Solvability of some nonlinear boundary value problems for singular integral equations of convolution type

Kh. A. Khachatryan

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 81 (2020) no. 1, pp. 3-40

Résumé

This work is devoted to questions about the existence and uniqueness of solutions to certain nonlinear boundary value problems for singular integral equations of convolution type on the whole straight line. It also looks at their asymptotic properties. Several particular cases of this problem have direct applications in $ p$-adic string theory, the mathematical theory of the geographic spread of epidemics, the kinetic theory of gases and radiative transfer theory. For the two classes of boundary value problems described by such equations, the existence of a nontrivial bounded continuous solution is proved, and the asymptotics of the solution that is constructed are investigated. In certain classes of functions which are bounded and continuous on the whole numerical axis, it is shown that no more than one solution exists. The results obtained are extended to certain nonlinear Urysohn-type equations and to Hammerstein-type equations with two nonlinearities. It is also proved that, in certain special cases, solutions to equations having a continuous convex nonlinearity have a series of important properties. Examples of applications of the above equations are given which illustrate the features of the results obtained.