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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 download by : skknchat@gmail.com 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 download by : skknchat@gmail.com 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 download by : skknchat@gmail.com 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 download by : skknchat@gmail.com 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 download by : skknchat@gmail.com 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 download by : skknchat@gmail.com 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 download by : skknchat@gmail.com 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 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 download by : skknchat@gmail.com D∗ F (¯ 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 download by : skknchat@gmail.com 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 download by : skknchat@gmail.com CONG HOA X HÔI CHn NGH*A VIET NAM Doclap-Ty -Hanh Phúe Xuan Hoa, ngày 06 tháng 09 nm 2021 NHAN XÉT LUAN ÁN TIÉN S* De tài: Nonlinear metric regularity of set-valued mappings on a fixed set and applications Chuyên ngành: Toán giäi tích Ma sơ: 946 01 02 Nghiên cru sinh: Ngoc Hân Nguoi nhan xét: PGS TS Nguy¿n Vän Tun Co quan cơng tác: Khoa Tốn, Trng EHSP Hà NÙi NOI DUNG NH¬N XÉT Tính câp thiêt (ý nghïa lý lu-n thye tien) cüa ê tài: DUC Khỏi niầm vờ tớnh chớnh quy metric cỗa mt ánh xa dugc båát dàu nghiên céu tir/ TRU- cuoi nm 1970 nhanh chóng thu hút dugc su quan tõm nghiờn cộru cỗa cỏc nha toanS U A P hoc thê gii vi nhing tng dung quan cùa Giäi tích biÃn phân lMHA thut toi uru Tính quy metric có liên quan ch·t ch d¿n két quå có iÃn Giäi tich hàm nhu jnh Iý hàm an, Nguyên lý ánh xa mß Banach-Schauder, Ënh lý Lyusternik-Graves dóng vai trị quan trong viÇe phát tri¿n quy tc tính tốn phép tnh vi phân suy rÙng, nghiên céu su tụn tĂi chọn sai sú v sr hi tồ cỗa thuat tốn tơi uu Cung voi su phát triên cỗa Giọi tich bien phõn v tở mt sẹ vỏn Á sinh thåc té, mơt só ểi biên tinh quy metric (dia phuong) dã duãc dra Trong dú, khỏi niầm tớnh echớnh khụng dja phuong cỗa loffe thu hỳt duoc su quan tõm Ãc biầt cỗa cỏc nhà tốn hoc Tinh quy metric khơng ja phrong dã dugc ing dung à nghiên céu su tôn tai cp iờm bõt ng cỗa ỏnh xa da trậ v nghiờn cỳu tớnh ụn dinh cỗa cỏc bi toỏn tụi uu phå thc tham sơ Ln án nghiên cíu vê cỏc dÃc trung cỗa mt sụ tớnh chõt chớnh quy không dja phurong ca ánh xa da trË úng dung Dây mÙt vân dê thÝi su, có ý nghïa khoa hÍc thu hút duoc su quan tâm cua nhiêu nhà tốn hÍc ngồi nC Sy