This paper develops a unified framework for analyzing transversality properties of set collections in nonlinear spaces, focusing on primal conditions and extending existing results to a broader nonlinear context.
Contribution
It introduces a general framework for quantitative analysis of transversality properties, unifying and extending previous linear and Holder case results in nonlinear settings.
Findings
01
Provides primal (metric and slope) characterizations of transversality.
02
Establishes quantitative relations between transversality and regularity properties.
03
Unifies existing results and extends them to a general nonlinear framework.
Abstract
The paper studies 'good arrangements' (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Holder case is given a special attention. Our main objective is not formally extending our earlier results from the Holder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and…
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11institutetext:
Nguyen Duy Cuong
Centre for Informatics and Applied Optimization, School of Science, Engineering and Information Technology, Federation University Australia, POB 663, Ballarat, Vic, 3350, Australia
Department of Mathematics, College of Natural Sciences, Can Tho University, Can Tho City, Vietnam
Centre for Informatics and Applied Optimization, School of Science, Engineering and Information Technology, Federation University Australia, POB 663, Ballarat, Vic, 3350, Australia
††thanks: The research was supported by the Australian Research Council, project DP160100854.
The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.
Nguyen Duy Cuong
Alexander Y. Kruger
(Received: date / Accepted: date)
Abstract
The paper studies ‘good arrangements’ (transversality properties)
of collections of sets
in a normed vector space
near a given point in their intersection.
We
target
primal (metric and slope) characterizations of transversality properties
in the nonlinear setting.
The Hölder case is given a special attention.
Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties.
The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic.
Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case.
Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as
nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed.
This paper continues a series of publications by the authors BuiCuoKru ; CuoKru4 ; CuoKru5 ; CuoKru6 ; ThaBuiCuoVer20 ; Kru09 ; Kru06 ; Kru18 ; KruTha13 ; Kru05 ; KruTha14 ; KruTha15 ; KruTha16 ; KruLukTha17 ; KruLukTha18 ; KruLop12.1 ; KruLop12.2 dedicated to studying ‘good arrangements’ of collections of sets in normed spaces near a point in their intersection.
Following Ioffe Iof17 , such arrangements are now commonly referred to as transversality properties.
Here we refer to transversality broadly as a group of ‘good arrangement’ properties, which includes semitransversality, subtransversality, transversality (a specific property) and some others.
The term regularity was extensively used for the same purpose in the earlier publications by the second author, and is still preferred by many authors.
Transversality (regularity) properties of collections of sets play
an important role in optimization and variational analysis, e.g., as constraint qualifications, qualification conditions in subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms.
Significant efforts have been invested into studying this class of properties and establishing their primal and dual necessary and/or sufficient characterizations in various settings (convex and nonconvex, finite and infinite dimensional, finite and infinite collections of sets).
In addition to the references provided above, we refer the readers to BauBor96 ; Iof00 ; LewLukMal09 ; NgaThe01 ; NgZan07 ; ZheNg08 ; BorLiTam17 ; BorLiYao14 ; DruLiWol17 ; ZheWeiYao10 ; HesLuk13 ; DruIofLew15 ; NolRon16 ; BakDeuLi05 ; BauBorLi99 ; Pen13 for results and historical comments.
Our aim is to develop a general framework for quantitative analysis of transversality properties of collections of sets.
In this paper, we focus on primal space conditions, and
establish metric characterizations and slope-type sufficient conditions for three closely related general nonlinear transversality properties: φ−semitransversality, φ−subtransversality and φ−transversality.
The slope sufficient conditions stem from applying the Ekeland variational principle to the definitions of the respective properties; the proofs are rather straightforward.
This type of conditions are often considered as just a first step on the way to producing more involved dual (subdifferential and normal cone) conditions, and the primal sufficient conditions remain hidden in the proofs.
We believe that primal conditions
(being in a sense analogues of very popular slope conditions for error bounds) can be of importance for
applications.
Moreover, subdividing the conventional regularity/transversality theory into the primal and dual parts clarifies the roles of the main tools employed within the theory: the Ekeland variational principle in the primal part and subdifferential sum rules in the dual part.
As a result, the proofs in both the primal and the ‘more involved’ dual parts become straightforward.
This observation goes beyond the transversality of collections of sets and applies also to the regularity of set-valued mappings and the error bound theory.
Unlike the earlier publications, here, besides estimates for the transversality moduli, we provide also quantitative estimates for the parameters δ’s involved in the definitions; cf. Definitions 1 and 2.
This can be of importance from the computational point of view.
We also examine quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding nonlinear regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe.
We would like to emphasize that our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a comprehensive theory of transversality.
The nonlinearity is just a simple setting, which allows us to unify the existing (and hopefully also future) results on the topic.
In fact, unlike the subtransversality property which has been well studied in the linear and Hölder settings (see, for instance, BauBorLi99 ; Iof00 ; BakDeuLi05 ; NgZan07 ; ZheNg08 ; HesLuk13 ; KruTha14 ; DruIofLew15 ; KruLukTha17 ), for the other two properties: semitransversality and transversality not many characterizations have been known even in the linear case; we fill this gap in the current paper.
Besides the conventional Hölder case, which is given a special attention in the paper, our general model covers also so called Hölder-type settings BolNguPeySut17 ; Li13 that have recently come into play in the closely related error bound theory due to their importance for applications.
Such nonlinear settings of transversality properties have not been studied before.
Some characterizations are new even in the linear setting.
Apart from being of interest on their own, the slope sufficient conditions for nonlinear transversality properties established in this paper lay the foundation for the dual sufficient conditions for the respective properties in Banach and Asplund spaces in CuoKru5 .
Primal and dual necessary conditions for the nonlinear transversality properties are
studied
in CuoKru4 ; CuoKru6 .
There exist strong connections between transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings.
In this paper, we establish quantitative relations between the two models in the general nonlinear setting.
Nonlinear regularity properties of set-valued mappings
and closely related error bound properties of (extended-)real-valued functions
have been intensively studied since 1980s; cf. FraQui12 ; GayGeoJea11 ; Kum09 ; LiMor12 ; BorZhu88 ; Fra87 ; Iof13 ; Kru16 ; Kru16.2 ; KruTha14 ; OuyZhaZhu19 ; ZheZhu16 ; AzeCor17 ; AzeCor14 ; CorMot08 ; YaoZhe16 .
The slope sufficient conditions for φ−subtransversality in Section 4 can be interpreted in terms of the corresponding conditions for nonlinear error bounds.
The semitransversality and transversality properties do not have exact counterparts within the conventional error bound theory.
As in most of our previous publications on the topic, our working model in this paper is a collection of n≥2 arbitrary subsets Ω1,…,Ωn of a normed vector space X, having a common point xˉ∈∩i=1nΩi.
The next definition introduces three most common Hölder transversality properties.
It is a modification of (KruTha14, , Definition 1).
Definition 1
Let
α>0 and q>0.
The collection {Ω1,…,Ωn} is
(i)
α−semitransversal of order q at xˉ if there exists a δ>0 such that
[TABLE]
for all ρ∈]0,δ[ and
xi∈X(i=1,…,n) with max1≤i≤n∥xi∥q<αρ;
2. (ii)
α−subtransversal of order q at xˉ if there exist δ1>0 and δ2>0 such that
[TABLE]
for all ρ∈]0,δ1[ and x∈Bδ2(xˉ) with
max1≤i≤ndq(x,Ωi)<αρ;
3. (iii)
α−transversal of order q at xˉ if there exist δ1>0 and δ2>0 such that
[TABLE]
for all ρ∈]0,δ1[, ωi∈Ωi∩Bδ2(xˉ) and xi∈X(i=1,…,n) with max1≤i≤n∥xi∥q<αρ.
The three properties in the above definition were referred to in KruTha14 as [q]−semiregularity, [q]−subregularity and [q]−regularity, respectively.
Property (ii) was defined in KruTha14 in a slightly different but equivalent way, under an additional
assumption
that q≤1.
When ∩i=1nΩi is closed and xˉ∈bd∩i=1nΩi, the condition q≤1 is indeed necessary for the α−subtransversality and α−transversality properties; see Remark 4.
At the same time, as observed in KruTha14 , the property of α−semitransversality can be meaningful with any positive q (and any positive α); see Example 1.
With q=1 (linear case), properties (i) and (iii) in Definition 1 were discussed in Kru06 (see also (Kru09, , Properties (R)S and (UR)S)), while property (ii) first appeared
in KruTha15 .
If ∩i=1nΩi is closed and xˉ∈bd∩i=1nΩi, then one can observe that properties (ii) and (iii) can only hold with α≤1; see Remark 4.
If q=1, when referring to the three properties in the above definition, we
talk simply about α−(semi-/sub-) transversality.
If a collection {Ω1,…,Ωn} is α−semitransversal (respectively, α−subtransversal or α−transversal) of order q at xˉ with some α>0 and δ>0 (or δ1>0 and δ2>0), we often simply say that {Ω1,…,Ωn} is semitransversal (respectively, subtransversal or transversal) of order q at xˉ.
The number α characterizes the corresponding property quantitatively.
The exact upper bound of all α>0 such that the property holds with some δ>0 (or δ1>0 and δ2>0) is called the modulus of this property.
We use the notations setrq[Ω1,…,Ωn](xˉ), strq[Ω1,…,Ωn](xˉ) and trq[Ω1,…,Ωn](xˉ) for the moduli of the respective properties.
If the property does not hold, then by convention the respective modulus equals 0.
If q<1, the Hölder transversality properties in Definition 1 are obviously weaker than the corresponding conventional linear properties and can be satisfied for collections of sets when the conventional ones fail.
This can happen in many natural situations (see examples in (KruTha14, , Section 2.3)), which explains the growing interest of researchers to studying the more subtle nonlinear transversality properties.
Our basic notation is standard, see, e.g., RocWet98 ; Mor06.1 ; DonRoc14 .
Throughout the paper, X and Y are either metric or, more often, normed vector spaces.
The open unit ball in any space is denoted by B, and Bδ(x) stands for the open ball with center x and radius δ>0.
If not explicitly stated otherwise, products of normed vector spaces are assumed to be equipped with the maximum norm ∥(x,y)∥:=max{∥x∥,∥y∥}, (x,y)∈X×Y.
The symbols R and R+ denote the real line (with the usual norm) and the set of all nonnegative real numbers, respectively.
Given a set Ω,
its interior and boundary
are denoted by intΩ and bdΩ, respectively.
The distance from a point x to Ω is defined by d(x,Ω):=infu∈Ω∥u−x∥, and we use the convention d(x,∅)=+∞.
The indicator function of Ω is defined as follows: iΩ(x)=0 if x∈Ω and iΩ(x)=+∞ if x∈/Ω.
For an extended-real-valued function f:X→R∪{+∞}, its domain and epigraph are defined,
respectively, by domf:={x∈X∣f(x)<+∞} and
epif:={(x,α)∈X×R∣f(x)≤α}.
The inverse of f (if it exists) is denoted by f−1.
A set-valued mapping F:X⇉Y between two sets X and Y is a mapping, which assigns to every x∈X a subset (possibly empty) F(x) of Y.
We use the notations gphF:={(x,y)∈X×Y∣y∈F(x)} and domF:={x∈X∣F(x)=∅}
for the graph and the domain of F, respectively, and F−1:Y⇉X for the inverse of F.
This inverse (which always exists with possibly empty values at some y) is defined by F−1(y):={x∈X∣y∈F(x)}, y∈Y.
Obviously domF−1=F(X).
The closed and open intervals between points x1 and x2 in a normed space are defined, respectively, by
[TABLE]
The semi-open intervals ]x1,x2] and [x1,x2[ are defined in a similar way.
The key tool in the proofs of the main results is the celebrated Ekeland variational principle; cf. Mor06.1 ; DonRoc14 ; Pen13 ; Iof17 .
Lemma 1
Suppose X is a complete metric space, f:X→R∪{+∞} is lower semicontinuous,
x∈X, ε>0 and λ>0.
If
[TABLE]
then there exists an x^∈X such that
(i)
d(x^,x)<λ;
2. (ii)
f(x^)≤f(x);
3. (iii)
f(u)+(ε/λ)d(u,x^)≥f(x^)* for all u∈X.*
The slopeDegMarTos80 and nonlocal slopeNgaThe08 ; Kru15 of a function f:X→R∪{+∞} on a metric space at x∈domf are defined, respectively, by
[TABLE]
where α+:=max{0,α} for any α∈R.
The limit ∣∇f∣(x) provides the rate of steepest descent
of f at x.
If X is a normed space, and f is Fréchet differentiable at x, then ∣∇f∣(x)=∥f′(x)∥.
When x∈/domf, we set ∣∇f∣(x)=∣∇f∣⋄(x):=+∞.
The next proposition is straightforward.
