Holomorphic solutions of a functional equation related to nonlinear difference systems (Q708722)

The author considers the functional equation \[ \Psi\big(X(x,\Psi(x)\big)=Y\big(x,\Psi(x)\big), \leqno(1) \] where \(X(x,y)\) and \(Y(x,y)\) are holomorphic functions of \((x,y)\in \mathbb {C}^2\) in a neighborhood of \((0,0)\) and have the following forms: \[ X(x,y)=x+y+\sum_{i+j\geq 2} c_{ij}x^iy^j; \quad Y(x,y)=y+\sum_{i+j\geq 2} d_{ij}x^iy^j, \] and shows the connections of equation (1) with the system of nonlinear difference equations \[ \left\{ \begin{aligned} x(t+1)&=X(x(y),y(t)),\\ y(t+1)&=Y(x(y),y(t)). \end{aligned} \right. \] Under quite intricate conditions on the coefficients \(c_{00}, c_{20}, d_{11}, d_{30}\), he proves an existence theorem for holomorphic (in a disc) solutions of equation (1). This paper is a continuation of former investigations on equation (1) under different conditions on \(X(x,y)\) and \(Y(x,y)\).