Transversality Properties: Primal Sufficient Conditions
aa r X i v : . [ m a t h . O C ] J un Noname manuscript No. (will be inserted by the editor)
Transversality Properties: Primal Sufficient Conditions
Nguyen Duy Cuong · Alexander Y. Kruger
Received: date / Accepted: date
Abstract
The paper studies ‘good arrangements’ (transversality properties) of collectionsof 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 H¨older case is given a special attention. Our main objective is not formally extendingour earlier results from the H¨older to a more general nonlinear setting, but rather to developa general framework for quantitative analysis of transversality properties. The nonlinearityis just a simple setting, which allows us to unify the existing results on the topic. Unlikethe well-studied subtransversality property, not many characterizations of the other twoimportant properties: semitransversality and transversality have been known even in thelinear case. Quantitative relations between nonlinear transversality properties and the cor-responding regularity properties of set-valued mappings as well as nonlinear extensions ofthe new transversality properties of a set-valued mapping to a set in the range space due toIoffe are also discussed.
Keywords
Transversality · Subtransversality · Semitransversality · Regularity · Subregu-larity · Semiregularity · Slope · Chain Rule
Mathematics Subject Classification (2010)
Primary 49J52 · · Secondary 49K40 · · This paper continues a series of publications by the authors [10, 14–16, 30–32, 36–44, 57]dedicated to studying ‘good arrangements’ of collections of sets in normed spaces neara point in their intersection. Following Ioffe [28], such arrangements are now commonlyreferred to as transversality properties. Here we refer to transversality broadly as a group
The research was supported by the Australian Research Council, project DP160100854. The second author ben-efited from the support of the FMJH Program PGMO and from the support of EDF.Nguyen Duy CuongCentre for Informatics and Applied Optimization, School of Science, Engineering and Information Technology,Federation University Australia, POB 663, Ballarat, Vic, 3350, AustraliaDepartment of Mathematics, College of Natural Sciences, Can Tho University, Can Tho City, VietnamEmail: [email protected], [email protected] Y. Kruger ( (cid:0) )Centre for Informatics and Applied Optimization, School of Science, Engineering and Information Technology,Federation University Australia, POB 663, Ballarat, Vic, 3350, AustraliaEmail: [email protected], Nguyen Duy Cuong, Alexander Y. Kruger of ‘good arrangement’ properties, which includes semitransversality , subtransversality , transversality (a specific property) and some others. The term regularity was extensivelyused for the same purpose in the earlier publications by the second author, and is stillpreferred by many authors.Transversality (regularity) properties of collections of sets play an important role inoptimization and variational analysis, e.g., as constraint qualifications, qualification con-ditions in subdifferential, normal cone and coderivative calculus, and convergence analysisof computational algorithms. Significant efforts have been invested into studying this classof properties and establishing their primal and dual necessary and/or sufficient character-izations in various settings (convex and nonconvex, finite and infinite dimensional, finiteand infinite collections of sets). In addition to the references provided above, we referthe readers to [3–5, 7, 8, 19, 20, 24, 25, 46, 50, 51, 53, 55, 59, 60] for results and historicalcomments.Our aim is to develop a general framework for quantitative analysis of transver-sality properties of collections of sets. In this paper, we focus on primal space con-ditions, and establish metric characterizations and slope-type sufficient conditions forthree closely related general nonlinear transversality properties: ϕ − semitransversality , ϕ − subtransversality and ϕ − transversality .The slope sufficient conditions stem from applying the Ekeland variational principleto the definitions of the respective properties; the proofs are rather straightforward. Thistype of conditions are often considered as just a first step on the way to producing moreinvolved dual (subdifferential and normal cone) conditions, and the primal sufficient con-ditions remain hidden in the proofs. We believe that primal conditions (being in a senseanalogues of very popular slope conditions for error bounds) can be of importance for ap-plications. Moreover, subdividing the conventional regularity/transversality theory into theprimal and dual parts clarifies the roles of the main tools employed within the theory: theEkeland variational principle in the primal part and subdifferential sum rules in the dualpart. As a result, the proofs in both the primal and the ‘more involved’ dual parts becomestraightforward. This observation goes beyond the transversality of collections of sets andapplies 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 1.1 and 2.1. This can be of importance from the computational point of view.We also examine quantitative relations between the nonlinear transversality properties ofcollections of sets and the corresponding nonlinear regularity properties of set-valued map-pings as well as nonlinear extensions of the new transversality properties of a set-valuedmapping to a set in the range space due to Ioffe.We would like to emphasize that our main objective is not formally extending ourearlier results from the H¨older to a more general nonlinear setting, but rather to developa comprehensive theory of transversality. The nonlinearity is just a simple setting, whichallows 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 H¨oldersettings (see, for instance, [3, 5, 19, 24, 25, 39, 42, 50, 59]), for the other two properties:semitransversality and transversality not many characterizations have been known even inthe linear case; we fill this gap in the current paper.Besides the conventional H¨older case, which is given a special attention in the paper,our general model covers also so called H¨older-type settings [6,47] that have recently comeinto 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. Somecharacterizations are new even in the linear setting.Apart from being of interest on their own, the slope sufficient conditions for nonlineartransversality properties established in this paper lay the foundation for the dual sufficient ransversality Properties: Primal Sufficient Conditions 3 conditions for the respective properties in Banach and Asplund spaces in [14]. Primal anddual necessary conditions for the nonlinear transversality properties are studied in [15,16].There exist strong connections between transversality properties of collections of setsand the corresponding regularity properties of set-valued mappings. In this paper, we es-tablish quantitative relations between the two models in the general nonlinear setting.Nonlinear regularity properties of set-valued mappings and closely related error boundproperties of (extended-)real-valued functions have been intensively studied since 1980s;cf. [1, 2, 9, 13, 21–23, 26, 34, 35, 42, 45, 48, 54, 58, 61]. The slope sufficient conditions for ϕ − subtransversality in Section 4 can be interpreted in terms of the corresponding condi-tions for nonlinear error bounds. The semitransversality and transversality properties donot have exact counterparts within the conventional error bound theory.As in most of our previous publications on the topic, our working model in this paperis a collection of n ≥ Ω , . . . , Ω n of a normed vector space X , havinga common point ¯ x ∈ ∩ ni = Ω i . The next definition introduces three most common H¨oldertransversality properties. It is a modification of [42, Definition 1]. Definition 1.1
Let α > q >
0. The collection { Ω , . . . , Ω n } is(i) α − semitransversal of order q at ¯ x if there exists a δ > n \ i = ( Ω i − x i ) ∩ B ρ ( ¯ x ) = /0 (1)for all ρ ∈ ] , δ [ and x i ∈ X ( i = , . . . , n ) with max ≤ i ≤ n k x i k q < αρ ;(ii) α − subtransversal of order q at ¯ x if there exist δ > δ > n \ i = Ω i ∩ B ρ ( x ) = /0 (2)for all ρ ∈ ] , δ [ and x ∈ B δ ( ¯ x ) with max ≤ i ≤ n d q ( x , Ω i ) < αρ ;(iii) α − transversal of order q at ¯ x if there exist δ > δ > n \ i = ( Ω i − ω i − x i ) ∩ ( ρ B ) = /0 (3)for all ρ ∈ ] , δ [ , ω i ∈ Ω i ∩ B δ ( ¯ x ) and x i ∈ X ( i = , . . . , n ) with max ≤ i ≤ n k x i k q < αρ .The three properties in the above definition were referred to in [42] as [ q ] − semi-regularity, [ q ] − subregularity and [ q ] − regularity, respectively. Property (ii) was definedin [42] in a slightly different but equivalent way, under an additional assumption that q ≤ ∩ ni = Ω i is closed and ¯ x ∈ bd ∩ ni = Ω i , the condition q ≤ α − subtransversality and α − transversality properties; see Remark 2.3. At the same time,as observed in [42], the property of α − semitransversality can be meaningful with anypositive q (and any positive α ); see Example 2.1.With q = S and (UR) S ]), while property (ii) first appeared in [43]. If ∩ ni = Ω i is closed and ¯ x ∈ bd ∩ ni = Ω i , then one can observe that properties (ii) and (iii) canonly hold with α ≤
1; see Remark 2.3. If q =
1, when referring to the three properties inthe above definition, we talk simply about α − (semi-/sub-) transversality .If a collection { Ω , . . . , Ω n } is α − semitransversal (respectively, α − subtransversal or α − transversal) of order q at ¯ x with some α > δ > δ > δ > { Ω , . . . , Ω n } is semitransversal (respectively, subtransversal or transversal ) of order q at ¯ x . The number α characterizes the corresponding propertyquantitatively. The exact upper bound of all α > Nguyen Duy Cuong, Alexander Y. Kruger δ > δ > δ >
0) is called the modulus of this property. We use the nota-tions s e tr q [ Ω , . . . , Ω n ]( ¯ x ) , str q [ Ω , . . . , Ω n ]( ¯ x ) and tr q [ Ω , . . . , Ω n ]( ¯ x ) for the moduli of therespective properties. If the property does not hold, then by convention the respective mod-ulus equals 0.If q <
1, the H¨older transversality properties in Definition 1.1 are obviously weakerthan the corresponding conventional linear properties and can be satisfied for collectionsof sets when the conventional ones fail. This can happen in many natural situations (see ex-amples in [42, Section 2.3]), which explains the growing interest of researchers to studyingthe more subtle nonlinear transversality properties.Our basic notation is standard, see, e.g., [18, 49, 56]. Throughout the paper, X and Y are either metric or, more often, normed vector spaces. The open unit ball in any space isdenoted by B , and B δ ( x ) stands for the open ball with center x and radius δ >
0. If notexplicitly stated otherwise, products of normed vector spaces are assumed to be equippedwith the maximum norm k ( x , y ) k : = max {k x k , k y k} , ( 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 int Ω and bd Ω , respectively.The distance from a point x to Ω is defined by d ( x , Ω ) : = inf u ∈ Ω k u − x k , and we use theconvention d ( x , /0 ) = + ∞ . The indicator function of Ω is defined as follows: i Ω ( x ) = x ∈ Ω and i Ω ( x ) = + ∞ if x / ∈ Ω .For an extended-real-valued function f : X → R ∪ { + ∞ } , its domain and epigraphare defined, respectively, by dom f : = { x ∈ X | f ( x ) < + ∞ } and epi f : = { ( x , α ) ∈ X × R | f ( x ) ≤ α } . The inverse of f (if it exists) is denoted by f − . 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 gph F : = { ( x , y ) ∈ X × Y | y ∈ F ( x ) } anddom F : = { x ∈ X | F ( x ) = /0 } for the graph and the domain of F , respectively, and F − : Y ⇒ X for the inverse of F . This inverse (which always exists with possibly empty valuesat some y ) is defined by F − ( y ) : = { x ∈ X | y ∈ F ( x ) } , y ∈ Y . Obviously dom F − = F ( X ) .The closed and open intervals between points x and x in a normed space are defined,respectively, by [ x , x ] : = { tx + ( − t ) x | t ∈ [ , ] } , ] x , x [ : = { tx + ( − t ) x | t ∈ ] , [ } . The semi-open intervals ] x , x ] and [ x , x [ are defined in a similar way.The key tool in the proofs of the main results is the celebrated Ekeland variationalprinciple; cf. [18, 28, 49, 55]. Lemma 1.1
Suppose X is a complete metric space, f : X → R ∪ { + ∞ } is lower semicon-tinuous, x ∈ X, ε > and λ > . If f ( x ) < inf X f + ε , then there exists an ˆ x ∈ X such that (i) d ( ˆ x , x ) < λ ; (ii) f ( ˆ x ) ≤ f ( x ) ; (iii) f ( u ) + ( ε / λ ) d ( u , ˆ x ) ≥ f ( ˆ x ) for all u ∈ X . The slope [17] and nonlocal slope [33,52] of a function f : X → R ∪ { + ∞ } on a metricspace at x ∈ dom f are defined, respectively, by | ∇ f | ( x ) : = lim sup u → x , u = x [ f ( x ) − f ( u )] + d ( x , u ) , | ∇ f | ⋄ ( x ) : = sup u = x [ f ( x ) − f + ( u )] + d ( x , u ) , ransversality Properties: Primal Sufficient Conditions 5 where α + : = max { , α } for any α ∈ R . The limit | ∇ f | ( x ) provides the rate of steepestdescent of f at x . If X is a normed space, and f is Fr´echet differentiable at x , then | ∇ f | ( x ) = k f ′ ( x ) k . When x / ∈ dom f , we set | ∇ f | ( x ) = | ∇ f | ⋄ ( x ) : = + ∞ . The next proposition isstraightforward. Proposition 1.1
Suppose X is a metric space, f : X → R ∪ { + ∞ } , and x ∈ X. (i) If f is not lower semicontinuous at x, then | ∇ f | ( x ) = + ∞ . (ii) If f ( x ) > , then | ∇ f | ( x ) ≤ | ∇ f | ⋄ ( x ) . When proving primal and dual characterizations of transversality properties in the non-linear setting we use chain rules for slopes and subdifferentials, respectively. The nextlemma provides a chain rule for slopes, which is used in Section 3. For its subdifferentialcounterparts we refer the reader to [14, Proposition 2.1].
Lemma 1.2
Let X be a metric space, f : X → R ∪ { + ∞ } , ϕ : R → R ∪ { + ∞ } , x ∈ dom fand f ( x ) ∈ dom ϕ . Suppose ϕ is nondecreasing on R and differentiable at f ( x ) with ϕ ′ ( f ( x )) > . 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 localminimum of ϕ ◦ f , and consequently, | ∇ ( ϕ ◦ f ) | ( x ) = | ∇ f | ( x ) = . Suppose x is not a lo-cal minimum of f . If f is not lower semicontinuous at x , i.e. α : = lim k → + ∞ f ( x k ) < f ( x ) for some sequence x k → x , then, in view of the assumptions, ϕ is strictly increasing near f ( x ) , and consequently, lim inf k → + ∞ ϕ ( f ( x k )) ≤ ϕ ( α ) < ϕ ( f ( x )) (with the conventionthat ϕ ( − ∞ ) = − ∞ ), i.e. ϕ ◦ f is not lower semicontinuous at x ; hence, in view of Propo-sition 1.1(i), | ∇ ( ϕ ◦ f ) | ( x ) = | ∇ f | ( x ) = + ∞ . Suppose f is lower semicontinuous at x , i.e.lim inf u → x , u = x f ( u ) = f ( x ) . Then, taking into account that x is not a local minimum of f , | ∇ ( ϕ ◦ f ) | ( x ) = lim sup u → x , u = x ϕ ( f ( x )) − ϕ ( f ( u )) d ( u , x )= lim sup u → x , u = xf ( u ) < f ( x ) ϕ ( f ( x )) − ϕ ( f ( u )) d ( u , x ) = lim sup u → x , u = xf ( u ) ↑ f ( x ) ϕ ( f ( x )) − ϕ ( f ( u )) d ( u , x )= lim sup u → x , u = xf ( u ) ↑ f ( x ) (cid:18) ϕ ( f ( x )) − ϕ ( f ( u )) f ( x ) − f ( u ) · f ( x ) − f ( u ) d ( u , x ) (cid:19) = ϕ ′ ( f ( x )) lim sup u → x , u = x f ( x ) − f ( u ) d ( u , x ) = ϕ ′ ( f ( x )) | ∇ f | ( x ) . The proof is complete. ⊓⊔ Remark 1.1 (i) The slope chain rule in Lemma 1.2 is a local result. Instead of assumingthat ϕ is defined on the whole real line, one can assume that ϕ is defined and finite on aclosed interval [ α , β ] around the point f ( x ) : α < f ( x ) < β . It is sufficient to define thecomposition 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.(ii) Lemma 1.2 slightly improves [2, Lemma 4.1], where f and ϕ are assumed lower semi-continuous and continuously differentiable, respectively.The rest of the paper is organized as follows. In Section 2, we discuss transversalityproperties of finite collections of sets in the nonlinear setting. In Section 3, we estab-lish metric characterizations of these properties. Section 4 is devoted to slope sufficientconditions for the nonlinear transversality properties. In Section 5, we discuss quantitative Nguyen Duy Cuong, Alexander Y. Kruger relations between nonlinear transversality of collections of sets and the corresponding non-linear regularity properties of set-valued mappings, and show that the two popular modelsare in a sense equivalent in the general nonlinear setting. As a consequence, we improvesome results established in [42] in the H¨older setting. We also briefly discuss nonlinear ex-tensions of the new transversality properties of a set-valued mapping to a set in the rangespace due to Ioffe [28].
