Luận văn thạc sĩ tính chính quy mêtric phi tuyến của ánh xạ đa trị trên một tập hợp 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 e 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 e 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, December 21, 2021 Advisors PhD student Assoc Prof Dr Habil Huynh Van Ngai Dao Ngoc Han i e 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 e 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 1.4 Metric regularity criteria in metric spaces 33 1.5 The infinitesimal criteria for metric regularity in metric spaces 36 1.3 iii e 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 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 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 regularity 90 Conclusions 98 4.3 Stability of generalized equations governed by composite multifunctions 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 iv e 5.3.2 5.4 Stability of implicit set-valued mappings associated to a composite mapping 124 Conclusions 130 General conclusions 131 List of Author’s Related Publications 133 References 134 Index 143 v e Table of Notations N R R+ ∅ Rn hx, yi ||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) b (¯ N x; Ω) N (¯ x; Ω) Ω x → x¯ F :X⇒Y F iΩ Jµ Jµε GraphF : : : : : : : : : : : : : : : : : : 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 closure of the mapping F the indicator function associated to the set Ω the µ-duality mapping the normalized ε-enlargement of the µ-duality mapping the graph of F e b ∗ F (¯ D x, y¯) ∗ D F (¯ x, y¯) ˆ (x) ∂f ∂f (x) |∇f |(x) |Γf |(x) ∇f (¯ x) EH : : : : : : : : SEH SH : : ϕFy (x) : ϕ∗F y (x) : ϕpT (x, y) : sur F surγ F reg F regγ F reg(γ,κ) F reg∗γ F reg∗(γ,κ) F : : : : : : : lip F lipγ F : the Lipschitz modulus of F : the γ-Lipschitz modulus of F the Fr´echet coderivative of F at (¯ x, y¯) the limiting coderivative of F at (¯ x, y¯) the Fr´echet subdifferential the mapping f at x the limiting subdifferential the mapping f at x the local slope of the mapping f at x the nonlocal slope of the mapping f at x the Fr´echet derivative of f : X → Y at x¯ the epigraphical set-valued mapping associated to the set-valued mapping H the solution mapping associated to EH 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-Milyutin 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 e 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 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)-metrically regular, 41 (µ, γ)-metrically regular∗ , 86 (µ, γ)-open∗ , 87 (µ, γ, κ)-metrically regular, 42 (µ, γ, κ)-metrically regular∗ , 86 B(A, r), 27 D∗ F , 19 Jµ , 21 Jµε , 22 DFix(F1 , F2 ), 78 Fix(F ), 78 dom F , 18 γ-Milyutin regular, 65 γ-Milyutin regular∗ , 90 Graph F , 18 lip F , 28 lipδ F , 29 SEH , 102 p EH , 102 F , 65 ∂f , 17 reg F , 28 reg(γ,κ) F , 65 reg∗(γ,κ) F , 91 regγ F , 29, 65 reg∗γ F , 90 sur F , 28 surγ F , 29 ϕFy , 44 ϕ∗F y , 88 p ϕT , 101 ϕpEH , 103 |Γf |, 23 |∇f |, 23 b ∗ F , 19 D b , 16 ∂f d(x, A), 27 iΩ , 19 SH , 103 Asplund space, 17 calm, 101 closed mapping, 18 epi-inner semicontinous, 106 epi-upper semicontinuity, 