On the mixed problem for quasilinear partial functional differential equations with unbounded delay

Tomasz Człapiński¹

¹

Annales Polonici Mathematici, Tome 72 (1999) no. 1, pp. 87-98.

Résumé

We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay $D_tz(t,x) = ∑_{i=1}^n f_i(t,x,z_{(t,x)})D_{x_i}z(t,x) + h(t,x,z_{(t,x)})$, where $z_{(t,x)} ∈ X̶_0$ is defined by $z_{(t,x)}(τ,s) = z(t+τ,x+s)$, $(τ,s) ∈ (-∞,0]×[0,r]$, and the phase space $X̶_0$ 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.

Détail

DOI : 10.4064/ap-72-1-87-98

Keywords: Carathéodory solutions, functional differential equation, bicharacteristics, fixed-point theorem, mixed problem, unbounded delay