The nonlinearity in the definitions of the transversality properties is determined by a con-tinuous strictly increasing function ϕ : R + → R + satisfying ϕ ( ) = t → + ∞ ϕ ( t ) =+ ∞ . The family of all such functions is denoted by C . We denote by C the subfamilyof functions from C which are differentiable on ] , + ∞ [ with ϕ ′ ( t ) > t >
0. Ob-viously, if ϕ ∈ C ( ϕ ∈ C ), then ϕ − ∈ C ( ϕ − ∈ C ). Observe that, for any α > q >
0, the function t α t q on R + belongs to C . Remark 2.1
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 Ω , . . . , Ω n are subsets of a normed space X and ¯ x ∈ ∩ ni = Ω i , if not explicitly stated otherwise, we assume from now on that ϕ ∈ C . Definition 2.1
The collection { Ω , . . . , Ω n } is(i) ϕ − semitransversal at ¯ x if there exists a δ > ρ ∈ ] , δ [ and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < ρ ;(ii) ϕ − subtransversal at ¯ x if there exist δ > δ > ρ ∈ ] , δ [ and x ∈ B δ ( ¯ x ) with ϕ ( max ≤ i ≤ n d ( x , Ω i )) < ρ ;(iii) ϕ − transversal at ¯ x if there exist δ > δ > ρ ∈ ] , δ [ , ω i ∈ Ω i ∩ B δ ( ¯ x ) and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < ρ .Observe that conditions (1) and (3) are trivially satisfied when x i = ( i = , . . . , n ) .Hence, in parts (i) and (iii) of Definition 2.1 (as well as Definition 1.1) one can addition-ally assume that max ≤ i ≤ n k x i k >
0. Similarly, in part (ii) of Definition 2.1 (as well asDefinition 1.1) one can assume that x / ∈ ∩ ni = Ω i .Each of the properties in Definition 2.1 is determined by a function ϕ ∈ C , and a num-ber δ > δ > δ > δ ’s is more technical: they control the size of the interval for the values of ρ and, inthe case of ϕ − subtransversality and ϕ − transversality in parts (ii) and (iii), the size of theneighbourhoods of ¯ x involved in the respective definitions. Of course, if a property is sat-isfied with some δ > δ >
0, it is satisfied also with the single δ : = min { δ , δ } inplace of both δ and δ . Unlike our previous publications on (linear and H¨older) transver-sality properties, we use in the current paper two different parameters to emphasise theirdifferent roles in the definitions and the corresponding characterizations. Moreover, we aregoing to provide quantitative estimates for the values of these parameters.Given a δ > δ > δ > ϕ ∈ C , it is obviously satisfied for any function ˆ ϕ ∈ C such that ˆ ϕ − ( t ) ≤ ϕ − ( t ) for all t ∈ ] , δ [ ( t ∈ ] , δ [ ), or equivalently, ˆ ϕ ( t ) ≥ ϕ ( t ) for all t ∈ ] , ϕ − ( δ )[ ( t ∈ ] , ϕ − ( δ )[ ). Thus, it makes sense looking for the smallest function in C (if it exists) ensuring the corresponding property for the given sets. Observe also thattaking a smaller δ > δ > δ >
0) may allow each of the properties tobe satisfied with a smaller ϕ . When the exact value of δ ( δ and δ ) in the definition of ransversality Properties: Primal Sufficient Conditions 7 the respective property is not important, it makes sense to look for the smallest functionensuring the corresponding property for some δ > δ and δ ).The most important realization of the three properties in Definition 2.1 corresponds tothe H¨older setting, i.e. ϕ being a power function, given for all t ≥ ϕ ( t ) : = α − t q withsome α > q >
0. In this case, Definition 2.1 reduces to Definition 1.1.Another important for applications class of functions is given by the so called
H¨older-type [6, 47] ones, i.e. functions of the form t α − ( t q + t ) , frequently used in the errorbound theory, or more generally, functions t α − ( t q + β t ) with some α > β > q >
0. Depending on the value of q , transversality properties determined by such functionscan be approximated by H¨older (if q <
1) or even linear (if q ≥
1) ones.
Proposition 2.1
Let ϕ ( t ) : = α − ( t q + β t ) with some α > , β > and q > . If the collec-tion { Ω , . . . , Ω n } is ϕ − (semi-/sub-)transversal at ¯ x, then it is α ′ − (semi-/sub-) transversalof order q ′ at ¯ x, where: (i) if q < , then q ′ = q and α ′ is any number in ] , α [ ; (ii) if q = , then q ′ = and α ′ : = α ( + β ) − ; (iii) if q > , then q ′ = and α ′ is any number in ] , αβ − [ .Proof The assertions follow from Definition 1.1 in view of the following observations:(i) if q < α ′ ∈ ] , α [ , then, for all sufficiently small t >
0, it holds α ′ ( + β t − q ) < α ,and consequently, ϕ ( t ) = α − ( + β t − q ) t q < ( α ′ ) − t q ;(ii) if q = α ′ = α ( + β ) − , then ϕ ( t ) = α − ( + β ) t = ( α ′ ) − t ;(iii) if q > α ′ ∈ ] , αβ − [ , then, for all sufficiently small t >
0, it holds α ′ ( β − t q − + ) < αβ − , and consequently, ϕ ( t ) = α − β ( β − t q − + ) t < ( α ′ ) − t . ⊓⊔ The next two propositions collect some simple facts about the properties in Defini-tion 2.1 and clarify relationships between them.
Proposition 2.2 (i) If Ω = . . . = Ω n , and there exists a δ > such that ϕ ( t ) ≥ t for allt ∈ ] , δ [ , then { Ω , . . . , Ω n } is ϕ − subtransversal at ¯ x with δ and any δ > . (ii) If { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with some δ > and δ > , then it is ϕ − se-mitransversal at ¯ x with δ and ϕ − subtransversal at ¯ x with any δ ′ ∈ ] , δ ] and δ ′ > such that ϕ − ( δ ′ ) + δ ′ ≤ δ . (iii) If ¯ x ∈ int ∩ ni = Ω i , then { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with some δ > and δ > .Proof (i) Let Ω : = Ω = . . . = Ω n . Then condition (2) becomes Ω ∩ B ρ ( x ) = /0. Thisinclusion is trivially satisfied if ϕ ( d ( x , Ω )) < ρ and ϕ ( ρ ) ≥ ρ .(ii) Let { Ω , . . . , Ω n } be ϕ − transversal at ¯ x with some δ > δ >
0. Since condition(1) is a particular case of condition (3) with ω i = ¯ x ( i = , . . . , n ), we can conclude that { Ω , . . . , Ω n } is ϕ − semitransversal at ¯ x with δ . Let δ ′ ∈ ] , δ ] and δ ′ > ϕ − ( δ ′ ) + δ ′ ≤ δ , and let ρ ∈ ] , δ ′ [ and x ∈ B δ ′ ( ¯ x ) with ϕ ( max ≤ i ≤ n d ( x , Ω i )) < ρ .Choose ω i ∈ Ω i ( i = , . . . , n ) such that ϕ ( max ≤ i ≤ n k x − ω i k ) < ρ . Then, for any i = , . . . , n , k ω i − ¯ x k ≤ k x − ω i k + k x k < ϕ − ( ρ ) + δ ′ < δ . Set x i : = x − ω i ( i = , . . . , n ) . We have ρ ∈ ] , δ [ , ω i ∈ Ω i ∩ B δ ( ¯ x ) ( i = , . . . , n ) and ϕ ( max ≤ i ≤ n k x i k ) < ρ . By Definition 2.1(iii), condition (3) is satisfied. This is equiv-alent to condition (2). In view of Definition 2.1(ii), { Ω , . . . , Ω n } is ϕ − subtransversalat ¯ x with δ ′ and δ ′ . Nguyen Duy Cuong, Alexander Y. Kruger (iii) Let ¯ x ∈ int ∩ ni = Ω i . Choose numbers δ > δ > δ : = ϕ − ( δ ) + δ , it holds B δ ( ¯ x ) ⊂ ∩ ni = Ω i . Then, for all ω i ∈ Ω i ∩ B δ ( ¯ x ) and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < δ , it holds 0 ∈ ∩ ni = ( Ω i − ω i − x i ) ,and consequently, condition (3) is satisfied with any ρ >
0. Hence, { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with δ and δ . ⊓⊔ Remark 2.2 (i) The inequality ϕ − ( δ ′ ) + δ ′ ≤ δ in Proposition 2.2(ii) and some state-ments below can obviously be replaced by the equality ϕ − ( δ ′ ) + δ ′ = δ providingin a sense the best estimate for the values of the parameters δ ′ and δ ′ .(ii) In the H¨older setting, parts (i) and (iii) of Proposition 2.2 recapture [42, Remarks 4 and3], respectively, while part (ii) improves [42, Remark 1].(iii) The nonlinear semitransversality and subtransversality properties are in general inde-pendent; see examples in [42, Section 2.3] and [43, Section 3.2]. Proposition 2.3
Let ∩ ni = Ω i be closed and ¯ x ∈ bd ∩ ni = Ω i . If { Ω , . . . , Ω n } is ϕ − subtrans-versal (in particular, if it is ϕ − transversal) at ¯ x with some δ > and δ > , then thereexists a ¯ t ∈ ] , min { δ , ϕ − ( δ ) } [ such that ϕ ( t ) ≥ t for all t ∈ ] , ¯ t ] .Proof Let { Ω , . . . , Ω n } be ϕ − subtransversal at ¯ x with some δ > δ >
0. Choosea point ˆ x / ∈ ∩ ni = Ω i such that k ˆ x − ¯ x k < min { ϕ − ( δ ) , δ } and set ¯ t : = d ( ˆ x , ∩ ni = Ω i ) . Then¯ t < min { ϕ − ( δ ) , δ } . Besides, ¯ t > ∩ ni = Ω i is closed. Thanks to the continuity ofthe function d ( · , ∩ ni = Ω i ) , for any t ∈ ] , ¯ t ] there is an x ∈ ] ¯ x , ˆ x ] such that d ( x , ∩ ni = Ω i ) = t .We have k x − ¯ x k ≤ k ˆ x − ¯ x k < δ and ϕ ( t ) ≤ ϕ ( ¯ t ) < δ . Take a ρ ∈ ] ϕ ( t ) , δ [ . Then ϕ ( max ≤ i ≤ n d ( x , Ω i )) ≤ ϕ ( t ) < ρ . By Definition 2.1(ii), t = d ( x , ∩ ni = Ω i ) < ρ , and let-ting ρ ↓ ϕ ( t ) , we arrive at t ≤ ϕ ( t ) . If { Ω , . . . , Ω n } is ϕ − transversal at ¯ x , the conclusionfollows in view of Proposition 2.2(ii). ⊓⊔ Remark 2.3
The conditions on ϕ in Proposition 2.3 in the H¨older setting can only besatisfied if either q <
1, or q = α ≤
1. This reflects the well known fact that theH¨older subtransversality and transversality properties are only meaningful when q ≤ q =
1) is only meaningful when α ≤
1; cf. [39, p. 705], [36,p. 118]. The extreme case q = α = d ( x , ∩ ni = Ω i ) = max ≤ i ≤ n d ( x , Ω i ) for all x near ¯ x .In accordance with Proposition 2.3, the ϕ − subtransversality and ϕ − transversalityproperties 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 2.1
Let R be equipped with the maximum norm, and let q > γ > Ω : = n ( ξ , ξ ) ∈ R | γ q ξ + | ξ | q ≥ o , Ω : = n ( ξ , ξ ) ∈ R | γ q ξ − | ξ | q ≤ o and ¯ x : =( , ) . Note that, when q >
1, the sets Ω and Ω are nonconvex. We claim that the pair { Ω , Ω } is ϕ − semitransversal at ¯ x with ϕ ( t ) : = γ t q ( t ≥ Proof
Given an r >
0, set x : = ( , − r ) and x : = ( , r ) . Then k x k = k x k = r and ( ± γ r q , ) ∈ ( Ω − x ) ∩ ( Ω − x ) . Moreover, it is easy to notice that either ( γ r q , ) or ( − γ r q , ) belongs to ( Ω − x ) ∩ ( Ω − x ) for any choice of vectors x , x ∈ R withmax {k x k , k x k} ≤ r . Hence, ( Ω − x ) ∩ ( Ω − x ) ∩ B ρ ( ¯ x ) = /0 for all such vectors x , x ∈ R as long as ρ > γ r q , and consequently, { Ω , Ω } is ϕ − semitransversal at ¯ x . ⊓⊔ The three transversality properties are defined in Definition 2.1 geometrically. We nowshow that they can be characterized in metric terms. These metric characterizations can beused as equivalent definitions of the respective properties. ransversality Properties: Primal Sufficient Conditions 9
Theorem 3.1
The collection { Ω , . . . , Ω n } is (i) ϕ − semitransversal at ¯ x with some δ > if and only ifd ¯ x , n \ i = ( Ω i − x i ) ! ≤ ϕ (cid:18) max ≤ i ≤ n k x i k (cid:19) (4) for all x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < δ ; (ii) ϕ − subtransversal at ¯ x with some δ > and δ > if and only if the following equiv-alent conditions hold: (a) for all x ∈ B δ ( ¯ x ) with ϕ ( max ≤ i ≤ n d ( x , Ω i )) < δ , it holdsd x , n \ i = Ω i ! ≤ ϕ (cid:18) max ≤ i ≤ n d ( x , Ω i ) (cid:19) ; (5)(b) for all x i ∈ X and ω i ∈ Ω i ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < δ and ω + x = . . . = ω n + x n ∈ B δ ( ¯ x ) , it holdsd , n \ i = ( Ω i − ω i − x i ) ! ≤ ϕ (cid:18) max ≤ i ≤ n k x i k (cid:19) ; (6)(iii) ϕ − transversal at ¯ x with some δ > and δ > if and only if inequality (6) holds forall ω i ∈ Ω i ∩ B δ ( ¯ x ) and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < δ .Proof (i) Let { Ω , . . . , Ω n } be ϕ − semitransversal at ¯ x with some δ >
