Integral Equation with the Toeplitz–Hankel Kernel and an Inhomogeneity in the Linear Part

In the cone Q0 = {u(x): u ∈ C[0, ∞), u(0) = 0 and u(x) > 0 at x > 0}, we consider the integral equation $${{u}^{\alpha }}(t)$$ = $$\int_0^x {[p(x - t)} $$ + $$q(x + t)]u(t)dt$$ + $$f(x)$$ with the Toeplitz–Hankel kernel p(x – t) + q(x + t) and the inhomogeneity f(x) in its linear part. Equations of this type with difference, total, and total-difference kernels arise when solving many problems in hydroaerodynamics, the theory of elasticity, population genetics, in the theory of radiative equilibrium and heat transfer by radiation, and others. At that, from the theoretical and applied points of view, non-negative continuous solutions from the cone Q0 are of particular interest. In the case of α > 1, conditions were found for the kernel and inhomogeneity under which the indicated integral equation has a unique solution in the entire class Q0. Without additional restrictions on the given functions, it is proved that this solution can be found by the method of successive approximations of the Picard type in some complete weighted metric space. For successive approximations, an estimate of the rate of their convergence to the exact solution is established in terms of the weight metric. In this case, two-sided a priori estimates of the solution obtained in the work play an important role. Examples are given to illustrate the results obtained. For 0 < α < 1, it is shown that this equation, as in the linear case (for α = 1) has no solutions in the cone Q0.