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

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 Analysis

Speciality 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

Declaration

This dissertation was completed at the Department of Mathematics and

Statistics, 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-reviewed

journals, others have not been published elsewhere. For using results from joint

papers I have gotten permissions from my co-authors.

Binh Dinh, December 21, 2021

Advisors PhD student

Assoc. Prof. Dr. Habil. Huynh Van Ngai Dao Ngoc Han

i

Acknowledgments

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 the

occasion of completing the thesis, I would like to express the deep gratitude to

Assoc. Prof. Dr. Habil. Huynh Van Ngai not only for his teaching and scientific

leadership, but also for the helping me access to the academic environment through

the workshops, mini courses that assist me in broadening my thinking to get the

entire view on the related issues in my research.

I wish to express my sincere gratitude to my second supervisor, Dr. Nguyen

Huu Tron, for accompanying me in study. Thanks to his enthusiastic guidance, I

approached the problems quickly, and this valuable support helps me to be more

mature in research.

A very special thank goes to the teachers at the Department of Mathematics

and Statistics who taught me wholeheartedly during the time of study, as well as

all the members of the Assoc. Prof. Huynh Van Ngai’s research group for their

valuable comments and suggestions on my research results. I would like to thank

the Department of Postgraduate Training, Quy Nhon University for creating the

best conditions for me to complete this work within the schedules.

I also want to thank my friends, PhD students and colleagues at Quy Nhon

University for their sharing and helping in the learning process. Especially, I am

grateful to Mrs. Pham Thi Kim Phung for her constant encouragement giving me

the motivation to overcome difficulties and pursue the PhD program.

I wish to acknowledge my mother, my parents in law for supporting me in

every decision. And, my enormous gratitude goes to my husband and sons for their

love and patience during the time I was working intensively to complete my PhD

program. Finally, my sincere thank goes to my father for guiding me to math and

this thesis is dedicated to him.

ii

Contents

Table of Notations 1

Introduction 3

1 Preliminaries 11

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

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2 Metric regularity on a fixed set: definitions and characterizations 38

2.1 Definitions and equivalence of the nonlinear metric regularity concepts 39

2.2 Characterizations of nonlinear metric regularity via slope . . . . . . 44

2.3 Characterizations of nonlinear metric regularity via coderivative . . 54

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Perturbation stability of Milyutin-type regularity and applications 64

3.1 Perturbation stability of Milyutin-type regularity . . . . . . . . . . 65

3.2 Application to fixed double-point problems . . . . . . . . . . . . . . 78

3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4 Star metric regularity 85

4.1 Definitions and characterizations of nonlinear star metric regularity 85

4.2 Stability of Milyutin-type regularity under perturbation of star reg￾ularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Stability of generalized equations governed by composite multi￾functions 99

5.1 Notation and some related concepts . . . . . . . . . . . . . . . . . . 100

5.2 Regularity of parametrized epigraphical and composition set-valued

mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3 Stability of implicit set-valued mappings . . . . . . . . . . . . . . . 120

5.3.1 Stability of implicit set-valued mappings associated to the

epigraphical set-valued mapping . . . . . . . . . . . . . . . . 120

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5.3.2 Stability of implicit set-valued mappings associated to a com￾posite mapping . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

General conclusions 131

List of Author’s Related Publications 133

References 134

Index 143

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Table of Notations

N : the set of natural numbers

R : the set of real numbers

R+ : the set of non-negative real numbers

∅ : the empty set

R

n

: the n-dimensional Euclidean vector space

hx, yi : the scalar product in an Euclidean space

||x|| : norm of a vector x

B(x, r) : the open ball centered x with radius r

B(x, r) : the closed ball centered x with radius r

BX : the open unit ball of X

BX : the closed unit ball of X

B(A, r) : the open ball around a set A with radius r > 0

e(A, B) : the excess of a set A over other one B

dom f : the domain of f

epi f : the epigraph of f

X∗

: the dual space of a Banach space X

X∗∗ : the dual space of X∗

A∗

: Y

∗ → X∗

: the adjoint operator of a bounded linear operator

A : X → Y

d(x, Ω) : the distance from x to a set Ω

Nb(¯x; Ω) : the Fr´echet normal cone of Ω at ¯x

N(¯x; Ω) : the Mordukhovich normal cone of Ω at ¯x

x

Ω

→ x¯ : x → x¯ and x ∈ Ω

F : X ⇒ Y : a set-valued map between X and Y

F : the closure of the mapping F

iΩ : the indicator function associated to the set Ω

Jµ : the µ-duality mapping

J

ε

µ

: the normalized ε-enlargement of the µ-duality mapping

GraphF : the graph of F

1

Db∗F(¯x, y¯) : the Fr´echet coderivative of F at (¯x, y¯)