0, and let x i ∈ X ( i = , . . . , n ) with ρ : = ϕ ( max ≤ i ≤ n k x i k ) < δ . Choose a ρ ∈ ] ρ , δ [ . By (1), d (cid:0) ¯ x , ∩ ni = ( Ω i − x i ) (cid:1) < ρ . Letting ρ ↓ ρ , we arrive at inequality (4).Conversely, let δ > x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < δ . For all ρ ∈ ] , δ [ and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < ρ , we have d (cid:0) ¯ x , ∩ ni = ( Ω i − x i ) (cid:1) < ρ , which implies condition (1).By Definition 2.1(i), { Ω , . . . , Ω n } is ϕ − semitransversal at ¯ x with δ .(ii) We first prove the equivalence between (a) and (b). Suppose condition (a) is satisfied.Let ω i ∈ Ω i and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < δ and x : = ω + x = . . . = ω n + x n ∈ B δ ( ¯ x ) . Then ϕ ( d ( x , Ω i )) = ϕ ( d ( ω i + x i , Ω i )) ≤ ϕ ( k x i k ) < δ ( i = , . . . , n ) , and consequently, inequality (5) is satisfied. Hence, d (cid:16) , n \ i = ( Ω i − ω i − x i ) (cid:17) = d (cid:16) x , n \ i = Ω i (cid:17) ≤ ϕ (cid:18) max ≤ i ≤ n d ( x , Ω i ) (cid:19) ≤ ϕ (cid:16) max ≤ i ≤ n k x i k (cid:17) . Suppose condition (b) is satisfied. Let x ∈ B δ ( ¯ x ) with ϕ ( max ≤ i ≤ n d ( x , Ω i )) < δ .Choose ω i ∈ Ω i ( i = , . . . , n ) such that ϕ ( max ≤ i ≤ n k x − ω i k ) < δ and set x ′ i : = x − ω i ( i = , . . . , n ) . Then x = x ′ i + ω i ∈ B δ ( ¯ x ) ( i = , . . . , n ) and ϕ ( max ≤ i ≤ n k x ′ i k ) < δ . Inview of inequality (6) with x ′ i in place of x i ( i = , . . . , n ) , we obtain d (cid:16) x , n \ i = Ω i (cid:17) ≤ ϕ (cid:16) max ≤ i ≤ n k x − ω i k (cid:17) . Taking infimum in the right-hand side over ω i ∈ Ω i ( i = , . . . , n ) , we arrive at inequal-ity (5). Next we show that ϕ − subtransversality is equivalent to condition (a). Let { Ω , . . . , Ω n } be ϕ − subtransversal at ¯ x with some δ > δ >
0, and let x ∈ B δ ( ¯ x ) with ρ : = ϕ ( max ≤ i ≤ n d ( x , Ω i )) < δ . Choose a ρ ∈ ] ρ , δ [ . By Definition 2.1(ii), ∩ ni = Ω i ∩ B ρ ( x ) = /0, and consequently, d (cid:0) x , ∩ ni = Ω i (cid:1) < ρ . Letting ρ ↓ ρ , we arriveat inequality (5). Conversely, let δ > δ >
0, and inequality (5) hold for all x ∈ B δ ( ¯ x ) with ϕ ( max ≤ i ≤ n d ( x , Ω i )) < δ . For any ρ ∈ ] , δ [ and x ∈ B δ ( ¯ x ) with ϕ ( max ≤ i ≤ n d ( x , Ω i )) < ρ , we have d (cid:0) x , ∩ ni = Ω i (cid:1) < ρ , which implies condition (2).By Definition 2.1(ii), { Ω , . . . , Ω n } is ϕ − subtransversal at ¯ x with δ and δ .(iii) Let { Ω , . . . , Ω n } be ϕ − transversal at ¯ x with some δ > δ >
0, and let ω i ∈ Ω i ∩ B δ ( ¯ x ) and x i ∈ X ( i = , . . . , n ) with ρ : = ϕ ( max ≤ i ≤ n k x i k ) < δ . Choose a ρ ∈ ] ρ , δ [ . By (3), d (cid:0) , ∩ ni = ( Ω i − ω i − x i ) (cid:1) < ρ . Letting ρ ↓ ρ , we arrive at inequal-ity (6).Conversely, let δ > δ >
0, and inequality (6) hold for all ω i ∈ Ω i ∩ B δ ( ¯ x ) and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < δ . For any ρ ∈ ] , δ [ , ω i ∈ Ω i ∩ B δ ( ¯ x ) and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < ρ , we have d (cid:0) , ∩ ni = ( Ω i − ω i − x i ) (cid:1) < ρ , which is equivalent to condition (3). By Defini-tion 2.1(iii), { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with δ and δ . ⊓⊔ Example 3.1
Let R be equipped with the maximum norm, and let Ω : = { ( ξ , ξ ) ∈ R | ξ ≥ } , Ω : = { ( ξ , ξ ) ∈ R | ξ ≤ ξ } and ¯ x : = ( , ) . Thus, Ω ∩ Ω = { ( ξ , ξ ) ∈ R | ≤ ξ ≤ ξ } , and no shift of the sets can make their intersection empty. We claim that thepair { Ω , Ω } is ϕ − semitransversal at ¯ x with ϕ ( t ) : = √ t ( t ≥ ) and δ : = Proof
Observe that, given any ε ≥
0, the vertical shifts of the sets determined by x ε : =( , − ε ) and x ε = ( , ε ) produce the largest ‘gap’ between them compared to all possibleshifts x and x with max {k x k , k x k} ≤ ε . Indeed, ( Ω − x ε ) ∩ ( Ω − x ε ) = { ( ξ , ξ ) ∈ R | ε ≤ ξ ≤ ξ − ε }⊂ ( Ω − x ) ∩ ( Ω − x ) , as long as max {k x k , k x k} ≤ ε . Observe also that ( √ ε , ε ) ∈ ( Ω − x ε ) ∩ ( Ω − x ε ) .Hence, for any x , x ∈ R with ε : = max {k x k , k x k} < ϕ − ( δ ) =
2, we have d ( ¯ x , ( Ω − x ) ∩ ( Ω − x )) ≤ k ( √ ε , ε ) k = √ ε = ϕ ( max {k x k , k x k} ) . In view of Theorem 3.1(i), { Ω , Ω } ϕ − semitransversal at ¯ x with δ . ⊓⊔ Example 3.2
Let R be equipped with the maximum norm, and let Ω : = { ( ξ , ξ ) ∈ R | ξ = ξ } , Ω : = { ( ξ , ξ ) ∈ R | ξ = − ξ } and ¯ x : = ( , ) . Thus, Ω ∩ Ω = { ¯ x } . Weclaim that, for any γ >
1, the pair { Ω , Ω } is ϕ − subtransversal at ¯ x with ϕ ( t ) : = γ √ t ( t ≥ ) and any δ > δ > δ + + q δ + < γ . Proof
Observe that, d ( x , Ω ∩ Ω ) = k x k for all x ∈ R and, given any ε ≥ x ε = ( , ε ) , one hasmin k x k = ε max { d ( x , Ω ) , d ( x , Ω ) } = d ( x ε , Ω ) = d ( x ε , Ω ) = min t ≥ max { ε − t , t } . It is easy to see that the minimum in the rightmost minimization problem is attained at t : = q ε + − satisfying ε − t = t . Thus,min k x k = ε max { d ( x , Ω ) , d ( x , Ω ) } = ε + − r ε + = ε ε + + q ε + . ransversality Properties: Primal Sufficient Conditions 11 Hence, for any x ∈ R with k x k < δ , we have d ( x , Ω ∩ Ω ) = k x k ≤ γ k x k r k x k + + q k x k + ≤ ϕ ( max { d ( x , Ω ) , d ( x , Ω ) } ) . In view of Theorem 3.1(ii), { Ω , Ω } is ϕ − subtransversal at ¯ x with δ and δ .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 δ and δ and have certain advantages, e.g., when establishing connections with metricregularity of set-valued mappings. The relations between the values of the parameters inthe two groups of metric characterizations can be estimated. Theorem 3.2
Let δ > and δ > . The following conditions are equivalent: (i) inequality (6) is satisfied for all x i ∈ X and ω i ∈ Ω i with ω i + x i ∈ B δ ( ¯ x ) ( i = , . . . , n ) and ϕ ( max ≤ i ≤ n k x i k ) < δ ; (ii) for all x i ∈ δ B ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n d ( ¯ x , Ω i − x i )) < δ , it holdsd (cid:16) ¯ x , n \ i = ( Ω i − x i ) (cid:17) ≤ ϕ (cid:16) max ≤ i ≤ n d ( ¯ x , Ω i − x i ) (cid:17) ; (7)(iii) for all x , x i ∈ X with x + x i ∈ B δ ( ¯ x ) ( i = , . . . , n ) and ϕ ( max ≤ i ≤ n d ( x , Ω i − x i )) < δ ,it holds d (cid:16) x , n \ i = ( Ω i − x i ) (cid:17) ≤ ϕ (cid:16) max ≤ i ≤ n d ( x , Ω i − x i ) (cid:17) . (8) Moreover, if { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with some δ > and δ > , thenconditions (i)–(iii) hold with any δ ′ ∈ ] , δ ] and δ ′ > satisfying ϕ − ( δ ′ ) + δ ′ ≤ δ inplace of δ and δ .Conversely, if conditions (i)–(iii) hold with some δ > and δ > , then { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with any δ ′ ∈ ] , δ ] and δ ′ > satisfying ϕ − ( δ ′ ) + δ ′ ≤ δ .Proof We first prove the equivalence of conditions (i)–(iii).(i) ⇒ (ii). Let x i ∈ δ B ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n d ( ¯ x , Ω i − x i )) < δ . Choose ω i ∈ Ω i ( i = , . . . , n ) such that ϕ ( max ≤ i ≤ n k ¯ x + x i − ω i k ) < δ . Set x ′ i : = ¯ x + x i − ω i ( i = , . . . , n ) . Then ω i + x ′ i ∈ B δ ( ¯ x ) ( i = , . . . , n ) and ϕ ( max ≤ i ≤ n k x ′ i k ) < δ . By (i),inequality (6) is satisfied with x ′ i in place of x i ( i = , . . . , n ) , i.e. d (cid:16) ¯ x , n \ i = ( Ω i − x i ) (cid:17) ≤ ϕ (cid:16) max ≤ i ≤ n k ¯ x + x i − ω i k (cid:17) . Taking the infimum in the righ-hand side over ω i ∈ Ω i ( i = , . . . , n ) , we arrive at inequality(7).(ii) ⇒ (iii). Let x , x i ∈ X with x + x i ∈ B δ ( ¯ x ) ( i = , . . . , n ) and ϕ ( max ≤ i ≤ n d ( x , Ω i − x i )) < δ . Set x ′ i : = x + x i − ¯ x ( i = , . . . , n ) . Then x ′ i ∈ δ B ( i = , . . . , n ) and ϕ ( max ≤ i ≤ n d ( ¯ x , Ω i − x ′ i )) < δ . By (ii), inequality (7) is satisfied with x ′ i in place of x i ( i = , . . . , n ) . This is equivalent to inequality (8).(iii) ⇒ (i). Let x i ∈ X and ω i ∈ Ω i with ω i + x i ∈ B δ ( ¯ x ) ( i = , . . . , n ) and ϕ ( max ≤ i ≤ n k x i k ) < δ . Set x ′ i : = ω i + x i − ¯ x ( i = , . . . , n ) . Then, ¯ x + x ′ i ∈ B δ ( ¯ x ) and ϕ ( max ≤ i ≤ n d ( ¯ x , Ω i − x ′ i )) ≤ ϕ ( k x i k ) < δ . By (iii), inequality (8) is satisfied with ¯ x and x ′ i in place of x and x i ( i = , . . . , n ) , respectively, i.e. d (cid:16) , n \ i = ( Ω i − ω i − x i ) (cid:17) ≤ ϕ (cid:16) max ≤ i ≤ n d ( , Ω i − ω i − x i ) (cid:17) . Since ω i ∈ Ω i ( i = , . . . , n ) , inequality (6) is satisfied.Suppose { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with some δ > δ >
0, and let δ ′ ∈ ] , δ ] and δ ′ > ϕ − ( δ ′ ) + δ ′ ≤ δ . Then, for all x i ∈ X and ω i ∈ Ω i with ω i + x i ∈ B δ ′ ( ¯ x ) ( i = , . . . , n ) and ϕ ( max ≤ i ≤ n k x i k ) < δ ′ , we have k ω i − ¯ x k ≤ k x i k + k ω i + x i − ¯ x k < ϕ − ( δ ′ ) + δ ′ ≤ δ ( i = , . . . , n ). By Theorem 3.1(iii), inequality (6) issatisfied, and consequently, condition (i) (as well as conditions (ii) and (iii)) holds with δ ′ and δ ′ .Conversely, suppose conditions (i)–(iii) hold with some δ > δ >
0, and let δ ′ ∈ ] , δ ] and δ ′ > ϕ − ( δ ′ ) + δ ′ ≤ δ . Then, for all ω i ∈ Ω i ∩ B δ ′ ( ¯ x ) and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < δ ′ , we have k ω i + x i − ¯ x k ≤ k x i k + k ω i − ¯ x k < ϕ − ( δ ′ ) + δ ′ ≤ δ ( i = , . . . , n ) . By (i), inequality (6) is satisfied, and consequently, { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with δ ′ and δ ′ according to Theorem 3.1(iii). ⊓⊔ Remark 3.1 (i) In the H¨older case, i.e. when ϕ ( t ) : = α − t q ( t ≥
0) for some α > q ∈ ] , ] , condition (8) served as the main metric characterization of transversality;cf. [42, 43]. In the linear case, condition (7) has been picked up recently in [10, 11].This condition seems an important advancement as it replaces an arbitrary point x in (8) with the given reference point ¯ x . Condition (6) in part (i) seems new. In viewof Theorem 3.1(iii), it is the most straightforward metric counterpart of the originalgeometric property (3).(ii) The metric characterizations of the three ϕ − transversality properties in the above the-orems look similar: each of them provides an upper error bound type estimate for thedistance from a point to the intersection of sets, which can be useful from the compu-tational point of view. For the account of nonlinear error bounds theory, we refer thereader to [1, 2, 13, 58].The next corollary provides qualitative metric characterizations of the three nonlineartransversality properties. They are direct consequences of Theorems 3.1 and 3.2. Corollary 3.1
The collection { Ω , . . . , Ω n } is (i) ϕ − semitransversal at ¯ x if and only if there exists a δ > such that inequality (4) holdsfor all x i ∈ δ B ( i = , . . . , n ) ; (ii) ϕ − subtransversal at ¯ x if and only if the following equivalent conditions hold: (a) there exists a δ > such that inequality (5) holds for all x ∈ B δ ( ¯ x ) ; (b) there exists a δ > such that inequality (6) holds for all ω i ∈ Ω i ∩ B δ ( ¯ x ) andx i ∈ δ B ( i = , . . . , n ) with ω + x = . . . = ω n + x n ; (iii) ϕ − transversal at ¯ x if and only if the following equivalent conditions hold: (a) there exists a δ > such that inequality (6) holds for all ω i ∈ Ω i ∩ B δ ( ¯ x ) andx i ∈ δ B ( i = , . . . , n ) ; (b) there exists a δ > such that inequality (7) holds for all x i ∈ δ B ( i = , . . . , n ) ; (c) there exists a δ > such that inequality (8) holds for all x ∈ B δ ( ¯ x ) and x i ∈ δ B ( i = , . . . , n ) .Remark 3.2 In the H¨older setting, i.e. when ϕ ( t ) : = α − t q ( t ≥ ) with some α > q >
