We prove a theorem on differential-functional inequalities in the Carathéodory sense. The proof is based on the faèt that we can solve a linear differential equation with first-order partial derivatives. We use also an integral Volterra-type inequality. We obtain a theorem on the uniqueness of generalized entropy solutions to the initial-value problem for non-linear partial differentialfunctional equations of the first order.
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Zdzisław Kamont, H. Leszczyński, Uniqueness Result for the Generalized Entropy Solutions to the Cauchy Problem for First-Order Partial Differential-Functional Equations. Z. Anal. Anwend. 13 (1994), no. 3, pp. 477–491DOI 10.4171/ZAA/500