Ekelands variational principle and some related issues

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 thelast half a century with ap

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

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

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

Chapter 1 A generalized distance and enhanced Ekelands variational

Chapter 2 On generalized Ekelands variational principle and

References 32

Chapter 3 On Ekelands variational principle for Pareto minima of

Chapter 5 More kinds of semicontinuity of set-valued maps and

3 Upper semicontinuity properties of solution maps 95

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 thelast half a century with applications in many fields of analysis, optimization and

number of equivalent formulations, all of which are well known with significantapplications and many of which were discovered independently, namely the Caristi-Kirk fixed-point theorem (Caristi 1976), the drop theorem of Daneˇs (Daneˇs 1972),the Takahashi theorem about the existence of minima (Takahashi 1991), the petaltheorem of Penot (Penot 1986), the Krasnoselski-Zabrejko theorem on solvability ofoperator 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 forequivalent 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 slightlyperturbed 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., tothe 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 vectorspace Extensions of X to the case of topological vector spaces, uniform spaces orL-spaces are investigated, e.g., in Khanh (1989), Hamel (2003, 2005), Hamel and

proposed and a minimal point with respect to (wrt) this order is proved to exist,

v

leading to a type of the EVP, see G¨opfer et al (2000) Smooth variants of the EVPare 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

small, has been attracted also much attention, see, e.g., Ha (2003, 2005, 2006), Baoand Mordukhovich (2007) Here various kinds of generalized derivatives are takeninto account: the Fre’chet, Clarke and Mordukhovich coderivatives; the Fre’chet,Clarke and Mordukhovich subdifferentials Fre’chet Hessians are also used to estab-lish 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 optimalsolutions in problems with noncompact feasible sets are dealt with in Ha (2003,2006), Bao and Mordukhovich (2007), El Amrouss (2006), using generalizations ofcoercivity assumptions, the Palais-Smale condition or the Cerami condition

One of the recent research interests is to consider the case where X is a metricspace but equipped with an additional generalized distance, based on which thesemicontinuity 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 InSuzuki (2001, 2005), τ -distance, which is more general than both afore-mentioneddistances, was introduced to improve the EVP τ -function was proposed and em-ployed in Lin and Du (2006)

The purpose of this thesis is to investigate the topic in both theory and tion aspects The thesis consists of five chapters

applica-The EVP says that, for a bounded from below and lower semicontinuous scalarfunction f on a complete metric space X, a slightly perturbed function has a strictminimum Examining the assumptions, we see that the completeness of X is crucialsince the principle relies on a convergence of a sequence to a desired point We canreplace 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 instancefunctions as regular as continuous linear ones are far from having a point as desired,

since they are unbounded from below However, in this thesis we also make efforts touse weaker, even to small extent, conditions like quasiboundedness or boundednessfrom below through some kinds of scalarization Such modifications may be morerelaxed than usual assumptions for vector functions, but they collapse to usualboundedness if the function in question is scalar Lower semicontinuity seems to bethe only assumption which may be relaxed somehow substantially We observe thatmany authors have recently used generalized distances together with the originalmetric of the space in order to weaken the lower semicontinuity condition imposed

by Ekeland This idea motivates our main commitment in Chapter 1, which isusing generalized distances to weaken this assumption In this thesis, we proposethe notion of weak τ -function, which is more general than all the aforementionedgeneralized distances Let (X, d) be a metric space

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) = 0 and limn→∞sup{p(xn, xm) : m >n} = 0, then limn→∞d(xn, yn) = 0;

(τ 4) p(x, y) = 0 and p(x, z) = 0 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→¯ yp(x, y) ≥ p(x, ¯y);

a weak τ -function becomes a τ -function introduced in Lin and Du (2006) Observefurther that the w-distance, Tataru’s distance, τ -distance and τ -function are allparticular 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 thementioned 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