D∗F(¯x, y¯) : the limiting coderivative of F at (¯x, y¯)

ˆ∂f(x) : the Fr´echet subdifferential the mapping f at x

∂f(x) : the limiting subdifferential the mapping f at x

|∇f|(x) : the local slope of the mapping f at x

|Γf|(x) : the nonlocal slope of the mapping f at x

∇f(¯x) : the Fr´echet derivative of f : X → Y at ¯x

EH : the epigraphical set-valued mapping associated to

the set-valued mapping H

SEH

: the solution mapping associated to EH

SH : the solution mapping associated to the set-valued

mapping H

ϕ

F

y

(x) : the lower semicontinuous envelop function of the

distance function d(y, F(x))

ϕ

∗F

y

(x) : the lower semicontinuous envelop function of the

distance function d(y, F(x) ∩ V )

ϕ

p

T

(x, y) : the lower semicontinuous envelop function of the

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

sur F : the modulus of openness of F

surγ F : the modulus of γ-openness of F

reg F : the modulus of metric regularity of F

regγ F : the modulus of γ-Milyutin regularity of F

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

reg∗

γ F : the modulus of γ-Milyutin regularity∗ of F

reg∗

(γ,κ)

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

lip F : the Lipschitz modulus of F

lipγ F : the γ-Lipschitz modulus of F

2

Introduction

In mathematics, solving many problems leads to the formation of equations

and 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

does the solution set change when the input data are perturbed? One of the

powerful frameworks to consider the existence of solutions of equations is metric

regularity. For equations of the form f(x) = y, where f : X → Y is a single-valued

mapping 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

between Banach spaces and strictly differentiable at ¯x, the problem of regularity of

f is reduced to that of its linear approximation ∇f(¯x) and the regularity criterion

is the surjectivity of ∇f(¯x). This result is obtained from the Lyusternik–Graves

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 or

systems 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 inclusions

are named as generalized equations or variational systems, which were initiated

by Robinson in 1970s, see [79, 80] for details. They cover many problems and

phenomena in mathematics and other science areas, such as equations, variational

inequalities, complementary problems, dynamical systems, optimal control, and

necessary/sufficient conditions for optimization and control problems, fixed point

theory, coincidence point theory and so on. Nowadays, generalized equations have

attracted the attention of many experts (see, for instance, [7, 28, 55, 61, 62, 79, 80]

and the references given therein). And thus, variational analysis has appeared in

response to the strong development.

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

behavior of the solution set F

−1

(y) of generalized equations when y and/ or F itself

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

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

F by simple objects, like linear operators (see, Ioffe [55]), and so the condition of

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

x near a given solution ¯x to the solution set F

−1

(y) (unknown quantity) of the

generalized equation through distance d(y, F(x)) from a point y near ¯y ∈ F(¯x) to

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 more

complicated. Then, F : X ⇒ Y satisfying the above estimation is said to be local

metric regularity around (¯x, y¯) if there exist some positive numbers τ, δ, ρ such that

d(x, F −1

(y)) ≤ τ d(y, F(x)), for all x ∈ B(¯x, δ) and y ∈ B(¯y, ρ).

Here, F

−1 denotes inverse mapping F

−1

(y) = {u ∈ X : y ∈ F(u)}. This property

is also called the linear local metric regularity since the quantity d(x, F −1

(y)) is

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

a given (¯x, y¯) ∈ Graph F. The constant τ measures the difference between two

quantities d(x, F −1

(y)) and d(y, F(x)). The infimum of such τ is called the modulus

of metric regularity of F at (¯x, y¯) and denoted by regF(¯x, y¯).

