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Holomorphic solutions of a functional equation and their applications to nonlinear second order difference equations. (English) Zbl 1130.39020

The author considers the functional equation \[ \psi(X(x,\psi(x))=Y(x,\psi(x)), (1) \] where \(X(x,y)\) and \(Y(x,y)\) are holomorphic functions of \((x,y)\in \mathbb C\) in a neighborhood of \((0,0)\) and have the following forms: \[ X(x,y)=\lambda x+y+\sum_{i+j\geq 2} c_{ij}x^iy^j \quad \text{or} \quad X(x,y)=\lambda x+\sum_{i+j\geq 2} c_{ij}x^iy^j \] and \[ Y(x,y)=\lambda y+\sum_{i+j\geq 2} d_{ij}x^iy^j. \] He proves existence and uniqueness theorems for holomorphic (in a disc) solutions of equation (1) in both cases, when \(\lambda\neq 0\), \(| \lambda| \neq 1\). The proofs make use of Schauder fixed point theorem. The results are applied for solving the following second order nonlinear difference equation: \[ u(t+2)=f(u(t),u(t+1)) \] where \[ f(x,y)=-\beta x-\alpha y+\sum_{i+j\geq 2} b_{ij}x^iy^j. \]

MSC:

39B32 Functional equations for complex functions
39A10 Additive difference equations
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