Singular integro-differential equations with Hilbert kernel and monotone nonlinearity

S. N. Askhabov

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 3, pp. 11-20

Résumé

In this paper applying methods of trigonometric series we establish that the singular integro-differential operator with the Hilbert kernel $(Gu)(x)=-\frac{1}{2\pi}\int\nolimits_{-\pi}^{\pi} u'(s)\, \mathrm{ctg}\,\frac{s-x}{2}\,ds$ with the domain $D(G)=\{u(x):\, u(x)$ absolutely continuous with $u'(x)\in L_{p'}(-\pi,\pi)$ and $u(-\pi)=u(\pi)=0\}$, where $p'=p/(p-1)$, ${1$, is a strictly positive, symmetric and potential. Using this result and the method of maximal monotone operators, we investigate three different classes of nonlinear singular integro-differential equations with the Hilbert kernel, containing an arbitrary parameter, in the class of $2\pi$-periodic real functions. The solvability and uniqueness theorems, covering also the linear case, are established under transparent restrictions. In contrast to previous papers devoted to other classes of nonlinear singular integro-differential equations with the Cauchy kernel, this one is based on inverting of the superposition operator generating the nonlinearity in the equations considered, and on the proof of the coercivity of this inverse operator. The corollaries are given that illustrate the obtained results.