phù hop gita dê tài lu-n án vói chuyên ngnh o tao: Ni dung cỗa tọi lu-n ỏn phù hop vÛi m só chuyên ngành download by : skknchat@gmail.com I¢ N download by : skknchat@gmail.com doy ono 9q 1oA Buonyo s yupua eyo oónp up ueni 'np W ugud IgosN "PI Yujyo 1oT oo ugyy ugo yuy Buen augq Kpq qusn onp ug ueng uy uỗnj eFo onyj suy a 1ou eno rin 1Ùy Ás Áes 'uhju u oia Qa '8 nn toi up01 Iq ogo 3uon uoi nq1 os Kèqu op 'quip uo yup nno ugrysu ohLa Suon Surip Sun ea nq upo Ieq "ugyd IA oNYI ugy oBq 'ugud ugrq onyi Buep ig ugoj Iq nyu pyn tho yus Qu ogo oyo Ánb qujyo qujj nFo ugIuõu pp Sunp dy oðnp 9yi 9o Águ pnb 19y oP ovo &üo * u Op 9o BA 1911 IYO Y VèYo Yoyo 1óu qujui Sunyo np up utnj 3uon pnb i9y op pp eno ugin ipud 3ugu puy uy uênj Buo) nau unj ipy Supqu a ugip uenj sunqu en» k¢o up op »ơs eoyy os o 'L nn oi 19Ányi I PA ugyd uprq yon gi3 Buon eyysu f 9o 19u do3 Suop Supyu i Keu pnb 9H opO doy in ep ex yug 19A 19N u!I 9s uey1 9o upyd ugiq ~s eno uÙIußu ëx yup eno quip uo eyo quy os 1Ùu ónp aunn oèp BA dós in ep ex yup eno ouou knb yujuo eFu qun 'os ueq oon1 rnyd t gp ugn ex quy eFo ounu ínb yujyo qun oyo np uÙpy nqip os 1óu oonp dèj 19ruL in ep èx yup deo 1óuI eFo Buóp gq ueip - 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Bàn tóm tt lu-n án uoe trinh bày mÙt cách khoa hoc ph£n ánh trung thre két quå nghiên céu trinh bày lu-n án 10 Chât lugng cüa nhïng báo khoa hÍe dã durye nghiên cru sinh cơng bĐ: Các kêt quà lu-n án thu roe mÛi, có ý ngh+a khoa hÍc thre ti¿n ä duoe cơng bơ 05 báo khoa hÍc, ó 01 dng tap chí chun ngành ISI có uy tin, 01 bi bỏo dọng trờn tap cỗa Truong ¡i hoc Quy Nhon tiên án phâm Cỏc cụng bụ ny ỗ iu kiần NCS bõo vÇ lu-n án tien s+ 11 Nhïng thiêu Nên 12 sót vê nÙi dung hinh théc cüa lu-n án bo sung phân kêt lu-n vào cuói Chuong 2-5 Kột lu-n chung: Luõn ỏn cỗa NCS dỏp ỳng ỗ cỏc yờu cu cỗa lu-n ỏn tiờn sù Toỏn hc chun ngành giäi tích tơi dơng ý ê lu-n án dugc dua bão vÇ tre hÙi dơng châm luân n câp Trrong Xác nhân cua co quan (Ký tên dóng dáu) Ngroi nh-n xét (Ký ghi rư ho tên) TIL HIEU TRUONG TRUONG PHĨNG TOCHÚC-HANH CHINH TRUONG DAI HOC SUPHAM HA NOr2 Nguyén Huy Hung Nyuyen Va download by : skknchat@gmail.com Tw Mau Nhan xét lugn án tiên sỵ cáp Truong cONG HOA X HOI CHU NGHÍA VIT NAM Doclap-Ty do-Hanh phúe Binh Dinh, ngày tháng.9 nam 2021 NHAN XÉT LUAN ÁN TIÈN SsÍ De tài Tinh chinh quy mêtric phi tuyên cüa ánh xa da tr/ mÙt Chuyên ngành: Ma so: Nghiên cúu sinh: Ngroi nhân xét: Co quan công tác: tap hop úng dång Tốn Giái tích 9460102 Dào Ngoc Hân TS Nguyên Vän Vk Truong D¡i hÍc Quy Nhon NOI DUNG NHAN XÉT Tính cáp thiêt (ý ngh+a lý lu-n v