In last decades, this property became a key concept in variational analysis

and plays a crucial role in many areas of applied mathematics. For a detailed

account, the reader is referred to the works by many researchers, for instance,

[2,5,11,15,16,24,28,32,35,36,40,43,46,48,50–53,55,56,58,69,77,81,83,90] and the

references given therein. Recently, some authors such as Gfrerer [43], Frankowska

and Quicampoix [40], Mordukhovich and Li [60], Penot [77], Kruger [58], Zheng

and Zhu [90], and Ngai, Tron, and Th´era [71] extended this property to the case of

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

d(x, F −1

(y)) ≤ τ d(y, F(x))α

, for all (x, y) ∈ B(¯x, δ) × B(¯y, ρ), (1)

or more general, to the nonlinear case

d(x, F −1

(y)) ≤ τµ(d(y, F(x))), for all (x, y) ∈ B(¯x, δ) × B(¯y, ρ), (2)

where µ : R+ → R+ is a function. These models have many applications in

optimization and variational analysis such as convergence analysis of optimization

4

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

establishing the optimality conditions, etc. Almost of these works concentrate

only on the local version of this property. The advantage of the local property is

that we are free to change the neighborhood of a given point without damaging

the property. However, these local versions do not meet many requirements in

mathematical problems, for instance, fixed point problem, Newton’s method. These

problems require that the set on which we work is fixed. Recently, several authors

sought to expand this property to nonlocal versions.

It was found that the nonlocal regularity can be started from well-known

Banach’s contraction map principle. Extension of this principle on closed ball in

a complete metric space established a connection between nonlocal regularity and

fixed point of maps. This was first observed by Arutyunov [5], Ioffe [50,51,55] and

some years before Dmitruk, Milyutin, and Osmolovskii [24] in connecting to the

extremal problems. The reader is referred to the works [3, 4, 29, 46, 47] for these

developments. In recent papers [50,55], Ioffe devoted a model of nonlocal regularity

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

i.e.,

d(x, F −1

(y)) ≤ τ d(y, F(x)), ∀(x, y) ∈ U × V and 0 < τ d(y, F(x)) < γ(x), (3)

where γ : X → R+ is positive on U.

The first purpose in this dissertation is to suggest some new models of non-local

nonlinear 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

product space X ×Y, with respect to a modulus function µ : R+ → R+, as well as a

gauge function γ : X → R+. The consideration of the metric regularity on a subset

W beyond usual box-sets U×V, is not only a natural generalization but also to meet

some practical applications for which the metric regularity on box-sets is violated.

For instance, it covers such as, some notions of directional metric regularity which

are used in the theory of optimality conditions and in sensitivity analysis (see,

e.g., [41, 42, 71, 72, 78] and the references given therein), and the notion of orbit

regularity considered in [55] related to fixed point problems. Precisely, we establish

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

5

analysis such as local slope, non-local slope and coderivatives. We also show that the

special case of these non-local models-the Milyutin regularity possesses a suitable

stability under small Lipschitz perturbation. It is worth to noting further that

the results established in here are new even regarding the case of box-sets. Our

approach inspired the earlier works ([68, 71]) for the local regularity, is based on

applying Ekeland’s variational principle (EVP, shortly) to the composition of the

modulus function and the lower semicontinuous envelope of the distance function

associated to the multifunction under consideration. The approaches based on the

(EVP) to problems related to the metric regularity of mappings and related topics

are now standard (see, e.g., the recent book by Ioffe [55]). The functions used in

this work to which (EVP) is applied, differently from ([50, 55]), are defined merely

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

of either the image space or the graph of the multifunction as in [50]. Still, since

we deal here with the regularity on a fixed set, controlled by a gauge function, the

way to fix parameters to apply (EVP) must be changed differently from the local

cases ([68, 71]).

