Frobenius Theorem in Banach Space and Generalized Inverse Analysis Method of Operators Under Small Perturbations

Let $\Lambda$ be an open set in Banach space $E$, $M(x)$ for $x\in \Lambda$ be a subspace in $E$, and $x_0$ be a point in $\Lambda$. We consider the family $\mathcal{F}=\{M(x):\forall x\in\Lambda\}$, but the dimension of $M(x)$ can be infinite, and investigate the necessary and sufficient conditions for $\mathcal{F}$ being $c^1$ integrable at $x_0$. Without new idea and method, it is difficult to generalize the classical Frobenius theorem in Euclid space to the infinite-dimensional $M (x)$ case. We first define the co-tailed set $J (x_0, E_ *)$ of $\mathcal{F}$ at $x_0$ so that for each $x$ in $J (x_0, E_ *)$, $M (x)$ has a unique operator value coordinate $\alpha(x)$ in $B(M (x_0), E_*),$ and prove that if $\mathcal{F}$ is integrable at $x_0$, $J (x_0, E_ *)$ must contain the integrable submanifold of $\mathcal{F}$ at $x_0$. Then, we present the desired necessary and sufficient conditions, which is the Frobenius theorem in the Banach space.It is well known that the classical Frobenius theorem is an important fundamental theorem in the fields of differential topology, differential geometry, differential equations, etc. However, they are all limited to cases where all $\mbox{dim}M(x)< \infty.$ It is now possible to generalize previous studies to the case of $\mbox{dim} M(x)=\infty.$ Using the generalized inverse analysis method of operators under small perturbations, we not only prove Frobenius theorem, but also give some applications to the initial value problem of differential equations with geometric significance, global analysis and the extremum principle under the submanifold constraint in Banach space. In particular, in the field of infinite dimensional geometric and functional analysis, these studies seem to belong to new results and are still in the preliminary stage.