Abstract
We study a class of nonlinear integral equations with a noncompact Hammerstein– Nemytskii operator on the entire line. Some special cases of such equations have specific applications in various fields of natural science. The combination of a method for constructing invariant cone segments for the corresponding nonlinear monotone operator with methods of the theory of functions of a real variable allows one to prove a constructive theorem on the existence of bounded positive solutions of equations of the class under consideration. The asymptotic behavior of the solution at \( \pm \infty \) is studied as well. In particular, we prove that the solution constructed in the paper is an integrable function on the negative half-line and that the difference between the limit at \(+\infty \) and the solution is integrable on the positive half-line. In one special case, we show that our solution generates a one-parameter family of bounded positive solutions. At the end of the paper, we give specific applied examples of nonlinearities to illustrate the results.
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Funding
The research by Kh.A. Khachatryan and H.S. Petrosyan was supported by the Russian Science Foundation, project no. 19-11-00223.
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Translated by V. Potapchouck
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Khachatryan, A.K., Khachatryan, K.A. & Petrosyan, H.S. On Positive Bounded Solutions of One Class of Nonlinear Integral Equations with the Hammerstein–Nemytskii Operator. Diff Equat 57, 768–779 (2021). https://doi.org/10.1134/S0012266121060069
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DOI: https://doi.org/10.1134/S0012266121060069