On the Darboux problem for linear hyperbolic functional-differential equations

Theorems on the Fredholm alternative and well-posedness of the Darboux problem ∂2u(t, x) ∂t ∂x = `(u)(t, x) + q(t, x), u(t, x0) = φ(t) for t ∈ [a, b], u(t0, x) = ψ(x) for x ∈ [c, d] are established, where ` : C(D;R) → L(D;R) is a linear bounded operator, q ∈ L(D;R), t0 ∈ [a, b], x0 ∈ [c, d], φ : [a, b]→ R, ψ : [c, d]→ R are absolutely continuous functions, and D = [a, b] × [c, d]. New sufficient conditions are also given for the existence and uniqueness of a Carathéodory solution to the problem considered. The general results are applied to a hyperbolic equation with argument deviations and, moreover, for the equation without argument deviations an integral representation of solutions to the Darboux problem is derived in this preprint.