Ekelands variational principle and some related issues

VIETNAM NATIONAL UNIVERSITY - HCMC UNIVERSITY OF SCIENCE Dinh Ngoc Quy EKELAND’S VARIATIONAL PRINCIPLE AND SOME RELATED ISSUES PhD THESIS IN MATHEMATICS Hochiminh City - 2012 VIETNAM NATIONAL UNIVERSITY - HCMC UNIVERSITY OF SCIENCE Dinh Ngoc Quy EKELAND’S VARIATIONAL PRINCIPLE AND SOME RELATED ISSUES Major: Mathematical Optimization Codes: 62 46 20 01 Referee 1: Assoc Prof Dr Nguyen Dinh Phu Referee 2: Assoc Prof Dr Mai Duc Thanh Referee 3: Dr Nguyen Ba Thi Independent Referee 1: Assoc Prof Dr Truong Xuan Duc Ha Independent Referee 2: Assoc Prof Dr Nguyen Dinh Huy SCIENTIFIC SUPERVISOR Prof D.Sc Phan Quoc Khanh Hochiminh City - 2012 Confirmation I confirm that all the results of this thesis come from my work under the supervision of Professor Phan Quoc Khanh They have never been published by other authors Hochiminh City, February 2012 The Author Dinh Ngoc Quy i Table of Contents Foreword v Chapter A generalized distance and enhanced Ekelands variational principle for vector functions 1 Introduction 2 Weak τ -functions and generalized lower semicontinuity of a function 3 Lower closed transitive relations and minimal elements The EVP The EVP for the case of range space being a normed space 12 Equivalent formulations 13 Particular cases 14 References 16 Chapter On generalized Ekelands variational principle and equivalent formulations for set-valued mappings 17 Introduction 18 Preliminaries 19 Weak τ -functions 21 Main results 24 Some implications for the single-valued case 29 ii Table of Contents References iii 32 Chapter On Ekelands variational principle for Pareto minima of set-valued mappings 34 Introduction 35 Notions and Preliminaries 36 Relaxed Lower Semicontinuity Properties of Set-Valued Mappings 37 Enhanced Versions of Ekelands Variational Principle 43 Concluding Remarks 50 References 51 Chapter Versions of Ekelands variational principle involving set perturbations 53 Introduction 54 Preliminaries 55 Ekelands Variational Principle for Kuroiwas Minimizers 61 Minimal Elements for Product Orders 70 Ekelands Variational Principle for Pareto Minimizers 75 Concluding Remarks 79 References 79 Chapter More kinds of semicontinuity of set-valued maps and stability of general inclusion problems 82 Introduction 84 More about semicontinuity of set-valued maps 87 Table of Contents iv Upper semicontinuity properties of solution maps 95 Lower semicontinuity properties of solution maps 99 Continuity of solution maps 104 Particular cases 106 6.1 Quasiequilibrium problems of type 106 6.2 Quasiequilibrium problems of type 109 6.3 A scalar equilibrium problem and Ekelands variational principle 112 References 116 Index 120 List of the papers related to the Thesis 122 List of Conference Reports 123 Conclusion 125 Acknowledgments 127 Foreword The celebrated Ekeland’s variational principle (Ekeland 1974) (EVP, from now on) is one of the most important results and cornerstones of nonlinear analysis in the last half a century with applications in many fields of analysis, optimization and operations research Its importance is emphasized by the fact that there are a number of equivalent formulations, all of which are well known with significant applications and many of which were discovered independently, namely the CaristiKirk fixed-point theorem (Caristi 1976), the drop theorem of Daneˇs (Daneˇs 1972), the Takahashi theorem about the existence of minima (Takahashi 1991), the petal theorem of Penot (Penot 1986), the Krasnoselski-Zabrejko theorem on solvability of operator equations (Zabreiko and Krasnoselski 1971), Phelps’ lemma (Phelps 1974), etc Over the last four decades a good deal of effort has been made to look for equivalent formulations or generalizations of the EVP The seminal EVP (Ekeland 1974) says roughly that, for a lower semicontinuous (lsc) and bounded from below function f on a complete metric