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

to them are also considered. Namely, in [49], base on the contraction mapping

principle, Ioffe explained the interconnection between regularity properties and

fixed points as well as implied the result of Milyutin’s theorem. Also, Arutyunov [5]

obtained two consequences of the coincidence theorem are the contraction

mapping principle and Milyutin’s theorem. These ideas help us to see the closed

relation between the nonlocal regularity of set-valued mappings and its fixed point

set on metric spaces. Thus, the inheritence of the models and characterizations of

metric regularity established in [84] have motivated our study to two related issues:

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 the

modulus estimation of perturbed map have been interested in and studied by many

authors in the community of variational analysis. Interested reader can refer to

the works [1, 2, 5, 19, 28, 34, 36, 49, 61, 69]. Quantitatively, if a regular mapping is

perturbed by a Lipschitz perturbation with Lipschitz constant smaller than the

6

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

be effected (see, Ioffe [55]). Indeed, according to [24], if a single-valued mapping F

from a complete metric space into Banach space is Milyutin regular with sur F ≥ r

and g is Lipschitz with lip g ≤ `, F + g is Milyutin regular with sur(F + g) ≥ r − `.

Local versions of the theorem achieved for set-valued mapping F in [28, 46, 87]

with the range space being a Banach space. Later, Ioffe [50] gave Milyutin’s

perturbation theorem in local context for the composition of a Milyutin regular

set-valued mapping and a Lipschitz single-valued mapping that is not required

linear structure in the range space.

The second purpose of the thesis is to establish nonlocal metric versions of the

Milyutin’s theorem under composite perturbation, so the additive perturbation by

the Lipschitz mapping with a sufficiently small constant also possesses the stability

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

of a fixed double-point of a pair of regularity set-valued mappings (F1, F2) between

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

[50, Theorem 6.1] and [49, Theorem 2], however, the two approaches are completely

different and the starting conditions to estimate the distance to the fixed double￾point set are also different. In Ioffe [49,50], the result of the Milyutin’s theorem can

be derived from the fixed double-point theorem; meanwhile, our proof of the fixed

double-point theorem is obtained by using the Milyutin’s perturbation theorem

established earlier. Furthermore, the choice of starting conditions different from

the ones in [50] may be more useful in some applications.

Another point worth noting here is that the extension of the local regularity

concepts to the fixed set version also leads to a concept of weak regularity know

as the star metric regularity. In Ioffe [50], the author proposed a star version of

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

U × V ⊂ X × Y as follows

d(x, F −1

(y)) ≤ τ d(y, F(x) ∩ V ), (4)

for all (x, y) ∈ U ×V such that 0 < τ d(y, F(x)∩V ) < γ(x). This version is strictly

weaker than the original one, and thus the use of star regular as assumptions in

7

principle will get better results.

Analogues of the results achieved, the third purpose of the thesis is to consider

some 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 establishing

the infinitesimal characterizations for the star regularity models, we also obtain

some versions of the perturbation theorem for models of Milyutin-type regularity

when the star regularity mapping is perturbed by a Lipschitz function with the

suitable Lipschitz constant.

Along with the metric regularity of a multifunction, the study of conditions

ensuring the metric regularity of parametric multifunctions, that is, implicit

multifunction theorems, plays an important role in investigating problems of

sensitivity analysis with respect to parameters. Regarding this issue, there are

many works by authors: Ioffe, Dontchev, Rockafellar, Dmitruk, Kruger, Durea,

Strugariu,... The results of the metric regularity were extended further to the case

of the sum of a set-valued mapping and a single-valued one by Arutyunov [3, 4],

Dontchev and Lewis [27], Lewis, Dontchev, and Rockafellar [26], Ioffe [55], and

Mordukhovich [61]. For the parametric case, metric regularity of the sum of

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

constant appears in Dmitruk and Kruger [25], Arag´on Artacho, Dontchev, Gaydu,

Geoffroy, and Veliov [2]. More generally, Ioffe [47] extended this result to the case of

the sum of a metrically regular multifunction and a Lipschitz one. In [30], Durea and

Strugariu established the openness for the sum of two set-valued mappings. Then

metric regularity of the sum of two set-valued mappings was studied by Ngai, Tron,

and Th´era [71]. In [89], Zheng and Ng obtained metric regularity of the composition

of a set-valued mapping and a single-valued one. Later, Durea and Strugariu [31,32]

established the openness for the generally nonparametric composition set-valued

mappings defined as in (5.1), and Durea, Ngai, Tron, and Strugariu [36] devoted

metric regularity for this mapping. Recently, the topic on stability of generalized

equations has also attracted the interest and the study of many experts and many

8