Criterion for the Existence of a 1-Lipschitz Selection from the Metric Projection onto the Set of Continuous Selections
- PDF / 194,839 Bytes
- 9 Pages / 594 x 792 pts Page_size
- 79 Downloads / 147 Views
CRITERION FOR THE EXISTENCE OF A 1-LIPSCHITZ SELECTION FROM THE METRIC PROJECTION ONTO THE SET OF CONTINUOUS SELECTIONS FROM A MULTIVALUED MAPPING A. A. Vasil’eva
UDC 515.126.83
Abstract. Let SF be the set of continuous bounded selections from the set-valued mapping F : T → 2H with nonempty convex closed values; here T is a paracompact Hausdorff topological space, and H is a Hilbert space. In this paper, we obtain a criterion for the existence of a 1-Lipschitz selection from the metric projection onto the set SF in C(T, H).
1. Introduction Let T be a paracompact Hausdorff topological space (in some sources (see, for example, [10]), the Hausdorff property is incorporated in the definition of a paracompact space), and let H be a Hilbert space. We denote by C(T, H) the space of continuous bounded functions f : T → H with norm f C(T,H) = sup f (t)H . t∈T
Consider the mapping F : T → 2H such that for any t ∈ T the set F (t) is nonempty, convex, and closed. In addition, we suppose that F is lower semicontinuous; i.e., for each open set V ⊂ H the set {t ∈ T : F (t) ∩ V = ∅} is open. We write SF = {f ∈ C(T, H) : ∀ t ∈ T f (t) ∈ F (t)}; i.e., SF is the set of continuous bounded selections from the set-valued mapping F . By Michael’s theorem [9], there exists a continuous selection from the mapping F . Therefore, if the space T is compact, then the set SF is nonempty. In [11, Theorem 2], the following result was proved: if T is compact, then the family of all sets SF coincides with the family of all nonempty closed spans in C(T, H) (here F is as above). Remark 1. If the mapping F is not lower semicontinuous and the set SF is nonempty, then there exists a lower semicontinuous mapping F˜ : T → 2H such that for any t ∈ T the set F˜ (t) is nonempty, convex, and closed, and SF = SF˜ (see [11, Proposition 1]; in the proof one may replace the compactness of T by the paracompactness). Let (X, · ) be a normed space, and let M ⊂ X, x ∈ X. We denote dist(x, M ) = inf{x − y : y ∈ M },
PM (x) = {z ∈ M : x − z = dist(x, M )}.
The mapping x → PM (x) is called the metric projection onto M . If for any x ∈ X the set PM (x) is nonempty, then we say that M is an existence set (or a proximinal set). If for any x ∈ X the set PM (x) is a singleton, then the set M is called a Chebyshev set. In this case, we denote by pM (x) the nearest to x point from M . Let M ⊂ X be an existence set. We say that the mapping sM : X → M is a selection from the metric projection if sM (x) ∈ PM (x) for any x ∈ X. It was proved by Khavinson [7] that if T is a Hausdorff compact, H = R, f1 , f2 ∈ C(T, R), f1 (t) ≤ f2 (t) for any t ∈ T , F (t) = [f1 (t), f2 (t)], then there exists a 1-Lipschitz (i.e., Lipschitz with constant 1) selection from the metric projection onto SF . In [11], it was shown that if T is a Hausdorff compact and H is Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 99–110, 2018.
454
c 2020 Springer Science+Business Media, LLC 1072–3374/20/2503–0454
a Hilbert space, then SF is proximinal (the compactness
Data Loading...