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 mulated as a theorem on minimal elements of a transitive relation by defining such asuitable relation So, theorems on minimal elements may be general formalizations

refor-of the EVP In this thesis we propose the notion refor-of lower closedness refor-of a generaltransitive relation on a metric space endowed with a weak τ -function p A transi-tive 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 oflower closed transitive relations in a metric space and establish a sufficient conditionfor the existence of minimal elements of such relations Note that we avoid here theusage of the Zorn lemma

Theorem 1.3.2 (Minimal Elements for Lower Closed Relations) Let < be a lower

X assume that the <-sector of x0, i.e S<(x0) = {x ∈ X : x<x0}, is nonempty and

by p (i.e limn→∞p(xn, xn+1) = 0) Then, there exists x ∈ S<(x0) such that

S<(x) = ∅ or S<(x) = {x}

Moreover, if < is reflexive, then S<(x) = {x}

Then, we focus on our second contribution, which is using theorem of minimalelements 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 theliterature As an example, we will now derive the main result, Theorem 3.8, of a

iff it satisfies the following conditions, for all x, y, z ∈ X,

(i) D(x, x) = {0} and D(x, y) 6= ∅, 0 6∈ D(x, y), ∀x 6= y;

We derive the following result which generalizes and improves the main Theorem

sv-K-metric Assume that

(A02) 0 6∈ clw∪p(x,y)≥δD(x, y), ∀δ > 0;

(A03) ≤D is lower closed;

Then, there exists v ∈ X such that

(a) [f (v) − f (x0) + αD(x0, v)] ∩ (−K) 6= ∅;

(b) [f (x) − f (v) + αD(v, x)] ∩ (−K) = ∅, ∀x ∈ X \ {v}

In Chapter 1, we consider two particular transitive relations related to functionsperturbed by a weak τ -function p The first one is defined via a vector function

Ψ : X × X → Y ∪ {+∞} satisfying the condition

(H) if Ψ(x, y) ∈ −K and Ψ(y, z) ∈ −K, then

We define a relation ≤k0 on X by setting

The following EVP is stronger than the classical principle even in the simple case

of f : R → R ∪ {+∞}

lower closed and that, for x0 ∈ X, Sk0(x0) is nonempty and ≤k0-complete, and either

of the following lower boundedness conditions holds

(a) Ψ(x0, Sk 0(x0)) is quasibounded from below;

(b) K is closed, proper and ϕ(Ψ(x0, Sk 0(x0))) is lower bounded

Then, there is v ∈ Sk0(x0) such that, for each x 6= v,

The following form of the EVP is more traditional

Theorem 1.4.6 Impose the assumptions of Theorem 1.4.1, except those on edness, and either of the following conditions, for some ε > 0 and all x ∈ X,

bound-(a) Ψ(x0, x) + εk0 6∈ −K

Then, for each λ > 0, there is v ∈ Sk0(x0) such that, for all x 6= v,

(ii) If K is closed, p satisfies (τ 2) and Ψ(x, ) is (k0, K)-lsc, then Sk0(x) is closed.Hence, if this is satisfied for all x ∈ X then ≤k0 is lower closed.

(iii) If K is closed, p satisfies (τ 2) and Ψ(x, ) is (k0, K)-lsca for all x ∈ X, then

≤k0 is lower closed

To the best of our knowledge, all the papers in the literature about the EVP for

a two-variable function Ψ : X × X → Y impose the condition

(H0) Ψ(x, z) ≤K Ψ(x, y) + Ψ(y, z) for all x, y, z ∈ X,

which is stronger than (H) They often conclude that the results include the sponding ones for one-variable function f : X → Y , since to prove the latter resultsone simply sets Ψ(x, y) = f (y) − f (x) and applies the results for Ψ In fact, the twocases are equivalent under condition (H’), if we impose even quasiboundedness andK-lower semicontinuity (or K-lower semicontinuity from above) assumptions (whichare more relaxed than the corresponding usual assumptions) Indeed, to derive a

quasi-boundedness and K-lower semicontinuity (or K-lower semicontinuity from above) of

for f a corresponding one for Ψ (see also Khanh and Quy (2011), Bao and Khanh(2003)) However, if we impose a still more general and relaxed assumption on the

