First Order Partial Functional Differential Equations with State Dependent Delays

Abstract

Initial problems for nonlinear equations are considered. A theorem on the local existence of semiclassical solutions is given. The method of bicharacteristics is used to transform the Cauchy problem into a system of integral functional equations of the Volterra type. Classical solutions of the system lead to semiclassical solutions of the original problem. A method of successive approximations and integral inequalities are used. The uniqueness result of the Perron type is proved by using a comparison technique. Differential equations with deviated variables and differential integral problems can be obtained from a general model by specializing given operators.