Volterra integral equation with power nonlinearity

With the help of an integral inequality generalizing, in particular, Chebyshev's inequality, we obtain sharp two-sided a priori estimates for the solution of the Volterra integral equation with a power nonlinearity and a general kernel in a cone consisting of all non-negative and continuous functions on the positive half-axis. On the basis of these estimates, a complete metric space is constructed that is invariant with respect to the nonlinear Volterra integral operator generated by this equation, and a global theorem on the existence, uniqueness, and method of finding a solution to the indicated equation is proved by the method of weighted metrics (analogous to the Belitsky method). It is shown that this solution can be found by the method of successive approximations of the Picard type and an estimate is given for the rate of their convergence in terms of the weight metric. It is shown that, in contrast to the linear case, the nonlinear homogeneous Volterra integral equation, in addition to the trivial solution, can also have a nontrivial solution. Conditions are indicated under which the homogeneous equation corresponding to a given nonlinear integral equation has only a trivial solution. At the same time, a refinement and generalization of some results obtained in the case of nonlinear integral equations with difference and sum kernels is given. Examples are given to illustrate the results obtained.