EVP stated under kinds of (relaxed) semicontinuity conditions for functions of twovariables is equivalent to that stated in terms of functions of one variable (we observe

in the literature that several authors argue that the former may be stronger thanthe latter), but under lower closedness of suitable transitive relations, the former is

in fact stronger

We derive general equivalent forms of our enhanced EVP as well It is knownthat the EVP has various equivalent formulations, many of which are stated in quitedifferent terms and/or were introduced independently In this section we derive onlysome equivalent formulations of Theorem 1.4.1, which are in terms of fixed points

or of more-general-than-fixed points Let the assumptions of Theorem 1.4.1 hold.Then, we have the following equivalent assertions, for any index set I

(i) (The EVP) There exists v ∈ Sk0(x0) such that, for x 6= v,

(ii) (Common fixed-point theorem for a family of multivalued maps) Let Ti: Sk0(x0) →

2X, i ∈ I, be such that Ti(x) 6= ∅, ∀x ∈ X, and for each x ∈ Sk0(x0) \ Ti(x) onehas y ∈ Sk0(x) \ {x} Then, there exists a common fixed point for Ti, i ∈ I, in

Sk0(x0)

(iii) (Caristi’s common fixed-point theorem for a family of multivalued maps) Let

x ∈ Sk0(x0) one has y ∈ Sk0(x) Then, the family {Ti}i∈I has a common fixedpoint in Sk0(x0)

(iv) (Caristi’s common fixed-point theorem for a family of single-valued maps) Let

Ti: Sk0(x0) → X, i ∈ I, be such that Ti(x) ∈ Sk0(x) for all x ∈ Sk0(x0) Then,{Ti}i∈I has a common fixed point in Sk0(x0)

{Ti}i∈I has a common invariant point in Sk0(x0)

(vi) (Maximal element theorem) Let Ti: Sk0(x0) → 2X, i ∈ I, be such that for each

x ∈ Sk0(x0) with Ti(x) 6= ∅, there exists y ∈ Sk0(x) \ {x} Then, there exists

x ∈ Sk0(x0) such that Ti(x) = ∅ for each i ∈ I

We prove the existence of p-strict minimizers of p-perturbations of the vectorfunction f in question, i.e., perturbations in terms of weak τ -functions instead ofthe metric We investigate also p-sharp minimizers of f and strong minimizers of

vector function f : X → Y if there is γ > 0 such that, ∀x 6= ¯x,

f (x) 6∈ f (¯x) + γp(¯x, x)BY − K,

the case p = d this is a well-recognized definition However, in the literature thesharp minimizer is also called strict, firm or isolated minimizer (see Khanh andTuan (2007), Khanh and Tuan (2008)) we also investigated firm minimizers of order1; but here we prefer to take the terminology in Bednarcruk and Przybyla (2007)among other references) The following corollary contains properly Theorem 4.1 andCorollary 4.6 of Bednarcruk and Przybyla (2007)

satisfy (τ 2), Y be a normed space partially ordered by a pointed closed convex cone

 > 0 If f (x) − f (x0) + k0 6∈ −K \ {0}, for all x ∈ X, then for each λ > 0, there

is v ∈ X such that, ∀x 6= v,

(i) p(x0, v) ≤ λ;

(ii) f (v) + ε

λp(x0, v)k0 ≤K f (x0);

(iii) v is a strict minimizer of f(v,k)(.) := f (.) + λp(v, )k for all k ∈ k0 − K

(iv) v is a p-sharp minimizer of order 1 of f ;

(v) v is a p-strong minimizer of f(v,k)

Chapter 1 considers also a particular case, where Y is a Banach space and the

for an arbitrary y∗ ∈ Y∗, ||y∗|| = 1, the Bishop-Phelps cone is

exists v ∈ X such that the following conditions hold

(i) p(x0, v) ≤ 1/g(k0);

(ii) f (v) + εp(x0, v)k ≤Kα f (x0);

(iii) v is a strict minimizer of f(v,k)(.) := f (.) + εd(v, )k for all k0 ≤Kα k, i.e

f(v,k)(x) 6∈ f(v,k)(v) − Kα, ∀x 6= v;