0, the above corollary improves [42, Theorem 1]. In the linear case, the equivalenceof the three characterizations of transversality in Corollary 3.1(iii) has been establishedin [10]. We refer the readers to [36, 39, 40] for more discussions and historical comments. ransversality Properties: Primal Sufficient Conditions 13
The next two propositions identify important situations when ‘restricted’ versions ofthe metric characterizations of nonlinear transversality properties in Theorem 3.1 canbe used: with all but one sets being translated in the cases of ϕ − semitransversalityand 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, whendealing with alternating (or cyclic) projections. The first proposition formulates simpli-fied necessary conditions for the transversality properties which are direct consequencesof the respective statements, while the second one gives conditions under which theseconditions become sufficient in the case of two sets. Proposition 3.1 (i) If { Ω , . . . , Ω n } is ϕ − semitransversal at ¯ x with some δ > , thend (cid:16) ¯ x , n − \ i = ( Ω i − x i ) ∩ Ω n (cid:17) ≤ ϕ (cid:16) max ≤ i ≤ n − k x i k (cid:17) for all x i ∈ X ( i = , . . . , n − ) with ϕ ( max ≤ i ≤ n − k x i k ) < δ . (ii) If { Ω , . . . , Ω n } is ϕ − subtransversal at ¯ x with some δ > and δ > , thend (cid:16) x , n \ i = Ω i (cid:17) ≤ ϕ (cid:16) max ≤ i ≤ n − d ( x , Ω i ) (cid:17) for all x ∈ Ω n ∩ B δ ( ¯ x ) with ϕ ( max ≤ i ≤ n − d ( x , Ω i )) < δ . (iii) If { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with some δ > and δ > , thend (cid:16) , n − \ i = ( Ω i − ω i − x i ) ∩ ( Ω n − ω n ) (cid:17) ≤ ϕ (cid:16) max ≤ i ≤ n − k x i k (cid:17) for all ω i ∈ Ω i ∩ B δ ( ¯ x ) ( i = , . . . , n ) and x i ∈ X ( i = , . . . , n − ) with ϕ ( max ≤ i ≤ n − k x i k ) < δ . Proposition 3.2
Let Ω , Ω be subsets of a normed space X, and ¯ x ∈ Ω ∩ Ω . Let α > , ¯ t > , ϕ ( t ) ≤ α t for all t ∈ ] , ¯ t ] , and α ′ : = ( + α ) − . (i) If for all x ∈ ¯ t B , d ( ¯ x , ( Ω − x ) ∩ Ω ) ≤ ϕ ( k x k ) , (9) then { Ω , Ω } is α ′ − semitransversal at ¯ x with δ : = (cid:0) α + (cid:1) ¯ t. (ii) If there exists a δ > such that, for all x ∈ Ω ∩ B δ ( ¯ x ) with d ( x , Ω ) < ¯ t,d ( x , Ω ∩ Ω ) ≤ ϕ ( d ( x , Ω )) , (10) then { Ω , Ω } is α ′ − subtransversal at ¯ x with δ : = (cid:0) α + (cid:1) ¯ t and δ . (iii) If there exists a δ > such that, for all ω i ∈ Ω i ∩ B δ ( ¯ x ) ( i = , ) and x ∈ ¯ t B ,d ( , ( Ω − ω − x ) ∩ ( Ω − ω )) ≤ ϕ ( k x k ) , then { Ω , Ω } is α ′ − transversal at ¯ x with δ : = (cid:0) α + (cid:1) ¯ t and δ .Proof (i) Let δ : = (cid:0) α + (cid:1) ¯ t , and inequality (9) be satisfied for all x ∈ ¯ t B . Let ρ ∈ ] , δ [ and x , x ∈ X with max {k x k , k x k} < α ′ ρ . Set x ′ : = x − x . Thus, k x ′ k ≤ {k x k , k x k} < α ′ δ = ¯ t . Hence, by (9) with x ′ in place of x , d ( ¯ x , ( Ω − x ) ∩ ( Ω − x )) ≤ k x k + d ( ¯ x − x , ( Ω − x ) ∩ ( Ω − x ))= k x k + d ( ¯ x , ( Ω − x ′ ) ∩ Ω ) ≤ k x k + ϕ ( k x ′ k ) ≤ k x k + α k x ′ k≤ ( + α ) max {k x k , k x k} < ρ . Hence, ( Ω − x ) ∩ ( Ω − x ) ∩ B ρ ( ¯ x ) = /0 and, by Definition 1.1(i), { Ω , Ω } is α ′ − semitransversal at ¯ x with δ . (ii) Let δ : = (cid:0) α + (cid:1) ¯ t , δ >
0, and inequality (10) be satisfied for all x ∈ Ω ∩ B δ ( ¯ x ) with d ( x , Ω ) < ¯ t . Let ρ ∈ ] , δ [ and x ∈ B δ ( ¯ x ) with max { d ( x , Ω ) , d ( x , Ω ) } < α ′ ρ .Choose a number γ > k x − ¯ x k < γ − δ and max { d ( x , Ω ) , d ( x , Ω ) } < γ − α ′ ρ , and a point x ′ ∈ Ω such that k x − x ′ k ≤ γ d ( x , Ω ) . Then k x ′ − ¯ x k ≤ k x − x ′ k + k x − ¯ x k ≤ γ d ( x , Ω ) + k x − ¯ x k≤ ( γ + ) k x − ¯ x k < ( + γ − ) δ < δ , d ( x ′ , Ω ) ≤ k x − x ′ k + d ( x , Ω ) ≤ ( γ + ) max { d ( x , Ω ) , d ( x , Ω ) } < ( + γ − ) α ′ δ < α ′ δ = ¯ t . Hence, by (10) with x ′ in place of x , d ( x , Ω ∩ Ω ) ≤ k x − x ′ k + d ( x ′ , Ω ∩ Ω ) ≤ k x − x ′ k + ϕ ( d ( x ′ , Ω )) ≤ k x − x ′ k + α d ( x ′ , Ω ) ≤ ( + α ) k x − x ′ k + α d ( x , Ω ) ≤ ( + α ) γ d ( x , Ω ) + α d ( x , Ω ) ≤ (( + α ) γ + α ) max { d ( x , Ω ) , d ( x , Ω ) } . Letting γ ↓
1, we arrive at d ( x , Ω ∩ Ω ) ≤ ( + α ) max { d ( x , Ω ) , d ( x , Ω ) } < ρ . Hence, Ω ∩ Ω ∩ B ρ ( x ) = /0 and, by Definition 1.1(ii), { Ω , Ω } is α ′ − subtransversalat ¯ x with δ and δ .(iii) The proof follows that of assertion (i) with the sets Ω − ω and Ω − ω in place of Ω and Ω , respectively. ⊓⊔ Remark 3.3 (i) In the linear case, Proposition 3.2(ii) recaptures [40, Theorem 1(iii)],while parts (i) and (iii) seem new.(ii) Restricted versions of the metric conditions in Theorem 3.2 can be produced in a sim-ilar way.Checking the metric estimates of the ϕ − subtransversality and ϕ − transversality canbe simplified as illustrated by the following proposition referring to condition (5) in The-orem 3.1(ii). Equivalent versions of conditions (7) and (8) in Theorem 3.2 look similar. Proposition 3.3
The following conditions are equivalent: (i) inequality (5) holds true; (ii) for all ω i ∈ Ω i ( i = , . . . , n ) , it holdsd (cid:16) x , n \ i = Ω i (cid:17) ≤ ϕ (cid:16) max ≤ i ≤ n k x − ω i k (cid:17) ; (11)(iii) inequality (11) holds true for all ω i ∈ Ω i with k ω i − ¯ x k < k x − ¯ x k + ϕ − ( k x − ¯ x k )( i = , . . . , n ) ; (iv) inequality (11) holds true for all ω i ∈ Ω i with ϕ ( k ω i − x k ) < k x − ¯ x k ( i = , . . . , n ) .Proof The equivalence (i) ⇔ (ii) and implications (ii) ⇒ (iii) ⇒ (iv) are straightforward.We next show that (iv) ⇒ (ii). Let condition (iv) hold true, ω i ∈ Ω i ( i = , . . . , n ) , and ϕ ( k ω i − x k ) ≥ k x − ¯ x k for some i . Then d (cid:16) x , n \ i = Ω i (cid:17) ≤ k x − ¯ x k ≤ ϕ (cid:16) max ≤ i ≤ n k x − ω i k (cid:17) , i.e. inequality (11) is satisfied, and consequently condition (ii) holds true. ⊓⊔ ransversality Properties: Primal Sufficient Conditions 15 In this section, we formulate slope sufficient conditions for the properties in Defini-tion 2.1. The conditions are straightforward consequences of the Ekeland variational prin-ciple (Lemma 1.1) applied to appropriate lower semicontinuous functions. Throughoutthis section, X is a Banach space, and the sets Ω , . . . , Ω n are closed. These are exactly theassumptions which ensure that the Ekeland variational principle is applicable. In view ofProposition 2.2(iii), it suffices to assume that ¯ x ∈ bd ∩ ni = Ω i .The sufficient conditions for the three properties follow the same pattern. We first es-tablish 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 sourceof more practical local (infinitesimal) conditions. The corresponding local slope sufficientconditions, their H¨older 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 X n + , we are going to use also the follow-ing norm depending on a parameter γ > k ( x , . . . , x n , x ) k γ : = max n k x k , γ max ≤ i ≤ n k x i k o , x , . . . , x n , x ∈ X . (12)4.1 Semitransversality Theorem 4.1
The collection { Ω , . . . , Ω n } is ϕ − semitransversal at ¯ x with some δ > if,for some γ > and any x i ∈ X ( i = , . . . , n ) satisfying < max ≤ i ≤ n k x i k < ϕ − ( δ ) , (13) there exists a λ ∈ ] ϕ ( max ≤ i ≤ n k x i k ) , δ [ such that sup u i ∈ Ω i ( i = ,..., n ) , u ∈ X ( u ,..., u n , u ) =( ω ,..., ω n , x ) ϕ (cid:16) max ≤ i ≤ n k ω i − x i − x k (cid:17) − ϕ (cid:16) max ≤ i ≤ n k u i − x i − u k (cid:17) k ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ for all x ∈ X and ω i ∈ Ω i ( i = , . . . , n ) satisfying k x − ¯ x k < λ , max ≤ i ≤ n k ω i − ¯ x k < λγ , (15)0 < max ≤ i ≤ n k ω i − x i − x k ≤ max ≤ i ≤ n k x i k . (16)The proof below employs two closely related nonnegative functions on X n + deter-mined by the given function ϕ ∈ C and vectors x , . . . , x n ∈ X : f ( u , . . . , u n , u ) : = ϕ (cid:16) max ≤ i ≤ n k u i − x i − u k (cid:17) , u , . . . , u n , u ∈ X , (17) b f : = f + i Ω × ... × Ω n . (18) Proof
Suppose { Ω , . . . , Ω n } is not ϕ − semitransversal at ¯ x with some δ >
0, and let γ > ρ ∈ ] , δ [ and x i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x i k ) < ρ such that ∩ ni = ( Ω i − x i ) ∩ B ρ ( ¯ x ) = /0. Thus, max ≤ i ≤ n k x i k >
0. Let λ ∈ ] ϕ ( max ≤ i ≤ n k x i k ) , δ [ and λ ′ : = min { λ , ρ } . Then λ ′ > ϕ ( max ≤ i ≤ n k x i k ) , ∩ ni = ( Ω i − x i ) ∩ B λ ′ ( ¯ x ) = /0, and consequently,max ≤ i ≤ n k u i − x i − u k > u i ∈ Ω i ( i = , . . . , n ) , u ∈ B λ ′ ( ¯ x ) . (19)Let f and b f be defined by (17) and (18), respectively, while X n + be equipped with themetric induced by the norm (12). We have b f ( ¯ x , . . . , ¯ x , ¯ x ) = ϕ ( max ≤ i ≤ n k x i k ) < λ ′ . Choosea number ε such that b f ( ¯ x , . . . , ¯ x , ¯ x ) < ε < λ ′ . Applying the Ekeland variational principle,we can find points ω i ∈ Ω i ( i = , . . . , n ) and x ∈ X such that k ( ω , . . . , ω n , x ) − ( ¯ x , . . . , ¯ x , ¯ x ) k γ < λ ′ ≤ λ , f ( ω , . . . , ω n , x ) ≤ f ( ¯ x , . . . , ¯ x , ¯ x ) , (20) f ( ω , . . . , ω n , x ) − f ( u , . . . , u n , u ) ≤ ελ ′ k ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ (21)for all ( u , . . . , u n , u ) ∈ Ω × . . . × Ω n × 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 4.1
The expression in the left-hand side of (14) is the nonlocal γ -slope [33, p. 60]at ( ω , . . . , ω n , x ) of the function (18).The next statement is a localized version of Theorem 4.1. Corollary 4.1 (i)
The collection { Ω , . . . , Ω n } is ϕ − semitransversal at ¯ x with some δ > if, for some γ > and any x i ∈ X ( i = , . . . , n ) satisfying (13) , there exists a λ ∈ ] ϕ ( max ≤ i ≤ n k x i k ) , δ [ such that lim sup u i Ω i → ω i ( i = ,..., n ) , u → x ( u ,..., u n , u ) =( ω ,..., ω n , x ) ϕ (cid:16) max ≤ i ≤ n k ω i − x i − x k (cid:17) − ϕ (cid:16) max ≤ i ≤ n k u i − x i − u k (cid:17) k ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ for all x ∈ X and ω i ∈ Ω i ( i = , . . . , n ) satisfying (15) and (16) . (ii) If ϕ ∈ C , then inequality (22) in part (i) can be replaced by ϕ ′ (cid:16) max ≤ i ≤ n k ω i − x i − x k (cid:17) × lim sup u i Ω i → ω i ( i = ,..., n ) , u → x ( u ,..., u n , u ) =( ω ,..., ω n , x ) max ≤ i ≤ n k ω i − x i − x k − max ≤ i ≤ n k u i − x i − u kk ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ . (23) Proof
The expression in the left-hand side of (22) is the γ -slope [33, p. 61] of the function(18) at ( ω , . . . , ω n , x ) . The first assertion follows from Theorem 4.1 in view of Proposi-tion 1.1(ii), while the second one is a consequence of Lemma 1.2 in view of Remark 1.1(i). ⊓⊔ In the H¨older setting, Theorem 4.1 and Corollary 4.1 yield the following statement.
