New study on the convergence of a formal transformation. II
The paper begins with a brief review of the results presented in part I [Funkc Ekvacioj, Ser. Int. 38, No. 3, 443-479 (1995; Zbl 0845.34007), see the preceding review]. The paper deals with a system of two nonlinear differential equations of the form \[ x^2 {dy\over dx}= (\mu+ \alpha x) y+ f(x, y, z),\quad x^2 {dz\over dx}= (- v+ \beta x) z+ g(x, y, z), \] where (i) \(x\) is an independent variable; (ii) \(\mu\) and \(v\) are positive numbers and their ratio is irrational; (iii) \(\alpha\) and \(\beta\) are complex constants and there is a positive quantity \(K\) satisfying the inequalities \(\mu+ \kappa {\mathfrak R} \alpha> 0\), \(- v+ \kappa {\mathfrak R} \beta> 0\); (iv) \(f(x, y, z)\) and \(g(x, y, z)\) are holomorphic and bounded functions of \((x, y, z)\) for \(|x|< a\), \(|y|< b\), \(|z|< b\), and their Taylor series expansions in \((y, z)\) contain neither the constant terms nor the linear terms, where \(a\) and \(b\) are small. Formal transformations in that earlier paper of the form \[ y= u+ \sum_{j+ k\geq 2} p_{jk}(x) u^j v^k,\quad z= v+ \sum_{j+ k\geq 2} q_{jk}(x) u^j v^k \] are rearranged, whereby the new transformations satisfy nonlinear differential equations of the form \[ x^2 {dP_0\over dx}= (\mu+ \alpha x) P_0+ f(x, P_0, V(x)+ Q_0),\;x^2 {dQ_0\over dx}= (- v+ \beta x) Q_0+ g(x, P_0, V(x)+ Q_0). \] A two parameter family of bounded solutions is constructed implementing fixed point techniques. The domains of holomorphy for the functions contained within the fixed point techniques are given by a family of the product of two circles over every point in the domain of the independent variable. Again this paper gives a very complete proof presenting several clever inequalities and proofs which are elegant.
two parameter family of bounded solutions
scientific article; zbMATH DE number 847736