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

Extract

Consider the Darboux problem

where φ,ψ:I→R^d (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × R^d→R^d (Q = I × I) satisfies the following hypotheses:

(A₁) f(.,.,z) is measurable for every z ∈ R^d;

(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 × R^d.