Geodesic
On positive solutions of a boundary value problem for a nonlinear integro-differential equation on a semi-infinite interval
Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 2, pp. 70-82 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The article is devoted to the study of a boundary value problem for a first order nonlinear integro-differential equation on the positive semi axis with a Hammerstein type noncompact integral operator. Such a problem arises in kinetic theory of plasma. In particular, this nonlinear integro-differential equation describes the problem of stationary distribution of electrons in semi infinite plasma in the presence of an external potential electric field. This boundary value problem can be derived from nonlinear Boltzmann model equation, where the role of unknown function plays the first coordinate of an electric field. Depending on a physical parameter, involved in the equation, some constructive existence theorems of one-parametric family of positive solutions in Sobolev's $W_1^1(\mathbb{R}^+)$ space are proved. The asymptotic behavior of the constructed solutions at infinity is also investigated. The proofs of the above statements are based on the construction of a one-parametric family of conic segments, which are invariant with respect to a convolution type nonlinear monotone operator. Further, using some a priori estimates, which are of independent interest, as well as some results from linear theory of conservative homogenous Wiener–Hopf integral equations, the asymptotic properties of obtained results are studied. At the end of the article, some important applications and examples are presented. 
@article{VMJ_2020_22_2_a6,
     author = {Kh. A. Khachatryan and H. S. Petrosyan},
     title = {On positive solutions of a boundary value problem for a nonlinear integro-differential equation on a semi-infinite interval},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {70--82},
     year = {2020},
     volume = {22},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2020_22_2_a6/}
}
TY  - JOUR
AU  - Kh. A. Khachatryan
AU  - H. S. Petrosyan
TI  - On positive solutions of a boundary value problem for a nonlinear integro-differential equation on a semi-infinite interval
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2020
SP  - 70
EP  - 82
VL  - 22
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VMJ_2020_22_2_a6/
LA  - ru
ID  - VMJ_2020_22_2_a6
ER  - 
%0 Journal Article
%A Kh. A. Khachatryan
%A H. S. Petrosyan
%T On positive solutions of a boundary value problem for a nonlinear integro-differential equation on a semi-infinite interval
%J Vladikavkazskij matematičeskij žurnal
%D 2020
%P 70-82
%V 22
%N 2
%U http://geodesic.mathdoc.fr/item/VMJ_2020_22_2_a6/
%G ru
%F VMJ_2020_22_2_a6
Kh. A. Khachatryan; H. S. Petrosyan. On positive solutions of a boundary value problem for a nonlinear integro-differential equation on a semi-infinite interval. Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 2, pp. 70-82. http://geodesic.mathdoc.fr/item/VMJ_2020_22_2_a6/

[1] Lifshitz E. M., Pitaevsky L. P., Statistical Physics, v. 10, Nauka, M., 1979 (in Russian)

[2] Abrikosov A. A., Fundamentals of the Theory of Metals, Elsevier Science Pub., 1988

[3] Khachatryan A. Kh., Khachatryan Kh. A., “On the Solvability of a Boundary Value Problem of Physical Kinetics”, Proceedings of the NAS Armenia: Mathematics, 41:6 (2006), 65–74 (in Russian) | MR

[4] Khachatryan Kh. A., “On Solvability in $W_1^1(\mathbb{R}^+)$ of a Nonlinear Integro-Differential Equation with a Noncompact Hammerstein-Nemytskii Operator”, St. Petersburg Mathematical Journal, 24 (2013), 167–183 | DOI | MR | Zbl

[5] Khachatryan Kh. A., Terdjyan T. E., Petrosyan H. S., “On the Solvability of One Class of Boundary-Value Problems for Non-Linear Integro-Differential Equation in Kinetic Theory of Plazma”, Journal of Siberian Federal University. Mathematics. Physics, 6:4 (2013), 451–461 | MR

[6] Kolmogorov A. N., Fomin S. V., Elements of Theory of Functions and Functional Analysis, Nauka, M., 1980 (in Russian) | MR

[7] Arabadzhyan L. G., Engibaryan N. B., “Convolution Equations and Nonlinear Functional Equations”, Journal of Mathematical Sciences, 36 (1987), 745–791 | DOI | MR | Zbl

[8] Budak B. M., Fomin S. V., Multiple Integrals and Series, Nauka, M., 1965 (in Russian) | MR

[9] Arabadzhyan L. G., Khachatryan A. S., “A Class of Integral Equations of Convolution Type”, Sbornik: Mathematics, 198:7 (2007), 949–966 | DOI | DOI | MR | Zbl