We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay , where is defined by , , and the phase space satisfies suitable axioms. Using the method of bicharacteristics and the fixed-point method we prove a theorem on the local existence and uniqueness of Carathéodory solutions of the mixed problem.
We consider viscosity solutions for first order differential-functional equations. Uniqueness theorems for initial, mixed, and boundary value problems are presented. Our theorems include some results for generalized ("almost everywhere") solutions.
This is the expanded text of a lecture about viscosity solutions of degenerate elliptic equations delivered at the XVI Congresso UMI. The aim of the paper is to review some fundamental results of the theory as developed in the last twenty years and to point out some of its recent developments and applications.
The paper is a supplement to a survey by J. Franců: Monotone operators, A survey directed to differential equations, Aplikace Matematiky, 35(1990), 257–301. An abstract existence theorem for the equation with a coercive weakly continuous operator is proved. The application to boundary value problems for differential equations is illustrated on two examples. Although this generalization of monotone operator theory is not as general as the M-condition, it is sufficient for many technical applications....