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