space X, a slightly perturbed function has a strict minimum Moreover, if X is a normed space and f is Gateaux differentiable, then its derivative can be made arbitrarily small We can first observe generalizations of the EVP to vector minimization, i.e., to the case where f is a mapping with a multidimensional range space Y , see e.g., Loridan (1984), Valyi (1985), Khanh (1989) Here Y may be even an ordered vector space Extensions of X to the case of topological vector spaces, uniform spaces or L-spaces are investigated, e.g., in Khanh (1989), Hamel (2003, 2005), Hamel and L¨ohne (2006), Qui (2005) In this research direction, a general partial order is often proposed and a minimal point with respect to (wrt) this order is proved to exist, v Foreword vi leading to a type of the EVP, see G¨ opfer et al (2000) Smooth variants of the EVP are studied, e.g., in Borwein and Preiss (1987), Li and Shi (2000) The second conclusion of Ekeland in the seminal work Ekeland (1974), that (when X is a normed space and f is Gateaux differentiable) f can be made arbitrarily small, has been attracted also much attention, see, e.g., Ha (2003, 2005, 2006), Bao and Mordukhovich (2007) Here various kinds of generalized derivatives are taken into account: the Fre’chet, Clarke and Mordukhovich coderivatives; the Fre’chet, Clarke and Mordukhovich subdifferentials Fre’chet Hessians are also used to establish the Ekeland principle for second-order optimality conditions (Arutyunov et al 1997) Stability results for the EVP are obtained, e.g., in Attouch and Riahi (1993), Huang (2001, 2002) In connection with the EVP, existence conditions for optimal solutions in problems with noncompact feasible sets are dealt with in Ha (2003, 2006), Bao and Mordukhovich (2007), El Amrouss (2006), using generalizations of coercivity assumptions, the Palais-Smale condition or the Cerami condition One of the recent research interests is to consider the case where X is a metric space but equipped with an additional generalized distance, based on which the semicontinuity assumption of Ekeland can be weakened w-distance was introduced in Kada et al (1996) and used also in Park (2000), Lin and Du (2007) In Tataru (1992), another distance was proposed to obtain a generalization of the EVP In Suzuki (2001, 2005), τ -distance, which is more general than both afore-mentioned distances, was introduced to improve the EVP τ -function was proposed and employed in Lin and Du (2006) The purpose of this thesis is to investigate the topic in both theory and application aspects The thesis consists of five chapters The EVP says that, for a bounded from below and lower semicontinuous scalar function f on a complete metric space X, a slightly perturbed function has a strict minimum Examining the assumptions, we see that the completeness of X is crucial since the principle relies on a convergence of a sequence to a desired point We can replace it only by a slightly weaker condition that the lower sector of a given point is complete The boundedness from below is, in nature, inevitable; for instance functions as regular as continuous linear ones are far from having a point as desired, Foreword vii since they are unbounded from below However, in this thesis we also make efforts to use weaker, even to small extent, conditions like quasiboundedness or boundedness from below through some kinds of scalarization Such modifications may be more relaxed than usual assumptions for vector functions, but they collapse to usual boundedness if the function in question is scalar Lower semicontinuity seems to be the only assumption which may be relaxed somehow substantially We observe that many authors have recently used generalized distances together with the original metric of the space in order to weaken the lower semicontinuity condition imposed by Ekeland This idea motivates our main commitment in Chapter 1, which is using generalized