25 F´echet coderivative, 19 143 e Fr´echet smooth space, 54 Fr´echet subdifferential, 16 gauge function, 30, 39, 64 inner semicontinous, 105 limiting coderivative, 19 limiting subdifferential, 17 Lipschitz-like, 100 local slope, 23 locally composition-stable, 103 lower semicontinous function, 14 metrically semiregular, 102 Milyutin regularity, 34 modulus function, 31, 39 nonlocal slope, 23 Robinson metrically regular, 125 upper semicontinous, 14 144 e CONG HÒA X HOI CHn NGH*A VIT NAM BO GIÁO DUC VÀ ÀO TAO Docl-p -Ty -Hanh phúc TRUONG DAIHOC QUY NHON Só: 1961/QÐ-HQN Binh Dinh, ngày 10 tháng nm 202 QUT INH vè viÇc thành l-p HÙi ơng ánh giá lu-n án tiên s+ câp Truròng doi voi nghiên céu sinh Dào Ngoe Hân HIEU TRNG TRNG DAI HOC QUY NHON Cn ct Quét dinh só 221/2003/QÐ-TTg ngày 30/1o/2003 ciùa Thii tng Chinh phi vé viÇc dói tên Truong Dai hÍc Su pham Quy Nhon thánh Iruong Dai hoc Quy Nhon; Can ciù nhiëm vå quyên han cua hiầu truong co sò giỏo dồc dai hoc quy dinh tai Khốn 3, Dièu 20 Lt só 34/2018/QH14 ngày 19/11/2018 cia Quúc hi nrúc Cong hũa x hi chỗ ngh+a ViÇt Nam khóa XTV sua dơi, bơ sung mÙt sô diêu cua Luât giáo dyc d¡i hoc; Cn ci Thóng te sĐ 08/2017/TT-BGDDT ngày 044/2017 cia BÙ trng B Giáo duc t¡o v¿ viÇc ban hành Quy chê tuyên sinh t¡o trinh dÙ tiên s+; Cn cú Out dinh só 1909/0Ð-DHON ngy 12/10/2017 cua HiÇu truong Truong Dai hoc Quy Nhon ve viÇc ban hành Quy djnh tuyên sinh t¡o trinh dÙ tiên s+; Can cu Quyét dËnh só 6775/0Ð-HON ngày 18/12/2015 cia Hieu truong Treong hoc Quy Nhon vê viÇc cong nhân nghiên cúu sinh, dê tài nghiên ciu nguuoi huúng dỏn nghiờn cỳu sinh; Theo de nghi cỗa Trurong phông Dào t¡o sau dai hoc QUYÉT ÐINH: DiÁu Thành l-p HÙi ông dánh giá lu-n án tiên si câp Truong vé dê tài: Tính quy mêtric phi tuyên cua ánh x¡ da tr/ mÙt tâp hãp v ỳng dồng" cỗa nghiờn ctu sinh: Do Ngoc Hõn, chun ngành: Tốn giäi tích, m sơ: 9460102, gơm thành viên có tên danh sách kèm theo Diêu HÙi ơng có nhiÇm vu ánh giá ln án tiờn s+ cỗa nghiờn cộru sinh theo dỳng cỏc quy chê, quy Ënh tuyên sinh t¡o trinh Ù tiên sï hiÇn hành ty giài thê sau hồn thành nhiÇm vå Các ơng (bà) Truong phịng t¡o sau ¡i hÍc, Hành Tơng Diêu hop, Kë hoach Tài chính, Truong khoa Khoa Tốn Thơng kë ơng (bà) - có tên & iêu chju trách nhiÇm thi hànhuydinh DU Noi nhan: Nhu Dieu 3; Luru: VT, - - DTSÐH HIEU TRUÖNG TRUONG DMHot QYAIHN e PGS TS Dô Ngoc My BO GIÁO DUC VÀ Å0 TAO TRUONG DAI HOC QUY NHON CONG HỊA X HOI CHn NGH*A VIỈT NAM Doelap-Ty - Hanh phúic DANH SÁCH HOI DÔNG DÁNH GIÁ LUÁN N TIấN S* CP TRUềNG Cho lu-n ỏn cỗa nghiờn céu sinh: Dào Ngoc Hân Co so t¡o: Truòng Dai hoe Quy Nhon Tap thê huóng dân: - PGS.TSKH Huynh V n - TS Nguyên Hïu Ngäi, Truong Dai hÍc Quy Nhon Tron, Truong ¡i hÍc Quy Nhon (Ban hành kèm theo Quyét dinh só 1961/QD-HQN 2021 cua T HiÇu trng Truong Ðai hoc Quy Nhon) Ho tên, hÍc hàm/hÍc v/ Chuyên ngành PGS.TS inh Thanh Drc Co quan công tác Chrc trách HÐD Giai tích TruongDH Quy Nhon Chù tich TS Le Quang Thuan Giài tích Truong H Quy Nhon Thu ký PGS.TS Phan Nh-t T+nh Ly thuyêt Truong ¡i hÍc Khoa hÍc Nguyên Huy Chiêu ru Lý Ly thuyet thuyét PGS.TS PGS.TS Pham TiÃn Son Ly