Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Khachatryan K. A., Sardaryan T. H. On Solvability of One Class of Urysohn Type Nonlinear Integral Equation on the Whole Line. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 1, pp. 40-50. DOI: 10.18500/1816-9791-2017-17-1-40-50, EDN: YNBYBB

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On Solvability of One Class of Urysohn Type Nonlinear Integral Equation on the Whole Line

Khachatryan Khachatur Agavardovich, Yerevan State University
Sardaryan Tigran Hrachaevich, Armenian National Agrarian University

In present work one class of Urysohn type nonlinear integral equation on whole line is studied. Equations observed have applications in various fields of mathematical physics. It is assumed that Hammerstein type nonlinear integral operator with a difference kernel serves local minorant in terms of M. A. Krasnoselskii for the Urysohn initial operator. Combination of construction methods of invariant cone segments for initial Urysohn nonlinear operator with the methods of monotone operator theory and convolution type conservative integral equations in the case of some restrictions on nonlinearity allows us to prove constructive existence theorems about one parametric positive solutions. A set of parameters is described and the behavior of constructed solutions at infinity is examined. At the еnd of the work specific examples are given for which conditions of formulated theorems are satisfied.

  1. Khachatryan A. Kh., Khachatryan Kh. A. Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases. Theoret. and Math. Phys., 2014, vol. 180, no. 2, pp. 990–1004. DOI: https://doi.org/10.1007/s11232-014-0194-6.
  2. Khachatryan Kh. A. On solvability of some classes of Urysohn nonlinear integral equations with noncompact operators. Ufimsk. Mat. Zh., 2010, vol. 2, no. 2, pp. 103–117.
  3. Khachatryan A. Kh., Khachatryan Kh. A. A nonlinear integral equation of Hammerstein type with a noncompact operator. Sb. Math., 2010, vol. 201, no. 4, pp. 595–606. DOI: https://doi.org/10.1070/SM2010v201n04ABEH004083.
  4. Diekman O. Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol., 1978, vol.6, no. 2, pp. 109–130. DOI: https://doi.org/10.1007/BF02450783.
  5. Vladimirov V. S., Volovich Ya. I. Nonlinear Dynamics equation in p-adic string theory. Theoret. and Math. Phys., 2004, vol. 138, no. 3, pp. 297–309. DOI: https://doi.org/10.1023/B:TAMP.0000018447.02723.29.
  6. Urysohn P. Ob odnom tipe nelineynix integralnix uravnenij [One type of nonlinear integral equations]. Mat. Sb. [Sbornik: Mathematics], 1923, vol. 31, no. 2, pp. 236—255 (in Russian).
  7. Zabreiko P. P., Pustyl’nik E. I. On continuity and complete continuity of nonlinear integral operators in Lp spaces. Uspekhi Mat. Nauk, 1964, vol. 19, iss. 2(116), pp. 204—205 (in Russian).
  8. Bobylev N. A., Ismailov I. G. Iterationnie proceduri v zadachax upravleniyai optimizacii [Itertive control problems in the procedures and optimization]. Pribory i sistemi upravleniya [Devices and control systems], 1997, no. 1, pp. 15–18 (in Russian). 
  9. Krasnoselskii M. A., Zabreiko P. P. Solvability of nonlinear operator equations. Funct. Anal. Appl., 1971, vol. 5, iss. 3, pp. 206–208. DOI: https://doi.org/10.1007/BF01078126.
  10. Khachatryan Kh. A. Sufficient conditions for the solvability of the Uryshon integral equation on a half-axis. Dokl. Math., 2009, vol. 79, iss. 2, pp. 246–249. DOI: https://doi.org/10.1134/S1064562409020264.
  11. Arabadzhyan L. G., Khachatryan A. S. A class of integral equations of convolution type. Sb. Math., 2007, vol. 198, no. 7, pp. 949–966. DOI: https://doi.org/10.1070/SM2007v198n07ABEH003868.
  12. Kolmogorov A. N., Fomin S. V. Elements of the Theory of Functions and Functional Analysis. Moscow, Nauka, 1981. 544 p. (in Russian).
  13. Yengibarian N. B. Renewal equation on the whole line. Stochastic Process. Apll., 2000, vol. 85, iss. 2, pp. 237–247. DOI: https://doi.org/10.1016/S0304-4149(99)00076-9.
  14. Engibaryan N. B. Conservative systems of integral convolution equations on the halfline and the entire line. Sb. Math., 2002, vol. 193, no. 6, pp. 847–867. DOI: https://doi.org/10.1070/SM2002v193n06ABEH000660.
  15. Rudin W. Functional analysis. McGraw-Hill, 1973. 397 p. (Russ. ed.: Rudin W. Funktsional’nyi analiz. Moscow, Mir, 1975. 449 p.)