St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

   
 

 

On solvability in $ W_1^1(\mathbb{R}^+)$ of a nonlinear integro-differential equation with a noncompact Hammerstein-Nemytskiĭ operator


Author: Kh. A. Khachatryan
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 24 (2012), nomer 1.
Journal: St. Petersburg Math. J. 24 (2013), 167-183
MSC (2010): Primary 45J05
Published electronically: November 15, 2012
MathSciNet review: 3013298
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. L. D. \cyr{L}andau and E. M. \cyr{L}ifshits, Teoreticheskaya fizika. Tom I: Mekhanika, Izdat. “Nauka”, Moscow, 1973 (Russian). Third edition, corrected and supplemented. MR 0353761
  • 2. 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
  • 3. 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
  • 4. 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
  • 5. 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
  • 6. 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
  • 7. A. N. Kolmogorov and S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, 5th ed., “Nauka”, Moscow, 1981 (Russian). With a supplement “Banach algebras” by V. M. Tikhomirov. MR 630899
    A. Kolmogorov, S. Fomine, and V. M. Tihomirov, Eléments de la théorie des fonctions et de l’analyse fonctionnelle, Éditions Mir, Moscow, 1974 (French). Avec un complément sur les algèbres de Banach, par V. M. Tikhomirov; Traduit du russe par Michel Dragnev. MR 0367598
  • 8. 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
  • 9. M. A. Krasnosel′skiĭ, P. P. Zabreĭko, E. I. Pustyl′nik, and P. E. Sobolevskiĭ, Integralnye operatory v prostranstvakh summiruemykh funktsii, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0206751
    M. A. Krasnosel′skiĭ, P. P. Zabreĭko, E. I. Pustyl′nik, and P. E. Sobolevskiĭ, Integral operators in spaces of summable functions, Noordhoff International Publishing, Leiden, 1976. Translated from the Russian by T. Ando; Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis. MR 0385645

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 45J05

Retrieve articles in all journals with MSC (2010): 45J05


Additional Information

Kh. A. Khachatryan
Affiliation: Institute of Mathematics, Armenian National Academy, Armenia
Email: Khach82@rambler.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2012-01235-5
Keywords: Sobolev space, Carathéodory condition, convergence of iterations, Hamerstein type equation
Received by editor(s): January 14, 2011
Published electronically: November 15, 2012
Article copyright: © Copyright 2012 American Mathematical Society