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

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 rf(¯ x) and the regularity criterion

is the surjectivity of rf(¯ 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

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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 Statis- tics, 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 presented 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, July 16, 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 iii 1.5 The infinitesimal criteria for metric regularity in metric spaces . . . . . . . . . . . . . . . . . . . . . . . 36 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 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 4 Star metric regularity 84 4.1 Definitions and characterizations of nonlinear star metric regularity 84 4.2 Stability of Milyutin-type regularity under perturbation of star reg- ularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5 Stability of generalized equations governed by composite multi- functions 97 5.1 Notation and some related concepts . . . . . . . . . . . . . . . . . . 98 5.2 Regularity of parametrized epigraphical and composition set-valued mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3 Stability of implicit set-valued mappings . . . . . . . . . . . . . . . 118 iv 5.3.1 Stability of implicit set-valued mappings associated to the epigraphical set-valued mapping . . . . . . . . . . . . . . . . 118 5.3.2 Stability of implicit set-valued mappings associated to a com- posite mapping . . . . . . . . . . . . . . . . . . . . . . . . . 122 Conclusions 128 List of Author’s Related Publications 130 References 131 v 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 Rn : the n-dimensional Euclidean vector space 〈x, y〉 : 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 N̂(x¯; Ω) : the Fre´chet 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 GraphF : the graph of F D̂∗F (x¯, y¯)(·) : the Fre´chet coderivative of F at (x¯, y¯) D∗F (x¯, y¯)(·) : the Mordukhovich coderivative of F at (x¯, y¯) ∇f(x¯) : the Fre´chet derivative of f : X → Y at x¯ EH : the epigraphical-type set-valued mapping associated to the set-valued mapping H SEH : the solution mapping associated to EH 1 SH : the solution mapping associated to the set-valued mapping H ϕFy (x) : the lower semicontinuous envelop function of the distance function d(y, F (x)) ϕ∗Fy (x) : the lower semicontinuous envelop function of the distance function d(y, F (x) ∩ V ) ϕpT (x, y) : the lower semicontinuous envelop function of the distance function d(y, T (x, p)) surF : the modulus of openness of F surγ F : the modulus of γ-openness of F regF : the modulus of metric regularity of F regγ F : the modulus of γ-metric 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 lipF : 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 [74, 75] 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, 27, 51, 56, 57, 74, 75] 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 3 functions or set-valued mappings, etc. Then, it is almost impossible to approximate F by simple objects, like linear operators (see, Ioffe [51]), 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. 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