Analytic solutions for nonlinear differential equations (Q1101863)

This is basically an extension of the previous work of the author [J. Differ. Equations 29, 86-104 (1978; Zbl 0416.34063)] to nonlinear equations. The equations under consideration here are of the form \((1)\quad (L+A)f(z)=G(z,f(z))\) where L is the differential operator \[ Lf(z) = z^ n\frac{d^ k}{dz^ k}f(z)\quad +\sum^{k}_{i=1}\phi_ i(z)\frac{d^{k-i}}{dz^{k-1}}f(z),\quad n=integer<k, \] defined in the space \(H_ 2(\Delta)\), the Hilbert space of analytic functions in the open disc \(\Delta =g\{z:| z| <1\}\), A is a bounded operator on \(H_ 2(\Delta)\), and G(z,f(z)) is an analytic function in some neighbourhood of zero. Essentially the object of the present paper is to present a method for the existence of analytic solutions of (1). The method reduces to the study of (1) in a Banach space \(H_ 1(\Delta)\), which is imbedded in \(H_ 2(\Delta)\) and predicts solutions which converge absolutely on the closed unit disc. The nonlinear operator G is Frechet differentiable in an open sphere of \(H_ 1(\Delta)\). The method enables one to use fixed point theory and bifurcation techniques.