Nonnegative solutions of the characteristic initial value problem for linear partial functional–differential equations of hyperbolic type

On the rectangle D=[a,b]×[c,d]D=[a,b]×[c,d], the problem on the existence and uniqueness of a nonnegative solution of the characteristic initial value problem for the equation ∂2u(t,x)∂t∂x=ℓ(u)(t,x)+q(t,x) is considered, where ℓ:C(D;R)→L(D;R)ℓ:C(D;R)→L(D;R) is a linear bounded operator and q∈L(D;R+)q∈L(D;R+).