Abstract For an integro-differential equation of the convolution type defined on the half-line [0, ∞) with a power nonlinearity and variable coefficient, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space C[0, ∞). It is shown that the solution can be found by a successive approximation method of the Picard type; an estimate for the rate of convergence of the approximations is produced. We present sharp two-sided a-priori estimates for the solutions. These estimates enable us to construct a complete metric space which is invariant under the nonlinear convolution operator considered here and to prove that the equation induced by this operator has a unique solution in this space as well as in the class of all non-negative continuous functions vanishing at the origin. Such equations with operators of fractional calculus as the Riemann-Liouville, Erdélyi-Kober, Hadamard fractional integrals arise, in particular, when describing the process of propagation of shock waves in gas-filled pipes, solving the problem about heating a half-infinite body in a nonlinear heat-transfer process, considering models of population genetics, and elsewhere.