Proposition 1
Suppose X is a metric space, f:X→R∪{+∞}, and x∈X.
(i)
If f is not lower semicontinuous at x, then ∣∇f∣(x)=+∞.
2. (ii)
If f(x)>0, then
∣∇f∣(x)≤∣∇f∣⋄(x).
When proving primal and dual characterizations of transversality properties in the nonlinear setting we use chain rules for slopes and subdifferentials, respectively.
The next lemma provides a chain rule for slopes, which is used in Section 3.
For its subdifferential counterparts we refer the reader to (CuoKru5, , Proposition 2.1).
Lemma 2
Let X be a metric space, f:X→R∪{+∞},
φ:R→R∪{+∞},
x∈domf and f(x)∈domφ.
Suppose φ is nondecreasing on R and differentiable at f(x) with φ′(f(x))>0.
Then
∣∇(φ∘f)∣(x)=φ′(f(x))∣∇f∣(x).
Proof
If x is a local minimum of f, then, thanks to the monotonicity of φ, it is also a local minimum of φ∘f, and consequently, ∣∇(φ∘f)∣(x)=∣∇f∣(x)=0.
Suppose x is not a local minimum of f.
If f is not lower semicontinuous at x, i.e. α:=limk→+∞f(xk)<f(x) for some sequence xk→x,
then, in view of the assumptions,
φ is strictly increasing near f(x),
and consequently,
liminfk→+∞φ(f(xk))≤φ(α)<φ(f(x)) (with the convention that φ(−∞)=−∞), i.e.
φ∘f is not lower semicontinuous at x; hence, in view of Proposition 1(i),
∣∇(φ∘f)∣(x)=∣∇f∣(x)=+∞.
Suppose f is lower semicontinuous at x, i.e.
liminfu→x,u=xf(u)=f(x).
Then, taking into account that x is not a local minimum of f,
[TABLE]
The proof is complete.
∎
Remark 1
(i)
The slope chain rule in Lemma 2 is a local result.
Instead of assuming that φ is defined on the whole real line, one can assume that φ is defined and finite on a closed interval [α,β] around the point f(x): α<f(x)<β.
It is sufficient to define the composition φ∘f for x with f(x)∈/[α,β] as follows:
(φ∘f)(x):=φ(α) if f(x)<α, and
(φ∘f)(x):=φ(β) if f(x)>β.
This does not affect the conclusion of the lemma.
2. (ii)
Lemma 2 slightly improves (AzeCor17, , Lemma 4.1), where f and φ are assumed lower semicontinuous and continuously differentiable, respectively.
The rest of the paper is organized as follows.
In Section 2, we discuss transversality properties of finite collections of sets in the nonlinear setting.
In Section 3, we establish metric characterizations of these properties.
Section 4 is devoted to slope sufficient conditions for the nonlinear transversality properties.
In Section 5, we discuss quantitative relations between nonlinear transversality of collections of sets and the corresponding nonlinear regularity properties of set-valued mappings, and show that the two popular models are in a sense equivalent in the general nonlinear setting.
As a consequence, we improve some results established in KruTha14 in the Hölder setting.
We also briefly discuss nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe Iof17 .
2 Definitions and Basic Relations
The nonlinearity in the definitions of the transversality properties is determined by a continuous strictly increasing function φ:R+→R+ satisfying φ(0)=0 and limt→+∞φ(t)=+∞.
The family of all such functions is denoted by C.
We denote by C1 the subfamily of functions from C which are differentiable on ]0,+∞[ with φ′(t)>0 for all t>0.
Obviously, if φ∈C (φ∈C1), then φ−1∈C (φ−1∈C1).
Observe that,
for any α>0 and q>0,
the function t↦αtq on R+ belongs to C1.
Remark 2
For the purposes of the paper, it is sufficient to assume that functions
φ∈C are defined and invertible near 0.
In addition to our standing assumption that
Ω1,…,Ωn are subsets of a normed space X and xˉ∈∩i=1nΩi,
if not explicitly stated otherwise,
we assume from now on
that φ∈C.
Definition 2
The collection {Ω1,…,Ωn} is
(i)
φ−semitransversal at xˉ if there exists a δ>0 such that condition (1) is satisfied
for all ρ∈]0,δ[ and xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<ρ;
2. (ii)
φ−subtransversal at xˉ if there exist δ1>0 and δ2>0 such that condition (2) is satisfied
for all ρ∈]0,δ1[ and x∈Bδ2(xˉ) with φ(max1≤i≤nd(x,Ωi))<ρ;
3. (iii)
φ−transversal at xˉ if there exist δ1>0 and δ2>0 such that condition (3) is satisfied for all ρ∈]0,δ1[, ωi∈Ωi∩Bδ2(xˉ) and xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<ρ.
Observe that conditions (1) and (3) are trivially satisfied when xi=0(i=1,…,n).
Hence, in parts (i) and (iii) of Definition 2 (as well as Definition 1) one can additionally assume that max1≤i≤n∥xi∥>0.
Similarly, in part (ii) of Definition 2 (as well as Definition 1) one can assume that x∈/∩i=1nΩi.
Each of the properties in Definition 2 is determined by a function φ∈C, and a number δ>0 in item (i) or numbers δ1>0 and δ2>0 in items (ii) and (iii).
The function plays the role of a kind of rate or modulus of the respective property, while the role of the δ’s is more technical: they control the size of the interval for the values of ρ and, in the case of φ−subtransversality and φ−transversality in parts (ii) and (iii), the size of the neighbourhoods of xˉ involved in the respective definitions.
Of course, if a property is satisfied with some δ1>0 and δ2>0, it is satisfied also with the single δ:=min{δ1,δ2} in place of both δ1 and δ2.
Unlike our previous publications on (linear and Hölder) transversality properties, we use in the current paper
two different parameters to emphasise their different roles in the definitions and the corresponding characterizations.
Moreover, we are going to provide quantitative estimates for the values of these parameters.
Given a δ>0 in item (i) (δ1>0 and δ2>0 in items (ii) and (iii)), if a property is satisfied for some function φ∈C, it is obviously satisfied for any function φ^∈C such that φ^−1(t)≤φ−1(t) for all t∈]0,δ[ (t∈]0,δ1[), or equivalently, φ^(t)≥φ(t) for all t∈]0,φ−1(δ)[ (t∈]0,φ−1(δ1)[).
Thus, it makes sense looking for the smallest function in C (if it exists) ensuring the corresponding property for the given sets.
Observe also that taking a smaller δ>0 (smaller δ1>0 and δ2>0) may allow each of the properties to be satisfied with a smaller φ.
When the exact value of δ (δ1 and δ2) in the definition of the respective property is not important,
it makes sense to look for the smallest function ensuring the corresponding property for some δ>0 (δ1 and δ2).
The most important realization of the three properties in Definition 2 corresponds to the Hölder setting, i.e. φ being a power function, given for all t≥0 by φ(t):=α−1tq with some α>0 and q>0.
In this case, Definition 2 reduces to Definition 1.
Another important for applications class of functions is given by the so called Hölder-typeBolNguPeySut17 ; Li13 ones, i.e. functions of the form t↦α−1(tq+t), frequently used in the error bound theory, or more generally, functions t↦α−1(tq+βt) with some α>0, β>0 and q>0.
Depending on the value of q, transversality properties determined by such functions can be approximated by Hölder (if q<1) or even linear (if q≥1) ones.
Proposition 2
Let φ(t):=α−1(tq+βt) with some α>0, β>0 and q>0.
If the collection {Ω1,…,Ωn} is φ−(semi-/sub-)transversal at xˉ,
then it is α′−(semi-/sub-) transversal of order q′ at xˉ, where:
(i)
if q<1, then q′=q and α′ is any number in ]0,α[;
2. (ii)
if q=1, then q′=1 and α′:=α(1+β)−1;
3. (iii)
if q>1, then q′=1 and α′ is any number in ]0,αβ−1[.
Proof
The assertions follow from Definition 1 in view of the following observations:
(i)
if q<1 and α′∈]0,α[, then, for all sufficiently small t>0, it holds α′(1+βt1−q)<α, and consequently,
φ(t)=α−1(1+βt1−q)tq<(α′)−1tq;
2. (ii)
if q=1 and α′=α(1+β)−1, then φ(t)=α−1(1+β)t=(α′)−1t;
3. (iii)
if q>1 and α′∈]0,αβ−1[, then, for all sufficiently small t>0, it holds α′(β−1tq−1+1)<αβ−1, and consequently, φ(t)=α−1β(β−1tq−1+1)t<(α′)−1t.
∎
The next two propositions collect some simple facts about the properties in Definition 2 and clarify relationships between them.
Proposition 3
(i)
*If Ω1=…=Ωn, and there exists a δ1>0 such that φ(t)≥t for all t∈]0,δ1[, then {Ω1,…,Ωn} is *φ−subtransversal at
xˉ with δ1 and any δ2>0.
2. (ii)
*If {Ω1,…,Ωn} is *φ−*transversal at xˉ with some δ1>0 and δ2>0, then it is *φ−*semitransversal at xˉ
with δ1 and *φ−subtransversal at xˉ
with any δ1′∈]0,δ1] and δ2′>0 such that φ−1(δ1′)+δ2′≤δ2.
3. (iii)
*If xˉ∈int∩i=1nΩi, then {Ω1,…,Ωn} is *φ−transversal at xˉ with some δ1>0 and δ2>0.
Proof
(i)
Let Ω:=Ω1=…=Ωn.
Then condition (2) becomes Ω∩Bρ(x)=∅.
This inclusion is trivially satisfied if φ(d(x,Ω))<ρ and φ(ρ)≥ρ.
2. (ii)
Let {Ω1,…,Ωn} be φ−transversal at xˉ
with some δ1>0 and δ2>0.
Since condition (1) is a particular case of condition (3) with ωi=xˉ (i=1,…,n), we can conclude that {Ω1,…,Ωn} is φ−semitransversal at xˉ
with δ1.
Let δ1′∈]0,δ1] and δ2′>0 be such that φ−1(δ1′)+δ2′≤δ2, and let
ρ∈]0,δ1′[ and x∈Bδ2′(xˉ) with φ(max1≤i≤nd(x,Ωi))<ρ.
Choose ωi∈Ωi(i=1,…,n) such that φ(max1≤i≤n∥x−ωi∥)<ρ.
Then, for any i=1,…,n,
[TABLE]
Set xi:=x−ωi(i=1,…,n).
We have ρ∈]0,δ1[, ωi∈Ωi∩Bδ2(xˉ)(i=1,…,n) and φ(max1≤i≤n∥xi∥)<ρ.
By Definition 2(iii), condition (3) is satisfied. This is equivalent to condition (2).
In view of Definition 2(ii), {Ω1,…,Ωn} is φ−subtransversal at xˉ with δ1′ and δ2′.
3. (iii)
Let xˉ∈int∩i=1nΩi.
Choose numbers δ1>0 and δ2>0 such that, with δ:=φ−1(δ1)+δ2, it holds
Bδ(xˉ)⊂∩i=1nΩi.
Then, for all ωi∈Ωi∩Bδ2(xˉ) and xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<δ1, it holds 0∈∩i=1n(Ωi−ωi−xi), and consequently, condition (3) is satisfied with any ρ>0.
Hence, {Ω1,…,Ωn} is φ−transversal at xˉ with δ1 and δ2.
∎
Remark 3
(i)
The inequality φ−1(δ1′)+δ2′≤δ2 in Proposition 3(ii) and some statements below can obviously be replaced by the equality φ−1(δ1′)+δ2′=δ2 providing in a sense the best estimate for the values of the parameters δ1′ and δ2′.
2. (ii)
In the Hölder setting, parts (i) and (iii) of Proposition 3 recapture (KruTha14, , Remarks 4 and 3), respectively, while part (ii) improves (KruTha14, , Remark 1).
3. (iii)
The nonlinear semitransversality and subtransversality properties are in general independent; see examples in (KruTha14, , Section 2.3) and (KruTha15, , Section 3.2).
Proposition 4
Let ∩i=1nΩi be closed and xˉ∈bd∩i=1nΩi.
If {Ω1,…,Ωn} is φ−subtransversal (in particular, if it is φ−transversal) at xˉ with some δ1>0 and δ2>0, then there exists a tˉ∈]0,min{δ2,φ−1(δ1)}[ such that φ(t)≥t for all t∈]0,tˉ].
Proof
Let {Ω1,…,Ωn} be φ−subtransversal at xˉ with some δ1>0 and δ2>0.
Choose a point x^∈/∩i=1nΩi such that ∥x^−xˉ∥<min{φ−1(δ1),δ2} and set tˉ:=d(x^,∩i=1nΩi).
Then tˉ<min{φ−1(δ1),δ2}.
Besides, tˉ>0 since ∩i=1nΩi is closed.
