Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


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Khachatryan K. A. The Solvability of a System of Nonlinear Integral Equations of Hammerstein Type on the Whole Line. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 2, pp. 164-181. DOI: 10.18500/1816-9791-2019-19-2-164-181, EDN: ZTOYKT

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
28.05.2019
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Russian
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Article
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517.968.4
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ZTOYKT

The Solvability of a System of Nonlinear Integral Equations of Hammerstein Type on the Whole Line

Autors: 
Khachatryan Khachatur Agavardovich, Yerevan State University
Abstract: 

In recent years, the interest has grown in nonlinear integral equations of convolution type in connection with their application in various fields of mathematical physics, inparticular, inthep-adic theory of an open-closed string, kinetic theory of gases, in the theory of radiation transfer in spectral lines. The paper is devoted to the questions of construction of nontrivial solutions and the study of their asymptotic behavior for one system of nonlinear integral equations of convolution type with a symmetric kernel on the whole axis. The results of the work are based on the combination of methods of invariant conical segments construction for the corresponding nonlinear monotone operator with methods of the theory of linear operators of convolution type. A constructive theore montheexistence oftwoasy mptotically different one-parameter familie sofpositive and bounded solutions was formulated and proved, which is the main difference from the previously obtained results. Moreover, from the structure of this system of nonline are quations follows that all possible shift softhe constructed solutionsal so satisfy the system. Special attention is paid to the study of the a symptoticbehavior of these solutions at the ends of the line. The limits of these solutions in ±∞ are calculated and it is proved that the constructed solutions belong to the L 1 (0,+∞) and L 1 (−∞,0) spaces respectively.

References: 
  1. 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/10.4213/tmf36
  2. Vladimirov V. S. The equation of the p-adic open string for the scalar tachyon field. Izv. Math., 2005, vol. 69, no. 3, pp. 487–512. DOI: https://doi.org/10.4213/im640
  3. Vladimirov V. S. Solutions of p-adic string equations. Theoret. and Math. Phys., 2011, vol. 167, no. 2, pp. 539–546. DOI: https://doi.org/10.4213/tmf6631
  4. Joukovskaya L. V. Iterative method for solving nonlinear integral equations describing rolling solutions in string theory. Theoret. and Math. Phys., 2006, vol. 146, no. 3, pp. 335–342. DOI: https://doi.org/10.4213/tmf2043
  5. Khachatryan Kh. A. On the solvability of certain classes of non-linear integral equations in p-adic string theory. Izv. Math., 2018, vol. 82, no. 2, pp. 407–427. DOI: https://doi.org/10.4213/im8580
  6. Engibaryan N. B. On a problem of nonlinear radiation transfer. Astrofizika, 1966, vol. 2, no. 1, pp. 31—36. DOI: https://doi.org/10.1007/BF01014505
  7. Khachatryan A. Kh., Khachatryan Kh. A. Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave. Theoret. and Math. Phys., 2016, vol. 189, no. 2, pp. 1609–1623. DOI: https://doi.org/10.1134/S0040577916110064
  8. Feller W. Introduction to Probability Theory and Its Applications. John Wiley and Sons, Inc., 1971. 669 p. (Russ. ed.: in 2 vols. Vol. 2. Moscow, Mir, 1984. 752 p.).
  9. Kendall D. G. Mathematical models of the spread of infection. Mathematics and Computer Science in Biology and Medicine, London, H. M. S. O., 1965, pp. 213–225.
  10. Diekmann O. Thresholds and Traveling Waves for the Geographical Spread of infection. Journal of Math. Biology, 1978, vol. 6, pp. 109–130. DOI: https://doi.org/10.1007/BF02450783
  11. Diеkmann O. Limiting behaviour in an epidemic model. Nonlinear Analysis: Theory, Methods & Applications, 1977, vol. 1, no. 5, pp. 459–470. DOI: https://doi.org/10.1016/0362-546X(77)90011-6
  12. Diekmann O. Run for your life. A note on the asymptotic speed of propagation of an epidemic. Journal of Differential Equations, 1979, vol. 33, iss. 1, pp. 58–73. DOI: https://doi.org/10.1016/0022-0396(79)90080-9
  13. 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
  14. Arabadzhyan L. G., Engibaryan N. B. Convolution equations and nonlinear functional equations. J. Soviet Math., 1987, vol. 36, iss. 6, pp. 745–791. DOI: https://doi.org/10.1007/BF01085507
  15. Sgibnev M. S. The matrix analogue of the Bleckwell renewal theorem on the real line. Sb. Math., 2006, vol. 197, no. 3, pp. 369–386. DOI: https://doi.org/10.1070/SM2006v197n03ABEH003762
  16. Khachatryan Kh. A. Positive solvability of some classes of non-linear integral equations of Hammerstein type on the semi-axis and on the whole line. Izv. Math., 2015, vol. 79, iss. 2, pp. 411–430. DOI: https://doi.org/10.1070/IM2015v079n02ABEH002748
  17. Lancaster P. Theory of Matrices. New York, Academic Press, 1969. 316 p. (Russ. ed.: Moscow, Nauka, 1982. 28 p.).
  18. Fikhtengol’ts G. M. The Fundamentals of Mathematical Analysis : International Series of Monographs in Pure and Applied Mathematics, vol. 2 (73). Pergamon Press, 1965. 540 p. (Russ. ed.: in 3 vols. Vol. 2. Moscow, Fizmatlit, 1966. 600 p.).
  19. Khachatryan Kh. A., Terjyan Ts. E., Avetisyan M. H. A one-parameter family of bounded solutions for a system of nonlinear integral equations on the whole line. Proceedings of the NAS Armenia: Mathematics, 2018, vol. 53, iss. 4, pp. 201–211 (in Russian).
  20. Kolmogorov A. N., Fomin S. V. Elements of the theory of functions and functional analysis: vol. I, II. Albany, New York, Graylock Press, 1957. 129 p.; 1961. 128 p. (Russ. ed.: Moscow, 1981. 544 p.).
Received: 
29.10.2018
Accepted: 
26.03.2019
Published: 
28.05.2019
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