Nonlinear metric regularity of set valued mappings on a fixed set and applications

MINISTRY OF EDUCATION AND TRAININGQUY NHON UNIVERSITY DAO NGOC HAN NONLINEAR METRIC REGULARITY OF SET-VALUED MAPPINGS ON A FIXED SET AND APPLICATIONS DOCTORAL THESIS IN MATHEMATICS Binh

MINISTRY OF EDUCATION AND TRAINING

QUY NHON UNIVERSITY

DAO NGOC HAN

NONLINEAR METRIC REGULARITY OF SET-VALUED MAPPINGS ON A FIXED SET AND APPLICATIONS

DOCTORAL THESIS IN MATHEMATICS

Binh Dinh - 2021

MINISTRY OF EDUCATION AND TRAINING

QUY NHON UNIVERSITY

DAO NGOC HAN

NONLINEAR METRIC REGULARITY OF SET-VALUED MAPPINGS ON A FIXED SET AND APPLICATIONS

Speciality: Mathematical AnalysisSpeciality code: 9 46 01 02

Reviewer 1: Assoc Prof Dr Phan Nhat Tinh

Reviewer 2: Assoc Prof Dr Nguyen Huy Chieu

Reviewer 3: Assoc Prof Dr Pham Tien Son

Supervisors:

Assoc Prof Dr Habil Huynh Van Ngai

Dr Nguyen Huu Tron

Binh Dinh - 2021

This dissertation was completed at the Department of Mathematics andStatistics, Quy Nhon University under the guidance of Assoc Prof Dr Habil.Huynh Van Ngai and Dr Nguyen Huu Tron I hereby declare that the results pre-sented in here are new and original Most of them were published in peer-reviewedjournals, others have not been published elsewhere For using results from jointpapers I have gotten permissions from my co-authors

Binh Dinh, December 21, 2021

The dissertation was carried out during the years I have been a PhD student

at the Department of Mathematics and Statistics, Quy Nhon University On theoccasion of completing the thesis, I would like to express the deep gratitude toAssoc Prof Dr Habil Huynh Van Ngai not only for his teaching and scientificleadership, but also for the helping me access to the academic environment throughthe workshops, mini courses that assist me in broadening my thinking to get theentire view on the related issues in my research

I wish to express my sincere gratitude to my second supervisor, Dr NguyenHuu Tron, for accompanying me in study Thanks to his enthusiastic guidance, Iapproached the problems quickly, and this valuable support helps me to be moremature in research

A very special thank goes to the teachers at the Department of Mathematicsand Statistics who taught me wholeheartedly during the time of study, as well asall the members of the Assoc Prof Huynh Van Ngai’s research group for theirvaluable comments and suggestions on my research results I would like to thankthe Department of Postgraduate Training, Quy Nhon University for creating thebest conditions for me to complete this work within the schedules

I also want to thank my friends, PhD students and colleagues at Quy NhonUniversity for their sharing and helping in the learning process Especially, I amgrateful to Mrs Pham Thi Kim Phung for her constant encouragement giving methe motivation to overcome difficulties and pursue the PhD program

I wish to acknowledge my mother, my parents in law for supporting me inevery decision And, my enormous gratitude goes to my husband and sons for theirlove and patience during the time I was working intensively to complete my PhDprogram Finally, my sincere thank goes to my father for guiding me to math andthis thesis is dedicated to him

1.1 Some related classical results 11

1.2 Basic tools from variational analysis and nonsmooth analysis 13

1.2.1 Ekeland’s variational principles 13

1.2.2 Subdifferentials and some calculus rules 15

1.2.3 Coderivatives of set-valued mappings 18

1.2.4 Duality mappings 20

1.2.5 Strong slope and some error bound results 22

1.3 Metric regularity and equivalent properties 27

1.3.1 Local metric regularity 27

1.3.2 Nonlocal metric regularity 29

1.3.3 Nonlinear metric regularity 30

1.4 Metric regularity criteria in metric spaces 33

1.5 The infinitesimal criteria for metric regularity in metric spaces 36

2 Metric regularity on a fixed set: definitions and characterizations 382.1 Definitions and equivalence of the nonlinear metric regularity concepts 392.2 Characterizations of nonlinear metric regularity via slope 442.3 Characterizations of nonlinear metric regularity via coderivative 542.4 Conclusions 63

