Geodesic
On a class of nonlinear integral equations of the Hammerstein--Volterra type on a semiaxis
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2023), pp. 75-86

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In this note, we study a class of nonlinear integral equations with a monotone Hammerstein-Volterra type operator in the critical case. This class of equations occurs in the kinetic theory of gases in the framework of the study of the nonlinear kinetic integro-differential model Boltzmann equation. The combination of methods for constructing invariant cone segments for a nonlinear monotone operator with the methods of the theory of functions of a real variable makes it possible, with the help of specially chosen successive approximations, to construct a positive summable and bounded solution on a non-negative semiaxis for the above class of equations. With an additional constraint on nonlinearity, it is also possible to prove the uniqueness of the solution in a certain class of positive and summable functions on the non-negative semiaxis. At the end, illustrative examples of nonlinearity and the kernel are given, which are of both theoretical and applied interest.
Mots-clés : kernel, convergence
Keywords: non-linearity, monotonicity, estimates, Caratheodory condition. 
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     author = {Kh. A. Khachatryan and H. S. Petrosyan},
     title = {On a class of nonlinear integral equations of the {Hammerstein--Volterra} type on a semiaxis},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {75--86},
     publisher = {mathdoc},
     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2023_1_a4/}
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Kh. A. Khachatryan; H. S. Petrosyan. On a class of nonlinear integral equations of the Hammerstein--Volterra type on a semiaxis. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2023), pp. 75-86. http://geodesic.mathdoc.fr/item/IVM_2023_1_a4/