Geodesic
Partial integral Fredholm equation in anisotropic classes of Lebesgue functions on $\mathbb{R}_2$
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings – XXXI". Voronezh, May 3-9, 2020, Tome 204 (2022), pp. 53-65

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In this paper, we propose a formula for representing the solution of a partial integral Fredholm equation of the second kind in the form of the corresponding Neumann series. We obtain conditions for the existence and uniqueness of this solution in the classes of Lebesgue functions $L_{\boldsymbol{p}}$, $\boldsymbol{p}=(p_1,p_2)$, defined in a finite rectangle $D=(a_1,b_1 )\times(a_2,b_2)$ of the Euclidean space $\mathbb{R}_2$.
Keywords: partial integral, Fredholm equation, resolvent, Neumann series, resonance theorem.
Mots-clés : anisotropic space 
@article{INTO_2022_204_a5,
     author = {L. N. Lyakhov and A. I. Inozemtsev},
     title = {Partial integral {Fredholm} equation in anisotropic classes of {Lebesgue} functions on $\mathbb{R}_2$},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {53--65},
     publisher = {mathdoc},
     volume = {204},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_204_a5/}
}
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L. N. Lyakhov; A. I. Inozemtsev. Partial integral Fredholm equation in anisotropic classes of Lebesgue functions on $\mathbb{R}_2$. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings – XXXI". Voronezh, May 3-9, 2020, Tome 204 (2022), pp. 53-65. http://geodesic.mathdoc.fr/item/INTO_2022_204_a5/