3 Perturbation stability of Milyutin-type regularity and applications 643.1 Perturbation stability of Milyutin-type regularity 653.2 Application to fixed double-point problems 783.3 Conclusions 84

4.1 Definitions and characterizations of nonlinear star metric regularity 854.2 Stability of Milyutin-type regularity under perturbation of star reg-ularity 904.3 Conclusions 98

5 Stability of generalized equations governed by composite

5.1 Notation and some related concepts 1005.2 Regularity of parametrized epigraphical and composition set-valuedmappings 1055.3 Stability of implicit set-valued mappings 1205.3.1 Stability of implicit set-valued mappings associated to the

epigraphical set-valued mapping 120

5.3.2 Stability of implicit set-valued mappings associated to a

com-posite mapping 1245.4 Conclusions 130

Table of Notations

A : X → Y

b

ˆ

the set-valued mapping H

mapping H

distance function d(y, F (x))

distance function d(y, F (x) ∩ V )

distance function d(y, T (x, p))

reg(γ,κ)F : the modulus of (γ, κ)-Milyutin regularity of F

reg∗(γ,κ)F : the modulus of (γ, κ)-Milyutin regularity∗ of F

In mathematics, solving many problems leads to the formation of equationsand solving them The basis question dealing with the equations is that whether

a solution exists or not If exists, how to identify a such solution? And, how

powerful frameworks to consider the existence of solutions of equations is metricregularity For equations of the form f (x) = y, where f : X → Y is a single-valuedmapping between metric spaces, the condition ensuring the existence of solutions

of equations is the surjectivity of f In the case of f being a single-valued mapping

theorem, which is considered as one of the main results of nonlinear analysis.Actually, a large amount of practical problems interested in outrun equations.For instance, systems of inequalities and equalities, variational inequalities orsystems of optimality conditions are covered by the solvability of an inclusion

y ∈ F (x),

where F : X ⇒ Y is a set-valued mapping between metric spaces These inclusionsare named as generalized equations or variational systems, which were initiated

phenomena in mathematics and other science areas, such as equations, variationalinequalities, complementary problems, dynamical systems, optimal control, andnecessary/sufficient conditions for optimization and control problems, fixed pointtheory, coincidence point theory and so on Nowadays, generalized equations have

and the references given therein) And thus, variational analysis has appeared inresponse to the strong development

A central issue of variational analysis is to investigate the existence and

are perturbed, where the mapping F may lack of smoothness: non-differentiable

functions or set-valued mappings, etc Then, it is almost impossible to approximate

surjectivity of the derivative mapping at the point in this case is not useful However,this can be replaced by giving an estimation of the distance from a certain point

the image of F at x In applications, the distance d(y, F (x)) is able to calculate

or estimate, meanwhile finding the exact solution set might be considerably morecomplicated Then, F : X ⇒ Y satisfying the above estimation is said to be local

estimated through a linear function of the distance d(y, F (x)) for all (x, y) near

In last decades, this property became a key concept in variational analysisand plays a crucial role in many areas of applied mathematics For a detailedaccount, the reader is referred to the works by many researchers, for instance,

higher order, i.e., for given a positive number α, one has the estimation

or more general, to the nonlinear case

optimization and variational analysis such as convergence analysis of optimization

algorithms, calculating high-order tangent cone, high-order error bounds, or

only on the local version of this property The advantage of the local property isthat we are free to change the neighborhood of a given point without damagingthe property However, these local versions do not meet many requirements inmathematical problems, for instance, fixed point problem, Newton’s method Theseproblems require that the set on which we work is fixed Recently, several authorssought to expand this property to nonlocal versions

It was found that the nonlocal regularity can be started from well-knownBanach’s contraction map principle Extension of this principle on closed ball in

a complete metric space established a connection between nonlocal regularity and

on a fixed set in the form of a box U × V , a subset of the product space X × Y ,i.e.,

