On nonlinear Volterra integral equations with state dependent delays in several variables (Q847019)

Consider \(G= [0,a]\subset\mathbb{R}^n\), \(a\in\mathbb{R}^n_+\), \(\Omega= [-r,a]\subset\mathbb{R}^n\), \(r\in\mathbb{R}^n_+\), \(B=\Omega\setminus G\subset \mathbb{R}^n\), \(E\) a Banach space, \(u:\Omega\to E\), a function defined by \(u_x(\tau)= u(x+\tau)\) for \(x\in G\) and \(\tau\in B\), \(F\in C(G\times E^m\times C(B,E),E)\) for \(m\in N\), \(f\in C(G^2\times C(B,E), E^m)\), \(\theta\in C(G\times C(B,E),G)\), \(\psi\in C(G\times C(B, E), G^m)\), \(\beta\in C(G,G)\), \(\alpha\in C(G,G^m)\), \(\varphi\in C(B,E)\) and \(H= (H_1,\dots, H_m): G\to \mathbb{R}^m\), where for \(1\leq i\leq m\) and \(x\in G\), \(H_i(x)\) is a subset of a \(p_i\)-dimensional hyperplane parallel to the coordinate axes, being Lebesgue measurable for \(1\leq p_i\leq n\). The authors consider the Cauchy problem \[ \begin{cases} u(x)= F(x, \int_{H(x)} f(x,s,u_{\psi(s,u_\alpha(s))})\,ds, u_{\theta(x, u_\beta(x))}),\quad & x\in G,\\ u(x)= \varphi(x),\quad x\in B,\end{cases}\tag{1} \] and prove that the Cauchy problem (1) has a unique solution in a class defined in Lemmas 2.2 and 2.3, under hypotheses given in these lemmas.