The paper deals with generalized solutions of first order partial functional differential equations. The unknown function is the functional variable in equations, the partial derivatives appear in a classical sense. As it is well known, solutions in the Carathéodory sense to initial or initial-boundary value problems for first order partial functional differential equations are not unique. Two natural subclasses are considered in the literature: generalized entropy solutions and solutions in the Cinquini Cibrario sense.
The paper deals with nonlinear equations \[ D_x z(x,y) =f\bigl(x,y,z_{(x,y)},\;D_y z(x,y)\bigr), \] where \(z_{(x,y)}\) is a function given by \(z_{(x,y)} (t,s)= z(x+t,y+s)\), \((t,s)\in D \subset \mathbb R^{1+n}\). Solutions in the Cinquini Cibrario sense of the Cauchy problem and the mixed problem are considered. The main result for the above equation can be characterized as follows. Suppose that there is a comparison function of the Perron type for \(f\) with respect to the functional variable. Then, under natural assumptions for \(f\) with respect to the last variable, the Cauchy problem or the mixed problem admits at most one solution.
The authors consider also almost linear functional differential systems and generalized solutions in the Friedrichs’ sense. The uniqueness result for the Cauchy problem is presented under the assumption of a nonlinear comparison function for the right hand side of the system. The method of differential inequalities is used. It is important that functional differential comparison problems are considered in the paper. An adequate example is given.