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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: 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 peerreviewed 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 i Dao Ngoc Han 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 scienti c 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 di culties 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 1.1 Some related classical results 1.2 Basic tools from variational analysis 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.3 Metric regularity and equivalent prop 1.3.1 1.3.2 1.3.3 1.4 Metric regularity criteria in metric spa iii 1.5 The regularity in metric spaces Metric regularity on a xed set: de nitions and characterizations 2.1 De nitions and equivalence of the non 2.2 Characterizations regularity via slope 2.3 Characterizations regularity via coderivative Perturbation stability of Milyutin-type regularity and applications 3.1 Perturbation regularity 3.2 Application to xed double-point prob Star metric regularity 4.1 De nitions and characterizations of n 4.2 Stability of Milyutin-type regularity un ularity Stability of generalized equations governed by composite multifunctions 5.1 Notation and some related concepts 5.2 Regularity of parametrized epigraphi mappings 5.3 Stability of implicit set-valued mappin 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 composite mapping 122 Conclusions List of Author’s Related Publications References v N R R+ ; Rn hx; yi jjxjj 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; ) N(x; ) b N(x; ) x!x F:X Y GraphF D F (x; y)( ) D b rf(x) EH S EH SH F ’y (x) F ’y (x) p ’ T (x; y) : the lower semicontinuous envelop function of the sur F sur F reg F reg F reg (;) F reg F reg( ; ) F lip F lip F (ii) T1 is metrically regular around (x; y1) with modulus m > 0; (iii) T2 is Lipschitz-like around ((x; p); y2) with respect to x, uniformly in p with modulus l > 0; (iv) T2 is Lipschitz-like around ((x; p); y2) with respect to p, uniformly in x with modulus > (v) T is metrically regular around ((y1; y2); 0) with respect to y1, uniformly in y2 with modulus > 0; (vi) (vii) with T is Lipschitz-like around ((y1; y2); 0) with respect to y2, uniformly in y1 modulus > 0; m l < and Then SH (0; ) is Robinson metrically regular around (x; p) with modulus SH (0; ) is Lipschitz-like around (x; p) with modulus Proof p Applying Theorem 5.2.1 yields that EH is metrically regular around (x; p; y1 ; y2 ; 0) with respect to 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; y2); 0), one obtains that SH (0; ) is Robinson metrically regular around (x; p) with modulus m : ml By the de nition of Robinson’s metric regularity of SH (0; ), we derive the existence of some > such that d(x; SH (0; p)) (5.46) for all (x; p) B((x; p); 1): By (iv), since T2 is Lipschitz-like around ((x; p); y2) with respect to p, uniformly in x with modulus > 0, there is > such that 0 T2(x; p) \ B(y2; 2) T2(x; p ) + d(p; p ) for all p; p B(p; 2); for all x B(x; 2): 126 Moreover, according to (vi), since T is Lipschitz-like around ((y 1; y2); 0) with respect to y2, uniformly in y1 with modulus > 0, there exists > such that 0 T (y1; y2) \ B(0; 3) T (y1; y2 ) + d(y2; y2 ) for all y1 B(y1; 3); for all y2; y2 B(y2; 3): Using the local composition-stability of the pair ((T1; T2); T ) around ((x; p); (y1; y2); 0) in (i), select > such that for every (x; p) B(x; 4) B(p; 4) and every z T (T1(x); T2(x; p)) \ B(0; 4) There such that z exists (y1 Set := minf 1; 2; 3; 4g; and take p; p B(p; ); and x SH (0; p)\B(x; ) This means that, T (T1(x); T2(x; p)) T (T1(x); T2(x; p)) \ B(0; 4) and x B(x; ): It follows that there exists (y1; y2) such that T (y1; y2): Consequently, for y2 So, by taking into account T (y 1; y2) and by using the estimations (5.49) and (5.50), we have This means SH d(x; which implies that SH (0; p) \ B(x; ) SH (0; p ) + that SH (0; ) is Lipschitz-like around (x; p) with modulus completes the proof m ml 127 , which Conclusions This dissertation obtained the following main results: Gave some new models of nonlinear metric regularity of set-valued mappings on a xed arbitrary-set as well as the equivalence of this one and Holder, openness properties Established in nitesimal characterizations for the nonlinear metric regularity models via slope and coderivative Achieved some versions of metric perturbation of Milyutin’s theorems on a xed 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 xed 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 xed 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 speci c variational problems, for instance, the variational inequality problem, the complementarity problem, the di erential inclusion, the optimal control, the parametric programming, Using the local metric regularity to study of existence and stability of global solutions of di erential inclusion in terms of x (t) and application to investigate the controllability of the dynamical system de ned by the di erential 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 xed set and application to xed point theorems", preprint Ngai H V., Tron N H., Han D N (2021), \Star metric regularity on a xed set", preprint Tron N H., Han D N., Ngai H V (2020), \Nonlinear metric regularity on xed sets", In revision, submitted to Optimization Tron N H., Han D N (2020), \Stability of generalized equations governed by composite multifunctions", Paci c 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 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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... parametrized epigraphi mappings 5.3 Stability of implicit set-valued mappin iv 5.3.1 Stability of implicit set-valued mappings associated to the epigraphical set-valued mapping... 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