Corollary 4.2
Let α > and q > . The collection { Ω , . . . , Ω n } is α − semitransversalof order q at ¯ x with some δ > if, for some γ > and any x i ∈ X ( i = , . . . , n ) with < max ≤ i ≤ n k x i k < ( αδ ) q , there exists a λ ∈ ] α − ( max ≤ i ≤ n k x i k ) q , δ [ such that sup u i ∈ Ω i ( i = ,..., n ) , u ∈ X ( u ,..., u n , u ) =( ω ,..., ω n , x ) (cid:16) max ≤ i ≤ n k ω i − x i − x k (cid:17) q − (cid:16) max ≤ i ≤ n k u i − x i − u k (cid:17) q k ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ α (24) ransversality Properties: Primal Sufficient Conditions 17 for all x ∈ X and ω i ∈ Ω i ( i = , . . . , n ) satisfying (15) and (16) , or all the more, such thatq (cid:16) max ≤ i ≤ n k ω i − x i − x k (cid:17) q − × lim sup u i Ω i → ω i ( i = ,..., n ) , u → x ( u ,..., u n , u ) =( ω ,..., ω n , x ) max ≤ i ≤ n k ω i − x i − x k − max ≤ i ≤ n k u i − x i − u kk ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ α . (25) Proof
The statement is a direct consequence of Theorem 4.1 and Corollary 4.1 with ϕ ( t ) : = α − t q for all t ≥
0. Observe that ϕ − ( t ) = ( α t ) q . ⊓⊔ Remark 4.2 (i) On top of the explicitly given restriction k ω i − ¯ x k < λ / γ in Theorem 4.1(and similar conditions in its corollaries) on the choice of the points ω i ∈ Ω i , whichinvolves γ , the other conditions implicitly impose another one: k ω i − ¯ x k ≤ k x − ¯ x k + k ω i − x i − x k + k x i k ≤ k x − ¯ x k + ≤ i ≤ n k x i k , and consequently, k ω i − ¯ x k < λ + ϕ − ( δ ) . This alternative restriction can be of im-portance when γ is small.(ii) The statements of Theorem 4.1 and its corollaries can be simplified (and weakened!)by dropping condition (16).(iii) Inequalities (14), (22)–(25), which are crucial for checking nonlinear semitransversal-ity, involve two groups of parameters: on one hand, sufficiently small vectors x i ∈ X ,not all zero, and on the other hand, points x ∈ X and ω i ∈ Ω i near ¯ x . Note an impor-tant difference between these two groups. The magnitudes of x i are directly controlledby the value of δ in the definition of ϕ − semitransversality: ϕ ( max ≤ i ≤ n k x i k ) < δ . At the same time, taking into account that λ can be made arbitrarily close to ϕ ( max ≤ i ≤ n k x i k ) , the magnitudes of x − ¯ x and ω i − ¯ x (as well as ω i − x i − x ) are de-termined by δ indirectly; they are controlled by max ≤ i ≤ n k x i k : cf. conditions (15) and(16).(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 ′ ∈ ] , γ [ .(v) Even in the linear setting, the characterizations in Corollary 4.2 are new.The next corollary provides a simplified (and weaker!) version of Theorem 4.1. Thesimplification comes at the expense of eliminating the difference between the two groupsof parameters highlighted in Remark 4.2(iii). Corollary 4.3
The collection { Ω , . . . , Ω n } is ϕ − semitransversal at ¯ x with some δ > if,for some γ > and any x i ∈ X ( i = , . . . , n ) satisfying (13) , inequality (14) holds for allx ∈ B δ ( ¯ x ) and ω i ∈ Ω i ∩ B δ / γ ( ¯ x ) ( i = , . . . , n ) satisfying (16) . Sacrificing the estimates for δ in Theorem 4.1, and Corollaries 4.1 and 4.3, we arriveat the following ‘ δ -free’ statement. Corollary 4.4
The collection { Ω , . . . , Ω n } is ϕ − semitransversal at ¯ x if, for some γ > and all x i ∈ X ( i = , . . . , n ) near with max ≤ i ≤ n k x i k > , x ∈ X and ω i ∈ Ω i ( i = , . . . , n ) near ¯ x satisfying (16) , inequality (14) holds true. Moreover, inequality (14) can be replacedby its localized version (22) , or by (23) if ϕ ∈ C . Theorem 4.2
The collection { Ω , . . . , Ω n } is ϕ − subtransversal at ¯ x with some δ > and δ > if, for some γ > and any x ′ ∈ X satisfying k x ′ − ¯ x k < δ , < max ≤ i ≤ n d ( x ′ , Ω i ) < ϕ − ( δ ) , (26) there exists a λ ∈ ] ϕ ( max ≤ i ≤ n d ( x ′ , Ω i )) , δ [ such that sup u i ∈ Ω i ( i = ,..., n ) , u ∈ X ( u ,..., u n , u ) =( ω ,..., ω n , x ) ϕ (cid:16) max ≤ i ≤ n k ω i − x k (cid:17) − ϕ (cid:16) max ≤ i ≤ n k u i − u k (cid:17) k ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ for all x ∈ X and ω i , ω ′ i ∈ Ω i ( i = , . . . , n ) satisfying k x − x ′ k < λ , max ≤ i ≤ n k ω i − ω ′ i k < λγ , (28)0 < max ≤ i ≤ n k ω i − x k ≤ max ≤ i ≤ n k ω ′ i − x ′ k < ϕ − ( λ ) . (29)The proof below follows the pattern of that of Theorem 4.1. It employs a continuousreal-valued function f : X n + → R + determined by the given function ϕ ∈ C : f ( u , . . . , u n , u ) : = ϕ (cid:16) max ≤ i ≤ n k u i − u k (cid:17) , u , . . . , u n , u ∈ X , (30)and its restriction to Ω × . . . × Ω n × X given by (18). Note that the function (30) is aparticular case of (17) corresponding to setting x i : = ( i = , . . . , n ) . We provide here theproof of Theorem 4.2 for completeness and to expose the differences in handling the twotransversality properties, but we skip the proofs of most of its corollaries. Proof
Suppose { Ω , . . . , Ω n } is not ϕ − subtransversal at ¯ x with some δ > δ > γ > ρ ∈ ] , δ [ and a point x ′ ∈ B δ ( ¯ x ) such that ϕ ( max ≤ i ≤ n d ( x ′ , Ω i )) < ρ and ∩ ni = Ω i ∩ B ρ ( x ′ ) = /0. Hence, x ′ / ∈∩ ni = Ω i and 0 < ϕ (cid:16) max ≤ i ≤ n d ( x ′ , Ω i ) (cid:17) < ρ ≤ d (cid:16) x ′ , n \ i = Ω i (cid:17) . Let λ ∈ ] ϕ ( max ≤ i ≤ n d ( x ′ , Ω i )) , δ [ . Choose numbers ε and λ ′ such that ϕ (cid:16) max ≤ i ≤ n d ( x ′ , Ω i ) (cid:17) < ε < λ ′ < min { λ , ρ } , and points ω ′ i ∈ Ω i ( i = , . . . , n ) such that ϕ ( max ≤ i ≤ n k ω ′ i − x ′ k ) < ε . Let f and b f bedefined by (30) and (18), respectively, while X n + be equipped with the metric induced bythe norm (12). We have b f ( ω ′ , . . . , ω ′ n , x ′ ) < ε . Applying the Ekeland variational principle,we can find points ω i ∈ Ω i ( i = , . . . , n ) and x ∈ X such that k ( ω , . . . , ω n , x ) − ( ω ′ , . . . , ω ′ n , x ′ ) k γ < λ ′ , f ( ω , . . . , ω n , x ) ≤ f ( ω ′ , . . . , ω ′ n , x ′ ) , (31) f ( ω , . . . , ω n , x ) − f ( u , . . . , u n , u ) ≤ ελ ′ k ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ (32)for all ( u , . . . , u n , u ) ∈ Ω × . . . × Ω n × X . Thanks to (31), we have k x − x ′ k < λ ′ , andconsequently, d (cid:16) x , n \ i = Ω i (cid:17) ≥ d (cid:16) x ′ , n \ i = Ω i (cid:17) − k x − x ′ k > d (cid:16) x ′ , n \ i = Ω i (cid:17) − λ ′ > . ransversality Properties: Primal Sufficient Conditions 19 Hence, x / ∈ ∩ ni = Ω i , and max ≤ i ≤ n k ω i − x k >
0. In view of the definitions of λ ′ and f , con-ditions (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 4.5 (i)
The collection { Ω , . . . , Ω n } is ϕ − subtransversal at ¯ x with some δ > and δ > if, for some γ > and any x ′ ∈ X satisfying (26) , there exists a λ ∈ ] ϕ ( max ≤ i ≤ n d ( x ′ , Ω i )) , δ [ such that lim sup u i Ω i → ω i ( i = ,..., n ) , u → x ( u ,..., u n , u ) =( ω ,..., ω n , x ) ϕ (cid:16) max ≤ i ≤ n k ω i − x k (cid:17) − ϕ (cid:16) max ≤ i ≤ n k u i − u k (cid:17) k ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ for all x ∈ X and ω i , ω ′ i ∈ Ω i ( i = , . . . , n ) satisfying (28) and (29) . (ii) If ϕ ∈ C , then inequality (33) in part (i) can be replaced by ϕ ′ (cid:16) max ≤ i ≤ n k ω i − x k (cid:17) × lim sup u i Ω i → ω i ( i = ,..., n ) , u → x ( u ,..., u n , u ) =( ω ,..., ω n , x ) max ≤ i ≤ n k ω i − x k − max ≤ i ≤ n k u i − u kk ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ . (34)In the H¨older setting, Theorem 4.2 and Corollary 4.5 yield the following statement. Inview of Remark 2.3, we assume that q ≤ Corollary 4.6
Let α > and q ∈ ] , ] . The collection { Ω , . . . , Ω n } is α − subtransversalof order q at ¯ x with some δ > and δ > if, for some γ > and any x ′ ∈ B δ ( ¯ x ) with < max ≤ i ≤ n d ( x ′ , Ω i ) < ( αδ ) q , there exists a λ ∈ (cid:3) α − (cid:0) max ≤ i ≤ n d ( x ′ , Ω i ) (cid:1) q , δ (cid:2) such that sup u i ∈ Ω i ( i = ,..., n ) , u ∈ X ( u ,..., u n , u ) =( ω ,..., ω n , x ) (cid:16) max ≤ i ≤ n k ω i − x k (cid:17) q − (cid:16) max ≤ i ≤ n k u i − u k (cid:17) q k ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ α , (35) for all x ∈ X and ω i , ω ′ i ∈ Ω i ( i = , . . . , n ) satisfying (28) and < max ≤ i ≤ n k ω i − x k ≤ max ≤ i ≤ n k ω ′ i − x ′ k < ( αλ ) q , or all the more, such thatq (cid:16) max ≤ i ≤ n k ω i − x k (cid:17) q − × lim sup u i Ω i → ω i ( i = ,..., n ) , u → x ( u ,..., u n , u ) =( ω ,..., ω n , x ) max ≤ i ≤ n k ω i − x k − max ≤ i ≤ n k u i − u kk ( u , . . . , u n , u ) − ( ω , . . . , ω n , x ) k γ ≥ α . (36) Remark 4.3 (i) The expressions in the left-hand sides of (27) and (33) are, respectively,the nonlocal γ -slope and the γ -slope at ( ω , . . . , ω n , x ) of the function (18). (ii) Under the conditions of Theorem 4.2, there are two ways for estimating k ω i − ¯ x k : k ω i − ¯ x k ≤ k x ′ − ¯ x k + k ω i − ω ′ i k + k ω ′ i − x ′ k < δ + λ / γ + ϕ − ( λ ) , k ω i − ¯ x k ≤ k x − ¯ x k + k ω i − x k≤ k x ′ − ¯ x k + k x − x ′ k + max ≤ i ≤ n k ω ′ i − x ′ k < δ + λ + ϕ − ( λ ) . The second estimate does not involve γ and is better than the first one when γ <
1. Asimilar observation can be made about Corollary 4.7.(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 choiceof x ′ : ϕ ( max ≤ i ≤ n d ( x ′ , Ω i )) < d (cid:0) x ′ , ∩ ni = Ω i (cid:1) .(iv) The statement of Theorem 4.2 and its corollaries can be simplified by dropping condi-tion (29).(v) Inequalities (27), (33)–(36), which are crucial for checking nonlinear subtransversality,involve points x ∈ X and ω i ∈ Ω i near ¯ x . Their distance from ¯ x is determined in Theo-rem 4.2 via other points: x ′ / ∈ ∩ ni = Ω i and ω ′ i ∈ Ω i ; cf. conditions (28) and (29). Only thedistance from x ′ to ¯ x and to the sets Ω i is directly controlled by the values of δ and δ in the definition of ϕ − subtransversality: x ′ ∈ B δ ( ¯ x ) and ϕ ( max ≤ i ≤ n d ( x ′ , Ω i )) < δ .All the other distances are controlled by λ , which can be made arbitrarily close to ϕ ( max ≤ i ≤ n d ( x ′ , Ω i )) .(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 ′ ∈ ] , γ [ .(vii) Corollary 4.6 strengthens [42, Proposition 6]. In the linear case, it improves [39, Propo-sition 10].The next corollary provides a simplified (and weaker!) version of Theorem 4.2; cf.Remark 4.3(v). Corollary 4.7
The collection { Ω , . . . , Ω n } is ϕ − subtransversal at ¯ x with some δ > and δ > if, for some γ > , inequality (27) holds for all x ∈ B δ + δ ( ¯ x ) and ω i ∈ Ω i ∩ B δ + δ / γ + ϕ − ( δ ) ( ¯ x ) ( i = , . . . , n ) satisfying < max ≤ i ≤ n k ω i − x k < ϕ − ( δ ) .Proof Let δ > δ > x ′ ∈ B δ ( ¯ x ) \ ∩ ni = Ω i , λ ∈ ] ϕ ( max ≤ i ≤ n d ( x ′ , Ω i )) , δ [ , andpoints x ∈ X and ω i , ω ′ i ∈ Ω i ( i = , . . . , n ) satisfy conditions (28) and (29). Then k x − ¯ x k ≤ k x − x ′ k + k x ′ − ¯ x k < λ + δ < δ + δ , k ω i − ¯ x k ≤ k x ′ − ¯ x k + k ω i − ω ′ i k + k ω ′ i − x ′ k < δ + λ / γ + ϕ − ( λ ) < δ + δ / γ + ϕ − ( δ ) , k ω i − x k < ϕ − ( λ ) < ϕ − ( δ ) , i.e. points x ∈ X and ω i ∈ Ω i ( i = , . . . , n ) satisfy all the conditions in the corollary. Hence,inequality (27) holds. It follows from Theorem 4.2 that { Ω , . . . , Ω n } is ϕ − subtransversalat ¯ x with δ and δ . ⊓⊔ Sacrificing the estimates for δ and δ in Theorem 4.2, and Corollaries 4.5 and 4.7, wecan formulate the following ‘ δ -free’ statement. Corollary 4.8
