A generalization of the inverse mapping theorem in infinite dimensions

Abstract: We present a generalization of the inverse mapping theorem, where variations of a weaker non-expansiveness property (referred to as property ${\sf A}$) replace the key $\mathsf{C}^1$ condition. We also obtain inverse mapping theorems that can be applied to non-smooth maps. Also as a by-product of the generalized inverse mapping theorem, we prove generalizations of the implicit function theorem and existence and uniqueness theorem of abstract PDE systems as well.

• The paper introduces property A to relax classical differentiability assumptions by leveraging fixed point theory in infinite-dimensional Banach spaces.
• It employs auxiliary maps that interpolate between contraction and non-expansiveness to establish local invertibility of nonlinear maps.
• The generalized results extend to implicit function theorem applications and abstract differential equations, highlighting practical analysis benefits.

Generalization of the Inverse Mapping Theorem in Infinite Dimensions

Introduction and Motivation

The classical inverse mapping theorem (IMT) is foundational in nonlinear analysis and differential geometry, guaranteeing local invertibility and smoothness of $C^1$ maps between finite- or infinite-dimensional Banach spaces under the assumption of everywhere invertible derivatives and continuous differentiability. However, optimal regularity conditions in the infinite-dimensional setting remain subtle, especially because local compactness—crucial in finite dimensions—fails for most Banach spaces of interest. Traditionally, generalizations of IMT proceed along three principal lines: employing stronger generalized derivatives (e.g., Clarke derivatives), strengthening non-degeneracy conditions for weaker notions of differentiability, or imposing contraction-type hypotheses amenable to fixed-point arguments. This paper by Sajjad Lakzian (2604.10002) develops the third approach, leveraging non-expansiveness properties considerably weaker than strict contraction and harnessing weak local compactness of Banach spaces to extend the inverse mapping theorem well beyond the classical $C^1$ setting.

Property $\mathsf{A}$ and the Fixed Point Approach

The principal innovation is the introduction of property $\sf A$, which encodes a spectrum of non-expansive-like conditions that interpolate between strong contraction and general non-expansivity. Specifically, for a map $f$ between Banach spaces, property $\sf A$ at a point $a$ stipulates, via a family of auxiliary maps constructed from local approximations (in analogy with Newton's method), that these auxiliary maps are contractive, non-expansive, or quasi-nonexpansive self-maps on suitable convex neighborhoods. Notably, this property need not require continuity or even full differentiability of $f$, and crucially, it decouples the regularity from differentiability assumptions. The framework supports both strong and weak variants, with the strong form corresponding to a genuine contraction and the weakest form allowing merely quasi-nonexpansiveness (existence, but not necessarily uniqueness, of fixed points with nonincreasing displacement from these points).

The mapping theorem is proved via fixed-point theory: in locally weakly compact Banach spaces, non-expansive maps on closed bounded convex subsets admit fixed points by the Schauder-Tychonoff theorem, while uniformly convex (or, more generally, $k$-uniformly rotund) Banach spaces guarantee even stronger fixed-point properties. By associating the solvability of the local equation $f(x) = y$ to the existence of fixed points for these auxiliary maps, openness and local surjectivity are established. Local injectivity is a consequence of the discreteness of preimages, which follows from the invertibility of the derivative and the convexity structure of the fixed point sets.

Main Theorems and Corollaries

The central theorem establishes that for an everywhere differentiable map $C^1$0 with invertible derivative, $C^1$1 is a local diffeomorphism under any one of several sets of hypotheses relating to property $C^1$2, weak topology, local weak compactness, and the fixed point property (FPP):

1. Strong property $C^1$3 on all scales: Classical Banach contraction mapping principles are invoked.
2. Weak-weak continuity, property $C^1$4, locally weak compactness: Schauder-type fixed-point theorems ensure surjectivity.
3. Locally weakly compact Banach space and FPP, property $C^1$5: Convexity and normal structure (notably abundant in reflexive or strictly convex spaces) yield the required fixed points.
4. Strict convexity with FPP, weak property $C^1$6: Further relaxation is possible, even to quasi-nonexpansive scenarios, leveraging geometric structure.

