Classical solutions of hyperbolic partial differential equations with implicit mixed derivative

Let f be a continuous function from ${\displaystyle [0,a]\times [0,\beta ]\times \left({ℝ}^n\right)⁴}$ into ${\displaystyle {ℝ}^n}$. Given ${\displaystyle u₀,v₀\in C⁰([0,\beta ],{ℝ}^n)}$, with f(0, x, ∫_0^x u₀(s)ds, ∫_0^x v₀(s)ds, u₀(x), v₀(x)) = v₀(x) for every x ∈ [0,β], consider the problem (P) { ∂²z/(∂t∂x) = f(t, x, z, ∂z/∂t, ∂z/∂x, ∂²z/(∂t∂x)), ${\displaystyle z(t,0)={\vartheta }_{{ℝ}^n}}$, ${\displaystyle z(0,x)={\int }_0^{x}u₀\left(s\right)ds}$, ∂²z(0,x)/(∂t∂x) = v₀(x). In this paper we prove that, under suitable assumptions, problem (P) has at least one classical solution that is local in the first variable and global in the other. As a consequence, we obtain a generalization of a result by P. Hartman and A. Wintner ([4], Theorem 1).