The collection { Ω , . . . , Ω n } is ϕ − subtransversal at ¯ x if inequality (27) holds true for some γ > and all x ∈ X near ¯ x and ω i ∈ Ω i ( i = , . . . , n ) near ¯ x satisfying max ≤ i ≤ n k ω i − x k > . Moreover, inequality (27) can be replaced by its localized version (33) , or by (34) if ϕ ∈ C . ransversality Properties: Primal Sufficient Conditions 21 ϕ − transversality is in a sense an overarching property covering both ϕ − semitrans-versality and ϕ − subtransversality (see Proposition 2.2(iii)), the next theorem containssome 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 { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with some δ > and δ > if, for some γ > and any ω ′ i ∈ Ω i ∩ B δ ( ¯ x ) ( i = , . . . , n ) and ξ ∈ ] , ϕ − ( δ )[ ,there exists a λ ∈ ] ϕ ( ξ ) , δ [ such that inequality (14) holds for all x , x i ∈ X and ω i ∈ Ω i ( i = , . . . , n ) satisfying k x − ¯ x k < λ , max ≤ i ≤ n k ω i − ω ′ i k < λγ , (37)0 < max ≤ i ≤ n k ω i − x i − x k ≤ max ≤ i ≤ n k ω ′ i − x i − ¯ x k = ξ . (38) Proof
Suppose { Ω , . . . , Ω n } is not ϕ − transversal at ¯ x with some δ > δ > γ > ρ ∈ ] , δ [ andpoints ω ′ i ∈ Ω i ∩ B δ ( ¯ x ) and x ′ i ∈ X ( i = , . . . , n ) with ϕ ( max ≤ i ≤ n k x ′ i k ) < ρ such that ∩ ni = ( Ω i − ω ′ i − x ′ i ) ∩ ( ρ B ) = /0. Thus, ξ : = max ≤ i ≤ n k x ′ i k > ξ < ϕ − ( ρ ) < ϕ − ( δ ) . Set x i : = ω ′ i + x ′ i − ¯ x ( i = , . . . , n ) . Thenmax ≤ i ≤ n k ω ′ i − x i − ¯ x k = max ≤ i ≤ n k x ′ i k = ξ . Let λ ∈ ] ϕ ( ξ ) , δ [ and λ ′ : = min { λ , ρ } . Then ∩ ni = ( Ω i − x i ) ∩ B λ ′ ( ¯ x ) = /0, and con-sequently, condition (19) holds true. Let f and b f be defined by (17) and (18), re-spectively, while X n + be equipped with the metric induced by the norm (12). Wehave b f ( ω ′ , . . . , ω ′ n , ¯ x ) = ϕ ( max ≤ i ≤ n k x ′ i k ) = ϕ ( ξ ) < λ ′ . Choose a number ε such that b f ( ω ′ , . . . , ω ′ n , ¯ x ) < ε < λ ′ . Applying the Ekeland variational principle, we can find points ω i ∈ Ω i ( i = , . . . , n ) and x ∈ X such that k ( ω , . . . , ω n , x ) − ( ω ′ , . . . , ω ′ n , ¯ x ) k γ < λ ′ , f ( ω , . . . , ω n , x ) ≤ f ( ω ′ , . . . , ω ′ n , ¯ x ) , (39)and condition (21) holds for all u ∈ X and u i ∈ Ω i ( i = , . . . , n ) . In view of (19) and thedefinitions 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 4.9 (i)
The collection { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with some δ > and δ > if, for some γ > and any ω ′ i ∈ Ω i ∩ B δ ( ¯ x ) ( i = , . . . , n ) and ξ ∈ ] , ϕ − ( δ )[ , there exists a λ ∈ ] ϕ ( ξ ) , δ [ such that inequality (22) holds for allx , x i ∈ X and ω i ∈ Ω i ( i = , . . . , n ) satisfying (37) and (38) . (ii) If ϕ ∈ C , then inequality (22) in part (i) can be replaced by (23) . In the H¨older setting, Theorem 4.3 and Corollary 4.9 yield the following statement. Inview of Remark 2.3, we assume that q ≤ Corollary 4.10
Let α > and q ∈ ] , ] . The collection { Ω , . . . , Ω n } is α − transversal oforder q at ¯ x with some δ > and δ > if, for some γ > and any ω ′ i ∈ Ω i ∩ B δ ( ¯ x )( i = , . . . , n ) and ξ ∈ ] , ( αδ ) q [ , there exists a λ ∈ ] α − ξ q , δ [ such that inequality (24) holds true for all x , x i ∈ X and ω i ∈ Ω i ( i = , . . . , n ) satisfying (37) and (38) , or all themore, such that inequality (25) holds true. Remark 4.4 (i) On top of the explicitly given restriction k ω i − ω ′ i k < λ / γ in Theorem 4.3(and similar conditions in its corollaries), which involves γ , the other conditions im-plicitly impose another one: k ω i − ω ′ i k ≤ k x − ¯ x k + k ω i − x i − x k + k ω ′ i − x i − ¯ x k≤ k x − ¯ x k + ξ < λ + ϕ − ( δ ) . This alternative restriction can be of importance when γ is small.(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 x i : ϕ ( ξ ) < d ( ¯ x , ∩ ni = ( Ω i − x i )) .(iii) The sufficient conditions for ϕ − semitransversality and ϕ − subtransversality in Theo-rems 4.1 and 4.2 are particular cases of those in Theorem 4.3, corresponding to setting ω ′ i : = ¯ x and x = . . . = x n , respectively.(iv) The statement of Theorem 4.3 and its corollaries can be simplified by dropping condi-tion (38).(v) Inequalities (14), (22)–(25), which are crucial for checking nonlinear transversality,involve a collection of parameters: x , x i ∈ X and ω i ∈ Ω i , which are related to anothercollection: a small number ξ > ω ′ i ∈ Ω i near ¯ x . The value of ξ and mag-nitudes of ω ′ i − ¯ x are directly controlled by the values of δ and δ in the definition of ϕ − transversality: ϕ ( ξ ) < δ and ω ′ i ∈ B δ ( ¯ x ) . At the same time, taking into accountthat λ can be made arbitrarily close to ϕ ( ξ ) , the magnitudes of x − ¯ x , ω i − ω ′ i and x i are determined by δ and δ indirectly; they are controlled by ξ : cf. conditions (37)and (38). Thus, the derived parameters x , x i ∈ X and ω i ∈ Ω i involved in (14) possessthe natural properties: when δ and δ are small, the points x and ω i are near ¯ x and thevectors x i are small.(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 ′ ∈ ] , γ [ .(vii) Even in the linear setting, the characterizations in Corollary 4.10 are new.The next corollary provides a simplified (and weaker!) version of Theorem 4.3; cf.Remark 4.4(v). Corollary 4.11
The collection { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with some δ > and δ > if, for some γ > , inequality (14) holds for all x ∈ B δ ( ¯ x ) , x i ∈ Xand ω i ∈ Ω i ∩ B δ + δ / γ ( ¯ x ) ( i = , . . . , n ) satisfying ϕ ( max ≤ i ≤ n d ( x i + ¯ x , Ω i )) < δ and < max ≤ i ≤ n k ω i − x i − x k < ϕ − ( δ ) .Proof Let δ > δ > ω ′ i ∈ Ω i ∩ B δ ( ¯ x ) , ξ ∈ ] , ϕ − ( δ )[ , λ ∈ ] ϕ ( ξ ) , δ [ , and points x , x i ∈ X and ω i ∈ Ω i ( i = , . . . , n ) satisfy conditions (37) and (38). Then k x − ¯ x k < λ < δ , k ω i − ¯ x k ≤ k ω ′ i − ¯ x k + k ω i − ω ′ i k < δ + λ / γ < δ + δ / γ , d ( x i + ¯ x , Ω i ) ≤ k x i + ¯ x − ω ′ i k ≤ ξ < ϕ − ( δ ) , < max ≤ i ≤ n k ω i − x i − x k ≤ ξ < ϕ − ( δ ) , i.e. points x , x i ∈ X and ω i ∈ Ω i ( i = , . . . , n ) satisfy all the conditions in the corol-lary. Hence, inequality (14) holds. It follows from Theorem 4.3 that { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with δ and δ . ⊓⊔ Sacrificing the estimates for δ and δ in Theorem 4.3, and Corollaries 4.9 and 4.11,we can formulate the following ‘ δ -free’ statement. ransversality Properties: Primal Sufficient Conditions 23 Corollary 4.12
The collection { Ω , . . . , Ω n } is ϕ − transversal at ¯ x if, for some γ > andall x ∈ X near ¯ x, x i ∈ X ( i = , . . . , n ) near and ω i ∈ Ω i ( i = , . . . , n ) near ¯ x satisfying max ≤ i ≤ n k ω i − x i − x k > , inequality (14) holds true. Moreover, inequality (14) can bereplaced by its localized version (22) , or by (23) if ϕ ∈ C .Remark 4.5 The sufficient conditions for ϕ − semitransversality and ϕ − transversality inTheorems 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 4.12 arestronger than the corresponding ones in Theorem 4.1 and Corollary 4.4, respectively, asthey 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 4.4 and 4.12 are almost identical: the only difference is theadditional condition max ≤ i ≤ n k ω i − x i − x k ≤ max ≤ i ≤ n k x i k in Corollary 4.4. The sufficient condition in Corollary 4.12 is still acceptable for char-acterizing ϕ − transversality, but the one in Corollary 4.4 seems a little too strong for ϕ − semitransversality. That is why we prefer not to oversimplify these sufficient condi-tions. In this section, we provide quantitative relations between the nonlinear transversality ofcollections of sets and the corresponding nonlinear regularity properties of set-valued map-pings. Besides, nonlinear extensions of the new transversality properties of a set-valuedmapping to a set in the range space due to Ioffe [28] are discussed.5.1 Regularity of Set-Valued MappingsOur model here is a set-valued mapping F : X ⇒ Y between metric spaces. We considerits local regularity properties near a given point ( ¯ x , ¯ y ) ∈ gph F . The nonlinearity in thedefinitions of the properties is determined by a function ϕ ∈ C .Regularity of set-valued mappings have been intensively studied for decades due totheir numerous important applications; see monographs [18, 28, 29, 49]. Nonlinear regu-larity properties have also been considered by many authors; cf. [9, 21–23, 26, 34, 35, 45,48, 54, 61]. The relations between transversality and regularity properties are well knownin the linear case [25, 27, 30–32, 36, 39, 40, 43] as well as in the H¨older setting [42]. Belowwe briefly discuss more general nonlinear models. Definition 5.1
The mapping F is(i) ϕ − semiregular at ( ¯ x , ¯ y ) if there exists a δ > d ( ¯ x , F − ( y )) ≤ ϕ ( d ( y , ¯ y )) for all y ∈ Y with ϕ ( d ( y , ¯ y )) < δ ;(ii) ϕ − subregular at ( ¯ x , ¯ y ) if there exist δ > δ > d ( x , F − ( ¯ y )) ≤ ϕ ( d ( ¯ y , F ( x ))) for all x ∈ B δ ( ¯ x ) with ϕ ( d ( ¯ y , F ( x ))) < δ ; (iii) ϕ − regular at ( ¯ x , ¯ y ) if there exist δ > δ > d ( x , F − ( y )) ≤ ϕ ( d ( y , F ( x ))) (40)for all x ∈ X and y ∈ Y with d ( x , ¯ x ) + d ( y , ¯ y ) < δ and ϕ ( d ( y , F ( x ))) < δ .The function ϕ ∈ C in the above definition plays the role of a kind of rate or modu-lus of the respective property. In the H¨older setting, i.e. when ϕ ( t ) : = α − t q with α > q >
0, we refer to the respective properties in Definition 5.1 as α − semiregularity, α − subregularity and α − regularity of order q . These regularity properties have been stud-ied in [22, 23, 34, 42, 45, 48]. It is usually assumed that q ≤
1. The exact upper bound ofall α > δ >
0, or δ > δ >
0, is called the modulus of this property. We use notations s e rg q [ F ]( ¯ x , ¯ y ) , srg q [ F ]( ¯ x , ¯ y ) and rg q [ F ]( ¯ x , ¯ y ) forthe moduli of the respective properties. If a property does not hold, then by conventionthe respective modulus equals 0. With q = semiregularity , subregularity and regularity , respectively; cf. [12, 18, 28, 32, 49, 56].The following assertion is a direct consequence of Definition 5.1. Proposition 5.1
If F is ϕ − regular at ( ¯ x , ¯ y ) with some δ > and δ > , then it is ϕ − se-miregular at ( ¯ x , ¯ y ) with δ : = min { δ , ϕ ( δ ) } and ϕ − subregular at ( ¯ x , ¯ y ) with δ and δ . Note the combined inequality d ( x , ¯ x ) + d ( y , ¯ y ) < δ employed in part (iii) of Defini-tion 5.1 instead of the more traditional separate conditions x ∈ B δ ( ¯ x ) and y ∈ B δ ( ¯ y ) . Thisreplacement does not affect the property of ϕ − regularity itself, but can have an effect onthe value of δ . Employing this inequality makes the property a direct analogue of themetric characterization of ϕ − transversality in Theorem 3.2 and is convenient for estab-lishing relations between the regularity and transversality properties. The next propositionprovides also an important special case when the point x in (40) can be fixed: x = ¯ x . Proposition 5.2
