Influenced by a recent note by M. Ivanov and N. Zlateva, we prove a statement in the style of Nash-Moser-Ekeland theorem for mappings from a Fréchet-Montel space with values in any Fréchet space (not necessarily standard). The mapping under consideration is supposed to be continuous and directionally differentiable (in particular Gateaux differentiable) with the derivative having a right inverse. We also consider an approximation by a graphical derivative and by a linear operator in the spirit of Graves’ theorem. Finally, we derive corollaries of the abstract results in finite dimensions. We obtain, in particular, sufficient conditions for the directional semiregularity of a mapping defined on a (locally) convex compact set in directions from a locally conic set; and also conditions guaranteeing that the nonlinear image of a convex set contains a prescribed ordered interval.

中文翻译：

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Fréchet-Montel空间中的逆映射定理

受M.Ivanov和N.Zlateva最近的注释影响，我们证明了Nash-Moser-Ekeland定理风格的陈述，该陈述用于从Fréchet-Montel空间映射任何Fréchet空间（不一定是标准空间）中的值。所考虑的映射被认为是连续的和方向可微的（特别是Gateaux可微的），导数具有右逆。根据Graves定理的精神，我们还考虑了图形导数和线性算子的近似值。最后，我们得出有限维度中抽象结果的推论。特别地，我们获得了在局部圆锥集的方向上（局部）凸紧集上定义的映射的方向半正则性的充分条件；