thuctiởn) cỗa ti; Tinh chớnh quy mờtric l mÙt khái niÇm chü chơt viÇc nghiên céu dáng diÇu cua ánh xa nghiÇm hÇ biên phân Ngn gơc cua có liên hÇ vÛi mÙt vài kêt qua kinh diên giai tích hàm nói riêng, giài tích nói chung Të doi chinh quy mờtric cựng vi cỏc biờn th cỗa nú ngy cng chéng tò vai trò quan cà vê mat lý thuyêt lán úng dung nôi bât linh vyre tói uu vâå giai tich khơng tron HiÇn nay, dây vân nhánh nghiÁn cúu thu hút nhiêu nhóm tác giá quan tam De tài luân án nghièn cru mơ hinh chinh quy mêtrie phi tun mà có thê xem nhu sy suy rÙng ty nhiên tië tinh chinh quy mêtric Chúng góp phân làm phong phú thêm cơng cu giai qut nhièu lóp tốn có liên quan mà o dó ký thu-t cua giai tich biờn phõn l cõu nụi quan Vi thờ chỗ dê quan tâm lu-n án có nhieu ý nghia cà vê khía canh lý thuyêt cüng nhu úng dung Sr phự hứp cỗa dờ ti vúi chuyờn ngnh t¡o; De tài luân án phù hop vÛi chun ngành Tốn giäi tích Su trùng läp hay khụng trựng lap cỗa d ti vúi cỏc cụng trinh khoa hoc dã nghiên cuu cịng bơ; Các kêt quầ trinh bày khn khó lu-n án khơng trùng l-p voi nhïng cơng trinh khoa hÍc dã cơng bơ Tinh trung thuc, ro ràng trích d§n ngn tài liÇu kham khào; download by : skknchat@gmail.com Danh muc tài liÇu tham khào uoc trinh bày rĐ ràng dây ù nguôn tra céru tin cay có giátrË chun ngành liên quan Tính hiÇn dĂi hop lý cỗa phuong phỏp nghiờn ciru Cỏch tiep cân ván è cua lu-n án phù hop vói mÙt sơ hróng nghiên céru duong d¡i ong chun ngành, sù dung công cu të giäi tich biên phân giäi tich da tr/ de giai quyêt toán d·t Các kêt quà d¡t duoc, nhïng óng góp mÛi giá r/ cua nhïng dóng góp mói ó De tài dã ua mÙt sơ mơ hinh quy phi tun tơng qt hóa khái niÇm vetinh quy ã duoe nghiên céru rÙng ri dåa mÙt sô kêt quà sâu sc gân day l+nh vrc giài tích biên phân Bng cơng cu phuong pháp phù hãgp, tác già dã dua nhïng Ãc trung dỗ mi cho mụ hinh nghiờn cỳu dụng thÝi kh£o sát có hÇ thơng tính ơn dËnh nhieu cỗa mụ hinh trờn mt sụ lúp ỏnh xa quan Các kêt quà héa hn cho nhïng úung dung mÙt sơ lóp tốn có liên quan, nhât ly thuyêt giai tich phi tuyên uu không tron Co so khoa hc, d tin cõy cỗa nhùng lu-n diờm v nhùrng kêt lu-n nêu lu-n án; khà nng phát triên cỗa dố ti: Dúng gop chinh tở lu-n ỏn dugc phát triên dra nên tng co sò lý thuyét cüng nhu ký thu-t ·c thù tië giài tích biên phân Do vay, kêt quå dê tài nhïng nguôn khão cúu có giá trË khoa hÍc cao Thêm nïa, ê tài cüng côn chúa dung nhiêu vân dë mÛ dáng quan tâm có thê xem xét chù dờ cỗa nhùng nghiờn cộu xa hon Bụ cuc v hinh cỗa lu-n ỏn; Luan ỏn dugc trinh by ro rng, õy dỗ vi vọn phong phự hop v bụ cuc hop lý Ban túm tt cỗa lu-n án có dam bão tinh khoa hÍc