Integral equation with the Toeplitz-Hankel kernel and inhomogeneity in the linear part

In the cone $Q_0 = \{u(x): u \in C[0, \infty), u(0) = 0$ and $u(x) > 0$ for $x > 0\}$ we consider the integral equation $u^\alpha(x) = \int_0^x [p(x - t) + q(x + t)]u(t)dt + f(x)$ with Toeplitz-Hankel kernel $p(x - t) + q(x + t)$ and inhomogeneity $f(x)$ in the linear part. Equations of this type with difference, total and total-difference kernels arise when solving many problems in hydroaerodynamics, elasticity theory, population genetics, in the theory of radiative equilibrium and heat transfer by radiation and others. In this case, from theoretical and applied points of view, non-negative continuous solutions from the cone $Q_0$ are of particular interest. In the case of $\alpha > 1$, conditions were found for the kernel and inhomogeneity under which the indicated integral equation has a unique solution in the entire class $Q_0$. 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, the 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 $0 < \alpha < 1$, it is shown that this equation, as in the linear case (for $\alpha = 1$), has no solutions in the cone $Q_0$.