Pointwise Estimation of the Difference of the Solutions of a Controlled Functional Operator Equation in Lebesgue Spaces

For a functional operator equation in Lebesgue space, we prove a statement on the pointwise estimate of the modulus of the increment of its global (on a fixed set $\Pi\subset\mathbb R^n$) solution under the variation of the control function appearing in this equation. As an auxiliary statement, we prove a generalization of Gronwall's lemma to the case of a nonlinear operator acting in Lebesgue space. The approach used here is based on methods from the theory of stability of existence of global solutions to Volterra operator equations.