Ein abstraktes nichtlineares Cauchy-Kowalewskaya-Theorem mit singulären Koeffizienten. I. (An abstract nonlinear Cauchy-Kowalewskaya theorem with singular coefficients. I) (Q1101879)

The paper is devoted to the singular Cauchy-Kowalewskaya problem \[ t\quad 1 dw/dt+cw=t^{1-\sigma} A(z,t,w)+B(z,t,w),\quad w(z,0)=0;\quad (0\leq \sigma <1,\quad 0\leq 1) \] for a Banach scale valued function \(w=w(z,t)\in B_ s\) \((0<s\leq 1\), \(z\in {\mathbb{C}}\), \(t\in {\mathbb{R}})\) with continuous nonlinear operators A, B satisfying certain Lipschitz-type conditions. Existence of a unique solution is proved in the case of weak singularity \(0\leq \ell <1\), analogous result under the condition t 1 dw/dt\(\to 0\) (as \(t\to 0)\) for the strong singularity \(1\leq \ell\) and Re c\(>0\).