distances to weaken this assumption In this thesis, we propose the notion of weak τ -function, which is more general than all the aforementioned generalized distances Let (X, d) be a metric space A function p : X × X → R+ is called a weak τ -function iff the following three conditions hold, for x, y, z ∈ X, (τ 1) (triangle inequality) p(x, z) ≤ p(x, y) + p(y, z); (τ 3) if xn , yn ∈ X satisfy limn→∞ p(xn , yn ) = and limn→∞ sup{p(xn , xm ) : m > n} = 0, then limn→∞ d(xn , yn ) = 0; (τ 4) p(x, y) = and p(x, z) = imply that y = z Note that with the following additional condition (τ 2) (lower semicontinuity) for all x ∈ X, p(x, ) is lower semicontinuous, i.e., liminfy→¯y p(x, y) ≥ p(x, y¯); a weak τ -function becomes a τ -function introduced in Lin and Du (2006) Observe further that the w-distance, Tataru’s distance, τ -distance and τ -function are all particular cases of the weak τ -function Example 2.1 in Khanh and Quy (2010) shows that being a weak τ -function may be strictly weaker than being a kind of the mentioned distances We omit the lower semicontinuity of p (i.e., condition (τ 2)), since in fact, we need lower semicontinuity properties of the mapping f + p, unlike Foreword viii in the case of using the metric d of X, where f needs be lower semicontinuous for f + d to be so It is well recognized that any form of Ekeland’s variational principle can be reformulated as a theorem on minimal elements of a transitive relation by defining such a suitable relation So, theorems on minimal elements may be general formalizations of the EVP In this thesis we propose the notion of lower closedness of a general transitive relation on a metric space endowed with a weak τ -function p A transitive relation on a metric space X is called lower closed iff for any -decreasing (i.e xn x2 x1 ) sequence converging to x one has x xn , ∀n ∈ N In Chapter 1, we contribute two developments We first propose a definition of lower closed transitive relations in a metric space and establish a sufficient condition for the existence of minimal elements of such relations Note that we avoid here the usage of the Zorn lemma Theorem 1.3.2 (Minimal Elements for Lower Closed Relations) Let be a lower closed transitive relation and p be a weak τ -function on a metric space X For x0 ∈ X assume that the -sector of x0 , i.e S (x0 ) = {x ∈ X : x x0 }, is nonempty and -complete Assume further that any -decreasing sequence {xn } in X is asymptotic by p (i.e limn→∞ p(xn , xn+1 ) = 0) Then, there exists x ∈ S (x0 ) such that S (x) = ∅ Moreover, if or S (x) = {x} is reflexive, then S (x) = {x} Then, we focus on our second contribution, which is using theorem of minimal elements as a basic tool to prove generalized EVP in various settings This theorem is general and seems to contain a large part of existing versions of the EVP in the literature As an example, we will now derive the main result, Theorem 3.8, of a very recent paper of Guti´errez, Jimener and Novo (2008) We recall the definition of a set-valued K-metric in Guti´errez, Jimener and Novo (2008) A set-valued map D : X ×X → 2K is called a set-valued K-metric (sv-K-metric) iff it satisfies the following conditions, for all x, y, z ∈ X, (i) D(x, x) = {0} and D(x, y) = ∅, ∈ D(x, y), ∀x = y; the assumptions in (i)-(iv), by setting F (x, y, λ) = f (x, y, λ) + d(x, y) − R+ it follows the 0-inclusion or 0-inclusion complement properties required in the these theorems We turn to lower semicontinuity By the implications (see Proposition 2.2): inner openness ⇒ moderate-inner openness ⇒ moderate-inner continuity ⇒ lower semicontinuity, actually the lower semicontinuity of Σ has been obtained in Corollary 6.8 as consequences of stronger properties However, the assumptions in this corollary (to guarantee stronger properties) may be too restrictive in some situations (see Example 4.3) To seek for other sufficient conditions, we first use the