Thanks to the continuity of the function d(⋅,∩i=1nΩi),
for any t∈]0,tˉ] there is an x∈]xˉ,x^]
such that d(x,∩i=1nΩi)=t.
We have ∥x−xˉ∥≤∥x^−xˉ∥<δ2 and φ(t)≤φ(tˉ)<δ1.
Take a ρ∈]φ(t),δ1[.
Then φ(max1≤i≤nd(x,Ωi))≤φ(t)<ρ.
By Definition 2(ii), t=d(x,∩i=1nΩi)<ρ, and letting ρ↓φ(t), we arrive at t≤φ(t).
If {Ω1,…,Ωn} is φ−transversal at xˉ, the conclusion follows
in view of Proposition 3(ii).
∎
Remark 4
The conditions on φ in Proposition 4 in the Hölder setting can only be satisfied if either q<1, or q=1 and α≤1.
This reflects the well known fact that the Hölder subtransversality and transversality properties are only meaningful when q≤1 and, moreover, the linear case (q=1) is only meaningful when α≤1; cf. (KruLukTha17, , p. 705), (Kru18, , p. 118).
The extreme case q=α=1 is in a sense singular for subtransversality as in this case Definition 1(ii) yields d(x,∩i=1nΩi)=max1≤i≤nd(x,Ωi) for all x near xˉ.
In accordance with Proposition 4, the φ−subtransversality and φ−transversality properties impose serious restrictions on the function φ.
This is not the case with the φ−semitransversality property: φ can be, e.g., any power function.
Example 1
Let R2 be equipped with the maximum norm, and let q>0, γ>0, Ω1:={(ξ1,ξ2)∈R2∣γq1ξ2+∣ξ1∣q1≥0},
Ω2:={(ξ1,ξ2)∈R2∣γq1ξ2−∣ξ1∣q1≤0}
and xˉ:=(0,0).
Note that, when q>1, the sets Ω1 and Ω2 are nonconvex.
We claim that the pair {Ω1,Ω2} is φ−semitransversal at xˉ with φ(t):=γtq (t≥0).
Proof
Given an r>0, set x1:=(0,−r) and x2:=(0,r).
Then ∥x1∥=∥x2∥=r and (±γrq,0)∈(Ω1−x1)∩(Ω2−x2).
Moreover, it is easy to notice that either (γrq,0) or (−γrq,0) belongs to (Ω1−x1)∩(Ω2−x2) for any choice of vectors x1,x2∈R2 with max{∥x1∥,∥x2∥}≤r.
Hence, (Ω1−x1)∩(Ω2−x2)∩Bρ(xˉ)=∅ for all such vectors x1,x2∈R2 as long as ρ>γrq, and consequently, {Ω1,Ω2} is φ−semitransversal at xˉ.
∎
3 Metric Characterizations
The three transversality properties are defined in Definition 2 geometrically.
We now show that they can be characterized in metric terms.
These metric characterizations can be used as equivalent definitions of the respective properties.
Theorem 3.1
The collection {Ω1,…,Ωn} is
(i)
φ−semitransversal at xˉ with some δ>0 if and only if
[TABLE]
for all xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<δ;
2. (ii)
φ−subtransversal at xˉ with some δ1>0 and δ2>0 if and only if the following equivalent conditions hold:
(a)
for all x∈Bδ2(xˉ) with φ(max1≤i≤nd(x,Ωi))<δ1, it holds
[TABLE]
2. (b)
for all xi∈X and ωi∈Ωi(i=1,…,n) with φ(max1≤i≤n∥xi∥)<δ1 and ω1+x1=…=ωn+xn∈Bδ2(xˉ), it holds
[TABLE]
3. (iii)
φ−transversal at xˉ with some δ1>0 and δ2>0 if and only if inequality (6) holds
for all ωi∈Ωi∩Bδ2(xˉ) and xi∈X(i=1,…,n) with
φ(max1≤i≤n∥xi∥)<δ1.
Proof
(i)
Let {Ω1,…,Ωn} be φ−semitransversal at xˉ with some δ>0, and let xi∈X(i=1,…,n) with ρ0:=φ(max1≤i≤n∥xi∥)<δ.
Choose a ρ∈]ρ0,δ[.
By (1), d(xˉ,∩i=1n(Ωi−xi))<ρ.
Letting ρ↓ρ0, we arrive at inequality (4).
Conversely, let δ>0
and inequality (4) hold for all xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<δ.
For all ρ∈]0,δ[ and xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<ρ, we have
d(xˉ,∩i=1n(Ωi−xi))<ρ, which implies condition (1). By Definition 2(i), {Ω1,…,Ωn} is φ−semitransversal at xˉ with δ.
2. (ii)
We first prove the equivalence between (a) and (b).
Suppose condition (a) is satisfied.
Let ωi∈Ωi and xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<δ1 and x:=ω1+x1=…=ωn+xn∈Bδ2(xˉ).
Then
[TABLE]
and consequently, inequality (5) is satisfied.
Hence,
[TABLE]
Suppose condition (b) is satisfied.
Let x∈Bδ2(xˉ) with φ(max1≤i≤nd(x,Ωi))<δ1.
Choose ωi∈Ωi(i=1,…,n) such that φ(max1≤i≤n∥x−ωi∥)<δ1 and
set xi′:=x−ωi(i=1,…,n).
Then x=xi′+ωi∈Bδ2(xˉ)(i=1,…,n) and φ(max1≤i≤n∥xi′∥)<δ1.
In view of inequality (6) with xi′ in place of xi(i=1,…,n), we obtain
[TABLE]
Taking infimum in the right-hand side over ωi∈Ωi(i=1,…,n), we arrive at inequality (5).
Next we show that φ−subtransversality is equivalent to condition (a).
Let {Ω1,…,Ωn} be φ−subtransversal at xˉ with some δ1>0 and δ2>0, and let x∈Bδ2(xˉ) with ρ0:=φ(max1≤i≤nd(x,Ωi))<δ1.
Choose a ρ∈]ρ0,δ1[.
By Definition 2(ii),
∩i=1nΩi∩Bρ(x)=∅, and consequently,
d(x,∩i=1nΩi)<ρ.
Letting ρ↓ρ0, we arrive at inequality (5).
Conversely, let δ1>0 and δ2>0, and inequality
(5) hold for all x∈Bδ2(xˉ) with φ(max1≤i≤nd(x,Ωi))<δ1.
For any ρ∈]0,δ1[ and x∈Bδ2(xˉ) with φ(max1≤i≤nd(x,Ωi))<ρ, we have
d(x,∩i=1nΩi)<ρ,
which implies condition (2).
By Definition 2(ii),
{Ω1,…,Ωn} is φ−subtransversal at xˉ with δ1 and δ2.
3. (iii)
Let {Ω1,…,Ωn} be φ−transversal at xˉ with some δ1>0 and δ2>0, and let ωi∈Ωi∩Bδ2(xˉ) and xi∈X(i=1,…,n) with ρ0:=φ(max1≤i≤n∥xi∥)<δ1.
Choose a ρ∈]ρ0,δ1[.
By (3),
d(0,∩i=1n(Ωi−ωi−xi))<ρ.
Letting ρ↓ρ0, we arrive at inequality (6).
Conversely, let δ1>0 and δ2>0, and inequality (6) hold for all ωi∈Ωi∩Bδ2(xˉ) and xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<δ1.
For any ρ∈]0,δ1[, ωi∈Ωi∩Bδ2(xˉ) and xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<ρ, we have
d(0,∩i=1n(Ωi−ωi−xi))<ρ,
which is equivalent to condition (3).
By Definition 2(iii), {Ω1,…,Ωn} is φ−transversal at xˉ with δ1 and δ2.
∎
Example 2
Let R2 be equipped with the maximum norm, and let Ω1:={(ξ1,ξ2)∈R2∣ξ2≥0},
Ω2:={(ξ1,ξ2)∈R2∣ξ2≤ξ12} and xˉ:=(0,0).
Thus, Ω1∩Ω2={(ξ1,ξ2)∈R2∣0≤ξ2≤ξ12}, and no shift of the sets can make their intersection empty.
We claim that the pair {Ω1,Ω2} is φ−semitransversal at xˉ with
φ(t):=2t(t≥0) and
δ:=2.
Proof
Observe that, given any ε≥0, the vertical shifts of the sets determined by x1ε:=(0,−ε) and x2ε=(0,ε) produce the largest ‘gap’ between them compared to all possible shifts x1 and x2 with max{∥x1∥,∥x2∥}≤ε.
Indeed,
[TABLE]
as long as max{∥x1∥,∥x2∥}≤ε.
Observe also that (2ε,ε)∈(Ω1−x1ε)∩(Ω2−x2ε).
Hence, for any x1,x2∈R2 with ε:=max{∥x1∥,∥x2∥}<φ−1(δ)=2, we have
[TABLE]
In view of Theorem 3.1(i), {Ω1,Ω2}φ−semitransversal at xˉ with δ.
∎
Example 3
Let R2 be equipped with the maximum norm, and let Ω1:={(ξ1,ξ2)∈R2∣ξ2=ξ12},
Ω2:={(ξ1,ξ2)∈R2∣ξ2=−ξ12}
and xˉ:=(0,0).
Thus, Ω1∩Ω2={xˉ}.
We claim that, for any γ>1, the pair {Ω1,Ω2} is φ−subtransversal at xˉ with
φ(t):=γt(t≥0) and any δ1>0 and δ2>0 satisfying δ2+21+δ2+41<γ2.
Proof
Observe that, d(x,Ω1∩Ω2)=∥x∥ for all x∈R2 and,
given any ε≥0 and the corresponding point xε=(0,ε), one has
[TABLE]
It is easy to see that the minimum in the rightmost minimization problem is attained at t:=ε+41−21 satisfying ε−t=t2.
Thus,
[TABLE]
Hence, for any x∈R2 with ∥x∥<δ2, we have
[TABLE]
In view of Theorem 3.1(ii), {Ω1,Ω2} is φ−subtransversal at xˉ with δ1 and δ2.
The next statement provides alternative metric characterizations of
φ−transversality.
These characterizations differ from the one in Theorem 3.1(iii) by values of the parameters δ1 and δ2 and have certain advantages, e.g., when establishing connections with metric regularity of set-valued mappings.
The relations between the values of the parameters in the two
groups of metric characterizations can be estimated.
Theorem 3.2
Let δ1>0 and δ2>0.
The following conditions are equivalent:
(i)
inequality (6) is satisfied
for all xi∈X and ωi∈Ωi with ωi+xi∈Bδ2(xˉ)(i=1,…,n) and φ(max1≤i≤n∥xi∥)<δ1;
2. (ii)
for all xi∈δ2B(i=1,…,n) with
φ(max1≤i≤nd(xˉ,Ωi−xi))<δ1, it holds
[TABLE]
3. (iii)
for all x,xi∈X with x+xi∈Bδ2(xˉ)(i=1,…,n) and φ(max1≤i≤nd(x,Ωi−xi))<δ1, it holds
[TABLE]
Moreover, if {Ω1,…,Ωn} is φ−transversal at xˉ with some δ1>0 and δ2>0, then conditions (i)–(iii) hold with any δ1′∈]0,δ1] and δ2′>0 satisfying φ−1(δ1′)+δ2′≤δ2 in place of δ1 and δ2.
Conversely, if conditions (i)–(iii) hold with some δ1>0 and δ2>0, then {Ω1,…,Ωn} is φ−transversal at xˉ with any δ1′∈]0,δ1] and δ2′>0 satisfying φ−1(δ1′)+δ2′≤δ2.
Proof
We first prove the equivalence of conditions (i)–(iii).
(i) \Rightarrow\(ii).
Let xi∈δ2B(i=1,…,n) with φ(max1≤i≤nd(xˉ,Ωi−xi))<δ1.
Choose ωi∈Ωi(i=1,…,n) such that φ(max1≤i≤n∥xˉ+xi−ωi∥)<δ1.
Set xi′:=xˉ+xi−ωi(i=1,…,n).
Then ωi+xi′∈Bδ2(xˉ)(i=1,…,n) and φ(max1≤i≤n∥xi′∥)<δ1.
By (i), inequality (6) is satisfied with xi′ in place of xi(i=1,…,n), i.e.
[TABLE]
Taking the infimum in the righ-hand side over ωi∈Ωi(i=1,…,n), we
arrive at inequality (7).
(ii) \Rightarrow\(iii).
Let x,xi∈X with x+xi∈Bδ2(xˉ)(i=1,…,n) and φ(max1≤i≤nd(x,Ωi−xi))<δ1.
Set xi′:=x+xi−xˉ(i=1,…,n).
Then xi′∈δ2B(i=1,…,n) and
φ(max1≤i≤nd(xˉ,Ωi−xi′))<δ1.
