On the Solvability of an Initial–Boundary Value Problem for a Functional Hyperbolic Equation With a Distributed Parameter

We study an initial–boundary value problem for a linear functional hyperbolic system with a constant time delay and an unknown distributed parameter depending on the spatial variable. The parameter enters both the differential equation and a nonlocal condition involving the spatial derivative at the initial and final times. By introducing auxiliary functions, the problem is reduced to a family of boundary value problems for first‐order delay differential systems, coupled with integral relations in the spatial variable. The Dzhumabaev parametrization method is applied to construct an algebraic system for a finite set of functional parameters. Sufficient conditions for the unique solvability of the auxiliary family and the convergence of the associated iterative algorithm are formulated in terms of the coefficients and the inverse of the resulting matrix. A successive approximation procedure based on the spatial integral relations is used to establish the unique solvability of the original problem. Solution bounds and estimates for the iterative approximations are obtained. A numerical experiment with a prescribed exact solution illustrates the construction of the solution, recovery of the distributed parameter, and convergence of the proposed algorithms.