At the end of Chapter 1, we restrict ourselves to the case of single-valued pings We show that our results improve - or include as special cases - those in Park(2000), Lin and Du (2006, 2007), Ansari (2007), Bianchi et al (2005, 2007), Gopfer

map-et al (2000)

Multivalued mappings (or set-valued mappings, multifunctions or multimaps)arise naturally in many situations Some frequently encountered examples are: thelevel sets and sublevel sets of a function, various subdifferentials of nonsmooth func-tions, the solution sets of an optimization problem depending on some parametersand the vector field of a control system Generalized EVP for multivalued map-pings is the aim of next three chapters In this thesis, we consider both Pareto andKuroiwa’s minima of for multivalued mappings Let X be a set, Y be a vector space,

x ∈ X is called Kuroiwa’s minimizer of F iff F (x) ⊆ F (x) + K, for some x ∈ X, plies F (x) ⊆ F (x) + K x is a Pareto minimizer of F iff there is y ∈ F (x) such that

im-F (X) ∩ (y − K) ⊆ y + K ∩ −K Hence, if im-F is single-valued, x is a Pareto minimizer

of F if and only if F (x) ∈ F (x) + K, for some x ∈ X, implies F (x) ∈ F (x) + K.Therefore, Kuroiwa’s minimizer may be roughly understood as a Pareto minimizer

of F considered as a single-valued mapping from X into the space of all subsets of

Y , i.e., each F (x) is now dealt with rougher as a point in this space However, inthe general set-valuedness case, the relation of the two above-mentioned minimizers

is more interesting and incomparable

In Chapter 2, using weak τ -functions, we investigate the EVP and equivalentformulations for Kuroiwa’s minimizer of a multivalued mapping F from a metricspace X into a Hausdorff topological vector space Y ordered by a convex cone K.Let (X, d) be a complete metric space, p be a weak τ -function on X, Y be a

semicontinuous (K-lsc) if the set {x ∈ X : F (x) ∩ (e − K) 6= ∅} is closed for all

e ∈ Y From the equality

it follows that F is K-lsc on X if and only if the set {x ∈ X : A ⊆ F (x) + K}

is closed for all A ⊆ Y In Chapter 2, we also propose a new definition F iscalled weak K-lower semicontinuous from above (w.K-lsca) at x ∈ X if for each

extended to a set, F is called w.K-lsca on A ⊆ X if F is K-lsca at all x ∈ A If

A = domF := {x ∈ X : F (x) 6= ∅} we omit ”on A” in the statement Of course if

x2 ≤k0 x1 ⇔ F (x1) ⊆ F (x2) + p(x1, x2)k0+ K

Then, ≤k0 is transitive If 0 ∈ K and p(x, x) = 0 for all x ∈ X, then ≤k0 is reflexive.For x ∈ X let S(x) = {x0 ∈ X : x0 ≤k0 x}

The first result for multivalued mappings below is a generalization of the EVP

x 6= v,

i.e v is a strict minimizer of F (.) + p(v, )k0 if p(x, x) = 0 for all x ∈ X

are collected in the following

(τ 2), then ≤k0 is lower closed

(iii) If K is closed, F : X → 2Y is K-lsc and K-closed valued and p satisfies (τ 2),

Since Kuroiwa’s minimizers and Pareto minimizers are incomparable, we see nodirect comparison between ours results in Chapter 2 with the results for Paretominimizers We observe only Ha (2005) which deals with Kuroiwa’s minimizers.For the special case where p = d, ours results strictly contain Theorem 3.1, themain result of Ha (2005), since F is assumed to be K-lsc instead of our assumptionabout lower closedness of ≤k0