The first purpose in this dissertation is to suggest some new models of non-localnonlinear regularity for set-valued mappings and to investigate them Concretely,

we consider the general notion of metric regularity on a fixed subset W of the

W beyond usual box-sets U ×V, is not only a natural generalization but also to meetsome practical applications for which the metric regularity on box-sets is violated.For instance, it covers such as, some notions of directional metric regularity whichare used in the theory of optimality conditions and in sensitivity analysis (see,

some characterizations for these regularity models based on the tools of variational

analysis such as local slope, non-local slope and coderivatives We also show that thespecial case of these non-local models-the Milyutin regularity possesses a suitablestability under small Lipschitz perturbation It is worth to noting further thatthe results established in here are new even regarding the case of box-sets Our

applying Ekeland’s variational principle (EVP, shortly) to the composition of themodulus function and the lower semicontinuous envelope of the distance functionassociated to the multifunction under consideration The approaches based on the(EVP) to problems related to the metric regularity of mappings and related topics

on the variable space This permits us to avoid the assumption on the completeness

we deal here with the regularity on a fixed set, controlled by a gauge function, theway to fix parameters to apply (EVP) must be changed differently from the local

In company with the expansion of the regularity concepts, problems related

principle, Ioffe explained the interconnection between regularity properties and

obtained two consequences of the coincidence theorem are the contractionmapping principle and Milyutin’s theorem These ideas help us to see the closedrelation between the nonlocal regularity of set-valued mappings and its fixed pointset on metric spaces Thus, the inheritence of the models and characterizations of

metric perturbation on a fixed set and its application to fixed point problems.The recent results on the stability of metric regularity or metric regularity-types

of set-valued mappings under single-valued or set-valued mappings as well as themodulus estimation of perturbed map have been interested in and studied by manyauthors in the community of variational analysis Interested reader can refer to

perturbed by a Lipschitz perturbation with Lipschitz constant smaller than the

rate of surjection of the unperturbed mapping then the regularity property cannot

from a complete metric space into Banach space is Milyutin regular with sur F ≥ rand g is Lipschitz with lip g ≤ `, F + g is Milyutin regular with sur(F + g) ≥ r − `

perturbation theorem in local context for the composition of a Milyutin regularset-valued mapping and a Lipschitz single-valued mapping that is not requiredlinear structure in the range space

The second purpose of the thesis is to establish nonlocal metric versions of theMilyutin’s theorem under composite perturbation, so the additive perturbation bythe Lipschitz mapping with a sufficiently small constant also possesses the stability

of Milyutin regularity And then, these results are applied to study the existence

Banach spaces X, Y : one from X to Y and other from Y to X A similar result

on the fixed double-point theorem was also established in the recent works by Ioffe

different and the starting conditions to estimate the distance to the fixed

be derived from the fixed double-point theorem; meanwhile, our proof of the fixeddouble-point theorem is obtained by using the Milyutin’s perturbation theoremestablished earlier Furthermore, the choice of starting conditions different from

Another point worth noting here is that the extension of the local regularityconcepts to the fixed set version also leads to a concept of weak regularity know

metric regularity of a set-valued mapping F between metric spaces X, Y on a box

U × V ⊂ X × Y as follows

for all (x, y) ∈ U × V such that 0 < τ d(y, F (x) ∩ V ) < γ(x) This version is strictlyweaker than the original one, and thus the use of star regular as assumptions in

principle will get better results.

Analogues of the results achieved, the third purpose of the thesis is to considersome models of nonlinear star metric regularity for set-valued mappings defined on

a fixed set of arbitrary form W ⊂ X × Y Specifically, in addition to establishingthe infinitesimal characterizations for the star regularity models, we also obtainsome versions of the perturbation theorem for models of Milyutin-type regularitywhen the star regularity mapping is perturbed by a Lipschitz function with thesuitable Lipschitz constant

Along with the metric regularity of a multifunction, the study of conditionsensuring the metric regularity of parametric multifunctions, that is, implicitmultifunction theorems, plays an important role in investigating problems ofsensitivity analysis with respect to parameters Regarding this issue, there aremany works by authors: Ioffe, Dontchev, Rockafellar, Dmitruk, Kruger, Durea,Strugariu, The results of the metric regularity were extended further to the case

parametric set-valued mappings were extended further to the case of the sum of

a metrically regular set-valued mapping and a single-valued mapping g(·, ·) which

is Lipschitz with respect to x, uniformly in p with a sufficiently small Lipschitz

