2\pi}ctg\frac{t-\tau}{2}\varphi(\tau)d\tau,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE(\chi\varphi)(t)=\frac{1}{2\pi}\int_{0}^{2\pi}K(t,\tau)\varphi(\tau)d\tau,NEWLINE\]NEWLINE and \( a(t),b(t)\) and \( K(t,\tau)\) are known \(2\pi\) periodic Hölder continuous functions with the exponent \(\alpha\) \(( 0<\alpha\leq1)\) and \( a^{2}(t)+b^{2}(t)\neq 0\) for all \(t\in [0,2\pi]\). The singular part of the operator \(R\) is approximated by the operators which preserve the main features of a singular operator and enable to get more accurate estimations from the point of view of the convergence rate than the earlier methods. Moreover, this method requires less calculation. In fact, in this work, the operator \( R \) is approximated by the sequence of operators of the form NEWLINE\[NEWLINE(R_{n}\varphi)(t)=\sum_{k=0}^{2n-1}\alpha_{k}^{(n)}(t)\varphi(t+\frac{nk}{n}),NEWLINE\]NEWLINE where \( \alpha_{k}^{(n)} (t) \) are \(2\pi \)-periodic Hölder continuous functions expressed by the given functions, \( k\in \{0,1,3,\dots\}\). It is proved that from the invertibility of the operator \(R\) in the space \( H_{\beta}\), \(0<\beta<\alpha \), the invertibility of the operator \(R_{n}\) (for large \(n\)) follows in this space and the estimation of the approximate solution of (*) in the space \(H_{\beta^{'}}\), \(0<\beta^{'}<\beta\), is given.NEWLINENEWLINEThe main result, the proof of which depends upon a series of seven lemmas, is stated in the following Theorem. Let the functions \( a(t)\), \( b(t)\) be continuously differentiable, the derivatives \(a'(t)\) and \( b'(t)\) belong to the class \( H_{\alpha}\) and \( a^{2}(t)+b^{2}(t)\neq 0 \) for \( t\in [0,2\pi] \). Let the operator \(R\) be invertible in the space \( H_{\beta} \), \( 0<\beta<\alpha \). Then, for large values of \( n \), the operators NEWLINE\[NEWLINE(R_{n}\varphi)(t)=a(t)\varphi(t)+b(t)(S_{n}\varphi)(t)+(K_{n}\varphi)(t)NEWLINE\]NEWLINE are also invertible in the space \(H_{\beta}\) and for \(f\in H_{\beta}\), the following estimation holds: NEWLINE\[NEWLINE\| R_{n}^{-1}f-R^{-1}f\|_{\beta'}\leq \frac{c_{30}+c_{31}}{n^{\beta-\beta'}}\| f \|_{\beta},\,\,0<\beta'<\beta,NEWLINE\]NEWLINE where \(c_{30}\) and \(c_{31}\) are constants.