thuyêt PGS.TS Nguyên Vn Tuyên Lythuyêt ngày 10 tháng nm TS Nguy¿n Vän Vk toi uu uu uu Lýtoithut uu -Dai hoc H Trurịng ¡i hÍc Vinh Truong Dai hÍc å Lat Truong Dai hÍc ph¡m Hà Noi Su Truong H Quy Nhon Phàn bien Phàn biÇn Phån bien3 Uy viên Uy viên Hoi dong gom 07 thành viên e cONG HOA X HOI CHn NGH*A VIET NAM DÙclap-Ty do-Hanh phúc Quy Nhon, ngày 25 tháng 08 nm 2021 NHAN XÉT LUAN ÁN TIÊN SÍi De tài: TÍNH CHÍNH QUY MÊTRIC PHI TUYÉN CUA ÁNH XA A TR TRÊN MOT TAP HOP VÀ ÚNG DUNG Chun ngành: Tốn Giäi tích M sơ: 94.60.01.02 Nghiên cúu sinh: Ngoc Hân Nguoi nh-n xét luân án: PGS TS inh Thanh Dúrc Chun ngành: Tốn Gi£i tích Dja chi: Truong hÍc Quy Nhon, 170 An Duong Vuong, TP Quy Nhon, Binh inh So diÇn tho¡i/Email: dinhthanhduc@qnu.edu.vn Co quan cơng tác: Truong hÍc Quy Nhon NOI DUNG NHAN XÉT Tinh câp thiêt (ý nghïa lý lu-n v thồrc tin) cỗa ti: Dờ ti Lu-n ỏn thuÙc li+nh vuc nghiên cru Giái tich toi uu Dè tài at vân à nghiên cru tính quy mêtric cỗa ỏnh x a trậ, dở xuõt mt sụ mử hinh mÛi cho tính quy metric quy metric, mt phiờn bn yờu cỗa tớnh chinh quy, ông thÝi ã dua iêu cân ü ôi vÛi mơ hinh quy ã Á cap Hon nïa lu-n án ã ê c-p ên tính quy metric cho ánh xa da trË d¡ng thi có tham sơ, tính nua quy cho ánh xa da trậ hop v nghiờn cộu tớnh ụn ậnh cỗa ỏnh xa da trË ân liên kêt vÛi ánh x¡ a trË dang thË có tham sĐ vå ánh xa da trË hãp Vân ê nghiên céu ã duoc nhiêu nhà tốn hoc thê giói nghiên có tinh khoa hÍc cao hiÇn ¡i, có ý ngh+a vê lý thuyêt toi uru úng dång thårc te Su phự hóp cỗa ti vúi chuyờn ngnh o tĂo: Noi dung cỗa Lu-n ỏn phự hop vÛi m so chun ngành Tốn giäi tích, Su trựng l-p cỗa ti vi cỏc cụng trinh khoa hÍc dã nghin ciéu cơng bơ: Các kêt q d¡t duoc Lu-n án hồn tồn mói, chúng th-t su mß rong cua kêt quå khác theo huróng này, tác già cÙng su ã cơng bơ tap chí uy tin thé giói Khơng trùng l-p vÛi k¿t q khác ã cơng bơ thê giói e Tinh trung thuc, Ù tin cây, rõràng trich dân nguôn tài liÇu kham kh£o: Các tài liÇu tham khào lu-n án duoc trích dân ày ù chi tiêt Tinh hiần Ăi, hóp lý cỗa phurong phỏp nghiờn cớru: Các phuong pháp nghiên céu lu-n án sir két hop cua giài tich a trË ly thuyêt tơi uu dåa tài liÇu tham kh£o có tinh thÝi så c-p nh-t, hiÇn dai Dánh giá kêt quå at duoc, nhïng óng góp mói v giỏ tr/ gúp dú, kh nng cỗa nhùng úng phỏt triờn cỗa ti: Luan ỏn l mt cụng trinh khoa hÍc nghiên céu mÙt cách có hÇ thĐng vƠn tớnh chớnh quy mờtric phi tuyờn cỗa ỏnh xa a trậ, tớnh ụn inh cỗa hm õn da tr/ Luan án có duoc kêt quå chât luong vê nghiên círu nghiên céu vê ánh x¡ a tr/ dËnh lýy hàm ân Chât lugng nhïng báo khoa hÍc duoc nghiên céu sinh cơng bơ0 Các kờt qu cỗa luõn ỏn duỗc cụng bú báo, dó có 01 báo tap chi qc tê uy tín (SCIE), 01 tap chí khoa hÍc QNU 01 dang gri ng Hai bài báo duói d¡ng reprint Kêt lu-n: Tơi ơng y cho NCS Dào Ngoc Hân bào vÇ lu-n án t¡i HÙi ông dánh giá lu¡n án câp TruÝng Xac ohan ua to quan _VNHIEU TRUONG Nguoi viêt nh-n xét DUC TRUON PHONC HANH CHÉVH TÖKG IKGP TRUONG DAI DAI NHON HOC QUY ak PGS TS Dinh Thanh Dúc ThS Tran Ngoc Anh e ... 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... e 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