Strugariu established the openness for the sum of two set-valued mappings Thenmetric regularity of the sum of two set-valued mappings was studied by Ngai, Tron,

established the openness for the generally nonparametric composition set-valued

metric regularity for this mapping Recently, the topic on stability of generalizedequations has also attracted the interest and the study of many experts and many

important results have been obtained We refer the reader to various contributions,

to establish metric regularity, semiregularity of the parametrized epigraphical andcomposition set-valued mapping and to derive the stability of the solution set

of generalized equations (called also implicit set-valued mappings) Precisely, wederive the calmness, Lipschitz-likeness, and Robinson metric regularity of implicitset-valued mappings associated to generalized equations

concentrate on the following issues:

In Chapter 1, we recall some classical results of the problem and some basicconcepts from variational analysis and nonsmooth analysis

In Chapter 2, we investigate some new models of nonlinear nonlocal regularity,

characterizations of metric regularity via strong slope of the lower semicontinuousenvelop and coderivative are established

Chapter 3 is devoted to the study of metric perturbation of Milyutin-typeregularity on a fixed set under composition of a set-valued mapping and a Lipschitz

achieved are applied to the fixed double-point problems

Chapter 4 is to introduce the concepts and characterizations of star metricregularity version, a weak one of metric regularity Furthermore, we also studythe stability of Milyutin-type regularity under perturbation of the star regularitymapping

We shall discuss in Chapter 5 the semiregularity of the parametric/nonparametric composite set-valued mapping, and the metric regularity of the

results, we study some kinds of regularity of the implicit multifunction associated

to the variational systems such as the calmness, Robinson metric regularity, metricregularity, and pseudo-Lipschitzness

1 The 16th Workshop on “Optimization and Scientific Computing”, Institute

of Mathematics, Vietnam Academy of Science and Technology (April 19–21, 2018,

Ba Vi, Hanoi)

2 Seminar at the Department of Mathematics and Statistics, Quy NhonUniversity

Chapter 1

Preliminaries

In this chapter some classical results, several concepts and tools of variationalanalysis are briefly summarized As a preparation for further investigation later, we

Mordukhovich coderivatives, rules of subdifferential calculus as well as obtainedresults of metric regularity

The concepts, tools, and results discussed in this chapter can be found in the

Recall that a map f : X → Y between topological spaces is said to be anopen mapping at x if for every open set V ⊂ X with x ∈ V , there exists an openset W ⊂ Y with f (x) ∈ W such that W ⊂ f (V ) The Banach’s open mappingprinciple gives us a sufficient condition to a continuous linear operator being anopen mapping

Banach spaces and A : X → Y be a continuous linear operator onto Y, that is

A(X) = Y Then A is an open mapping.

Remark 1.1.1 A is an open mapping if and only if there exists r > 0 such that

A : X → Y be a continuous linear operator The quantity

is called the Banach constant of A

From the Banach’s open mapping theorem, we obtain the metric regularity ofthe linear single-valued mapping F

Proposition 1.1.1 Let X, Y be Banach spaces and F : X → Y be a continuouslinear operator Then the following statements are equivalent:

(i) F is surjective

(ii) F is an open mapping

(iii) There exists k > 0 such that

Proof The implication (i) ⇒ (ii) is obvious by the Banach’s open mapping theorem

To show (ii) ⇒ (i), let y ∈ Y be arbitrary If y = 0, then 0 = F (0) ∈ F (X)

surjective For (ii) ⇒ (iii), since F is an open mapping, then there exists r > 0

It follows that y = F (u) + F (x) = F (u + x), so u + x ∈ F−1(y) One has

To show (iii) ⇒ (ii), we assume that there exists k > 0 such that (iii) occurs Then,

1

In the case of the differentiable nonlinear single-valued mapping F , the metric

assertions are equivalent:

(i) There exist τ, ε > 0 such that

non-smooth analysis

1.2.1 Ekeland’s variational principles

Introduced by Ekeland in 1974, Ekeland’s variational principle is one of themost important techniques of variational analysis and has many profound

applications in mathematics as well as other areas of science including optimization,nonsmooth analysis, economics, control theory, etc In our research, this principle

is an important tool

Let f be an extended real-valued function which can assume values ±∞ along

of f and epi f = {(x, r) ∈ X × R : f (x) ≤ r} the epigraph of f Recall that

x→x 0

f (x), f is called lower semicontinuous if f is lower semicontinuous

semicontinous

Let us recall that a lower semicontinuous function on a compact metric space

X attains its minimum The compactness is in fact a necessary property for theinfimum of an arbitrary lower semicontinuous function f to be realized at a point

¯

x In its absence therefore, such a minimizer does not exist in general On the

In this case, the variational principle of Ekeland asserts that even in the absence

of compactness, it is possible to show that, in a complete space, there is a small

f : X → R ∪ {+∞} be a proper lower semicontinuous function bounded frombelow Suppose that ε > 0 and z ∈ X satisfy

(iii) f (x) + εd(x, y) ≥ f (y), for all x ∈ X.

