We study the initial value problem for a class of coupled Euler–Poisson–Darboux–Tricomi equations with nonlinear source terms. Such systems arise in fluid dynamics, plasma physics, transonic flow, and wave propagation in inhomogeneous media. Using new integral representations of the solutions and appropriate fixed point theorems in Banach spaces, we establish the existence of at least one, at least two, and at least three nonnegative classical solutions under suitable conditions on the parameters and initial data. An illustrative example is provided to demonstrate the applicability of the main results.