By (ii), inequality (7) is satisfied with xi′ in place of xi(i=1,…,n).
This is equivalent to inequality (8).
(iii) \Rightarrow\(i).
Let xi∈X and ωi∈Ωi with ωi+xi∈Bδ2(xˉ)(i=1,…,n) and φ(max1≤i≤n∥xi∥)<δ1.
Set xi′:=ωi+xi−xˉ(i=1,…,n).
Then, xˉ+xi′∈Bδ2(xˉ) and
φ(max1≤i≤nd(xˉ,Ωi−xi′))≤φ(∥xi∥)<δ1.
By (iii), inequality (8) is satisfied with xˉ and xi′ in place of x and xi(i=1,…,n), respectively,
i.e.
[TABLE]
Since ωi∈Ωi(i=1,…,n),
inequality (6) is satisfied.
Suppose {Ω1,…,Ωn} is φ−transversal at xˉ with some δ1>0 and δ2>0, and let δ1′∈]0,δ1] and δ2′>0 be such that φ−1(δ1′)+δ2′≤δ2.
Then, for all xi∈X and ωi∈Ωi with ωi+xi∈Bδ2′(xˉ)(i=1,…,n) and φ(max1≤i≤n∥xi∥)<δ1′, we have ∥ωi−xˉ∥≤∥xi∥+∥ωi+xi−xˉ∥<φ−1(δ1′)+δ2′≤δ2 (i=1,…,n).
By Theorem 3.1(iii), inequality (6) is satisfied, and consequently, condition (i) (as well as conditions (ii) and (iii)) holds with δ1′ and δ2′.
Conversely, suppose conditions (i)–(iii) hold with some δ1>0 and δ2>0, and let δ1′∈]0,δ1] and δ2′>0 be such that φ−1(δ1′)+δ2′≤δ2.
Then, for all ωi∈Ωi∩Bδ2′(xˉ) and xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<δ1′, we have ∥ωi+xi−xˉ∥≤∥xi∥+∥ωi−xˉ∥<φ−1(δ1′)+δ2′≤δ2(i=1,…,n).
By (i), inequality (6) is satisfied, and consequently, {Ω1,…,Ωn} is φ−transversal at xˉ with δ1′ and δ2′ according to Theorem 3.1(iii).
∎
Remark 5
(i)
In the Hölder case, i.e. when φ(t):=α−1tq (t≥0) for some α>0 and q∈]0,1],
condition (8) served as the main metric characterization of transversality; cf. KruTha15 ; KruTha14 .
In the linear case, condition (7) has been picked up recently in BuiKru19 ; BuiCuoKru .
This condition seems an important advancement as it replaces an arbitrary point x in (8) with the given reference point xˉ.
Condition (6) in part (i) seems new.
In view of Theorem 3.1(iii), it is the most straightforward metric counterpart of the original geometric property (3).
2. (ii)
The metric characterizations of the three φ−transversality properties in the above theorems look similar: each of them provides an upper error bound type estimate for the distance from a point to the intersection of sets, which can be useful from the computational point of view.
For the account of nonlinear error bounds theory, we refer the reader to AzeCor17 ; AzeCor14 ; CorMot08 ; YaoZhe16 .
The next corollary provides qualitative metric characterizations of the three nonlinear transversality properties.
They are direct consequences of Theorems 3.1 and 3.2.
Corollary 1
The collection {Ω1,…,Ωn} is
(i)
φ−semitransversal at xˉ if and only if
there exists a δ>0 such that inequality (4) holds for all xi∈δB(i=1,…,n);
2. (ii)
φ−subtransversal at xˉ
if and only if the following equivalent conditions hold:
(a)
there exists a δ>0 such that inequality (5) holds for all x∈Bδ(xˉ);
2. (b)
there exists a δ>0 such that inequality (6) holds for all ωi∈Ωi∩Bδ(xˉ) and xi∈δB(i=1,…,n) with ω1+x1=…=ωn+xn;
3. (iii)
φ−transversal at xˉ if and only if the following equivalent conditions hold:
(a)
there exists a δ>0 such that inequality (6) holds for all ωi∈Ωi∩Bδ(xˉ) and xi∈δB(i=1,…,n);
2. (b)
there exists a δ>0 such that inequality (7) holds for all xi∈δB(i=1,…,n);
3. (c)
there exists a δ>0 such that inequality (8) holds for all x∈Bδ(xˉ) and xi∈δB(i=1,…,n).
Remark 6
In the Hölder setting, i.e. when
φ(t):=α−1tq(t≥0) with some α>0 and q>0, the above corollary improves (KruTha14, , Theorem 1).
In the linear case, the equivalence of the three characterizations of transversality in Corollary 1(iii) has been established in BuiCuoKru .
We refer the readers to KruLukTha17 ; KruLukTha18 ; Kru18 for more discussions and historical comments.
The next two propositions identify important situations when ‘restricted’ versions of the metric characterizations of nonlinear transversality properties in
Theorem 3.1 can be used: with all but one sets being translated in the cases of φ−semitransversality and φ−transversality, and with the point x restricted to one of the sets in the case of φ−subtransversality.
The latter restricted version is of importance, for instance, when dealing with alternating (or cyclic) projections.
The first proposition formulates simplified necessary conditions for the transversality properties which are direct consequences of the respective statements, while the second one gives conditions under which these conditions become sufficient in the case of two sets.
Proposition 5
(i)
*If {Ω1,…,Ωn} is *φ−semitransversal at xˉ with some δ>0, then
[TABLE]
for all xi∈X(i=1,…,n−1) with φ(max1≤i≤n−1∥xi∥)<δ.
2. (ii)
*If {Ω1,…,Ωn} is *φ−subtransversal at xˉ with some δ1>0 and δ2>0, then
[TABLE]
for all x∈Ωn∩Bδ2(xˉ) with φ(max1≤i≤n−1d(x,Ωi))<δ1.
3. (iii)
*If {Ω1,…,Ωn} is *φ−transversal at xˉ with some δ1>0 and δ2>0, then
[TABLE]
for all ωi∈Ωi∩Bδ2(xˉ)(i=1,…,n) and
xi∈X(i=1,…,n−1) with φ(max1≤i≤n−1∥xi∥)<δ1.
Proposition 6
Let Ω1,Ω2 be subsets of a normed space X, and xˉ∈Ω1∩Ω2.
Let α>0, tˉ>0,
φ(t)≤αt for all t∈]0,tˉ],
and α′:=(1+2α)−1.
(i)
If
for all x∈tˉB,
[TABLE]
*then {Ω1,Ω2} is *α′−semitransversal at xˉ with δ:=(α+21)tˉ.
2. (ii)
If there exists a δ2>0 such that,
for all x∈Ω2∩B2δ2(xˉ) with d(x,Ω1)<tˉ,
[TABLE]
*then {Ω1,Ω2} is *α′−subtransversal at xˉ with δ1:=(α+21)tˉ and δ2.
3. (iii)
If there exists a δ2>0 such that,
for all ωi∈Ωi∩Bδ2(xˉ)(i=1,2) and x∈tˉB,
[TABLE]
*then {Ω1,Ω2} is *α′−transversal at xˉ with δ1:=(α+21)tˉ and δ2.
Proof
(i)
Let δ:=(α+21)tˉ, and inequality (9) be satisfied for all x∈tˉB.
Let ρ∈]0,δ[ and x1,x2∈X with max{∥x1∥,∥x2∥}<α′ρ.
Set x′:=x1−x2.
Thus,
∥x′∥≤2max{∥x1∥,∥x2∥}<2α′δ=tˉ.
Hence, by (9) with x′ in place of x,
[TABLE]
Hence, (Ω1−x1)∩(Ω2−x2)∩Bρ(xˉ)=∅ and, by Definition 1(i),
{Ω1,Ω2} is
α′−semitransversal at xˉ with δ.
2. (ii)
Let δ1:=(α+21)tˉ, δ2>0, and
inequality (10) be satisfied
for all x∈Ω2∩B2δ2(xˉ) with d(x,Ω1)<tˉ.
Let ρ∈]0,δ1[ and x∈Bδ2(xˉ) with max{d(x,Ω1),d(x,Ω2)}<α′ρ.
Choose a number γ>1 such that
[TABLE]
and a point x′∈Ω2 such that ∥x−x′∥≤γd(x,Ω2).
Then
Hence, Ω1∩Ω2∩Bρ(x)=∅ and, by Definition 1(ii),
{Ω1,Ω2} is
α′−subtransversal at xˉ with δ1 and δ2.
3. (iii)
The proof follows that of assertion (i) with the sets Ω1−ω1 and Ω2−ω2 in place of Ω1 and Ω2, respectively.
∎
Remark 7
(i)
In the linear case, Proposition 6(ii) recaptures (KruLukTha18, , Theorem 1(iii)), while
parts (i) and (iii) seem new.
2. (ii)
Restricted versions of the metric conditions in Theorem 3.2 can be produced in a similar way.
Checking the metric estimates of the φ−subtransversality and φ−transversality can be simplified as illustrated by the following proposition referring to condition (5) in Theorem 3.1(ii).
Equivalent versions of conditions (7) and (8) in Theorem 3.2 look similar.
Proposition 7
The following conditions are equivalent:
(i)
inequality (5) holds true;
2. (ii)
for all ωi∈Ωi(i=1,…,n), it holds
[TABLE]
3. (iii)
inequality (11) holds true for all ωi∈Ωi with ∥ωi−xˉ∥<∥x−xˉ∥+φ−1(∥x−xˉ∥)(i=1,…,n);
4. (iv)
inequality (11) holds true for all ωi∈Ωi with φ(∥ωi−x∥)<∥x−xˉ∥(i=1,…,n).
Proof
The equivalence (i) \Leftrightarrow\(ii) and implications (ii) \Rightarrow\(iii) \Rightarrow\(iv) are straightforward.
We next show that (iv) \Rightarrow\(ii).
Let condition (iv) hold true, ωi∈Ωi(i=1,…,n), and φ(∥ωi−x∥)≥∥x−xˉ∥ for some i.
Then
[TABLE]
i.e. inequality (11) is satisfied, and consequently condition (ii) holds true.
∎
4 Slope Sufficient Conditions
In this section, we formulate slope sufficient conditions for the properties in Definition 2.
The conditions are straightforward consequences of the Ekeland variational principle (Lemma 1) applied to appropriate lower semicontinuous functions.
Throughout this section, X is a Banach space, and the sets Ω1,…,Ωn are closed.
These are exactly the assumptions which ensure that the Ekeland variational principle is applicable.
In view of Proposition 3(iii), it suffices to assume that xˉ∈bd∩i=1nΩi.
The sufficient conditions for the three properties follow the same pattern.
We first establish nonlocal slope sufficient conditions arising from the Ekeland variational principle.
These nonlocal conditions are largely of theoretical interest (unless the sets are convex): they encapsulate the application of the Ekeland variational principle and serve as a source of more practical local (infinitesimal) conditions.
The corresponding local slope sufficient conditions, their Hölder as well as simplified δ-free versions are formulated as corollaries.
This way we expose the hierarchy of this type of conditions.
Along with the standard maximum norm on Xn+1, we are going to use also the following norm depending on a parameter γ>0:
[TABLE]
4.1 Semitransversality
Theorem 4.1
The collection {Ω1,…,Ωn} is φ−semitransversal at xˉ with some δ>0 if, for some γ>0 and any xi∈X(i=1,…,n) satisfying
[TABLE]
there exists a λ∈]φ(max1≤i≤n∥xi∥),δ[ such that
[TABLE]
for all x∈X and ωi∈Ωi(i=1,…,n) satisfying
[TABLE]
The proof below employs two closely related nonnegative functions on Xn+1 determined by the given function φ∈C and vectors
x1,…,xn∈X:
[TABLE]
Proof
Suppose {Ω1,…,Ωn} is not φ−semitransversal at xˉ with some δ>0, and
let γ>0 be given.
By Definition 2(i), there exist a ρ∈]0,δ[ and xi∈X(i=1,…,n) with φ(max1≤i≤n∥xi∥)<ρ such that ∩i=1n(Ωi−xi)∩Bρ(xˉ)=∅.
Thus, max1≤i≤n∥xi∥>0.
Let λ∈]φ(max1≤i≤n∥xi∥),δ[ and λ′:=min{λ,ρ}.
Then λ′>φ(max1≤i≤n∥xi∥),
∩i=1n(Ωi−xi)∩Bλ′(xˉ)=∅, and consequently,
[TABLE]
Let f and f be defined by (17) and (18), respectively, while Xn+1 be equipped with the metric induced by the norm (12).
We have f(xˉ,…,xˉ,xˉ)=φ(max1≤i≤n∥xi∥)<λ′.
Choose a number ε such that f(xˉ,…,xˉ,xˉ)<ε<λ′.