phàn ánh trung thuc két quà nghièen céu trinh bày lu-n án hay khơng? Ban tóm tt lu-n án phàn ánh tuong dơi dây ù rõ nét nhïng óng góp nÙi dung ln án 10.Chât luong nhïng báo khoa hoc dã duoe nghiên cru sinh công bĐ; Két q cua dê tài bao gơm ân phâàm khoa hÍc, dó dã có cơng trinh durgc xt bàn tap chí qc tè có uy tín 11.Nhùng thiờu sút vờ ni dung v hinh thỳc cỗa lu-n án; Mac dù luan án da duoe trinh bày hồn chinh, nhiên duong nhu vån cịn lõi an lốt vàdơi chĐ nÙi dung có thê cân duoc làm rơ thêm 12 Kêt lu-n chung 12.1 Muc dó dỏp ng cỏc yờu cõu cỗa mt lu-n ỏn tiờn si: Luan án nói chung ã dáp éng dây dü yờu cõu 12.2 Nhùng diờm cõn sỗa chùa, bụ sung; De nghË tác giä xem xét hiÇu dinh lõi ân lốt hay thiêu sót vê nÙi dung (nêu cân thiêt) download by : skknchat@gmail.com 12.3 Ki¿n nghË Õi vÛi HÙi Ĩng vÁ viÇc cơng nh-n trinh dÙ câp bäng tiên si cho nghiên céu sinh ê nghË HÙi ông công nh-n kêt quà à tài câp bng tiên sỵ vÛi nghiên cru sinh thrc hien lu-n án Xác nhân cüa co Nguoi nhân xét (Ký ghi rö ho tên) quan TL.HICUTRUONG u TƯNC HOP RNG PHĨNG HÄNITCHÍNH LT Nguyen V n TRUDHG EDAN4QE QU NHON/ ThS Tran Ngoe Anh download by : skknchat@gmail.com Vk download by : skknchat@gmail.com ugp iuony ugin BupiyD TgN UBA yuánH HYSL'SOd uouL PA ugoL eOyy SupniL 'YuiuI SugI P1 SL SOd BuoniI is ugn up uenj pia yuep doy ugiyd yuey ugn gp ugI ngip np Suop rÙH UOYN Án) DÓs TèG BuoNiL eFo is un Ùp yun oëj oep yuip Knd 'oëI oPG BA onp opiO ÓA eFo is ugii p yuiy oëj ogp 9uo án) LOA neIYp 1OG uop 1ÓH uQIA Yueyi L eno ieu 9o ns toA IZ07 ugu 60 Supyi sz Kgsu 00 Qis 80 onj oRA onyi quyo dos ugIUd 'T uoniI s uen ug uènj pis yuep Buop 1ÓH 3pnIg ug uènj ýa ogq oonp ueH 0N oPG SON deyd oyo euu Suou1 Bugp 1OH uouN ÁnD OOy reg BuoniL eNo ngo ngk oeyi 191p ugo oi nyj oo pA ug uenj upq ii upos ep ueH o05N OpG Yurs nno ueINT ONQG IĨH VNO NAIX A ugIA AN ugAnL ueA ugÁnN S.L SOd | T1ON PH ueyd n_ ooy Ieg RuoniL DAUEA ugínN SLL uoUN ÁnD ooy Ieg Buon uoS uL wBYd 'SL SOd | S ng!y nH ugÁnN S.L SDd| t enH Oy I uOIq ueYd uoyN And ooy IeG SugniL Suop 1OH uouN nd ooy ieg 3ugn1L Suop 19H nyL yoi nyo yuLL 1eUN UEYd SL SDd| ¬ uenuI Suenò ?T SL ona yuey yuig SL Sd| oy Bugo uenb o u EA ÓH LL :ugIA Yuu4 L uQs uouN KnO ooy eg SuonuI Supny náIH BNo 1Z0T/80/01 Agõu NÒHd-GÒ/1961 9s yuip 12knD o9u1 dej qupya oonp upH N OPG SON OY0 s 0o deo js ugn ug uenj PI5 yuep Sugp 1OH yon gis upoJ quBu ugAnyy TO109t6 0s eN uPH 0N OPG :Yuis nno ugusN Sunp Sun Ea (610T- SIO) E eOUX doy di iÙu ugn /an vp tx quy eno ugáni Iuyd oingu ánb quiyo quiL up uenj ei PG ONQNAL dyo S NJIL NY NÝNT y1D HNYG ONQG IÓHdÓH NYS NJIA onyd yueH -op i -dÇi oÙG WVN LAIA VIHON QHD IƯH VX YOH ON00 B NDHN ANO NOHNANịS VUDNON OVL OVG VAndoyoos download by : skknchat@gmail.com ug ugn unp 1Ùu wi uo1 Áeq yuun ugH o03N Opa SON S 0o SuouyX SON Bno oej ogp qun pnbpa oos eOYpN Yoij ÁI Pa np ureyp ronõuBupqu "Buop 10H ugIA Yup4n opo eno uprA A Opq oonp 9p ujiy neip np Kp gA Buer o1 ooy POYY Yoi A 0o quis nno ugiuN A OPq oðnp yuis nno ugruôu 2p oyyy 19Iy1 uo uÙpy neip opo PA UPH 008N OPG SON eno oóu eouy ypij oop Suop 104 nyn 'uensL Suend T SL ue uènj sA OPq o0np $ON P 19II ugo ugIy ng1p ogo pa SON eno Ooy Boyy yoij át 0op Suop IÓH nYL E opq ionq yuuy Buonyoenb BuoyL - 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epyo ens uns oq ugo u p BunuN oy ugoj IS ugIn ug uènj iou eFo no ngí opo np Kpp Sun dep ug ueng S ugn ug uènj jÙuu eFo no ngs ogo Sun dep op onNS BO GIÁO DJC VÅ ÀO TAO TRUONGDAI HOC QUY NHON cONG HOÀ X HQI CHn NGH*A VIET NAM DÙc lap- Tr - Hanh phúc Binh Dinh, ngày 21 tháng 12 nm 2021 BAN GIÅI TRÌNH CHÍNH SÚA LUAN ÁN TIÉN si THEO KÈT LUAN CÜA HOI ÔNG ÁNH GIÁ LUAN ÁN TIÉN S* CÂP TRUÖNG Ho tên nghiên cúu sinh: ÅO NGOC HÄN De tài lu-n án: Tính chinh quy mêtric phi tuyộn cỗa ỏnh xa da trậ trờn mt tõp høp urng dung Chuyên ngành: Toán Giái tich Mã sơ: 9460102 Nguoi/Tap thà hng dn khoa hÍc: PGS TSKH HUYNH VN NG - Truong hÍc Quy Nhon TS NGUYÈN HÜU TRON - Truong Dai hÍc Quy Nhon - Cn Cn cúr Biên b£n Nghi quyêt cuc hop Hi ụng cỳ nh-õn xột cỗa cỏc thnh viên HÙi dông ánh ánh giá lu-n án tiên sï câp Trrong; giá lu-n án tiên s+ câp Truong Nghiên cru sinh d chinh sua: 1) Bô sung phân kêt luân vào cuôi chuong 2, chuong 3, chuong chuong 5: Chúng tơi dã tiép thu góp ý v bụ sung theo ý kiờn cỗa hi ụng 2) Bồn túm tt lu-n ỏn cõn chinh sỗa cõn th-n: Chúng tơi th-n hai bàn tóm tt tiêng ViÇt tiêng Anh 3) Bô sung phân chi måc: Chúng tụi ọtiờp r soỏt v chinh sỗa cõn thu gúp ý v bụ sung theo ý kiờn cỗa nguoi nhõn xét 4) MÙt sô l6i chinh tà trinh bày: Chúng tơi ã rà sốt tồn bÙ lu-n án ckng nhu ban tóm tt lu-n án sua chïa nhïng chÑ cân thi¿t Nghiên cru sinh ê ngh/ bào luru: Khôông XÁC NHAN CÙA TAP THÉ HUNG DÅN NGHIÊN CÚU SINH TM T-p thà hurÛng dân yRal a download by : skknchat@gmail.com Ngoe Hân XÁC NHAN CÙA HOI ÔNG ÁNH GIÁ LUAN ÁN CÁP TRUÒNG THU'KÝ HOI HONGG CHU T.CH HOI ÔNNG Cuang Thuan XÁC NHAN CÙA CO Ssß ÀO TA0 TLHIRU TRUONG CTRNRHONGDÃO TO SAU DAI HOC TRUONG DAIDAI HOC "e uu QUY NHON PGS.TS.Ho Xuân Quang download by : skknchat@gmail.com ... 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... skknchat@gmail.com 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
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