auxiliary problem (EPλ ): find x¯ ∈ X such that, for each y ∈ X, f (¯ x, y, λ) + d(¯ x, y) > ¯ (This is problem (QVIPλ ) for this situation.) Let Σ(λ) be the solution set of ¯ (EPλ ) Then clearly Σ(λ) ⊆ Σ(λ) ¯ Corollary 6.9 Assume for problem (EPλ ) that X is compact and Σ(λ) = ∅ in a ¯ and that neighborhood of λ ¯ y ∈ X and from (yn , λn ) → (y, λ) ¯ there exists an index n0 ¯ λ), (i) for x ∈ Σ( such that f (x, yn0 , λn0 ) + d(x, yn0 ) > 0; ¯ ⊆ clΣ( ¯ ¯ λ) (ii) Σ(λ) ¯ Then, Σ is lsc at λ Proof Set K1 ≡ K2 = X and F (x, y, λ) = f (x, y, λ)+d(x, y)−R+ , which implies ¯ E = X Note that x ∈ Σ(λ) if and only if ∈ intF (x, y, λ) for all y ∈ X By (i), ¯ for all x ∈ Σ( ¯ ¯ λ) intF (x, , ) has the 0-inclusion complement property in X × {λ} ¯ implies that Σ ¯ is moderate-inner continuous at λ, ¯ According to Theorem 4.2, Σ ¯ is lsc at λ ¯ and (ii), Σ is lsc at λ, ¯ since ¯ at λ By the lower semicontinuity of Σ ¯ ⊆ clΣ( ¯ ⊆ liminf λ→λ¯ Σ(λ) ¯ ¯ λ) ¯ Σ(λ) ⊆ liminf λ→λ¯ Σ(λ) 113 Remark 6.1 Assumption (i) in Corollary 6.9 can be replaced by the lower semi¯ for all x ∈ Σ( ¯ ¯ λ) continuity of f (x, , ) in X × {λ} Indeed, let y ∈ X and ¯ Since x ∈ Σ( ¯ then f (x, y, λ) ¯ + d(x, y) > By the lower ¯ λ), (yn , λn ) → (y, λ) semicontinuity of f (x, , ), we have ¯ + d(x, y) ≤ < f (x, y, λ) lim [f (x, yn , λn ) + d(x, yn )] ¯ (yn ,λn )→(y,λ) Then, there exists an index n0 such that f (x, yn0 , λn0 ) + d(x, yn0 ) > To explain the need of developing still another sufficient condition for lower semicontinuity, let us consider the following example Example 6.1 Let X = [0, ], Λ = (0, +∞) and g : X → R given by   x if x ∈ [0, 2], g(x) =  2 if x ∈ (2, ] Let f : X × X × Λ → R defined by f (x, y, λ) = λ1 (g(y) − g(x)) ¯ Corollary 6.9 cannot be in use since Σ(λ) = ∅ for all λ ∈ (0, +∞) Moreover, ¯ ∈ (0, 2), Corollary 6.8(iii) and (iv) give us nothing, since these condifor any λ ¯− tions there not hold Indeed, let yn = λ ¯ λ , 6n ¯− λn = λ ¯ λ 2n ¯ y = λ ¯ and x = λ, ¯2 ¯ λ) ¯ and x = λ ¯ ∈ Σ(λ), ¯ since Σ(λ) ¯ = [0, λ] ¯ ∪ [ λ + , ] But Then, (yn , λn ) → (λ, ¯ 2λ we have ¯ ¯ ¯ ¯ ¯ ¯ − λ ) + d(λ, ¯ λ ¯− λ )= ¯ − λ ) − g(λ)) ¯ + λ ¯ λ ¯ − λ ,λ (g( λ f (λ, ¯ − λ¯ ) 6n 2n 6n 6n 6n (λ 2n ¯ − λ¯ )2 λ ¯2 ¯ (λ λ 5λ 6n = ( − ) + =− < ¯ λ ¯ 2 6n 36n(2n − 1) (λ − 2n ) i.e., the condition in Corollary 6.8(iv) is not satisfied It follows that the condition in Corollary 6.8(iii) is not fulfilled either To check directly lower semicontinuity of the solution we compute it to have, for all λ ∈ (0, +∞), Σ(λ) = [0, λ] ∪ [ and hence Σ is lsc in (0, 1) ∪ (1, ∞) 114 λ2 + , ] 2λ Now we try to employ the following auxiliary problem called a parametric Ekeland’s variational problem, for λ ∈ Λ, (EVPλ ) find x¯ ∈ X such that, ∀y ∈ X \ {¯ x}, f (¯ x, y, λ) + d(¯ x, y) > ˆ Let Σ(λ) stand for its solution set Note that, if f (x, x, λ) = for all x ∈ X, then ˆ Σ(λ) ⊆ Σ(λ) The name of this problem is justified as follows Set f (x, y, λ) = ˆ g(y, λ) − g(x, λ) Then, x¯ ∈ Σ(λ) means that, for all y ∈ X \ {¯ x}, g(y, λ) + d(¯ x, y) > g(¯ x, λ) Thus, the assertion of the existence of a solution x¯ is just the conclusion of (parametric) Ekeland’s variational principle The interested reader is referred to [25-28] for discussions on one-variable and two-variable versions of Ekeland’s variational principle The following result about the solution existence of (EVPλ ) is an immediate consequence of Theorem 2.1 of [24] and Lemma 3.8(iii) and Theorem 4.1 of [25] Proposition 6.10 Assume for problem (EVPλ ), for all λ and x, y, z ∈ X, (i) f (x, y, λ) + f (y, z, λ) ≥ f (x, z, λ) and f (x, x, λ) = 0; (ii) f (x, , λ) is bounded from below; (iii) f (x, , λ) is lsc ˆ ˆ Then, Σ(λ) = ∅ Moreover, for each x ∈ X, there exists x¯ ∈ Σ(λ) such that f (x, x¯, λ) + d(x, x¯) ≤ Applying this proposition we obtain the following sufficient condition for the lower semicontinuity of the solution map Σ of (EPλ ) ¯ ∈ Λ, impose the assumptions