Applying the Ekeland variational principle, we can find points ωi∈Ωi(i=1,…,n) and x∈X such that
[TABLE]
for all (u1,…,un,u)∈Ω1×…×Ωn×X.
In view of (19) and the definitions of λ′ and f, conditions (20) yield (15) and (16).
Since ε/λ′<1, condition (21) contradicts (14). ∎
Remark 8
The expression in the left-hand side of (14) is the nonlocal γ-slope (Kru15, , p. 60)
at (ω1,…,ωn,x)
of the function (18).
The next statement is a localized version of Theorem 4.1.
Corollary 2
(i)
*The collection {Ω1,…,Ωn} is *φ−semitransversal at xˉ with some δ>0 if, for some γ>0 and any xi∈X(i=1,…,n) satisfying
(13), there exists a λ∈]φ(max1≤i≤n∥xi∥),δ[ such that
[TABLE]
for all x∈X and ωi∈Ωi(i=1,…,n) satisfying (15) and (16).
2. (ii)
If φ∈C1,
then inequality (22) in part (i) can be replaced by
[TABLE]
Proof
The expression in the left-hand side of (22) is the γ-slope (Kru15, , p. 61) of the function (18) at (ω1,…,ωn,x).
The first assertion follows from Theorem 4.1 in view of Proposition 1(ii), while the second one is a consequence of Lemma 2 in view of Remark 1(i).
∎
In the Hölder setting, Theorem 4.1 and Corollary 2 yield the following statement.
Corollary 3
Let
α>0 and q>0.
The collection {Ω1,…,Ωn} is α−semitransversal of order q at xˉ with some δ>0 if, for some γ>0 and any xi∈X(i=1,…,n) with 0<max1≤i≤n∥xi∥<(αδ)q1, there exists a λ∈]α−1(max1≤i≤n∥xi∥)q,δ[ such that
[TABLE]
for all x∈X and ωi∈Ωi(i=1,…,n) satisfying (15) and (16), or all the more, such that
[TABLE]
Proof
The statement is a direct consequence of Theorem 4.1 and Corollary 2 with φ(t):=α−1tq for all t≥0.
Observe that φ−1(t)=(αt)q1.
∎
Remark 9
(i)
On top of the explicitly given restriction ∥ωi−xˉ∥<λ/γ in Theorem 4.1 (and similar conditions in its corollaries) on the choice of the points ωi∈Ωi, which involves γ, the other conditions implicitly impose another one:
[TABLE]
and consequently, ∥ωi−xˉ∥<λ+2φ−1(δ).
This alternative restriction can be of importance when γ is small.
2. (ii)
The statements of Theorem 4.1 and its corollaries can be simplified (and weakened!) by dropping condition (16).
3. (iii)
Inequalities (14), (22)–(25), which are crucial for checking nonlinear semitransversality, involve two groups of parameters: on one hand, sufficiently small vectors xi∈X, not all zero, and on the other hand, points x∈X and ωi∈Ωi near xˉ.
Note an important difference between these two groups.
The magnitudes of xi are directly controlled by the value of δ in the definition of φ−semitransversality: φ(max1≤i≤n∥xi∥)<δ.
At the same time, taking into account that λ can be made arbitrarily close to φ(max1≤i≤n∥xi∥), the magnitudes of x−xˉ and ωi−xˉ (as well as ωi−xi−x) are determined by δ indirectly; they are controlled by max1≤i≤n∥xi∥: cf. conditions (15) and (16).
4. (iv)
In view of the definition of the parametric norm (12), if any of the inequalities (14), (22)–(25) holds true for some γ>0, then it also holds for any γ′∈]0,γ[.
5. (v)
Even in the linear setting, the characterizations in Corollary 3 are new.
The next corollary provides a simplified (and weaker!) version of Theorem 4.1.
The simplification comes at the expense of eliminating the difference between the two groups of parameters highlighted in Remark 9(iii).
Corollary 4
The collection {Ω1,…,Ωn} is φ−semitransversal at xˉ with some δ>0 if, for some γ>0 and any xi∈X(i=1,…,n) satisfying
(13), inequality (14) holds
for all x∈Bδ(xˉ) and ωi∈Ωi∩Bδ/γ(xˉ)(i=1,…,n) satisfying (16).
Sacrificing the estimates for δ in Theorem 4.1, and Corollaries 2 and 4, we arrive at the following ‘δ-free’ statement.
Corollary 5
The collection {Ω1,…,Ωn} is φ−semitransversal at xˉ if,
for some γ>0 and all xi∈X(i=1,…,n) near [math] with max1≤i≤n∥xi∥>0, x∈X and ωi∈Ωi(i=1,…,n) near xˉ satisfying
(16), inequality (14) holds true.
Moreover,
inequality (14) can be replaced by its localized version (22), or by (23) if φ∈C1.
4.2 Subtransversality
Theorem 4.2
The collection {Ω1,…,Ωn} is φ−subtransversal at xˉ with some δ1>0 and δ2>0 if, for some γ>0 and any x′∈X satisfying
[TABLE]
there exists a λ∈]φ(max1≤i≤nd(x′,Ωi)),δ1[ such that
[TABLE]
for all x∈X and ωi,ωi′∈Ωi(i=1,…,n) satisfying
[TABLE]
The proof below follows the pattern of that of Theorem 4.1.
It employs
a continuous real-valued function f:Xn+1→R+ determined by the given function φ∈C:
[TABLE]
and its restriction to Ω1×…×Ωn×X given by (18).
Note that the function (30) is a particular case of (17) corresponding to setting xi:=0(i=1,…,n).
We provide here the proof of Theorem 4.2 for completeness and to expose the differences in handling the two transversality properties, but we skip the proofs of most of its corollaries.
Proof
Suppose {Ω1,…,Ωn} is not φ−subtransversal at xˉ with some δ1>0 and δ2>0, and
let γ>0 be given.
By Definition 2(ii), there exist a number ρ∈]0,δ1[ and a point x′∈Bδ2(xˉ) such that φ(max1≤i≤nd(x′,Ωi))<ρ and ∩i=1nΩi∩Bρ(x′)=∅.
Hence, x′∈/∩i=1nΩi and
[TABLE]
Let λ∈]φ(max1≤i≤nd(x′,Ωi)),δ1[.
Choose numbers ε and λ′ such that
[TABLE]
and points ωi′∈Ωi(i=1,…,n) such that
φ(max1≤i≤n∥ωi′−x′∥)<ε.
Let f and f be defined by (30) and (18), respectively, while Xn+1 be equipped with the metric induced by the norm (12).
We have f(ω1′,…,ωn′,x′)<ε.
Applying the Ekeland variational principle, we can find points ωi∈Ωi(i=1,…,n) and x∈X such that
[TABLE]
for all (u1,…,un,u)∈Ω1×…×Ωn×X.
Thanks to (31), we have
∥x−x′∥<λ′, and consequently,
[TABLE]
Hence, x∈/∩i=1nΩi, and max1≤i≤n∥ωi−x∥>0.
In view of the definitions of λ′ and f, conditions (31) together with the last inequality yield (28) and (29).
Since ε/λ′<1, condition (32) contradicts (27).
∎
The next statement is a localized version of Theorem 4.2.
Corollary 6
(i)
*The collection {Ω1,…,Ωn} is *φ−subtransversal at xˉ with some δ1>0 and δ2>0 if, for some γ>0 and any x′∈X satisfying (26),
there exists a λ∈]φ(max1≤i≤nd(x′,Ωi)),δ1[ such that
[TABLE]
for all x∈X and ωi,ωi′∈Ωi(i=1,…,n) satisfying (28) and (29).
2. (ii)
If φ∈C1, then inequality (33) in part (i) can be replaced by
[TABLE]
In the Hölder setting, Theorem 4.2 and Corollary 6 yield the following statement.
In view of Remark 4, we assume that q≤1.
Corollary 7
Let α>0 and q∈]0,1].
The collection {Ω1,…,Ωn} is α−subtransversal of order q at xˉ with some δ1>0 and δ2>0 if, for some γ>0 and any x′∈Bδ2(xˉ) with 0<1≤i≤nmaxd(x′,Ωi)<(αδ1)q1,
there exists a \lambda\in\big{]}\alpha^{-1}\big{(}\max\limits_{1\leq i\leq n}d(x^{\prime},\Omega_{i})\big{)}^{q},\delta_{1}\big{[} such that
[TABLE]
for all x∈X and ωi,ωi′∈Ωi(i=1,…,n) satisfying (28) and
[TABLE]
or all the more, such that
[TABLE]
Remark 10
(i)
The expressions in the left-hand sides of (27) and (33) are, respectively, the nonlocal γ-slope and the γ-slope
at (ω1,…,ωn,x)
of the function (18).
2. (ii)
Under the conditions of Theorem 4.2, there are two ways for estimating ∥ωi−xˉ∥:
[TABLE]
The second estimate does not involve γ and is better than the first one when γ<1.
A similar observation can be made about Corollary 8.
3. (iii)
It can be observed from the proof of Theorem 4.2 that the sufficient conditions for φ−subtransversality can be strengthened by adding another restriction on the choice of x′: φ(max1≤i≤nd(x′,Ωi))<d(x′,∩i=1nΩi).
4. (iv)
The statement of Theorem 4.2 and its corollaries can be simplified by dropping condition (29).
5. (v)
Inequalities (27), (33)–(36), which are crucial for checking nonlinear subtransversality, involve points x∈X and ωi∈Ωi near xˉ.
Their distance from xˉ is determined in Theorem 4.2 via other points: x′∈/∩i=1nΩi and ωi′∈Ωi; cf. conditions (28) and (29).
Only the distance from x′ to xˉ and to the sets Ωi is directly controlled by the values of δ1 and δ2 in the definition of φ−subtransversality: x′∈Bδ2(xˉ) and φ(max1≤i≤nd(x′,Ωi))<δ1.
All the other distances are controlled by λ, which can be made arbitrarily close to φ(max1≤i≤nd(x′,Ωi)).
6. (vi)
In view of the definition of the parametric norm (12), if any of the inequalities (27), (33)–(36) holds true for some γ>0, then it also holds for any γ′∈]0,γ[.
7. (vii)
Corollary 7 strengthens (KruTha14, , Proposition 6).
In the linear case, it improves (KruLukTha17, , Proposition 10).
The next corollary provides a simplified (and weaker!) version of Theorem 4.2; cf.
Remark 10(v).
Corollary 8
The collection {Ω1,…,Ωn} is φ−subtransversal at xˉ with some δ1>0 and δ2>0 if, for some γ>0,
inequality (27) holds
for all x∈Bδ1+δ2(xˉ) and ωi∈Ωi∩Bδ2+δ1/γ+φ−1(δ1)(xˉ)(i=1,…,n) satisfying
0<max1≤i≤n∥ωi−x∥<φ−1(δ1).
Proof
Let δ1>0 and δ2>0, x′∈Bδ2(xˉ)∖∩i=1nΩi, λ∈]φ(max1≤i≤nd(x′,Ωi)),δ1[, and points x∈X and ωi,ωi′∈Ωi(i=1,…,n) satisfy conditions (28) and (29).
Then
[TABLE]
i.e. points x∈X and ωi∈Ωi(i=1,…,n) satisfy all the conditions in the corollary.
Hence, inequality (27) holds.
It follows from Theorem 4.2 that {Ω1,…,Ωn} is φ−subtransversal at xˉ with δ1 and δ2.
∎
Sacrificing the estimates for δ1 and δ2 in Theorem 4.2, and Corollaries 6 and
8, we can formulate the following ‘δ-free’ statement.
Corollary 9
The collection {Ω1,…,Ωn} is φ−subtransversal at xˉ if inequality (27) holds true
for some γ>0 and all x∈X near xˉ and ωi∈Ωi(i=1,…,n) near xˉ satisfying
max1≤i≤n∥ωi−x∥>0.
Moreover,
inequality (27) can be replaced by its localized version (33), or by (34) if φ∈C1.
4.3 Transversality
Since φ−transversality is in a sense an overarching property covering both φ−semitransversality and φ−subtransversality (see Proposition 3(iii)), the next theorem contains some elements of both Theorems 4.1 and 4.2, and its proof goes along the same lines.
Similar to the proof of Theorem 4.1,
it employs functions (17) and (18).
Theorem 4.3
The collection {Ω1,…,Ωn} is φ−transversal at xˉ with some δ1>0 and δ2>0 if, for some γ>0 and any ωi′∈Ωi∩Bδ2(xˉ)(i=1,…,n) and ξ∈]0,φ−1(δ1)[, there exists a λ∈]φ(ξ),δ1[ such that
inequality (14) holds
for all x,xi∈X and ωi∈Ωi(i=1,…,n) satisfying
[TABLE]
Proof
Suppose {Ω1,…,Ωn} is not φ−transversal at xˉ with some δ1>0 and δ2>0, and let γ>0 be given.
