A Generalization of Bihari's Lemma to the Case of Volterra Operators in Lebesgue Spaces

For operators acting in the Lebesgue space $L_q(\Pi)$, $1<q<\infty$, an abstract analog of Bihari's lemma is stated and proved. We show that it can be used to derive a uniform pointwise estimate of the increment of the solution of a controlled functional-operator equation in the Lebesgue space. The procedure of reducing controlled initial boundary-value problems to this equation is illustrated by the Goursat–Darboux problem.