Let δ > and δ > . Consider the following conditions: (a) inequality (40) holds for all x ∈ B δ ( ¯ x ) and y ∈ B δ ( ¯ y ) with ϕ ( d ( y , F ( x ))) < δ ; (b) inequality (40) holds for all x ∈ X and y ∈ Y with d ( x , ¯ x ) + d ( y , ¯ y ) < δ and ϕ ( d ( y , F ( x ))) < δ ; (c) d ( ¯ x , F − ( y )) ≤ ϕ ( d ( y , F ( ¯ x ))) for all y ∈ B δ ( ¯ y ) with ϕ ( d ( y , F ( ¯ x ))) < δ .Then (i) (a) ⇒ (b) ⇒ (c) . Moreover, condition (b) implies (a) with δ ′ : = δ / in place of δ . (ii) If X is a normed space, Y = X n for some n ∈ N , ¯ y = ( ¯ x , . . . , ¯ x n ) and F : X ⇒ X n isgiven by F ( x ) : = ( Ω − x ) × . . . × ( Ω n − x ) , x ∈ X , (41) where Ω , . . . , Ω n ⊂ X, then (b) ⇔ (c) .Proof (i) All the implications are straightforward.(ii) In view of (i), we only need to prove (c) ⇒ (b). Suppose condition (c) is satisfied.Let x ∈ X , y = ( x , . . . , x n ) ∈ X n , k x − ¯ x k + k y − ¯ y k < δ and ϕ ( d ( y , F ( x ))) < δ . Set x ′ i : = x i + x − ¯ x ( i = , . . . , n ) and y ′ : = ( x ′ , . . . , x ′ n ) . Then k y ′ − ¯ y k ≤ k y ′ − y k + k y − ¯ y k = k x − ¯ x k + k y − ¯ y k < δ , d ( x , F − ( y )) = d ( x , ∩ ni = ( Ω i − x i )) = d ( ¯ x , ∩ ni = ( Ω i − x ′ i )) = d ( ¯ x , F − ( y ′ )) , d ( y , F ( x )) = max ≤ i ≤ n d ( x i , Ω i − x ) = max ≤ i ≤ n d ( x ′ i , Ω i − ¯ x ) = d ( y ′ , F ( ¯ x )) . and, thanks to (c), d ( x , F − ( y )) ≤ ϕ ( d ( y , F ( x ))) . ⊓⊔ ransversality Properties: Primal Sufficient Conditions 25 The set-valued mapping (41) plays the key role in establishing relations between theregularity and transversality properties. It was most likely first used by Ioffe in [25].Observe that F − ( x , . . . , x n ) = ( Ω − x ) ∩ . . . ∩ ( Ω n − x n ) for all x , . . . , x n ∈ X and, if¯ x ∈ ∩ ni = Ω i , then ( , . . . , ) ∈ F ( ¯ x ) . Theorem 5.1
Let Ω , . . . , Ω n be subsets of a normed space X, ¯ x ∈ ∩ ni = Ω i , ϕ ∈ C , and Fbe defined by (41) . (i) The collection { Ω , . . . , Ω n } is ϕ − semitransversal at ¯ x with some δ > if and only ifF is ϕ − semiregular at ( ¯ x , ( , . . . , )) with δ . (ii) The collection { Ω , . . . , Ω n } is ϕ − subtransversal at ¯ x with some δ > and δ > ifand only if F is ϕ − subregular at ( ¯ x , ( , . . . , )) with δ and δ . (iii) If { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with some δ > and δ > , then F is ϕ − regular at ( ¯ x , ( , . . . , )) with any δ ′ ∈ ] , δ ] and δ ′ > satisfying ϕ − ( δ ′ ) + δ ′ ≤ δ .Conversely, if F is ϕ − regular at ( ¯ x , ( , . . . , )) with some δ > and δ > ,then { Ω , . . . , Ω n } is ϕ − transversal at ¯ x with any δ ′ ∈ ] , δ ] and δ ′ > satisfying ϕ − ( δ ′ ) + δ ′ ≤ δ .Proof (i) and (ii) follow from Theorem 3.1(i) and (ii), respectively, while (iii) is a conse-quence of Theorem 3.2. ⊓⊔ The next corollary provides δ − free versions of the assertions in Theorem 5.1. Corollary 5.1
Let Ω , . . . , Ω n be subsets of a normed space X, ¯ x ∈ ∩ ni = Ω i , ϕ ∈ C , and Fbe defined by (41) . The collection { Ω , . . . , Ω n } is (i) ϕ − semitransversal at ¯ x if and only if F is ϕ − semiregular at ( ¯ x , ( , . . . , )) ; (ii) ϕ − subtransversal at ¯ x if and only if F is ϕ − subregular at ( ¯ x , ( , . . . , )) ; (iii) ϕ − transversal at ¯ x if and only if F is ϕ − regular at ( ¯ x , ( , . . . , )) .Remark 5.1 (i) In the H¨older setting Corollary 5.1 reduces to [42, Proposition 9].(ii) Apart from the mapping F defined by (41), in the case of two sets other set-valuedmappings can be used to ensure similar equivalences between the transversality andregularity properties; see [28].In view of Theorem 5.1, the nonlinear transversality properties of collections of setscan be viewed as particular cases of the corresponding nonlinear regularity properties ofset-valued mappings. We are going to show that the two popular models are in a senseequivalent.Given an arbitrary set-valued mapping F : X ⇒ Y between metric spaces and a point ( ¯ x , ¯ y ) ∈ gph F , we can consider the two sets: Ω : = gph F , Ω : = X × { ¯ y } (42)in the product space X × Y . Note that ( ¯ x , ¯ y ) ∈ Ω ∩ Ω = F − ( ¯ y ) × { ¯ y } . To establish therelationship between the two sets of properties, we have to assume in the next two theoremsthat X and Y are normed vector spaces. Theorem 5.2
Let X and Y be normed spaces, F : X ⇒ Y , ( ¯ x , ¯ y ) ∈ gph F, and ϕ ∈ C . Let Ω and Ω be defined by (42) , and ψ ( t ) : = ϕ ( t ) + t for all t ≥ . (i) If F is ϕ − semiregular at ( ¯ x , ¯ y ) with some δ > , then { Ω , Ω } is ψ − semitransversalat ( ¯ x , ¯ y ) with δ ′ : = δ + ϕ − ( δ ) / . (ii) If F is ϕ − subregular at ( ¯ x , ¯ y ) with some δ > and δ > , then { Ω , Ω } is ψ − sub-transversal at ( ¯ x , ¯ y ) with any δ ′ > and δ ′ > such that ϕ ( ψ − ( δ ′ )) ≤ δ and ψ − ( δ ′ ) + δ ′ ≤ δ . (iii) If F is ϕ − regular at ( ¯ x , ¯ y ) with some δ > and δ > , then { Ω , Ω } is ψ − trans-versal at ( ¯ x , ¯ y ) with any δ ′ > and δ ′ > such that ϕ ( ψ − ( δ ′ )) ≤ δ and ψ − ( δ ′ ) + δ ′ ≤ δ / .Proof Observe that ψ ∈ C , ϕ ( ψ − ( t )) + ψ − ( t ) = t and ψ ( ϕ − ( t ) / ) = t + ϕ − ( t ) / t ≥ F be ϕ − semiregular at ( ¯ x , ¯ y ) with some δ >
0. Set δ ′ : = δ + ϕ − ( δ ) / = ψ ( ϕ − ( δ ) / ) . Let ρ ∈ ] , δ ′ [ and ( u , v ) , ( u , v ) ∈ ψ − ( ρ ) B . Set y ′ : = ¯ y + v − v .Observe that ( Ω − ( u , v )) ∩ ( Ω − ( u , v )) = ( gph F − ( u , v )) ∩ ( X × { ¯ y − v } )= (cid:0) F − ( y ′ ) − u (cid:1) × { ¯ y − v } . We have k y ′ − ¯ y k = k v − v k ≤ k v k + k v k < ψ − ( ρ ) , and consequently, ϕ ( k y ′ − ¯ y k ) < ϕ ( ψ − ( ρ )) < ϕ ( ψ − ( δ ′ )) = δ . By Definition 5.1(i), d ( ¯ x , F − ( y ′ ) − u ) ≤ d ( ¯ x , F − ( y ′ )) + k u k≤ ϕ ( k y ′ − ¯ y k ) + k u k < ϕ ( ψ − ( ρ )) + ψ − ( ρ ) = ρ , and consequently, d (( ¯ x , ¯ y ) , ( Ω − ( u , v )) ∩ ( Ω − ( u , v ))) ≤ max { d ( ¯ x , F − ( y ′ ) − u ) , k v k} < max { ρ , ψ − ( ρ ) } = ρ ;hence, ( Ω − ( u , v )) ∩ ( Ω − ( u , v )) ∩ B ρ ( ¯ x , ¯ y ) = /0 . (43)By Definition 2.1(i), { Ω , Ω } is ϕ − semitransversal at ( ¯ x , ¯ y ) with δ ′ .(ii) Let F be ϕ − subregular at ( ¯ x , ¯ y ) with some δ > δ >
0. Choose numbers δ ′ > δ ′ > ϕ ( ψ − ( δ ′ )) ≤ δ and ψ − ( δ ′ ) + δ ′ ≤ δ . Let ρ ∈ ] , δ ′ [ and ( x , y ) ∈ B δ ′ ( ¯ x , ¯ y ) with ψ ( max { d (( x , y ) , Ω ) , d (( x , y ) , Ω ) } ) < ρ , i.e. k y − ¯ y k < ψ − ( ρ ) and there exists a point ( x , y ) ∈ gph F such that k ( x , y ) − ( x , y ) k < ψ − ( ρ ) . Then k x − ¯ x k ≤ k x − x k + k x − ¯ x k < ψ − ( δ ′ ) + δ ′ ≤ δ , d ( ¯ y , F ( x )) ≤ k y − ¯ y k ≤ k y − ¯ y k + k y − y k < ψ − ( ρ ) , and consequently, ϕ ( d ( ¯ y , F ( x ))) < ϕ ( ψ − ( ρ )) < ϕ ( ψ − ( δ ′ )) ≤ δ . Choose a pos-itive ε < ψ − ( ρ ) − d ( ¯ y , F ( x )) . By Definition 5.1(ii), there exists an x ′ ∈ F − ( ¯ y ) suchthat k x ′ − x k < ϕ ( d ( ¯ y , F ( x )) + ε ) < ϕ ( ψ − ( ρ )) . Hence, ( x ′ , ¯ y ) ∈ Ω ∩ Ω and k x − x ′ k ≤ k x − x ′ k + k x − x k < ϕ ( ψ − ( ρ )) + ψ − ( ρ ) = ρ , k y − ¯ y k < ψ − ( ρ ) < ρ . Thus, Ω ∩ Ω ∩ B ρ ( x , y ) = /0. By Definition 2.1(ii), { Ω , Ω } is ψ − subtransversal at ( ¯ x , ¯ y ) with δ ′ and δ ′ .(iii) Let F be ϕ − regular at ( ¯ x , ¯ y ) with some δ > δ >
0. Choose numbers δ ′ > δ ′ > ϕ ( ψ − ( δ ′ )) ≤ δ and ψ − ( δ ′ ) + δ ′ ≤ δ /
2. Let ρ ∈ ] , δ ′ [ , ( x , y ) ∈ gph F ∩ B δ ′ ( ¯ x , ¯ y ) , x ∈ B δ ′ ( ¯ x ) and ( u , v ) , ( u , v ) ∈ ψ − ( ρ ) B . Set y ′ : = y + v − v .Then k x − ¯ x k + k y ′ − ¯ y k ≤ k v k + k v k + k x − ¯ x k + k y − ¯ y k < ψ − ( δ ′ ) + δ ′ ≤ δ , ϕ ( d ( y ′ , F ( x ))) ≤ ϕ ( k y ′ − y k ) ≤ ϕ ( k v k + k v k ) < ϕ ( ψ − ( δ ′ )) ≤ δ . ransversality Properties: Primal Sufficient Conditions 27 Choose a positive ε < (cid:0) ψ − ( ρ ) − max {k v k , k v k} (cid:1) . By Definition 5.1(iii), thereexists an x ′ ∈ F − ( y ′ ) such that k x − x ′ k < ϕ ( k y ′ − y k + ε ) ≤ ϕ ( {k v k , k v k} + ε ) < ϕ ( ψ − ( ρ )) . Denote ˆ x : = x ′ − x − u and ˆ y : = y ′ − y − v . Thus, ( x ′ , y ′ ) ∈ Ω and ( ˆ x , ˆ y ) ∈ Ω − ( x , y ) − ( u , v ) . At the same time, ˆ y = − v and ( ˆ x , ˆ y ) ∈ Ω − ( x , ¯ y ) − ( u , v ) .Moreover, k ˆ x k ≤ k x ′ − x k + k u k < ϕ ( ψ − ( ρ )) + ψ − ( ρ ) = ρ , k ˆ y k = k v k < ψ − ( ρ ) < ρ ;hence ( x ′ , y ′ ) ∈ ρ B . By Definition 2.1(iii), { Ω , Ω } is ψ − transversal at ( ¯ x , ¯ y ) with δ ′ and δ ′ . ⊓⊔ Theorem 5.3
Let X and Y be normed spaces, F : X ⇒ Y , ( ¯ x , ¯ y ) ∈ gph F, and ϕ ∈ C . Let Ω and Ω be defined by (42) , and ψ ( t ) : = ϕ ( t / ) for all t ≥ . (i) If { Ω , Ω } is ϕ − semitransversal at ( ¯ x , ¯ y ) with some δ > , then F is ψ − semiregularat ( ¯ x , ¯ y ) with δ . (ii) If { Ω , Ω } is ϕ − subtransversal at ( ¯ x , ¯ y ) with some δ > and δ > , then F is ψ − subregular at ( ¯ x , ¯ y ) with δ ′ : = min { δ , ψ ( δ ) } and δ . (iii) If { Ω , Ω } is ϕ − transversal at ( ¯ x , ¯ y ) with some δ > and δ > , then F is ψ − re-gular at ( ¯ x , ¯ y ) with any δ ′ ∈ ] , δ ] and δ ′ > such that ψ − ( δ ′ ) + δ ′ ≤ δ .Proof Observe that ψ ∈ C .(i) Let { Ω , Ω } be ϕ − semitransversal at ( ¯ x , ¯ y ) with some δ >
0. By Definition 2.1(i),condition (43) is satisfied for all ρ ∈ ] , δ [ and ( u , v ) , ( u , v ) ∈ ϕ − ( ρ ) B . Let y ∈ Y with ρ : = ψ ( k y − ¯ y k ) < δ . Choose a ρ ∈ ] ρ , δ [ and observe that ϕ (cid:18)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) , y − ¯ y (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:19) = ϕ (cid:18) k y − ¯ y k (cid:19) = ψ ( k y − ¯ y k ) < ρ . In view of (43), we can find ( x , y ) ∈ gph F and x ∈ X such that ( x , y ) − (cid:18) , y − ¯ y (cid:19) = ( x , ¯ y ) − (cid:18) , ¯ y − y (cid:19) ∈ B ρ ( ¯ x , ¯ y ) . Hence, y = ¯ y + y − ¯ y = y , x ∈ F − ( y ) , k x − ¯ x k < ρ , and consequently, d ( ¯ x , F − ( y )) < ρ . Letting ρ ↓ ρ , we obtain d ( ¯ x , F − ( y )) ≤ ψ ( k y − ¯ y k ) . By Definition 5.1(i), F is ψ − semiregular at ( ¯ x , ¯ y ) with δ .(ii) Let { Ω , Ω } be ϕ − subtransversal at ( ¯ x , ¯ y ) with some δ > δ >
0. By Defini-tion 2.1(ii), gph F ∩ ( X × { ¯ y } ) ∩ B ρ ( x , y ) = /0 for all ρ ∈ ] , δ [ and ( x , y ) ∈ B δ ( ¯ x , ¯ y ) with ϕ ( d (( x , y ) , gph F )) < ρ and ϕ ( k y − ¯ y k ) < ρ . Set δ ′ : = min { δ , ψ ( δ ) } . Let x ∈ B δ ( ¯ x ) and ψ ( d ( ¯ y , F ( x ))) < δ ′ . Choose a y ∈ F ( x ) such that ρ : = ψ ( k ¯ y − y k ) < δ ′ , and a ρ ∈ ] ρ , δ ′ [ . Set ˆ y : = y + ¯ y . Observe that k ˆ y − y k = k ˆ y − ¯ y k = k ¯ y − y k = ψ − ( ρ ) < ψ − ( ρ ) = ϕ − ( ρ ) , k ˆ y − ¯ y k < ψ − ( ρ ) < ψ − ( δ ′ ) ≤ δ . Thus, ρ ∈ ] , δ [ , ( x , ˆ y ) ∈ B δ ( ¯ x , ¯ y ) , ϕ ( d (( x , ˆ y ) , gph F )) ≤ ϕ ( k ˆ y − y k ) < ρ and ϕ ( k ˆ y − ¯ y k ) < ρ . Hence, gph F ∩ ( X × { ¯ y } ) ∩ B ρ ( x , ˆ y ) = /0, and consequently, d ( x , F − ( ¯ y )) < ρ .Letting ρ ↓ ρ , we obtain d ( x , F − ( ¯ y )) ≤ ψ ( k ¯ y − y k ) . Taking the infimum in the right-hand side of this inequality over y ∈ F ( x ) , we conclude that F is ψ − subregular at ( ¯ x , ¯ y ) with δ ′ and δ in view of Definition 5.1(ii). (iii) Let { Ω , Ω } be ϕ − transversal at ( ¯ x , ¯ y ) with some δ > δ >
0, i.e.for all ρ ∈ ] , δ [ , ( x ′ , y ′ ) ∈ gph F ∩ B δ ( ¯ x , ¯ y ) , u ∈ X and v , v ∈ Y with ϕ ( max {k u k , k v k , k v k} ) < ρ , it holds (cid:0) gph F − ( x ′ , y ′ ) − ( u , v ) (cid:1) ∩ ( X × {− v } ) ∩ ( ρ B ) = /0 , or equivalently, d (cid:0) x ′ + u , F − ( y ′ + v − v ) (cid:1) < ρ . In other words, d (cid:0) x , F − ( y ) (cid:1) < ρ for all ρ ∈ ] , δ [ , ( x ′ , y ′ ) ∈ gph F ∩ B δ ( ¯ x , ¯ y ) , x ∈ X and y ∈ Y with k x − x ′ k < ϕ − ( ρ ) and k y − y ′ k < ϕ − ( ρ ) . Choose numbers δ ′ ∈ ] , δ ] and δ ′ > ψ − ( δ ′ ) + δ ′ ≤ δ . Let x ∈ X and y ∈ Y with k x − ¯ x k + k y − ¯ y k < δ ′ and ψ ( d ( y , F ( x ))) < δ ′ .Choose a y ′ ∈ F ( x ) such that ρ : = ψ ( k y − y ′ k ) < δ ′ and a ρ ∈ ] ρ , δ ′ [ . Then ρ ∈ ] , δ [ , ( x , y ′ ) ∈ gph F , k x − ¯ x k < δ ′ < δ , k y ′ − ¯ y k ≤ k y ′ − y k + k y − ¯ y k < ψ − ( δ ′ ) + δ ′ ≤ δ and k y − y ′ k < ψ − ( ρ ) = ϕ − ( ρ ) . Hence, d (cid:0) x , F − ( y ) (cid:1) < ρ . Letting ρ ↓ ρ , weobtain d ( x , F − ( y )) ≤ ψ ( k y − y ′ k ) . Taking the infimum in the right-hand side of thisinequality over y ′ ∈ F ( x ) , we conclude that F is ψ − regular at ( ¯ x , ¯ y ) with δ ′ and δ ′ inview of Definition 5.1(iii). ⊓⊔ The next corollary of Theorems 5.2 and 5.3 provides qualitative relations between theregularity and transversality properties.