By Definition 2(iii), there exist a number ρ∈]0,δ1[ and points ωi′∈Ωi∩Bδ2(xˉ) and xi′∈X(i=1,…,n) with φ(max1≤i≤n∥xi′∥)<ρ such that
∩i=1n(Ωi−ωi′−xi′)∩(ρB)=∅.
Thus, ξ:=max1≤i≤n∥xi′∥>0 and
ξ<φ−1(ρ)<φ−1(δ1).
Set xi:=ωi′+xi′−xˉ(i=1,…,n).
Then
[TABLE]
Let λ∈]φ(ξ),δ1[ and λ′:=min{λ,ρ}.
Then ∩i=1n(Ωi−xi)∩Bλ′(xˉ)=∅, and consequently, condition (19) holds true.
Let f and f be defined by (17) and (18), respectively, while Xn+1 be equipped with the metric induced by the norm (12).
We have f(ω1′,…,ωn′,xˉ)=φ(max1≤i≤n∥xi′∥)=φ(ξ)<λ′.
Choose a number ε such that
f(ω1′,…,ωn′,xˉ)<ε<λ′.
Applying the Ekeland variational principle, we can find points ωi∈Ωi(i=1,…,n) and x∈X such that
[TABLE]
and condition (21) holds
for all u∈X and ui∈Ωi(i=1,…,n).
In view of (19) and the definitions of λ′ and f, conditions (39) yield (37) and (38).
Since ε/λ′<1, condition (21) contradicts (14).
∎
The next statement is a localized version of Theorem 4.3.
Corollary 10
(i)
*The collection {Ω1,…,Ωn} is *φ−transversal at xˉ with some δ1>0 and δ2>0 if, for some γ>0 and any ωi′∈Ωi∩Bδ2(xˉ)(i=1,…,n) and ξ∈]0,φ−1(δ1)[, there exists a λ∈]φ(ξ),δ1[ such that
inequality (22) holds for all x,xi∈X and ωi∈Ωi(i=1,…,n) satisfying
(37) and (38).
2. (ii)
If φ∈C1, then inequality (22) in part (i) can be replaced by (23).
In the Hölder setting, Theorem 4.3 and Corollary 10 yield the following statement.
In view of Remark 4, we assume that q≤1.
Corollary 11
Let α>0 and q∈]0,1].
The collection {Ω1,…,Ωn} is α−transversal of order q at xˉ with some δ1>0 and δ2>0 if, for some γ>0 and any ωi′∈Ωi∩Bδ2(xˉ)(i=1,…,n) and ξ∈]0,(αδ1)q1[, there exists a λ∈]α−1ξq,δ1[ such that inequality (24) holds true for all x,xi∈X and ωi∈Ωi(i=1,…,n) satisfying (37) and (38), or all the more, such that inequality (25) holds true.
Remark 11
(i)
On top of the explicitly given restriction ∥ωi−ωi′∥<λ/γ in Theorem 4.3 (and similar conditions in its corollaries), which involves γ, the other conditions implicitly impose another one:
[TABLE]
This alternative restriction can be of importance when γ is small.
2. (ii)
It can be observed from the proof of Theorem 4.3 that the sufficient conditions for φ−transversality can be strengthened by adding another restriction on the choice of ξ and xi:
φ(ξ)<d(xˉ,∩i=1n(Ωi−xi)).
3. (iii)
The sufficient conditions for φ−semitransversality and φ−subtransversality in Theorems 4.1 and 4.2 are particular cases of those in Theorem 4.3,
corresponding to setting ωi′:=xˉ and x1=…=xn, respectively.
4. (iv)
The statement of Theorem 4.3 and its corollaries can be simplified by dropping condition (38).
5. (v)
Inequalities (14), (22)–(25), which are crucial for checking nonlinear transversality, involve a collection of parameters: x,xi∈X and ωi∈Ωi, which are related to another collection: a small number ξ>0 and points ωi′∈Ωi near xˉ.
The value of ξ and magnitudes of ωi′−xˉ are directly controlled by the values of δ1 and δ2 in the definition of φ−transversality: φ(ξ)<δ1 and ωi′∈Bδ2(xˉ).
At the same time, taking into account that λ can be made arbitrarily close to φ(ξ), the magnitudes of x−xˉ, ωi−ωi′ and xi are determined by δ1 and δ2 indirectly; they are controlled by ξ: cf. conditions (37) and (38).
Thus, the derived parameters x,xi∈X and ωi∈Ωi involved in (14) possess the natural properties: when δ1 and δ2 are small, the points x and ωi are near xˉ and the vectors xi are small.
6. (vi)
In view of the definition of the parametric norm (12), if any of the inequalities (14), (22)–(25) holds true for some γ>0, then it also holds for any γ′∈]0,γ[.
7. (vii)
Even in the linear setting, the characterizations in Corollary 11 are new.
The next corollary provides a simplified (and weaker!) version of Theorem 4.3; cf. Remark 11(v).
Corollary 12
The collection {Ω1,…,Ωn} is φ−transversal at xˉ with some δ1>0 and δ2>0 if, for some γ>0, inequality (14) holds for all x∈Bδ1(xˉ), xi∈X and ωi∈Ωi∩Bδ2+δ1/γ(xˉ)(i=1,…,n) satisfying φ(max1≤i≤nd(xi+xˉ,Ωi))<δ1 and 0<max1≤i≤n∥ωi−xi−x∥<φ−1(δ1).
Proof
Let δ1>0, δ2>0, ωi′∈Ωi∩Bδ2(xˉ), ξ∈]0,φ−1(δ1)[, λ∈]φ(ξ),δ1[, and points x,xi∈X and ωi∈Ωi(i=1,…,n) satisfy conditions (37) and (38).
Then
[TABLE]
i.e. points x,xi∈X and ωi∈Ωi(i=1,…,n) satisfy all the conditions in the corollary.
Hence, inequality (14) holds.
It follows from Theorem 4.3 that {Ω1,…,Ωn} is φ−transversal at xˉ with δ1 and δ2.
∎
Sacrificing the estimates for δ1 and δ2 in Theorem 4.3, and Corollaries 10 and
12, we can formulate the following ‘δ-free’ statement.
Corollary 13
The collection {Ω1,…,Ωn} is φ−transversal at xˉ if,
for some γ>0 and all x∈X near xˉ, xi∈X(i=1,…,n) near [math] and ωi∈Ωi(i=1,…,n) near xˉ satisfying max1≤i≤n∥ωi−xi−x∥>0, inequality (14) holds true.
Moreover,
inequality (14) can be replaced by its localized version (22), or by (23) if φ∈C1.
Remark 12
The sufficient conditions for φ−semitransversality and φ−transversality in Theorems 4.1 and 4.3 and their corollaries use the same (slope) inequalities (14), (22) and (23).
Nevertheless, the sufficient conditions in Theorem 4.3 and Corollary 13 are stronger than the corresponding ones in Theorem 4.1 and Corollary 5, respectively, as they require the inequalities to be satisfied on a larger set of points.
This is natural as φ−transversality is a stronger property than φ−semitransversality.
At the same time, the ‘δ-free’ versions in Corollaries 5 and 13 are almost identical: the only difference is the additional condition
[TABLE]
in Corollary 5.
The sufficient condition in Corollary 13 is still acceptable for characterizing φ−transversality, but the one in Corollary 5 seems a little too strong for φ−semitransversality.
That is why we prefer not to oversimplify these sufficient conditions.
5 Transversality and Regularity
In this section, we provide quantitative relations between the nonlinear transversality of collections of sets and the corresponding nonlinear regularity properties of set-valued mappings.
Besides, nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe Iof17 are discussed.
5.1 Regularity of Set-Valued Mappings
Our model here is a set-valued mapping F:X⇉Y between metric spaces.
We consider its local regularity properties near a given point (xˉ,yˉ)∈gphF.
The nonlinearity in the definitions of the properties is determined by a function φ∈C.
φ−semiregular at (xˉ,yˉ) if there exists a δ>0 such that
[TABLE]
for all y∈Y with φ(d(y,yˉ))<δ;
2. (ii)
φ−subregular at (xˉ,yˉ) if there exist δ1>0 and δ2>0 such that
[TABLE]
for all x∈Bδ2(xˉ) with φ(d(yˉ,F(x)))<δ1;
3. (iii)
φ−regular at (xˉ,yˉ) if there exist δ1>0 and δ2>0 such that
[TABLE]
for all x∈X and y∈Y with d(x,xˉ)+d(y,yˉ)<δ2 and φ(d(y,F(x)))<δ1.
The function φ∈C in the above definition plays the role of a kind of rate or modulus of the respective property.
In the Hölder setting, i.e. when
φ(t):=α−1tq with α>0 and q>0, we refer to the respective properties in Definition 3 as α−semiregularity, α−subregularity and α−regularity of order q.
These regularity properties have been studied in KruTha14 ; FraQui12 ; GayGeoJea11 ; Kum09 ; Kru16.2 ; LiMor12 .
It is usually assumed that q≤1.
The exact upper bound of all α>0 such that a property holds with some δ>0, or δ1>0 and δ2>0, is called the modulus of this property.
We use notations sergq[F](xˉ,yˉ), srgq[F](xˉ,yˉ) and rgq[F](xˉ,yˉ) for the moduli of the respective properties.
If a property does not hold, then by convention the respective modulus equals 0.
With q=1 (linear case), the properties are called metric semiregularity, subregularity and regularity, respectively; cf. Mor06.1 ; DonRoc14 ; RocWet98 ; Iof17 ; Kru09 ; CibFabKru19 .
The following assertion is a direct consequence of Definition 3.
Proposition 8
If F is φ−regular at (xˉ,yˉ) with some δ1>0 and δ2>0, then it is φ−semiregular at (xˉ,yˉ) with δ:=min{δ1,φ(δ2)} and φ−subregular at (xˉ,yˉ) with δ1 and δ2.
Note the combined inequality d(x,xˉ)+d(y,yˉ)<δ2 employed in part (iii) of Definition 3 instead of the more traditional separate conditions x∈Bδ2(xˉ) and y∈Bδ2(yˉ).
This replacement does not affect the property of φ−regularity itself, but can have an effect on the value of δ2.
Employing this inequality makes the property a direct analogue of the metric characterization of φ−transversality in Theorem 3.2 and is convenient for establishing relations between the regularity and transversality properties.
The next proposition provides also an important special case when the point x in (40) can be fixed: x=xˉ.
Proposition 9
Let δ1>0 and δ2>0.
Consider the following conditions:
(a)
inequality (40) holds
for all x∈Bδ2(xˉ) and y∈Bδ2(yˉ) with φ(d(y,F(x)))<δ1;
2. (b)
inequality (40) holds
for all x∈X and y∈Y with
d(x,xˉ)+d(y,yˉ)<δ2 and φ(d(y,F(x)))<δ1;
3. (c)
d(xˉ,F−1(y))≤φ(d(y,F(xˉ)))*
for all y∈Bδ2(yˉ) with φ(d(y,F(xˉ)))<δ1.*
Then
(i)
(a) \Rightarrow\(b) \Rightarrow\(c).
Moreover, condition (b) implies (a) with δ2′:=δ2/2 in place of δ2.
2. (ii)
If X is a normed space, Y=Xn for some n∈N, yˉ=(xˉ1,…,xˉn) and F:X⇉Xn is given by
[TABLE]
where Ω1,…,Ωn⊂X,
then (b) \Leftrightarrow\(c).
Proof
(i)
All the implications are straightforward.
2. (ii)
In view of (i), we only need to prove (c) \Rightarrow\(b).
Suppose condition (c) is satisfied.
Let x∈X, y=(x1,…,xn)∈Xn, ∥x−xˉ∥+∥y−yˉ∥<δ2 and φ(d(y,F(x)))<δ1.
Set xi′:=xi+x−xˉ(i=1,…,n) and y′:=(x1′,…,xn′).
Then
[TABLE]
and, thanks to (c), d(x,F−1(y))≤φ(d(y,F(x))).
∎
The set-valued mapping (41) plays the key role in establishing relations between the regularity and transversality properties.
It was most likely first used by Ioffe in Iof00 .
Observe that
F−1(x1,…,xn)=(Ω1−x1)∩…∩(Ωn−xn) for all x1,…,xn∈X and,
if xˉ∈∩i=1nΩi, then (0,…,0)∈F(xˉ).
Theorem 5.1
Let Ω1,…,Ωn be subsets of a normed space X, xˉ∈∩i=1nΩi, φ∈C, and F be defined by (41).
(i)
*The collection {Ω1,…,Ωn} is *φ−*semitransversal at xˉ with some δ>0 if and only if F is *φ−semiregular at (xˉ,(0,…,0)) with δ.
