A convolution type nonlinear integro-differential equation with a variable coefficient and an inhomogeneity in the linear part

We study a Volterra integro-differential equation of convolution type with a power nonlinearity, variable coefficient $a(x)$ and an inhomogeneity $f(x)$ in the linear part, which is closely related to the corresponding nonlinear integral equation, arising in the study of fluid infiltration from a cylindrical reservoir\eject into an isotropic homogeneous porous medium, when describing the process of propagation of shock waves in gas-filled pipes, when solving the problem about heating a half-infinite body in a nonlinear heat-transfer process, in models of population genetics, and others. It is important to note that in relation to the above-mentioned and other applications, of special interest are continuous positive (for $x > 0$) solutions of the integral equation. Based on the obtained exact lower and upper a priori estimates for the solution of the integral equation, we construct a weighted complete metric space $P_b$, invariant with respect to the nonlinear integral convolution operator generated by this equation, and, using the method of weighted metrics (an analogue of Belitsky's method), we prove the global existence theorem and the uniqueness of the solution of the nonlinear integro-differential equation under study both in the space $P_b$ and in the whole class $Q_0^1$ of continuously differentiable functions positive for $x>0$. It is shown that the solution can be found in the $P_b$ space by a successive approximation method of the Picard type. Estimates for the rate of convergence of the successive approximations to the exact solution in terms of the weight metric of the space $P_b$ are derived. In particular, for $f(x)=0$, this theorem implies that the corresponding homogeneous nonlinear integro-differential equation, in contrast to the linear case, has a nontrivial solution. Examples are also given to illustrate the results obtained.