Corollary 5.2
Let X and Y be normed spaces, F : X ⇒ Y , ( ¯ x , ¯ y ) ∈ gph F, and ϕ ∈ C . Let Ω and Ω be defined by (42) , ψ ( t ) : = ϕ ( t ) + t and ψ ( t ) : = ϕ ( t / ) for all t ≥ . (i) If F is ϕ − (semi-/sub-)regular at ( ¯ x , ¯ y ) , then { Ω , Ω } is ψ − (semi-/sub-)transversalat ( ¯ x , ¯ y ) . (ii) If { Ω , Ω } is ϕ − (semi-/sub-)transversal at ( ¯ x , ¯ y ) , then F is ψ − (semi-/sub-)regularat ( ¯ x , ¯ y ) . The next statement addresses the H¨older setting. It is a consequence of Theorems 5.2and 5.3 with ϕ ( t ) : = α − t q for some α > q > t ≥ Corollary 5.3
Let X and Y be normed spaces, F : X ⇒ Y , ( ¯ x , ¯ y ) ∈ gph F, α > and q > .Let Ω and Ω be defined by (42) , α : = − q α , α : = q α , and ψ ( t ) : = α − t q + t for allt ≥ . (i) If F is α − semiregular of order q at ( ¯ x , ¯ y ) with some δ > , then { Ω , Ω } is ψ − se-mitransversal at ( ¯ x , ¯ y ) with δ ′ : = δ + ( αδ ) q / .If { Ω , Ω } is α − semitransversal of order q at ( ¯ x , ¯ y ) with some δ > , then F is α − semiregular of order q at ( ¯ x , ¯ y ) with δ . (ii) Let q ≤ . If F is α − subregular of order q at ( ¯ x , ¯ y ) with some δ > and δ > ,then { Ω , Ω } is ψ − subtransversal at ( ¯ x , ¯ y ) with any δ ′ > and δ ′ > such that ( ψ − ( δ ′ )) q ≤ αδ and ψ − ( δ ′ ) + δ ′ ≤ δ .If { Ω , Ω } is α − subtransversal of order q at ( ¯ x , ¯ y ) with some δ > and δ > , thenF is α − subregular of order q at ( ¯ x , ¯ y ) with δ ′ : = min { δ , α − δ q } and δ . (iii) Let q ≤ . If F is α − regular of order q at ( ¯ x , ¯ y ) with some δ > and δ > ,then { Ω , Ω } is ψ − transversal at ( ¯ x , ¯ y ) with any δ ′ > and δ ′ > such that ( ψ − ( δ ′ )) q ≤ αδ and ψ − ( δ ′ ) + δ ′ ≤ δ / .If { Ω , Ω } is α − transversal of order q at ( ¯ x , ¯ y ) with some δ > and δ > , then Fis α − regular of order q at ( ¯ x , ¯ y ) with any δ ′ ∈ ] , δ ] and δ ′ > such that ( αδ ′ ) q + δ ′ ≤ δ . In view of Corollary 5.3, H¨older transversality properties of { Ω , Ω } imply the corre-sponding H¨older regularity properties of F , while H¨older regularity properties of F implycertain ‘H¨older-type’ transversality properties of { Ω , Ω } determined by the function ψ .Utilizing Proposition 2.1, they can be approximated by proper H¨older (or even linear)transversality properties. ransversality Properties: Primal Sufficient Conditions 29 Corollary 5.4
Let X and Y be normed spaces, F : X ⇒ Y , ( ¯ x , ¯ y ) ∈ gph F, α > and q > .Let Ω and Ω be defined by (42) and α : = − q α . If F is α − (semi-/sub-) transversal at ( ¯ x , ¯ y ) , then { Ω , Ω } is α ′ − (semi-/sub-)transversal of order q ′ at ¯ x, where: (i) if q < , then q ′ = q and α ′ is any number in ] , α [ ; (ii) if q = , then q ′ = and α ′ : = ( + α − ) − ; (iii) if q > , then q ′ = and α ′ is any number in ] , [ . Thanks to Corollaries 5.3 and 5.4, in the case q ∈ ] , ] we have full equivalence be-tween the two sets of properties. The following corollary recaptures [42, Proposition 10]. Corollary 5.5
Let X and Y be normed spaces, F : X ⇒ Y , ( ¯ x , ¯ y ) ∈ gph F, and q ∈ ] , ] .Let Ω and Ω be defined by (42) . (i) { Ω , Ω } is α − semitransversal of order q at ( ¯ x , ¯ y ) if and only if F is semiregular oforder q at ( ¯ x , ¯ y ) . Moreover, s e rg q [ F ]( ¯ x , ¯ y ) s e rg q [ F ]( ¯ x , ¯ y ) + q ≤ s e tr q [ Ω , Ω ]( ¯ x ) ≤ s e rg q [ F ]( ¯ x , ¯ y ) q . (ii) { Ω , Ω } is α − subtransversal of order q at ( ¯ x , ¯ y ) if and only if F is subregular oforder q at ( ¯ x , ¯ y ) . Moreover, srg q [ F ]( ¯ x , ¯ y ) srg q [ F ]( ¯ x , ¯ y ) + q ≤ str q [ Ω , Ω ]( ¯ x ) ≤ srg q [ F ]( ¯ x , ¯ y ) q . (iii) { Ω , Ω } is α − transversal of order q at ( ¯ x , ¯ y ) if and only if F is regular of order q at ( ¯ x , ¯ y ) . Moreover, rg q [ F ]( ¯ x , ¯ y ) rg q [ F ]( ¯ x , ¯ y ) + q ≤ tr q [ Ω , Ω ]( ¯ x ) ≤ rg q [ F ]( ¯ x , ¯ y ) q . transversality properties of a set-valued mapping to a set in the range space due to Ioffe[27,28]. For geometric and subdifferential/normal cone characterizations of the properties,we refer the reader to [14–16]. In the rest of this section, F : X ⇒ Y is a set-valued mappingbetween normed spaces, ( ¯ x , ¯ y ) ∈ gph F , S is a subset of Y , ¯ y ∈ S , and ϕ ∈ C . Definition 5.2
The mapping F is(i) ϕ − semitransversal to S at ( ¯ x , ¯ y ) if { gph F , X × S } is ϕ − semitransversal at ( ¯ x , ¯ y ) , i.e.there exists a δ > ( gph F − ( u , v )) ∩ ( X × ( S − v )) ∩ B ρ ( ¯ x , ¯ y ) = /0for all ρ ∈ ] , δ [ , u ∈ X , v , v ∈ Y with ϕ ( max {k u k , k v k , k v k} ) < ρ ;(ii) ϕ − subtransversal to S at ( ¯ x , ¯ y ) if { gph F , X × S } is ϕ − subtransversal at ( ¯ x , ¯ y ) , i.e. thereexist δ > δ > F ∩ ( X × S ) ∩ B ρ ( x , y ) = /0for all ρ ∈ ] , δ [ and ( x , y ) ∈ B δ ( ¯ x , ¯ y ) with ϕ ( max { d (( x , y ) , gph F ) , d ( y , S ) } ) < ρ ; (iii) ϕ − transversal to S at ( ¯ x , ¯ y ) if { gph F , X × S } is ϕ − transversal at ( ¯ x , ¯ y ) , i.e. there exist δ > δ > ( gph F − ( x , y ) − ( u , v )) ∩ ( X × ( S − y − v )) ∩ ( ρ B ) = /0for all ρ ∈ ] , δ [ , ( x , y ) ∈ gph F ∩ B δ ( ¯ x , ¯ y ) , y ∈ S ∩ B δ ( ¯ y ) , u ∈ X , v , v ∈ Y with ϕ ( max {k u k , k v k , k v k} ) < ρ .The two-set model { gph F , X × S } employed in Definition 5.2 is an extension of themodel (42), which corresponds to the case when S is a singleton: S : = { ¯ y } .The metric characterizations of the properties in the next two statements are conse-quences of Theorems 3.1 and 3.2, respectively. Each characterization can be used as anequivalent definition for the respective property. Corollary 5.6
The mapping F is (i) ϕ − semitransversal to S at ( ¯ x , ¯ y ) with some δ > if and only ifd (cid:0) ( ¯ x , ¯ y ) , ( gph F − ( x , y )) ∩ ( X × ( S − y )) (cid:1) ≤ ϕ ( max {k x k , k y k , k y k} ) for all x ∈ X, y , y ∈ Y with ϕ ( max {k x k , k y k , k y k} ) < δ ; (ii) is ϕ − subtransversal to S at ( ¯ x , ¯ y ) with some δ > and δ > if and only if thefollowing equivalent conditions hold: (a) for all ( x , y ) ∈ B δ ( ¯ x , ¯ y ) with ϕ (cid:0) max { d (( x , y ) , gph F ) , d ( y , S ) } (cid:1) < δ , it holdsd (( x , y ) , gph F ∩ ( X × S )) ≤ ϕ ( max { d (( x , y ) , gph F ) , d ( y , S ) } ) ;(b) for all ( x , y ) ∈ gph F ∩ B δ ( ¯ x , ¯ y ) , y ∈ S ∩ B δ ( ¯ y ) and u ∈ X, v , v ∈ Y with ϕ ( max {k u k , k v k , k v k} ) < δ and x + u ∈ B δ ( ¯ x ) , y + v = y + v ∈ B δ ( ¯ y ) ,it holdsd (( , ) , ( gph F − ( x , y ) − ( u , v )) ∩ ( X × ( S − y − v ))) ≤ ϕ ( max {k u k , k v k , k v k} ) ; (44)(iii) ϕ − transversal to S at ( ¯ x , ¯ y ) with some δ > and δ > if and only if inequality (44) holds for all ( x , y ) ∈ gph F ∩ B δ ( ¯ x , ¯ y ) , y ∈ S ∩ B δ ( ¯ y ) and u ∈ X, v , v ∈ Y with ϕ ( max {k u k , k v k , k v k} ) < δ . Corollary 5.7
Let δ > and δ > . The following conditions are equivalent: (i) for all ( x , y ) ∈ gph F ∩ B δ ( ¯ x , ¯ y ) , y ∈ S ∩ B δ ( ¯ y ) and u ∈ X, v , v ∈ Y with x + u ∈ B δ ( ¯ x ) , y + v , y + v ∈ B δ ( ¯ y ) and ϕ ( max {k u k , k v k , k v k} ) < δ , inequality (44) holds true; (ii) for all x , y , y ∈ δ B with ϕ ( max { d (( ¯ x , ¯ y ) , gph F − ( x , y )) , d ( ¯ y , S − y ) } ) < δ , itholdsd (( ¯ x , ¯ y ) , ( gph F − ( x , y )) ∩ ( X × ( S − y ))) ≤ ϕ ( max { d (( ¯ x , ¯ y ) , gph F − ( x , y )) , d ( ¯ y , S − y ) } ) ; (45)(iii) for all x , x ∈ X, y , y , y ∈ Y such that x + x ∈ B δ ( ¯ x ) , y + y , y + y ∈ B δ ( ¯ y ) and ϕ ( max { d (( x , y ) , gph F − ( x , y )) , d ( y , S − y ) } ) < δ , it holdsd (cid:0) ( x , y ) , ( gph F − ( x , y )) ∩ ( X × ( S − y )) (cid:1) ≤ ϕ (cid:0) max { d (( x , y ) , gph F − ( x , y )) , d ( y , S − y ) } (cid:1) . ransversality Properties: Primal Sufficient Conditions 31 Moreover, if F is ϕ − transversal to S at ( ¯ x , ¯ y ) with some δ > and δ > , thenconditions (i)–(iii) hold with any δ ′ ∈ ] , δ ] and δ ′ > satisfying ϕ − ( δ ′ ) + δ ′ ≤ δ inplace of δ and δ .Conversely, if conditions (i)–(iii) hold with some δ > and δ > , then F is ϕ − tra-nsversal to S at ( ¯ x , ¯ y ) with any δ ′ ∈ ] , δ ] and δ ′ > satisfying ϕ − ( δ ′ ) + δ ′ ≤ δ .Remark 5.2 In the linear case, i.e. when ϕ ( t ) : = α t for some α > t ≥
0, in viewof Corollaries 5.6(ii)(a) and 5.7(iii), the properties in parts (ii) and (iii) of Definition 5.2reduce, respectively, to the ones in [28, Definitions 7.11 and 7.8]. The property in part (i)is new.The set-valued mapping (41), crucial for establishing equivalences between trans-versality 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: G ( x , y ) : = (cid:0) gph F − ( x , y ) (cid:1) × (cid:0) X × ( S − y ) (cid:1) , ( x , y ) ∈ X × Y . (46)Observe that G − ( x , y , x , y ) = (cid:0) gph F − ( x , y ) (cid:1) ∩ (cid:0) X × ( S − y ) (cid:1) for all x , x ∈ X , y , y ∈ Y and, if ( ¯ x , ¯ y ) ∈ gph F , ¯ y ∈ S , then (cid:0) ( , ) , ( , ) (cid:1) ∈ G ( ¯ x , ¯ y ) .The relationships between the nonlinear transversality and regularity properties in thenext statement are direct consequences of Theorem 5.1. Theorem 5.4
Let G be defined by (46) . (i) F is ϕ − semitransversal to S at ( ¯ x , ¯ y ) with some δ > if and only if G is ϕ − semiregularat (cid:0) ( ¯ x , ¯ y ) , ( , ) , ( , ) (cid:1) with δ . (ii) F is ϕ − subtransversal to S at ( ¯ x , ¯ y ) with some δ > and δ > if and only if G is ϕ − subregular at (cid:0) ( ¯ x , ¯ y ) , ( , ) , ( , ) (cid:1) with δ and δ . (iii) If F is ϕ − transversal to S at ( ¯ x , ¯ y ) with some δ > and δ > , then G is ϕ − regularat (cid:0) ( ¯ x , ¯ y ) , ( , ) , ( , ) (cid:1) with any δ ′ ∈ ] , δ ] and δ ′ > satisfying δ ′ + ϕ − ( δ ′ ) ≤ δ .Conversely, if G is ϕ − regular at (cid:0) ( ¯ x , ¯ y ) , ( , ) , ( , ) (cid:1) with some δ > and δ > ,then F is ϕ − transversal to S at ( ¯ x , ¯ y ) with any δ ′ ∈ ] , δ ] and δ ′ > satisfying δ ′ + ϕ − ( δ ′ ) ≤ δ .Remark 5.3 It is easy to see that the set-valued mapping (46) can be replaced in our con-siderations by the truncated mapping G : X × Y ⇒ X × Y × Y defined by G ( x , y ) : = (cid:0) gph F − ( x , y ) (cid:1) × ( S − y ) , ( x , y ) ∈ X × Y . The last mapping admits a simple representation G ( x , y ) = gph F − ( x , y , y ) , where theset-valued mapping F : X ⇒ Y × Y is defined by F ( x ) : = F ( x ) × S , x ∈ X . It was shown in [28, Theorems 7.12 and 7.9] that in the linear case the subtransversalityand transversality of F to S at ( ¯ x , ¯ y ) are equivalent to the metric subregularity and regular-ity, respectively, of the mapping ( x , y ) F ( x ) − ( y , y ) at (( ¯ x , ¯ y ) , ) . Acknowledgement
The authors wish to thank the referee and the handling editor for their careful reading ofthe manuscript and valuable comments and suggestions.
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