In addition, Ekeland’s variational principle is also equivalently stated in thefollowing form:

f : X → R ∪ {+∞} be a proper lower semicontinuous function bounded frombelow Suppose that ε > 0 and z ∈ X satisfy

1.2.2 Subdifferentials and some calculus rules

The notion of gradient of a function f : X → R on a Banach space X, denoted

∇f , plays a fundamental role in all sciences Geometrically, the gradient of f at

entirely below the epigraph of f ; here, a function f : X → R ∪ {+∞} on a vectorspace is called convex if

The notion of subgradient generalizes the notion of gradient: a subgradient of a

For convex functions, the subdifferential is a good substitute to the notion ofgradient: it exists at points where the function is not differentiable, and it ispossessed of properties similar to differential such as the Fermat Principle

“0 ∈ ∂f (x) iff x is a minimizer of f ”, the existence of exact calculus rules, e.g., thesum rule “∂(f + g)(x) = ∂f (x) + ∂g(x)”, However, not all functions studied inoptimization are convex, and sometimes we need to deal with generalized

subdifferential to nonconvex functions, and each of these types is appropriate to

derivative and subdifferential in sense of convex analysis

were interested in and have proven to be useful tools for the analysis of nonsmoothfunctions on Asplund spaces, a very important subclass of general Banach spaces,which are wide enough to cover variational and optimal problems appearing in

Let X be a real Banach space and f be a lower semicontinous function from

The limiting subdifferential (or, Mordukhovich subdifferential) at ¯x is constructed

and the symbol F : X ⇒ Y means “F is a set-valued mapping between metric spaces

X, Y ”, that is a correspondence associates every x a set F (x), possibly empty

Recall that a Banach space is said to be Asplund if every convex continuous

(i) ˆ∂(f1+f2)(x0) = ˆ∂f1(x0)+f20(x0) when f2 is continuously Fr´echet differentiable

composition of a differentiable and a lower semicontinuous function

lower semicontinuous function and γ is differentiable on (0, +∞) then for every

x ∈ dom f such that f (x) > 0, one has

ˆ

1.2.3 Coderivatives of set-valued mappings

The coderivative concept was introduced by Mordukhovich in 1980, as ageneral duality concept of the classic derivative concept of mappings betweenBanach spaces Coderivative is considered as a useful tool to develop the dualapproach to optimization and equiliblium problems The reader is referred to the

and related problems

Let X, Y be metric spaces For every set-valued mapping F : X ⇒ Y , weassociate two sets, the graph and the domain:

Then,

A set-valued mapping F is closed valued if for any x the set F (x) is closed;

F is a closed mapping if its graph is a closed set; F is locally closed at a point ofthe graph if there is a closed neighborhood of the point whose intersection with the

graph is closed The map is locally closed if it is locally closed at any point of thegraph.

Let Ω be a nonempty subset of a real Banach space X Given x ∈ Ω and

ε ≥ 0 The set of ε-normals to Ω at x is given by

all k ∈ N The collection of such normals

b

We put D∗F (¯x, ¯y)(y∗) = ∅ for all y∗ ∈ Y∗ if (¯x, ¯y) /∈ Graph F The relationshipsbetween coderivatives and derivatives of single-valued differentiable mappings are

b

neighborhood

1.2.4 Duality mappings

The concept of duality mapping was introduced by Beurling and Livingston

of this concept and characterized the duality mappings via the subdifferential of

is called the duality map with function φ, where X is any normed space In the

As a consequence of the following lemma, we will work on normed linear spaces

empty for any x in X

duality mapping concept as q-duality mapping and its normalized enlargements

In this subsection, we present these concepts for the more general cases in which

where the duality mapping J : X ⇒ X∗ for a Banach space X is defined by

In particular, in the case of X being a Hilbert space, we have J (x) =

xkxk

if

x 6= 0

Remark 1.2.1 The set J (x), for every x ∈ X, defined as above agrees with to theconvex subdifferential of the norm function k · k at x, i.e.,

