Tính chính quy mêtric phi tuyến của ánh xạ đa trị trên một tập và ứng dụng

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: 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 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 Introduction 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 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 Metric regularity criteria in metric spaces 33 1.3 1.4 iii 1.5 The infinitesimal criteria for metric regularity in metric spaces 36 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 Characterizations of nonlinear metric regularity via coderivative 54 2.3 Perturbation stability of Milyutin-type regularity and applications 64 3.1 3.2 Perturbation stability of Milyutin-type regularity 65 Application to fixed double-point problems 78 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 regularity 89 Stability of generalized equations governed by composite multifunctions 97 5.1 Notation and some related concepts 5.2 Regularity of parametrized epigraphical and composition set-valued mappings 103 5.3 Stability of implicit set-valued mappings 118 iv 98 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 composite mapping 122 Conclusions 128 List of Author’s Related Publications 130 References 131 v Table of Notations N R R+ ∅ Rn x, y ||x|| B(x, r) B(x, r) BX BX B(A, r) e(A, B) dom f epi f X∗ X ∗∗ A∗ : Y ∗ → X ∗ d(x, Ω) N (¯ x; Ω) N (¯ x; Ω) Ω x → x¯ F :X⇒Y GraphF D∗ F (¯ x, y¯)(·) D∗ F (¯ x, y¯)(·) ∇f (¯ x) EH SEH : : : : : : : : : : : : : : : : : : the set of natural numbers the set of real numbers the set of non-negative real numbers the empty set the n-dimensional Euclidean vector space the scalar product in an Euclidean space norm of a vector x the open ball centered x with radius r the closed ball centered x with radius r the open unit ball of X the closed unit ball of X the open ball around a set A with radius r > the excess of a set A over other one B the domain of f the epigraph of f the dual space of a Banach space X the dual space of X ∗ the adjoint operator of a bounded linear operator A:X→Y : the distance from x to a set Ω : the Fr´echet normal cone of Ω at x¯ : the Mordukhovich normal cone of Ω at x¯ x → x¯ and x ∈ Ω a set-valued map between X and Y the graph of F the Fr´echet coderivative of F at (¯ x, y¯) the Mordukhovich coderivative of F at (¯ x, y¯) the Fr´echet derivative of f : X → Y at x¯ the epigraphical-type set-valued mapping associated to the set-valued mapping H : the solution mapping associated to EH : : : : : : : sur F surγ F reg F regγ F reg(γ,κ) F reg∗γ F reg∗(γ,κ) F : the solution mapping associated to the set-valued mapping H : the lower semicontinuous envelop function of the distance function d(y, F (x)) : the lower semicontinuous envelop function of the distance function d(y, F (x) ∩ V ) : the lower semicontinuous envelop function of the distance function d(y, T (x, p)) : the modulus of openness of F : the modulus of γ-openness of F : the modulus of metric regularity of F : the modulus of γ-metric regularity of F : the modulus of (γ, κ)-Milyutin regularity of F : the modulus of γ-Milyutin regularity∗ of F : the modulus of (γ, κ)-Milyutin regularity∗ of F lip F lipγ F : the Lipschitz modulus of F : the γ-Lipschitz modulus of F SH ϕFy (x) ϕ∗F y (x) ϕpT (x, y) 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 Since z ∈ T (y1 , y2 ), (y1 , y2 ) ∈ T1 (x ) × T2 (x , p), one has z ∈ Hp (x ) x ∈ Hp−1 (z) and as a result, Thus, d(x, Hp−1 (z)) ≤ τ d(z, T (T1 (x) ∩ B(¯ y1 , r), T2 (x, p) ∩ B(¯ y2 , r))) + ε for all (x, p, z) ∈ B(¯ x, r) × V × B(0, r) Letting ε → 0, we get the conclusion Proposition 5.3.2 Let T1 : X ⇒ Y1 , T2 : X × P ⇒ Y2 and T : Y1 × Y2 ⇒ Z, X, Y1 , Y2 be complete metric spaces, P be a topological space, Z be a normed linear space satisfying conditions (a), (b), (c) in Lemma 5.2.3 around (¯ x, p¯, y¯1 , y¯2 , 0) ∈ X × P × Y1 × Y2 × Z If there exist neighborhoods B(¯ x, r), V, B(¯ y1 , r), B(¯ y2 , r) of the points x, p, y1 , y2 , respectively such that d(x, SH (0, p)) ≤ τ d(0, T (T1 (x)∩B(¯ y1 , r), T2 (x, p)∩B(¯ y2 , r))), ∀ (x, p) ∈ B(¯ x, r)×V, (5.45) and ((T1 , T2 ), T ) is locally composition-stable around ((¯ x, p¯), (¯ y1 , y¯2 ), 0), then SH (0, ·) is Robinson metrically regular around (¯ x, p¯) with modulus τ Proof Suppose that (5.45) holds for every (x, p) ∈ B(¯ x, r) × V Since ((T1 , T2 ), T ) is locally composition-stable around ((¯ x, p¯), (¯ y1 , y¯2 ), 0), then there exists δ > such that every (x, p) ∈ B(¯ x, r) × V and every z ∈ (T ◦ (T1 , T2 ))(x, p) ∩ B(0, δ), there is (y1 , y2 ) ∈ (F1 (x) ∩ B(¯ y1 , δ)) × (F2 (x, p) ∩ B(¯ y2 , δ)) such that z ∈ T (y1 , y2 ) Taking δ smaller if necessary, we may assume that δ < r/2 Fixing now (x, p) B( x, /2)ìV, we consider two cases: ã Case d(0, Hp (x)) < δ/2 Choose γ > small enough in order to get d(0, Hp (x)) + γ < δ/2 It follows that there is a point t ∈ Hp (x) such that t < d(0, Hp (x)) + γ < δ/2 Therefore, t ∈ (T ◦ (T1 , T2 ))(x, p) ∩ B(0, δ) and thus by the local composition stability, there is (y1 , y2 ) ∈ (F1 (x) ∩ B(¯ y1 , δ)) × (F2 (x, p) ∩ B(¯ y2 , δ)) such that t ∈ T (y1 , y2 ) As a result, d(x, SH (0, p)) ≤ τ d(0, T (T1 (x) ∩ B(¯ y1 , r), T2 (x, p) ∩ B(¯ y2 , r))) ≤ τ t < τ d(0, Hp (x)) + γ Since γ > is arbitrarily small, one gets that d(x, SH (0, p)) ≤ τ d(0, Hp (x)) 124 Since (x, p) is arbitrary in B( x, /2) ì V, we obtain the conclusion ã Case d(0, Hp (x)) ≥ δ/2 By Lemma 5.2.3, (i) one has that the set-valued mapping p ⇒ Hp (¯ x) is inner semicontinuous at p¯ for Hence, the distance function p → d(0, Hp (¯ x)) is upper semicontinuous at p¯, and therefore there exists a neighborhood W of p¯ such that d(0, Hp (¯ x)) ≤ δ/4, ∀ p ∈ W Shrinking W if necessary, one may assume that W ⊂ V and choose < δ1 < min{δ, τ δ/4} Taking (x, p) ∈ B(¯ x, δ1 ) × W , then, by (5.45), for every ε > 0, there is u ∈ SH (0, p) such that d(¯ x, u) < (1 + ε)τ d(0, Hp (¯ x)) Consequently, d(x, u) ≤d(x, x¯) + d(¯ x, u) < δ1 + (1 + ε)τ d(0, Hp (¯ x)) < τ δ/4 + (1 + ε)τ δ/4 ≤ τ /2d(0, Hp (x)) + τ /2(1 + ε)d(0, Hp (x)) Since ε is arbitrary, one gets that d(x, SH (0, p)) ≤ τ d(0, Hp (x)), establishing the proof Using these propositions along with Theorem 5.2.1, one obtains Robinson’s metric regularity and Lipschitz-likeness of the map SH (0, ·) given in Theorem 5.3.3 below Theorem 5.3.3 Let X, Y1 , Y2 be complete metric spaces, P be a metric space, Z be a normed linear space, and let T1 : X ⇒ Y1 , T2 : X × P ⇒ Y2 and T : Y1 × Y2 ⇒ Z satisfy conditions (a), (b), (c) in Lemma 5.2.3 around (¯ x, p¯, y¯1 , y¯2 , 0) ∈ X × P × Y1 × Y2 × Z Suppose that (i) ((T1 , T2 ), T ) is locally composition-stable around ((¯ x, p¯), (¯ y1 , y¯2 ), 0); 125 (ii) T1 is metrically regular around (¯ x, y¯1 ) with modulus m > 0; (iii) T2 is Lipschitz-like around ((¯ x, p¯), y¯2 ) with respect to x, uniformly in p with modulus l > 0; (iv) T2 is Lipschitz-like around ((¯ x, p¯), y¯2 ) with respect to p, uniformly in x with modulus θ > (v) T is metrically regular around ((¯ y1 , y¯2 ), 0) with respect to y1 , uniformly in y2 with modulus λ > 0; (vi) T is Lipschitz-like around ((¯ y1 , y¯2 ), 0) with respect to y2 , uniformly in y1 with modulus γ > 0; (vii) λmγl < Then SH (0, ·) is Robinson metrically regular around (¯ x, p¯) with modulus γθmλ SH (0, ·) is Lipschitz-like around (¯ x, p¯) with modulus 1−mλlγ mλ 1−mλlγ and p Proof • Applying Theorem 5.2.1 yields that EH is metrically regular around mλ (¯ x, p¯, y¯1 , y¯2 , 0) with respect to (x, y1 , y2 ) uniformly in p with modulus 1−mλlγ Then, by Proposition 5.3.1, one obtains the estimation (5.42) In this estimation, by replacing z by 0, one has (5.45), and from Proposition 5.3.2 along with the local composition stability of ((T1 , T2 ), T ) around ((¯ x, p¯), (¯ y1 , y¯2 ), 0), one obtains that mλ SH (0, ·) is Robinson metrically regular around (¯ x, p¯) with modulus 1−mλlγ • By the definition of Robinson’s metric regularity of SH (0, ·), we derive the existence of some δ1 > such that mλ mλ d(0, H(x, p)) = d (0, T (T1 (x), T2 (x, p))) − mλlγ − mλlγ (5.46) for all (x, p) ∈ B((¯ x, p¯), δ1 ) By (iv), since T2 is Lipschitz-like around ((¯ x, p¯), y¯2 ) with respect to p, uniformly in x with modulus θ > 0, there is δ2 > such that d(x, SH (0, p)) ≤ T2 (x, p) ∩ B(¯ y2 , δ2 ) ⊂ T2 (x, p ) + θd(p, p )B Y2 for all p, p ∈ B(¯ p, δ2 ), for all x ∈ B(¯ x, δ2 ) 126 (5.47) Moreover, according to (vi), since T is Lipschitz-like around ((¯ y1 , y¯2 ), 0) with respect to y2 , uniformly in y1 with modulus γ > 0, there exists δ3 > such that T (y1 , y2 ) ∩ B(0, δ3 ) ⊂ T (y1 , y2 ) + γd(y2 , y2 )B Z (5.48) y2 , δ3 ) Using the local composition-stability for all y1 ∈ B(¯ y1 , δ3 ), for all y2 , y2 ∈ B(¯ of the pair ((T1 , T2 ), T ) around ((¯ x, p¯), (¯ y1 , y¯2 ), 0) in (i), select δ4 > such that for every (x, p) ∈ B(¯ x, δ4 ) × B(¯ p, δ4 ) and every z ∈ T (T1 (x), T2 (x, p)) ∩ B(0, δ4 ) There exists (y1 , y2 ) ∈ T1 (x) ∩ B y¯1 , min{δ2 , δ3 } × T2 (x, p) ∩ B y¯2 , min{δ2 , δ3 } such that z ∈ T (y1 , y2 ) Set α := min{δ1 , δ2 , δ3 , δ4 }, and take p, p ∈ B(¯ p, α), and x ∈ SH (0, p)∩B(¯ x, α) This means that, ∈ T (T1 (x), T2 (x, p)) ⊂ T (T1 (x), T2 (x, p)) ∩ B(0, δ4 ) and x ∈ B(¯ x, α) It follows that there exists (y1 , y2 ) ∈ T1 (x) ∩ B y¯1 , min{δ2 , δ3 } × T2 (x, p) ∩ B y¯1 , min{δ2 , δ3 } such that ∈ T (y1 , y2 ) Consequently, for y2 ∈ T2 (x, p ), mλ d 0, T (T1 (x), T2 (x, p )) − mλlγ mλ ≤ d(0, T (y1 , y2 ))) − mλlγ d(x, SH (0, p )) ≤ (5.49) (5.50) So, by taking into account ∈ T (y1 , y2 ) and by using the estimations (5.49) and (5.50), we have mλ e (T (y1 , y2 ), T (y1 , y2 )) − mλlγ θγmλ γmλ ≤ d(y2 , y2 ) ≤ d(p, p ), − mλlγ − mλlγ d(x, SH (0, p )) ≤ which implies that SH (0, p) ∩ B(¯ x, α) ⊂ SH (0, p ) + θγmλ d(p, p )B X − mλlγ This means that SH (0, ·) is Lipschitz-like around (¯ x, p¯) with modulus completes the proof 127 γθmλ 1−mλlγ , which Conclusions This dissertation obtained the following main results: • Gave some new models of nonlinear metric regularity of set-valued mappings on a fixed arbitrary-set as well as the equivalence of this one and Hăolder, openness properties ã Established infinitesimal characterizations for the nonlinear metric regularity models via slope and