On the local Cauchy problem for first order partial differential functional equations

Abstract

A theorem on the existence of weak solutions of the Cauchy problem for first order functional differential equations defined on the Haar pyramid is proved. The initial problem is transformed into a system of functional integral equations for the unknown function and for its partial derivatives with respect to spatial variables. The method of bicharacteristics and integral inequalities are applied. Differential equations with deviated variables and differential integral equations can be obtained from the general theory by specializing given operators. 1. Introduction. For any metric spaces X and Y we denote by C(X, Y) the class of all continuous functions from X into Y. We will use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components.

Weak solutions of functional differential inequalities with first-order partial derivatives

“…We get from Theorem 2.1 that (21 holds. Inequalities (20), (21), imply (18), which completes the proof of the lemma.…”

“…It is clear that there are functional differential equations which satisfy Assumptions H [F] and they do not satisfy the assumptions of the existence theorem presented in [18]. …”

Global solutions of initial problems for hyperbolic functional differential systems

“…Existence results for functional differential problems and viscosity solutions can be found in [18], [19]. For further bibliography concerning existence results for functional differential equations, see [12], [13], [15].…”

On the quasilinear Cauchy problem for a hyperbolic functional differential equation

“…Existence results for initial problems for semilinear equations can be found in [5]. Sufficient conditions for the existence of classical solutions defined on the Haar pyramid are given in [15,16]. Classical solutions and differentiability with respect to initial data for Volterra type equations are studied in [14].…”

“…The results presented in [1,[3][4][5][6][10][11][12][13][14][15][16]19,20] have the following property: functional differential equations or systems considered in these papers satisfy the Volterra condition. Until now there have not been any results on functional differential equations of the form (1.3), which do not satisfy the Volterra condition.…”