Integro-differential equations of the convolution on a finite interval with a kernel having logarithmic singularity

I. V. Andronov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 22, Tome 203 (1992), pp. 5-11

Résumé

Integro-differential equations of the convolution are examined $$ \frac{d^{2n}}{dx^{2n}}\int^1_{-1}\left(a((x-t)^2)\ln|x-t|+b((x-t)^2)\right)\varphi(t)\,dt=f(x). $$ Here functions $a(s)$ and $b(s)$ belong to $C^\infty$ and decrease at infinity. The Fourier transform of the kernel is supposed to be sectorial, i.e. it has a positive projection on some direction in complex plane. The theorem of existence and uniqueness of solutions in spaces defined by the representation $$ \varphi(t)=(1-t^2)^{\delta_n}\psi(t)\qquad\delta_n=n-1+\varepsilon,\quad\varepsilon>0,\quad\psi\in C^1[-1,1], $$ is proved. The proprieties of continuity of solutions are established.