2. (ii)
*The collection {Ω1,…,Ωn} is *φ−*subtransversal at xˉ with some δ1>0 and δ2>0 if and only if F is *φ−subregular at (xˉ,(0,…,0)) with δ1 and δ2.
3. (iii)
*If {Ω1,…,Ωn} is *φ−*transversal at xˉ with some δ1>0 and δ2>0,
then F is *φ−regular at (xˉ,(0,…,0)) with
any δ1′∈]0,δ1] and δ2′>0 satisfying φ−1(δ1′)+δ2′≤δ2.
*Conversely, if F is *φ−*regular at (xˉ,(0,…,0)) with some δ1>0 and δ2>0, then {Ω1,…,Ωn} is *φ−transversal at xˉ with any δ1′∈]0,δ1] and δ2′>0 satisfying φ−1(δ1′)+δ2′≤δ2.
Proof
(i) and (ii) follow from Theorem 3.1(i) and (ii), respectively, while (iii) is a consequence of Theorem 3.2.
∎
The next corollary provides δ−free versions of the assertions in Theorem 5.1.
Corollary 14
Let Ω1,…,Ωn be subsets of a normed space X, xˉ∈∩i=1nΩi, φ∈C, and F be defined by (41).
The collection {Ω1,…,Ωn} is
(i)
φ−*semitransversal at xˉ if and only if F is *φ−semiregular at (xˉ,(0,…,0));
2. (ii)
φ−*subtransversal at xˉ if and only if F is *φ−subregular at (xˉ,(0,…,0));
3. (iii)
φ−*transversal at xˉ if and only if F is *φ−regular at (xˉ,(0,…,0)).
Remark 13
(i)
In the Hölder setting Corollary 14 reduces to (KruTha14, , Proposition 9).
2. (ii)
Apart from the mapping F defined by (41), in the case of two sets other set-valued mappings can be used to ensure similar equivalences between the transversality and regularity properties; see Iof17 .
In view of Theorem 5.1, the nonlinear transversality properties of collections of sets can be viewed as particular cases of the corresponding nonlinear regularity properties of set-valued mappings.
We are going to show that the two popular models are in a sense equivalent.
Given an arbitrary set-valued mapping F:X⇉Y between metric spaces and a point (xˉ,yˉ)∈gphF, we can consider the two sets:
[TABLE]
in the product space X×Y.
Note that (xˉ,yˉ)∈Ω1∩Ω2=F−1(yˉ)×{yˉ}.
To establish the relationship between the two sets of properties, we have to assume in the next two theorems that X and Y are normed vector spaces.
Theorem 5.2
Let X and Y be normed spaces, F:X⇉Y, (xˉ,yˉ)∈gphF, and φ∈C.
Let Ω1 and Ω2 be defined by (42), and ψ(t):=φ(2t)+t for all t≥0.
(i)
*If F is *φ−*semiregular at (xˉ,yˉ) with some δ>0, then {Ω1,Ω2} is *ψ−semitransversal at (xˉ,yˉ) with δ′:=δ+φ−1(δ)/2.
2. (ii)
*If F is *φ−*subregular at (xˉ,yˉ) with some δ1>0 and δ2>0, then {Ω1,Ω2} is *ψ−subtransversal at (xˉ,yˉ) with any δ1′>0 and δ2′>0 such that φ(2ψ−1(δ1′))≤δ1 and
ψ−1(δ1′)+δ2′≤δ2.
3. (iii)
*If F is *φ−*regular at (xˉ,yˉ) with some δ1>0 and δ2>0, then {Ω1,Ω2} is *ψ−transversal at (xˉ,yˉ) with any δ1′>0 and δ2′>0 such that
φ(2ψ−1(δ1′))≤δ1 and
ψ−1(δ1′)+δ2′≤δ2/2.
Proof
Observe that ψ∈C, φ(2ψ−1(t))+ψ−1(t)=t and ψ(φ−1(t)/2)=t+φ−1(t)/2 for all t≥0.
(i)
Let F be φ−semiregular at (xˉ,yˉ) with some δ>0.
Set δ′:=δ+φ−1(δ)/2=ψ(φ−1(δ)/2).
Let ρ∈]0,δ′[ and (u1,v1),(u2,v2)∈ψ−1(ρ)B.
Set y′:=yˉ+v1−v2.
Observe that
[TABLE]
We have
∥y′−yˉ∥=∥v1−v2∥≤∥v1∥+∥v2∥<2ψ−1(ρ),
and consequently, φ(∥y′−yˉ∥)<φ(2ψ−1(ρ))<φ(2ψ−1(δ′))=δ.
By Definition 3(i),
[TABLE]
and consequently,
[TABLE]
hence,
[TABLE]
By Definition 2(i), {Ω1,Ω2} is φ−semitransversal at (xˉ,yˉ) with δ′.
2. (ii)
Let F be φ−subregular at (xˉ,yˉ) with some δ1>0 and δ2>0.
Choose numbers δ1′>0 and δ2′>0 such that
φ(2ψ−1(δ1′))≤δ1 and
ψ−1(δ1′)+δ2′≤δ2.
Let ρ∈]0,δ1′[ and (x,y)∈Bδ2′(xˉ,yˉ) with ψ(max{d((x,y),Ω1),d((x,y),Ω2)})<ρ, i.e. ∥y−yˉ∥<ψ−1(ρ) and there exists a point (x1,y1)∈gphF such that ∥(x,y)−(x1,y1)∥<ψ−1(ρ).
Then
[TABLE]
and consequently,
φ(d(yˉ,F(x1)))<φ(2ψ−1(ρ))<φ(2ψ−1(δ1′))≤δ1.
Choose a positive ε<2ψ−1(ρ)−d(yˉ,F(x1)).
By Definition 3(ii), there exists an x′∈F−1(yˉ) such that ∥x′−x1∥<φ(d(yˉ,F(x1))+ε)<φ(2ψ−1(ρ)).
Hence, (x′,yˉ)∈Ω1∩Ω2 and
[TABLE]
Thus, Ω1∩Ω2∩Bρ(x,y)=∅.
By Definition 2(ii), {Ω1,Ω2} is ψ−subtransversal at (xˉ,yˉ) with δ1′ and δ2′.
3. (iii)
Let F be φ−regular at (xˉ,yˉ) with some δ1>0 and δ2>0.
Choose numbers δ1′>0 and δ2′>0 such that
φ(2ψ−1(δ1′))≤δ1 and
ψ−1(δ1′)+δ2′≤δ2/2.
Let ρ∈]0,δ1′[, (x1,y1)∈gphF∩Bδ2′(xˉ,yˉ), x2∈Bδ2′(xˉ) and (u1,v1),(u2,v2)∈ψ−1(ρ)B.
Set y′:=y1+v1−v2.
Then
[TABLE]
Choose a positive ε<2(ψ−1(ρ)−max{∥v1∥,∥v2∥}).
By Definition 3(iii), there exists an x′∈F−1(y′) such that
[TABLE]
Denote x^:=x′−x1−u1 and y^:=y′−y1−v1.
Thus, (x′,y′)∈Ω1 and (x^,y^)∈Ω1−(x1,y1)−(u1,v1).
At the same time, y^=−v2 and (x^,y^)∈Ω2−(x2,yˉ)−(u2,v2).
Moreover,
[TABLE]
hence (x′,y′)∈ρB.
By Definition 2(iii), {Ω1,Ω2} is ψ−transversal at (xˉ,yˉ) with δ1′ and δ2′.
∎
Theorem 5.3
Let X and Y be normed spaces, F:X⇉Y, (xˉ,yˉ)∈gphF, and φ∈C.
Let Ω1 and Ω2 be defined by (42), and ψ(t):=φ(t/2) for all t≥0.
(i)
*If {Ω1,Ω2} is *φ−*semitransversal at (xˉ,yˉ) with some δ>0, then F is *ψ−semiregular at (xˉ,yˉ) with δ.
2. (ii)
*If {Ω1,Ω2} is *φ−*subtransversal at (xˉ,yˉ) with some δ1>0 and δ2>0, then F is *ψ−subregular at (xˉ,yˉ) with δ1′:=min{δ1,ψ(2δ2)} and δ2.
3. (iii)
*If {Ω1,Ω2} is *φ−*transversal at (xˉ,yˉ) with some δ1>0 and δ2>0, then F is *ψ−regular at (xˉ,yˉ) with any δ1′∈]0,δ1] and δ2′>0 such that
ψ−1(δ1′)+δ2′≤δ2.
Proof
Observe that ψ∈C.
(i)
Let {Ω1,Ω2} be φ−semitransversal at (xˉ,yˉ) with some δ>0.
By Definition 2(i), condition (43) is satisfied for all ρ∈]0,δ[ and (u1,v1),(u2,v2)∈φ−1(ρ)B.
Let y∈Y with ρ0:=ψ(∥y−yˉ∥)<δ.
Choose a ρ∈]ρ0,δ[ and observe that
[TABLE]
In view of (43), we can find (x1,y1)∈gphF and x2∈X such that
[TABLE]
Hence, y1=yˉ+22y−yˉ=y, x1∈F−1(y), ∥x1−xˉ∥<ρ, and consequently, d(xˉ,F−1(y))<ρ.
Letting ρ↓ρ0, we obtain d(xˉ,F−1(y))≤ψ(∥y−yˉ∥).
By Definition 3(i), F is ψ−semiregular at (xˉ,yˉ) with δ.
2. (ii)
Let {Ω1,Ω2} be φ−subtransversal at (xˉ,yˉ) with some δ1>0 and δ2>0.
By Definition 2(ii), gphF∩(X×{yˉ})∩Bρ(x,y)=∅ for all ρ∈]0,δ1[ and (x,y)∈Bδ2(xˉ,yˉ) with φ(d((x,y),gphF))<ρ and φ(∥y−yˉ∥)<ρ.
Set δ1′:=min{δ1,ψ(2δ2)}.
Let x∈Bδ2(xˉ) and ψ(d(yˉ,F(x)))<δ1′.
Choose a y∈F(x) such that
ρ0:=ψ(∥yˉ−y∥)<δ1′, and a
ρ∈]ρ0,δ1′[.
Set y^:=2y+yˉ.
Observe that
[TABLE]
Thus, ρ∈]0,δ1[, (x,y^)∈Bδ2(xˉ,yˉ), φ(d((x,y^),gphF))≤φ(∥y^−y∥)<ρ and φ(∥y^−yˉ∥)<ρ.
Hence, gphF∩(X×{yˉ})∩Bρ(x,y^)=∅, and consequently, d(x,F−1(yˉ))<ρ.
Letting ρ↓ρ0, we obtain d(x,F−1(yˉ))≤ψ(∥yˉ−y∥).
Taking the infimum in the right-hand side of this inequality over y∈F(x), we conclude that
F is ψ−subregular at (xˉ,yˉ) with δ1′ and δ2 in view of Definition 3(ii).
3. (iii)
Let {Ω1,Ω2} be φ−transversal at (xˉ,yˉ) with some δ1>0 and δ2>0, i.e. for all ρ∈]0,δ1[, (x′,y′)∈gphF∩Bδ2(xˉ,yˉ), u1∈X and v1,v2∈Y with φ(max{∥u1∥,∥v1∥,∥v2∥})<ρ, it holds
[TABLE]
or equivalently,
d(x′+u1,F−1(y′+v1−v2))<ρ.
In other words, d(x,F−1(y))<ρ
for all ρ∈]0,δ1[, (x′,y′)∈gphF∩Bδ2(xˉ,yˉ), x∈X and y∈Y with ∥x−x′∥<φ−1(ρ) and ∥y−y′∥<2φ−1(ρ).
Choose numbers δ1′∈]0,δ1] and δ2′>0 such that ψ−1(δ1′)+δ2′≤δ2.
Let x∈X and y∈Y with ∥x−xˉ∥+∥y−yˉ∥<δ2′ and ψ(d(y,F(x)))<δ1′.
Choose a y′∈F(x) such that ρ0:=ψ(∥y−y′∥)<δ1′ and a ρ∈]ρ0,δ1′[.
Then ρ∈]0,δ1[, (x,y′)∈gphF,
∥x−xˉ∥<δ2′<δ2, ∥y′−yˉ∥≤∥y′−y∥+∥y−yˉ∥<ψ−1(δ1′)+δ2′≤δ2 and ∥y−y′∥<ψ−1(ρ)=2φ−1(ρ).
Hence, d(x,F−1(y))<ρ.
Letting ρ↓ρ0, we obtain d(x,F−1(y))≤ψ(∥y−y′∥).
Taking the infimum in the right-hand side of this inequality over y′∈F(x), we conclude that F is ψ−regular at (xˉ,yˉ) with δ1′ and δ2′ in view of Definition 3(iii).
∎
The next corollary of Theorems 5.2 and 5.3 provides qualitative relations between the regularity and transversality properties.
