On a class of convolution type nonlinear integral equations with an even kernel

Abstract

The work is devoted to the study and solution of one class of nonlinear integral equations with monotone and noncompact Hammerstein-type operators on whole line. Equations of this kind arise in many branches of natural science. In particular, such equations are applied in the dynamic theory of [Formula: see text]-adic open–closed strings, in the kinetic theory of gases, in the mathematical theory of the spatial-temporal spread of pandemic. A constructive theorem for existence of non-negative nontrivial and essentially bounded solution is proved. The asymptotic behavior of constructed solution at [Formula: see text] is studied. It is also proved that the difference between the limit in [Formula: see text] and the solution represents summable function on [Formula: see text] Using results obtained a non-negative integrable and bounded solution for new class of integral equations on whole line with monotone and sign-alternating nonlinearity is constructed. Under additional restrictions on nonlinearity, a uniqueness theorem for a solution in a certain class of continuous and bounded functions is also proved. At the end of the work, the specific examples of above equations, satisfying all conditions of the formulated theorem are given.