An approximate solution of the Fredholm type equation of the second kind for any $\lambda\ne 0$

Alexander Kouleshoff

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2005), pp. 77-93

Résumé

Consider the following equation \begin{equation*} \bigl((I-\lambda K)\varphi\bigr)(s)=\varphi(s)-\lambda\int\limits_a^bK(s,t)\varphi(t)\,dt=f(s). \end{equation*} Assume that the complex-valued kernel $K(s,t)$ is defined on $(a-\varepsilon,b+\varepsilon)\times(a-\varepsilon,b+\varepsilon)$ for some $\varepsilon>0$ and \begin{gather*} \|K\|_2^2=\int\limits_a^b|K(s,t)|^2\,ds\,dt, \\ p(s,t)=\lambda K(s,t)+\overline{\lambda}\,\overline{K(t,s)}-|\lambda|^2\int\limits_a^b\overline{K(\xi,s)}K(\xi,t)\,d\xi. \end{gather*} Consider the following mapping \begin{equation*} f\colon[a,b]\ni\xi\to p(s,\xi)p(\xi,t)\in L_2([a,b]\times[a,b]). \end{equation*} If the function $f$ is integrable according to definition of the Riemann integral (as the function with values in the space $L_2([a,b]\times[a,b])$), then the kernel of the square of the integral operator \begin{equation*} (P\varphi)(s)=\int\limits_a^bp(s,t)\varphi(t)\,dt \end{equation*} can be approximated by a finite dimensional kernel. The formula $(I-P)^+=(I-P^2)^+(I+P)$ and the persistency of the operator $(I-P^2)^+$ with respect to perturbations of a special type are proved. For any $\lambda\neq 0$ we find approximations of the function $\varphi$ which minimizes functional $\|(I-\lambda K)\varphi-f\|_2$ and has the least norm in $L_2[a,b]$ among all functions minimizing the above mentioned functional. Simultaneously we find approximations of the kernel and orthocomplement to the image of the operator $I-\lambda K$ if $\lambda\neq 0$ is a characteristic number. The corresponding approximation errors are obtained.