Cite this article as:
Khachatryan K. A. The Solvability of a System of Nonlinear Integral Equations of Hammerstein Type on the Whole Line. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2019, vol. 19, iss. 2, pp. 164-181. DOI: https://doi.org/10.18500/1816-9791-2019-19-2-164-181
The Solvability of a System of Nonlinear Integral Equations of Hammerstein Type on the Whole Line
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.
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