On solvability in 𝑊₁¹(ℝ⁺) of a nonlinear integro-differential equation with a noncompact Hammerstein–Nemytskiĭ operator

Khachatryan, Kh.

doi:10.1090/S1061-0022-2012-01235-5

On solvability in $W_1^1(\mathbb {R}^+)$ of a nonlinear integro-differential equation with a noncompact Hammerstein–Nemytskiĭ operator
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by Kh. A. Khachatryan
Translated by: A. Plotkin

St. Petersburg Math. J. 24 (2013), 167-183

DOI: https://doi.org/10.1090/S1061-0022-2012-01235-5

Published electronically: November 15, 2012

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Abstract:

A nonlinear first order integro-differential equation is studied; this equation has important applications to the kinetic theory of metals. Under some conditions on nonlinearity, the existence of a positive solution in the Sobolev class $W_1^1(\mathbb {R}^+)$ is proved.

References

L. D. Landau and E. M. Lifshits, Teoreticheskaya fizika. Tom I: Mekhanika, Izdat. “Nauka”, Moscow, 1973 (Russian). Third edition, corrected and supplemented. MR 0353761
A. V. Latyshev and A. A. Yushkanov, An electron plasma in a metal half-space in an alternating electric field, Zh. Vychisl. Mat. Mat. Fiz. 41 (2001), no. 8, 1229–1241 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 41 (2001), no. 8, 1169–1181. MR 1865012
Kh. A. Khachatryan, Integrodifferential equations of physical kinetics, Izv. Nats. Akad. Nauk Armenii Mat. 39 (2004), no. 3, 72–80 (2005) (Russian, with English and Russian summaries); English transl., J. Contemp. Math. Anal. 39 (2004), no. 3, 49–57 (2005). MR 2169422
A. Kh. Khachatryan and Kh. A. Khachatryan, On the solvability of a boundary value problem of physical kinetics, Izv. Nats. Akad. Nauk Armenii Mat. 41 (2006), no. 6, 65–74 (Russian, with English and Russian summaries); English transl., J. Contemp. Math. Anal. 41 (2006), no. 6, 47–56 (2007). MR 2357133
L. G. Arabadzhyan, An integral equation of transport theory in an inhomogeneous medium, Differentsial′nye Uravneniya 23 (1987), no. 9, 1618–1622, 1653–1654 (Russian). MR 911377
L. G. Arabadzhyan and N. B. Engibaryan, Convolution equations and nonlinear functional equations, Mathematical analysis, Vol. 22, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 175–244, 248 (Russian). MR 780564
A. N. Kolmogorov and S. V. Fomin, Èlementy teorii funktsiĭ i funktsional′nogo analiza, 5th ed., “Nauka”, Moscow, 1981 (Russian). With a supplement “Banach algebras” by V. M. Tikhomirov. MR 630899
A. Kh. Khachatryan and Kh. A. Khachatryan, On solvability of one class of Hammerstein nonlinear integral equations, Bul. Acad. Ştiinţe Repub. Mold. Mat. 2 (2010), 67–83. MR 2743053
M. A. Krasnosel′skiĭ, P. P. Zabreĭko, E. I. Pustyl′nik, and P. E. Sobolevskiĭ, Integral′nye operatory v prostranstvakh summiruemykh funktsiĭ, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0206751