Given ε ≥ 0, the normalized ε-enlargement of the µ-duality mapping is defined by

1.2.5 Strong slope and some error bound results

The concept of infinitesimal in metric spaces is expressed by a quantity called(strong) slope The notion of slope was first introduced in 1980 by De Giorgi,

on metric spaces Slope is proven to be an useful tool in the characterization of errorbounds, one of the most important regularity properties, providing an estimatefor the distance of a point from the solution set Based on the approach from

results on existence of an error bound is used to derive metric regularity results formultifunctions

(i) The quantity defined by |∇f |(x) = 0 if x is a local minimum of f ; otherwise

|∇f |(x) = lim sup

u→x,u6=x

f (x) − f (u)d(x, u)

is called the local slope of the function f at x ∈ dom f

(ii) The quantity

|Γf |(x) := sup

u6=x

d(x, u)

is called the nonlocal slope of the function f at x ∈ dom f

all x ∈ X

As regards to geometric meaning, the slope of a function at a given point is

a maximal speed descent of the function from this one In particular, if X is a

Since f attains the (global) minimum at x = 0, |∇f |(0) = |Γf |(0) = 0 For x 6= 0,

f is differentiable, so we have |∇f |(x) = 2x if x > 0 and |∇f |(x) = 1 if x < 0.Furthermore, if x > 0, one has

which yields |Γf |(x) = 2x If x < 0, one has

which follows that |Γf |(x) = 1 Hence, |Γf |(x) = max{2x, 1}, if x 6= 0

We next recall the concept of error bounds Let X be a metric space and

f : X → R ∪ {+∞} For α ∈ R, β ∈ R ∪ {+∞}, we denote by:

the closed and open sublevel sets of f respectively, and if α < β, we further denote

by [α < f < β] := [f < β]\[f ≤ α] the slice between α and β We shall say that fadmits a global error bound at the level α if there exists a real τ > 0 such that

for lower semicontinuous functions defined on a complete metric space in terms ofslope

a lower semicontinuous function, and α ∈ R, β ∈ R ∪ {+∞} with α < β

a) Assume that [f < β] 6= ∅ and

b) Conversely, assume that (1.7) holds for some τ > 0, then

definition of epi-upper semicontinuity of a family of functions

We say that a function f : X × P → R ∪ {+∞} is epi-upper semicontinuity at

p→¯ p

inf

p→¯ p

inf

x∈B(¯ x,ε)f (x, p)



Remark 1.2.2 In the similar argument, we can see that the conclusion of

In order to apply the result above, it is necessary to give an estimation ofslope The next propositon gives a below estimation of local slope by using asubdifferential operator Here, the notion of subdifferential is mentioned as follows

(P1) if f is convex then

local minimum point of f + g then, for every ε > 0 there exist x, y ∈ X,

operator such that properties (P 1) and (P 2) are satisfied Then, for every lowersemicontinuous function f : X → R ∪ {+∞} and every x ∈ X, we have

(y,f (y))→(x,f (x))d∗(0, ∂∗f (y))

As an immediate consequence of Theorem1.2.4and Proposition1.2.1, we thushave the following corollary:

such that (P 1) and (P 2) hold, f : X → R ∪ {+∞} be a lower semicontiuousfunction, and α ∈ R, β ∈ R ∪ {+∞} with α < β and [f < β] 6= ∅ Assume that,for some τ > 0, one has

inf

x∈[α<f <β]d∗(0, ∂∗f (x)) ≥ τ

Then, for all α ≤ γ < β

In this section, the concept of metric regularity for set-valued mappings isintroduced We shall see that in what follows, this concept is a natural extension

1.3.1 Local metric regularity

Firstly, we show the most popular case of local regularity near a certain point

And, given a nonempty subset A of X, the symbol B(A, r) denotes the open ballaround A with radius r > 0 defined by B(A, r) = {x ∈ X : d(x, A) < r}, whered(x, A) = inf{d(x, y) : y ∈ A}

mapping F is

(i) open or covering at a linear rate near (¯x, ¯y) if there are r, ε > 0 such that

such that

All three properties of the definition refer to the same phenomenon Morespecifically, one has the next proposition