coderivative • Achieved some versions of metric perturbation of Milyutin’s theorems on a fixed set under composite perturbation of mappings between metric spaces, and then their particular case about additive perturbation of a set-valued mapping between Banach spaces by Lipschitz single-valued mapping is also contained • Showed the existence of fixed double-point of a pair of set-valued mappings by using the result on the perturbation stability of Milyutin-type regular • Provided some versions of nonlinear star metric regularity of multifunctions on a fixed set and proposed characterizations for these models via nonlocal and local slope • Indicated the metric regularity of the parametrized epigraphical set-valued mappings and semiregularity of compositions set-valued mappings • Attained the stability of implicit set-valued mappings associated to epighraphical set-valued mapping such as Lipschitz-likeness, calmness • Obtained the stability of implicit set-valued mappings associated to a composite mapping such as Robinson’s metric regularity, Lipschitz-likeness 128 Future investigation In the future, we intend to continue investigation in the following directions: From our general model, it can be applied to the study of regular models for specific variational problems, for instance, the variational inequality problem, the complementarity problem, the differential inclusion, the optimal control, the parametric programming, Using the local metric regularity to study of existence and stability of global solutions of differential inclusion in terms of x (t) ∈ F (t, x(t)), t ∈ [a, +∞), (5.51) and application to investigate the controllability of the dynamical system defined by the differential conclusion above Application to the problem of coincidence point theorem Application to the study of convergence of Newton-type method to solve the optimization problems Application to the study the sensitivity and the stability of solutions for optimization problems when the data are perturbed 129 List of Author’s Related Publications Ngai H V., Tron N H., Han D N (2021), “Metric perturbation of Milyutin regularity on a fixed set and application to fixed point theorems”, preprint Ngai H V., Tron N H., Han D N (2021), “Star metric regularity on a fixed set”, preprint Tron N H., Han D N., Ngai H V (2020), “Nonlinear metric regularity on fixed sets”, In revision, submitted to Optimization Tron N H., Han D N (2020), “Stability of generalized equations governed by composite multifunctions”, Pacific Journal of Optimization, 16 (4), 641–662 Tron N H., Han D N (2020), “On the Milyutin regularity of set-valued mappings”, Journal of Science - Quy Nhon University, 14 (3), 37–45 130 References [1] Adly S., Ngai H V., Vu N V (2017), “Stability of metric regularity with set-valued perturbations and application to Newton’s method for solving generalized equation”, Set-Valued Var Anal., 25 (3), 543–567 [2] Arag´on Artacho F J., Dontchev A L., Gaydu M., Geoffroy M H., Veliov V M (2011), “Metric regularity of Newton’s iteration”, SIAM J Control Optim., 49 (2), 339–362 [3] Arutyunov A V (2007), “Covering mapping in metric spaces, and fixed points”, Dokl Math., 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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... 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... 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