On the structure of the set of solutions of the Darboux problem for hyperbolic equations

Abstract

Consider the Darboux problemwhere φ,ψ:I→Rd (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × Rd→Rd (Q = I × I) satisfies the following hypotheses:(A1) f(.,.,z) is measurable for every z ∈ Rd;(A2) f(x, y,.) is continuous for a.a. (almost all) (x, y) ∈ Q;(A3) there exists an integrable function α:Q →[0, + ∞) such that |f(x, y, z)|≦α(x, y) for every (x, y, z)∈ Q × Rd.