of Theorem 6.11 For each λ in neighborhood of λ Proposition 6.10 and assume further that X is compact and ¯ ˆ λ) (a) f (x, , ) is lsc for all x ∈ Σ( 115 ¯ ⊆ clΣ( ¯ ˆ λ) (b) Σ(λ) ¯ Then, Σ is lsc at (λ) ¯ Indeed, suppose to the contrary that ˆ is lsc at λ Proof First we prove that Σ ¯ and λn → λ ¯ such that, for any xn ∈ Σ(λ ˆ λ) ˆ n ), xn → x Without there are x ∈ Σ( ˆ n ) for all n, i.e., for some yn = x, loss of generality, we may assume that x ∈ Σ(λ f (x, yn , λn ) + d(x, yn ) ≤ ˆ n ) such that For each yn and λn , Proposition 6.10 yields xn ∈ Σ(λ f (yn , xn , λn ) + d(yn , xn ) ≤ The above two inequalities together with (i) of Proposition 6.10 imply that f (x, xn , λn ) + d(x, xn ) ≤ (f (x, yn , λn ) + f (yn , xn , λn )) + (d(x, yn ) + d(yn , xn )) = (f (x, yn , λn ) + d(x, yn )) + (f (yn , xn , λn ) + d(yn , xn )) ≤ As X is compact, one has xn → x¯ (taking a subsequence if necessary) By (a), ¯ + d(x, x¯) ≤ By the contradiction the last inequality implies that f (x, x¯, λ) ¯ ˆ λ), assumption, we have x¯ = x Hence, as x ∈ Σ( ¯ + d(x, x¯) > f (x, x¯, λ) ¯ Since f (x, x, λ) = for all x ∈ X, ˆ is lsc at λ This contradiction shows that Σ ¯ ⊆ Σ(λ) ¯ By the lower semicontinuity of Σ ¯ since ˆ λ) ˆ and (b), Σ is lsc at λ, then Σ( ¯ ⊆ clΣ( ¯ ⊆ liminf λ→λ¯ Σ(λ) ˆ λ) ˆ ⊆ liminf λ→λ¯ Σ(λ) Σ(λ) Now we apply Theorem 6.11 to consider Example 6.1 We can check that for ¯ ∈ (0, 1)∪(1, ∞) the assumptions of Theorem 6.11 are fulfilled Consequently, all λ ¯ = Theorem 6.11 says nothing, Σ is lower semicontinuous in this set (Only at λ ˆ since Σ(1) = [0, 1] does not contain Σ(1) = [0, 1] ∪ { }.) 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Acta Math Vietnam 28 (2003) 345-350 118 [28] P.Q Khanh, D.N Quy, On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings, J Glob Optim 49 (2011) 381-396 119 Index  -approximate minimizer, 13 (e, K ) -lower semicontinuous ( (e, K ) -lsc), (e, K ) -lower semicontinuous from above ( (e, K ) -lsca), K -bounded from below ( K -bd fb), 37 K -bounded from below by scalarizations, K -closed, 21, 58 K -epiclosed, 38 K -level closed, 38 K -lower semicontinuous ( K -lsc), 3, 20, 38 K -lower semicontinuous from above ( K lsca), 3, 20, 38, 58 K  (positive polar of a convex cone K ), 5, 36 K # (quasi-interior of K  ), 36 Y * (the topological dual of Y ), R -sector, R -complete, R -decreasing,  -distance,  -function, liminfo *x x G( x) , 87 liminfo x x G( x) , 85 e-level set, 38 p-sharp minimizer of order 1, 12 p-strong minimizer, 12  -distance,  -nomal, (k0 , K ) -lower semicontinuous ( (k0 , K ) -lsc), 41 (k0 , K ) -lower semicontinuous from above ( (k0 , K ) -lsca), 41 asymptotic, base of K , based (well-based), Bishop-Phelps cone, 13 bounded from below, bounded set, Caristi's common fixed-point theorem, 13 Cauchy sequence, clB (closure of B), coB (convex hull of B), coneB (cinical hull of B), common fixed-point theorem, 13 common invariant -point theorem, 13 condition (  2' ), condition (  3' ), * liminf x x G( x) , 87 liminf x x G( x) , 84 condition (  4' ), condition (  5' ), condition (  ), condition (  ), condition (  ), condition (  ), limsupo*x x G( x) , 87 limsupox x G( x) , 85 limsup*x x G( x) , 87 limsup x x G( x) , 84  -inclusion complement property, 94  -inclusion property, 94  D-minimizer, 66  k0-minimizer, 47  -minimizer, 13 4 3 3 condition ( A2' ), condition ( A3' ), condition ( A1 ), 120 pseudomonotone, 104 QEP1 , 106 condition ( A2 ), condition ( A3 ), condition ( A4 ), condition (H'), 11 condition (H), domF, 38 domf, dominantion property, 37 EP , 112 EVP, EVP , 115 firm minimizer of order 1, 12 H-continuous (Hausdorff continuous), 84 H-lsc (Hausdorff lower semicontinuous), 84 H-usc (Hausdorff upper semicontinuous), 84 Hausdorff locally convex space, inferior open, 85 inner continuous, 84 inner open, 85 invariant point, Kuroiwa's minimizer, 56 limiting monotonicity condition, 50 lower closed, lower semicontinuous (lsc), 84 maximal element theorem, 13 moderate-inner continuous, 89 moderate-inner open, 89 moderate-outer continuous, 89 moderate-outer open, 89 normal, outer continuous, 84 outer open, 85 Painlevé-Kuratowski inferior limits, 84 Painlevé-Kuratowski superior limits, 84 parametric Ekeland's variational problem, 115 parametric quasiequilibrium problem, 109 parametric quasivariational problem, 86 parametric vector quasiequilibrium problem, 106 Pareto minimizer, 56 QEP2 , 109 quasibounded from below, 3, 21 quasimonotone, 104 quasivariational inclusion problem, 86 quasivariational relation problem, 86 QVIP1 , 86 QVIP , 86 121 QVRP , 86 scalar parametric uncontrained equilibrium problem, 112 scalarizing function, semi-limit, 84 set-valued K -metric (sv-K-metric), SQEP1 , 106 strict domination, 37 strict Kuroiwa’s minimizer, 56 strict Pareto minimizer, 56 strictly monotone, 44 strong D -minimizer, 66 strongly K-bounded from below (str K-bd fb), 37 strongly K-quasibounded from below (str K-qbd fb), 37 sublinear scalarizing function, 43 superior open limits, 84 theorem on minimal elements, topological dual, topological vector space, transitive relation, triangle inequality, upper K-continuous (uKc), 38 upper semicontinuous (usc), 38, 84 weak  -function, weak K-lower semicontinuous from below (w.Klsca), 58 weak lower semicontinuity, weak topology, weakly K-bounded from below (w.K-bd fb), 38 List of the papers related to the Thesis [1] Khanh P.Q., Quy D.N (2010), A generalized distance and enhanced Ekeland’s variational principle for vector functions, Nonlinear Analysis 73, pp 22452259 [2] Khanh P.Q., Quy D.N (2011), On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings, Journal of Global Optimization 49, pp 381-396 [3] Khanh P.Q., Quy D.N (2012), On generalized Ekeland’s variational principle for Pareto minima of set-valued mappings, Journal of Optimization Theory and Applications, 153, pp 280-297 [4] Khanh P.Q., Quy D.N., Versions of Ekeland’s variational principle involving sets of perturbations, Journal of Global Optimization, submitted for publication [5] Anh L.Q., Khanh P.Q., Quy D.N., More kinds of semicontinuity of set-valued maps and stability of quasivariational inclusions, Journal of Mathematical Analysis and Applications, submitted for publication 122 List of Conference Reports [1] Anh L.Q., Khanh P.Q, Quy D.N, Lower and upper semicontinuity of solutions to general multivalued vector quasiequilibrium problems, 3th Workshop on Optimization and Scientific Computing, April 20-24 (2005), Hanoi, Vietnam [2] Anh L.Q., Khanh P.Q, Quy D.N, On Holder continuity of the unique solution to multivalued equilibrium problems in metric spaces, 4th Workshop on Optimization and Scientific Computing, April 26-29 (2006), Ba Vi, Vietnam [3] Khanh P.Q, Quy D.N, On Ekelands ε-variational principle with various efficiency notions for set-valued maps, 5th Workshop on Optimization and Scientific Computing, May 16-19 (2007), Ba Vi, Vietnam [4] Khanh P.Q., Quy D.N., A generalized distance and Ekeland’s variational principle for multivalued mappings, The 6th Vietnam-Korea Workshop ”Mathematical Optimization Theory and Applications”, February 25-29 (2008), Nhatrang, Vietnam [5] Khanh P.Q., Quy D.N., On generalized Ekeland’s variational principle for Pareto extreme of multimaps, The 7th International Workshop on Mathematical Optimization Theory and Applications, July 31- August (2008), Hanoi, Vietnam [6] Khanh P.Q., Quy D.N., Generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings, The 7th Vietnam Mathematical Congress, August 04-08 (2008), Quy Nhon, Vietnam [7] Khanh P.Q., Quy D.N., Enhanced versions of Ekelands variational principle for approximate Pareto minima of set-valued mappings, International Sympo123 List of Conference Reports 124 sium on Variational Analysis and Optimization Dedicated to Professor Boris Mordukhovich on his 60th Birthday, November 28-30 (2008), Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan [8] Quy D.N., Versions of Ekeland’s variational principle involving sets of perturbations, The 8th Vietnam-Korea Workshop ”Mathematical Optimization Theory and Applications”, December 8-10 (2011), Dalat, Vietnam Conclusion Ekeland’s variational principle is one of the most important results in nonlinear analysis and optimization for the last four decades During this period it has been being developed, generalized and applied to many fields of mathematics by many authors around the world This thesis is to investigate this principle and some related issues The thesis consists of five chapters In Chapter we propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation We introduce the notion of a weak τ -function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced EVP for various settings, under relaxed lower semicontinuity assumptions These principles conclude the existence not only of p-strict minimizers of p-perturbations of the considered vector function, but also p-sharp and p-strong minimizers In Chapter 2, using weak τ -functions we discuss the EVP for Kuroiwa minimizers of a set-valued mapping and equivalent results Chapter includes relaxed lower semicontinuity properties for set-valued mappings in terms of weak τ -functions, and enhanced EVP for Pareto minimizers of set-valued mappings and underlying minimal-element principles with relaxed lower semicontinuity In Chapter 4, we develop EVP with perturbations by a closed bounded convex subset D of Y , instead of perturbations in a direction k0 , considering both Pareto and Kuroiwa’s minima of a set-valued map A corresponding minimal element theorem for a product order is also proved as a underlying fact for the EVP 125 Conclusion 126 We introduce in Chapter several new kinds of inferior and superior limits and corresponding kinds of semicontinuity of a set-valued map Together with the known concepts of semicontinuity, they can be used to have a clearer insight of local behaviors of maps Then, we investigate all major semicontinuity properties of solution maps to a general quasivariational inclusion Consequences are derived for several particular problems, including some connections to Ekeland’s variational principle In each chapter of this thesis, comparisons between our results and recent known ones, including even comparisons when applied to particular cases, are provided Numerous corollaries and examples are also given to illustrate the main results Acknowledgments I am indebted to many kind people who have significantly contributed to the thesis First of all, I am deeply grateful to Professor Phan Quoc Khanh, my supervisor, for his kind, continuous guidance and encouragement My warmest thanks are addressed to Professor Samir Adly, my French cotutelle supervisor, for his fruitful suggestions and helps from a remote distance I am grateful to the anonymous referees for their valuable remarks which helped to improve the previous version of the thesis I would like to thank very much the University of Science of Hochiminh City and the University of Limoges for accepting me as a cotutelle PhD student and providing favorable conditions and facilities for my study, and the professors and lecturers who taught me The last but not least thanks are devoted to my family, friends and especially my colleagues from the seminar of the Section of Optimization and System Theory, headed by Professor Phan Quoc Khanh, especially Dr Lam Quoc 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Functional Anal Appl 5, 206-208 (1971) Chapter 1 A generalized distance and enhanced Ekeland’s variational principle for vector functions 1 Nonlinear Analysis 73 (2010) 2245–2259 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A generalized distance and enhanced Ekeland’s variational principle for vector functions Phan Quoc Khanh a,∗ , Dinh Ngoc Quy