Two corollaries extend these local results globally by appealing to topological criteria: properness or compactness of the domain guarantees that $C^1$7 is a finite-sheeted covering map, and the classical Hadamard-Levy criterion for global invertibility is also encompassed within this generalized setting (if the integral of the inverse norm of derivatives diverges, $C^1$8 is globally invertible).

A further generalization addresses cases where $C^1$9 is only locally injective on a dense set or is not everywhere differentiable. By introducing segmentally dense sets and notions of weak directional non-degeneracy, the results are extended to guarantee "almost diffeomorphism"—local homeomorphism and differentiability on dense sets—in cases substantially below $\mathsf{A}$0 regularity.

Generalized Implicit Function Theorem

A key corollary is a generalized implicit function theorem: for a function $\mathsf{A}$1 differentiable with respect to $\mathsf{A}$2 and satisfying property $\mathsf{A}$3 in $\mathsf{A}$4, the level sets $\mathsf{A}$5 locally admit unique differentiable parametrizations as graphs over $\mathsf{A}$6. This significantly broadens the scope of the classical implicit function theorem by relaxing the differentiability requirement and enabling application to non-smooth, non-Lipschitz scenarios, provided the fixed-point property is satisfied.

Applications to Abstract Differential Equations

The generalization is applied to abstract ODEs/PDEs in Banach spaces. Specifically, if $\mathsf{A}$7 is as above and the associated $\mathsf{A}$8 (defined via inverses of partial derivatives of $\mathsf{A}$9) may fail to satisfy Lipschitz or even continuity conditions, the Cauchy problem

$\sf A$0

admits unique local solutions given implicitly by the level sets $\sf A$1. This provides existence and uniqueness results in regimes far outside classical Picard-Lindelöf theory.

Theoretical and Practical Implications

This generalization has several theoretical ramifications. It underscores that, in infinite dimensions, local invertibility can be established under non-expansiveness conditions without relying on the continuity of the derivative, provided sufficient convexity and compactness properties of the Banach space are available. This significantly widens the class of nonlinear maps amenable to local and global analysis. The connection between invertibility and geometric properties of Banach spaces (uniform rotundity, strict convexity, FPP) is brought to the forefront.

Practically, the results support the analysis of partial differential operators, control flows in infinite-dimensional systems, and other settings where smoothness is limited but underlying non-expansive behaviors or weak continuity can be exploited. The techniques facilitate treatment of equations with right-hand sides lacking standard regularity, relevant, for example, in optimization, functional analysis, or abstract evolution equations on Banach manifolds.

Prospects and Future Developments

Potential future directions include refining property $\sf A$2 to encompass broader classes of non-smooth maps (e.g., merely Lipschitz or measurable mappings), investigating analogous results in non-locally convex or even non-linear spaces, and extending the framework to settings with partial or set-valued inverses. Another avenue is to apply these techniques to concrete classes of maps arising in applications (e.g., non-smooth PDEs, critical point theory, or infinite-dimensional dynamical systems) and potentially to quantum or stochastic analysis, where differentiability is rare and non-expansive dynamics are ubiquitous.

Conclusion

This paper generalizes the inverse mapping theorem to infinite-dimensional Banach spaces by substituting non-expansiveness conditions for classical $\sf A$3 assumptions and leveraging fixed-point theorems intrinsic to the geometry of the spaces involved. The framework encompasses non-smooth, non-continuous, and even non-Lipschitz maps, with significant implications for both nonlinear analysis and applications. The synthesis of geometric, topological, and analytic perspectives offers a flexible foundation for ongoing advances in infinite-dimensional analysis and its applications (2604.10002).