### A generalization of Schauder's theorem and its application to Cauchy-Kovalevskaya problem.

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We consider an abstract version of the Cauchy-Kowalewski Problem with the right hand side being free from the Lipschitz type conditions and prove the existence theorem.

We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations (SE) f(x,u,Dx u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(0,0,{\xi}^{0})\in {\u2102}_{x}^{n}\times {\u2102}_{u}\times {\u2102}_{\xi}^{n}({\xi}^{0}={D}_{x}u\left(0\right))$ and $f(0,0,{\xi}^{0})=0$. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $\left(\xi \in {\u2102}^{n}\right)$. The criterion of convergence of a formal solution $u\left(x\right)={\sum}_{\left|\alpha \right|\ge 1}{u}_{\alpha}{x}^{\alpha}$ of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal...

We study convergence of formal power series along families of formal or analytic vector fields. One of our results says that if a formal power series converges along a family of vector fields, then it also converges along their commutators. Using this theorem and a result of T. Morimoto, we prove analyticity of formal solutions for a class of nonlinear singular PDEs. In the proofs we use results from control theory.

On établit des estimations de l’intégrale singulière de Cauchy et des opérateurs du potentiel dans des échelles d’Ovjannikov de fonctions analytiques. Ces estimations sont utilisées pour obtenir des résultats d’existence locale en temps de solutions analytiques pour certains problèmes à frontière libre dans le plan.

In this paper, we calculate the formal Gevrey index of the formal solution of a class of nonlinear first order totally characteristic type partial differential equations with irregular singularity in the space variable. We also prove that our index is the best possible one in a generic case.

The question how many real analytic affine connections exist locally on a smooth manifold $M$ of dimension $n$ is studied. The families of general affine connections with torsion and with skew-symmetric Ricci tensor, or symmetric Ricci tensor, respectively, are described in terms of the number of arbitrary functions of $n$ variables.

The question how many real analytic equiaffine connections with arbitrary torsion exist locally on a smooth manifold $M$ of dimension $n$ is studied. The families of general equiaffine connections and with skew-symmetric Ricci tensor, or with symmetric Ricci tensor, respectively, are described in terms of the number of arbitrary functions of $n$ variables.