On the solvability of a class of nonlinear integral equations in the problem of a spread of an epidemic

On the solubility of a class of two-dimensional integral equations on a quarter plane with monotone nonlinearity

“…First of all, we should single out the problems of mathematical physics and mathematical biology. So, very important in practical terms is a special case of the equation when ρ j (u, v) = u + v, j = 1, 2, (u, v) ∈ R + 2 with specific representations of the kernel P and the nonlinearity G. Such equations arise in the dynamical theory of p-adic open-closed strings for the scalar field of tachyons, in the mathematical theory of space-time (geographical) propagation of pandemics, in the kinetic theory of gases, in the theory of radiative transfer in inhomogeneous media [1][2][3][4][5][6][7][8].…”

“…In the particular case ρ j (u, v) = u + v, j = 1, 2, (u, v) ∈ R + 2 , when the functions G and P do not depend on the variables (x 1 , x 2 ), the equation (1) was studied in [8][9][10] under various restrictions on nonlinearity. It should be noted that in the one-dimensional case the corresponding nonlinear integral equation with the difference kernel P(x − y) on the semiaxis, for various representations of the nonlinearity was studied in detail in the papers [11][12][13].…”

On summable solutions of a class of nonlinear integral equations on the whole line

“…Under conditions a)-d),(2) and (25), if ξ ∈ (M 0 , 1), then equation (26) cannot have more than one solution in the following class of measurable and bounded functions on R:M := {f (x) : q ξ (x) ⩽ f (x) ⩽ Γ(x), x ∈ R},(27)whereΓ(x) := ηe σx , x ⩽ 0, η, x > 0.Proof. Assume the opposite: equation (26) has two solutions f and f from the class M. Below we verify that κ := sup x∈R e −σx |f (x) − f (x)| < +∞,…”