On a perturbation approach to open mapping theorems

Abstract

The open covering property, also known in the literature as metric regularity, is investigated for certain classes of set-valued maps in a Banach space setting. The focus of the present study is on a perturbation approach for deriving open covering criteria, which stems from a theorem due to Milyutin, and which is developed here by means of an abstract notion of first-order approximation for single-valued maps between normed spaces. As a result, a criterion is achieved for parametric set-valued maps in terms of open covering of their strict approximation. Two applications are presented: the first one relates to Robinson-type theorems in the context of quasidifferential analysis, whereas the second concerns the estimation of the distance from the solution set to a nonsmooth problem in parametric convex optimization.

Keywords:

• open covering
• metric regularity
• Banach constant
• Lyusternik–Graves theorem
• Banach lemma
• strict γ-approximation

AMS Subject Classification :

• Primary: 49J53
• Secondary: 46A30
• 47H04
• 49J52
• 90C31
• 90C25

Acknowledgements

The author would like to express his gratitude to an anonymous referee for several useful remarks and an important comment, to Professor Boris S. Mordukhovich for helpful discussions on the subject and to Professor Diethard Pallaschke for his encouragement.

Notes

‘Metric’ in the sense that it is not a merely set-theoretical or topological property, but it requires a metric space setting and includes radius estimations. Clearly, the author looks beyond the reductive view according to which an open mapping theorem serves only to claim that a given map carries open sets to open sets.

In the original formulation of the above result [13, Theorem 1.3], map S is supposed to be b-contractive instead of Lipschitz continuous with a constant b. Nonetheless, consider Remark 1 for a comparison of these assumptions.

More precisely, denoting by the space of all n×n matrices with real entries, endowed with the operator norm, and by the subset of singular matrices, the Eckart–Young theorem says that for every , it holds .

Since the convention is accepted, the above assumption requires that , namely that .

An ordering cone is referred to as being normal if there exists a base of neighbourhood V at 0 such that . The ordering cone of any Banach lattice is normal 6.

Set ∂h(0) to be operationally convex means that for every M, N∈∂h(0) and for every , with , one has .