Moreover, under the convention that 0 · ∞ = 1,

1.3.2 Nonlocal metric regularity

In the definition of local regularity, we speak about the satisfaction of regular

instrument for proving many practical problems, for instance, existence theorems

in a given set

Let X, Y be metric spaces, U ⊂ X and V ⊂ Y (we usually assume that U , Vare open), let F : X ⇒ Y , and let γ(·) and δ(·) be extended real-valued functions

on X and Y , assumed positive values (possibily infinite) on U and V respectively

mapping F is

(i) γ-open (or γ-covering) at a linear rate on U × V if there is a r > 0 such that

B(F (x), rt) ∩ V ⊂ F (B(x, t))

modulus of γ-openness of F on U × V

(ii) γ-metrically regular on U × V if there is a K > 0 such that

(iii) δ-pseudo-Lipschitz property on U × V if there is a K > 0 such that

d(y, F (x)) ≤ Kd(x, u)

the δ-Lipschitz modulus of F on U × V

The functions γ, δ are not needed for local regularity because the neighborhood

essential elements of the definition They determine how far we shall reach from

gauge functions The use of the gauge functions permits us to control the distancefrom the exact solution to the approximate solution so that this distance can bearbitrarily small

The equivalence theorem is going to be presented in the following as a principalresult of the regularity theory which emphasizes the equivalence in metric nature

of the just defined three regularity phenomena

of metric spaces X, Y , any F : X ⇒ Y , any U ⊂ X and V ⊂ Y and any extended

(i) F is γ-open at a linear rate on U × V

(ii) F is γ-metrically regular on U × V

Moreover, under the convention that 0 · ∞ = 1,

1.3.3 Nonlinear metric regularity

of balls around x in X in the case of openness are not proportional to the radius ofthe neighborhood of F (x) covered by the image of the balls with center x under F Despite that, the use of proven techniques similar to those in the linear case alsogives us the same results for these models In fact, this is a replacement of rt in

of the other properties; specifically, this nonlinear function is a modulus function µwhich is nonnegative, strictly increasing on [0, +∞), µ(0) = 0 and lim

spaces Let as before, U ⊂ X and V ⊂ Y be open sets, γ(·) be a function on Xwhich is positive on U , and δ(·) be a function on Y which is positive on V Assumefinally that we are given three modulus functions µ(·), ν(·) and η(·)

(i) F is γ- open on U × V with functional modulus not smaller than µ(·) if theinclusion

B(F (x), µ(t)) ∩ V ⊂ F (B(x, t))holds whenever x ∈ U and t < γ(x)

(ii) F is γ-metrically regular on U × V with functional modulus not greater thanν(·) if the inequality

holds whenever x ∈ U, y ∈ V, ν(d(y, F (x))) < γ(x)

inequality

d(y, F (x)) ≤ η(d(x, u))holds provided x ∈ U, y ∈ V ∩ F (u) and η(d(x, u)) < δ(y)

The equivalence theorem is also extended to this model

are equivalent:

(i) F is γ-open on U × V with functional modulus not smaller than µ

(ii) F is γ-metrically regular on U × V with functional modulus not greater than

(iii) F−1 is γ-H¨older on V × U with functional modulus not greater than µ−1.

Now, we concentrate on a special case of modulus functions, the most

with the concept of nonlinear regularity of order k

Fix as usual an F : X ⇒ Y with complete graph, open sets U ⊂ X, V ⊂ Yand a function γ on X, and let k ≥ 1

(i) F is γ-open of order k on U × V if there is an r > 0 such that

order k of F on U × V

(ii) F is γ-metrically regular of order k on U × V if there is a κ > 0 such that

γ-metric regularity of order k of F on U × V

We shall further also define for F the γ-rate of surjection of order k at (¯x, ¯y),

between the rates for local regularity is given by the following proposition

will serve us as a basic for obtaining various qualitative and quantitativecharacterizations of regularity

Throughout this section and subsequent chapters we work with the followingtwo functions associated with F : X ⇒ Y and y ∈ Y as follows:

Let ξ > 0 be given, we define ξ-metric on X × Y by

Assume that U ⊂ X is an open set and γ is a positive function on U , we set

x∈U

B(x, γ(x))

open sets, and let F : X ⇒ Y with complete graph in the product metric Let