Corollary 15
Let X and Y be normed spaces, F:X⇉Y, (xˉ,yˉ)∈gphF, and φ∈C.
Let Ω1 and Ω2 be defined by (42), ψ1(t):=φ(2t)+t and ψ2(t):=φ(t/2) for all t≥0.
(i)
If F is φ−(semi-/sub-)regular at (xˉ,yˉ), then {Ω1,Ω2} is ψ1−(semi-/sub-)transversal at (xˉ,yˉ).
2. (ii)
If {Ω1,Ω2} is φ−(semi-/sub-)transversal at (xˉ,yˉ), then F is ψ2−(semi-/sub-)regular at (xˉ,yˉ).
The next statement addresses the Hölder setting.
It is a consequence of Theorems 5.2 and 5.3 with φ(t):=α−1tq for some α>0, q>0 and all t≥0.
Corollary 16
Let X and Y be normed spaces, F:X⇉Y, (xˉ,yˉ)∈gphF, α>0 and q>0.
Let Ω1 and Ω2 be defined by (42), α1:=2−qα, α2:=2qα, and ψ(t):=α1−1tq+t for all t≥0.
(i)
*If F is *α−*semiregular of order q at (xˉ,yˉ) with some δ>0, then {Ω1,Ω2} is *ψ−semitransversal at (xˉ,yˉ) with δ′:=δ+(αδ)q1/2.
*If {Ω1,Ω2} is *α−*semitransversal of order q at (xˉ,yˉ) with some δ>0, then F is *α2−semiregular of order q at (xˉ,yˉ) with δ.
2. (ii)
*Let q≤1.
If F is *α−*subregular of order q at (xˉ,yˉ) with some δ1>0 and δ2>0, then {Ω1,Ω2} is *ψ−subtransversal at (xˉ,yˉ) with any δ1′>0 and δ2′>0 such that (2ψ−1(δ1′))q≤αδ1 and
ψ−1(δ1′)+δ2′≤δ2.
*If {Ω1,Ω2} is *α−*subtransversal of order q at (xˉ,yˉ) with some δ1>0 and δ2>0,
then F is *α2−subregular of order q at (xˉ,yˉ) with δ1′:=min{δ1,α−1δ2q} and δ2.
3. (iii)
*Let q≤1.
If F is *α−*regular of order q at (xˉ,yˉ) with some δ1>0 and δ2>0, then {Ω1,Ω2} is *ψ−transversal at (xˉ,yˉ) with any δ1′>0 and δ2′>0 such that
(2ψ−1(δ1′))q≤αδ1 and
ψ−1(δ1′)+δ2′≤δ2/2.
*If {Ω1,Ω2} is *α−*transversal of order q at (xˉ,yˉ) with some δ1>0 and δ2>0,
then F is *α2−regular of order q
at (xˉ,yˉ) with
any δ1′∈]0,δ1] and δ2′>0 such that 2(αδ1′)q1+δ2′≤δ2.
In view of Corollary 16, Hölder transversality properties of {Ω1,Ω2} imply the corresponding Hölder regularity properties of F, while Hölder regularity properties of F imply certain ‘Hölder-type’ transversality properties of {Ω1,Ω2} determined by the function ψ.
Utilizing Proposition 2, they can be approximated by proper Hölder (or even linear) transversality properties.
Corollary 17
Let X and Y be normed spaces, F:X⇉Y, (xˉ,yˉ)∈gphF, α>0 and q>0.
Let Ω1 and Ω2 be defined by (42) and α1:=2−qα.
If
F is α−(semi-/sub-) transversal at (xˉ,yˉ), then {Ω1,Ω2} is α′−(semi-/sub-)transversal of order q′ at xˉ, where:
(i)
if q<1, then q′=q and α′ is any number in ]0,α1[;
2. (ii)
if q=1, then q′=1 and α′:=(1+α1−1)−1;
3. (iii)
if q>1, then q′=1 and α′ is any number in ]0,1[.
Thanks to Corollaries 16 and 17, in the case q∈]0,1] we have full equivalence between the two sets of properties.
The following corollary recaptures (KruTha14, , Proposition 10).
Corollary 18
Let X and Y be normed spaces, F:X⇉Y, (xˉ,yˉ)∈gphF, and q∈]0,1].
Let Ω1 and Ω2 be defined by (42).
(i)
{Ω1,Ω2}* is *α−semitransversal of order q at (xˉ,yˉ) if and only if F is semiregular of order q at (xˉ,yˉ).
Moreover,
[TABLE]
2. (ii)
{Ω1,Ω2}* is *α−subtransversal of order q at (xˉ,yˉ) if and only if F is subregular of order q at (xˉ,yˉ).
Moreover,
[TABLE]
3. (iii)
{Ω1,Ω2}* is *α−transversal of order q at (xˉ,yˉ) if and only if F is regular of order q at (xˉ,yˉ).
Moreover,
[TABLE]
5.2 Transversality of a Mapping to a Set in the Range Space
Finally, we briefly discuss metric characterizations of nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe Iof16 ; Iof17 .
For geometric and subdifferential/normal cone characterizations of the properties, we refer the reader to CuoKru4 ; CuoKru5 ; CuoKru6 .
In the rest of this section, F:X⇉Y is a set-valued mapping between normed spaces, (xˉ,yˉ)∈gphF, S is a subset of Y, yˉ∈S, and φ∈C.
Definition 4
The mapping F is
(i)
φ−semitransversal to S at (xˉ,yˉ) if {gphF,X×S} is φ−semitransversal at (xˉ,yˉ), i.e. there exists a δ>0 such that
[TABLE]
for all ρ∈]0,δ[, u1∈X, v1,v2∈Y with φ(max{∥u1∥,∥v1∥,∥v2∥})<ρ;
2. (ii)
φ−subtransversal to S at (xˉ,yˉ) if {gphF,X×S} is φ−subtransversal at (xˉ,yˉ), i.e. there exist δ1>0 and δ2>0 such that
[TABLE]
for all ρ∈]0,δ1[ and (x,y)∈Bδ2(xˉ,yˉ) with φ(max{d((x,y),gphF),d(y,S)})<ρ;
3. (iii)
φ−transversal to S at (xˉ,yˉ) if {gphF,X×S} is φ−transversal at (xˉ,yˉ), i.e. there exist δ1>0 and δ2>0 such that
[TABLE]
for all ρ∈]0,δ1[, (x1,y1)∈gphF∩Bδ2(xˉ,yˉ), y2∈S∩Bδ2(yˉ), u1∈X, v1,v2∈Y with φ(max{∥u1∥,∥v1∥,∥v2∥})<ρ.
The two-set model {gphF,X×S} employed in Definition 4 is an extension of the model (42), which corresponds to the case when S is a singleton: S:={yˉ}.
The metric characterizations of the properties in the next two statements are consequences of Theorems 3.1 and 3.2, respectively.
Each characterization can be used as an equivalent definition for the respective property.
Corollary 19
The mapping F is
(i)
φ−semitransversal to S at (xˉ,yˉ) with some δ>0 if and only if
[TABLE]
for all x1∈X, y1,y2∈Y with φ(max{∥x1∥,∥y1∥,∥y2∥})<δ;
2. (ii)
*is *φ−subtransversal to S at (xˉ,yˉ) with some δ1>0 and δ2>0 if and only if
the following equivalent conditions hold:
(a)
for all (x,y)∈Bδ2(xˉ,yˉ) with \varphi\big{(}\max\left\{d((x,y),{\rm gph}\,F),d(y,S)\right\}\big{)}<\delta_{1}, it holds
[TABLE]
2. (b)
for all (x1,y1)∈gphF∩Bδ2(xˉ,yˉ), y2∈S∩Bδ2(yˉ) and u1∈X, v1,v2∈Y with φ(max{∥u1∥,∥v1∥,∥v2∥})<δ1 and x1+u1∈Bδ2(xˉ),
y1+v1=y2+v2∈Bδ2(yˉ),
it holds
[TABLE]
3. (iii)
φ−transversal to S at (xˉ,yˉ) with some δ1>0 and δ2>0 if and only if inequality (44) holds for all (x1,y1)∈gphF∩Bδ2(xˉ,yˉ), y2∈S∩Bδ2(yˉ) and u1∈X, v1,v2∈Y with φ(max{∥u1∥,∥v1∥,∥v2∥})<δ1.
Corollary 20
Let δ1>0 and δ2>0.
The following conditions are equivalent:
(i)
for all (x1,y1)∈gphF∩Bδ2(xˉ,yˉ), y2∈S∩Bδ2(yˉ) and u1∈X, v1,v2∈Y with x1+u1∈Bδ2(xˉ), y1+v1,y2+v2∈Bδ2(yˉ) and φ(max{∥u1∥,∥v1∥,∥v2∥})<δ1, inequality (44) holds true;
2. (ii)
for all x1,y1,y2∈δ2B with
φ(max{d((xˉ,yˉ),gphF−(x1,y1)),d(yˉ,S−y2)})<δ1, it holds
[TABLE]
3. (iii)
for all x,x1∈X, y,y1,y2∈Y such that x+x1∈Bδ2(xˉ), y+y1,y+y2∈Bδ2(yˉ) and φ(max{d((x,y),gphF−(x1,y1)),d(y,S−y2)})<δ1, it holds
[TABLE]
Moreover, if F is φ−transversal to S at (xˉ,yˉ) with some δ1>0 and δ2>0, then conditions (i)–(iii) hold with any δ1′∈]0,δ1] and δ2′>0 satisfying φ−1(δ1′)+δ2′≤δ2 in place of δ1 and δ2.
Conversely, if conditions (i)–(iii) hold with some δ1>0 and δ2>0, then F is φ−transversal to S at (xˉ,yˉ) with any δ1′∈]0,δ1] and δ2′>0 satisfying φ−1(δ1′)+δ2′≤δ2.
Remark 14
In the linear case, i.e. when φ(t):=αt for some α>0 and all t≥0,
in view of Corollaries 19(ii)(a) and 20(iii),
the properties in parts (ii) and (iii) of Definition 4 reduce, respectively, to the ones in (Iof17, , Definitions 7.11 and 7.8).
The property in part (i) is new.
The set-valued mapping (41), crucial for establishing equivalences between transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings, in the setting considered here translates into the mapping G:X×Y⇉(X×Y)×(X×Y) of the following form:
[TABLE]
Observe that
G^{-1}(x_{1},y_{1},x_{2},y_{2})=\big{(}{\rm gph}\,F-(x_{1},y_{1})\big{)}\cap\big{(}X\times(S-y_{2})\big{)} for all x1,x2∈X, y1,y2∈Y and,
if (xˉ,yˉ)∈gphF, yˉ∈S, then \big{(}(0,0),(0,0)\big{)}\in G(\bar{x},\bar{y}).
The relationships between the nonlinear transversality and regularity properties in the next statement are direct consequences of Theorem 5.1.
F* is *φ−*semitransversal to S at (xˉ,yˉ) with some δ>0 if and only if G is *φ−semiregular at \big{(}(\bar{x},\bar{y}),(0,0),(0,0)\big{)} with δ.
2. (ii)
F* is *φ−*subtransversal to S at (xˉ,yˉ) with some δ1>0 and δ2>0 if and only if G is *φ−subregular at \big{(}(\bar{x},\bar{y}),(0,0),(0,0)\big{)} with δ1 and δ2.
3. (iii)
*If F is *φ−*transversal to S at (xˉ,yˉ) with some δ1>0 and δ2>0, then G is *φ−regular at \big{(}(\bar{x},\bar{y}),(0,0),(0,0)\big{)} with
any δ1′∈]0,δ1] and δ2′>0 satisfying δ2′+φ−1(δ1′)≤δ2.
*Conversely, if G is *φ−*regular at \big{(}(\bar{x},\bar{y}),(0,0),(0,0)\big{)} with some δ1>0 and δ2>0, then F is *φ−transversal to S at (xˉ,yˉ) with any δ1′∈]0,δ1] and δ2′>0 satisfying δ2′+φ−1(δ1′)≤δ2.
Remark 15
It is easy to see that the set-valued mapping (46) can be replaced in our considerations by the truncated mapping G:X×Y⇉X×Y×Y defined by
[TABLE]
The last mapping admits a simple representation G(x,y)=gphF−(x,y,y), where
the set-valued mapping F:X⇉Y×Y is defined by
[TABLE]
It was shown in (Iof17, , Theorems 7.12 and 7.9) that in the linear case the subtransversality and transversality of F to S at (xˉ,yˉ) are equivalent to the metric subregularity and regularity, respectively, of the mapping (x,y)↦F(x)−(y,y) at ((xˉ,yˉ),0).
Acknowledgement
The authors wish to thank the referee and the handling editor for their careful reading of the manuscript and valuable comments and suggestions.
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