Hyperbolic problems in three independent variables: Mathematical analysis and constructive methods

We consider initial-boundary value problems associated with nonlinear hyperbolic partial differential equations in three independent variables in a unified setting, wherein the forcing function is a sum of two monotonic functions. After deriving the required mathematical theory of hyperbolic partial differential inequalities, we employ natural lower–upper solutions and two types of coupled lower–upper solutions, to derive linear iterative schemes which converge uniformly and monotonically to the minimal–maximal solutions or the coupled minimal–maximal solutions of the nonlinear problem. Examples are given to illustrate the results.