To have a generalization similar to the extended real line R ∪ {+∞}, we extend

Y by an additional element, denoted also by +∞, with the usual rules for additionand multiplication with reals We avoid indeterminate expressions like 0.(+∞) and

and denote domf := {x ∈ X : f (x) 6= +∞} We say that f is proper if domf 6= ∅

semicontinuous (K-lsc) at ¯x (or K-level closed at ¯x) iff, for any e ∈ Y and xn → ¯x,

e-level set We say that F is K-lsc or K-level closed iff F is K-lsc at all points in

domF := {x ∈ X : F (x) 6= ∅}, or what is the same iff, for each e ∈ Y , the e-levelset is closed If F is K-epiclosed (i.e., epiF := {(x, y) ∈ X × Y : y ∈ F (x) + K}

is closed in X × Y ), then F is clearly K-level closed, but not vice versa Observethat if K is a closed convex cone and F is uKc, then F is K-lsc A more relaxednotion of K-lower semicontinuity from above at a point x ∈ X was proposed inChen et al (2002) for single-valued scalar functions In Chapter 2, we extend

it to set-valued mappings, suitably for considering a Kuroiwa’s minimum which

set, F is called K-lsca on A ⊆ X iff F is K-lsca at all x ∈ A If A = domF , weomit ”on A” in the statement In Chapter 3, we modify the mentioned definition ofK-lower semicontinuity from above in a natural way, replacing the inclusion relation

of subsets by another transitive relation (suitable for investigating Pareto minima)

for each n ∈ N Note that being K-lsca is more relaxed than satisfying the limitingmonotonicity condition defined in Bao and Mordukhovich (2010) It is easy to seethat F is K-lsca iff grF := {(x, y) ∈ X × Y : y ∈ F (x)} satisfies condition (H2) inGopfert et al (2000) Next we propose two relaxed lower semicontinuity definitions

(x2, y2) ≤k0 (x1, y1) iff y2+ p(x1, x2)k0 ≤K y1

(x, y) ∈ grF , each sequence {(xn, yn)} in grF with xn → x and (xn, yn) ≤k0 (x, y),there exists y ∈ F (x) such that (x, y) ≤k0 (x, y) F is said to be (k0, K)-lsca at x iff

such that (x, y) ≤k0 (xn, yn), for each n ∈ N Note that F is (k0, K)-lsc iff the set{x0 ∈ X : ∃y0 ∈ F (x0), (x0, y0) ≤k0 (x, y)} is closed, for each (x, y) ∈ grF For the

case where p = d, F is (k0, K)-lsca at x iff grF satisfies condition (H1) stated inGopfert et al (2000) Relations of all kinds of semicontinuity are collected in thefollowing propositions.

K-lsca if at least one of the following conditions holds

(i) F is K-lsc and sequentially-compact-valued;

compact for every x ∈ domF ;

se-quentially compact for every x ∈ domF ;

(iv) F is uKc and sequentially-compact-valued

sequen-tially compact for every x ∈ domF ;

sequentially compact for every x ∈ domF

(i) if F is K-lsc at ¯x, then F is (k0, K)-lsc at ¯x;

(ii) if F is K-lsca at ¯x, then F is (k0, K)-lsca at ¯x

In this Chapter 3, using the weak τ -function, we first go into details to have aninsight of relaxed lower semicontinuity properties of a set-valued mapping Then,imposing these semicontinuities we establish sufficient conditions for the existence

of minimal elements and strict minimal elements of a set From these underlyingprinciples, we get enhanced versions of the EVP for Pareto minimizers of a set-valued mapping.

(a) F is str K-qbd fb (i.e., F (X) is str K-qbd fb);

(b) K is proper, closed and F is K-bd fb (i.e., F (X) is K-bd fb)

Assume further that (x0, y0) ∈ grF and SgrF(x0, y0) be nonempty and ≤k0-complete

in X Then, for any ε > 0 and λ > 0, there exists (¯x, ¯y) ∈ grF with

(i) ¯y + ε

λp(x0, ¯x)k0 ≤K y0;

(ii) y + ε

λp(¯x, x)k0 6≤K y for all (x, y) ∈ grF with x 6= ¯¯ x

If (x0, y0) is an εk0-minimizer of F (i.e., y + εk0 6≤K y0, ∀y ∈ ImF := {y ∈ Y : y ∈

F (x), for some x ∈ X}), then ¯x can be chosen to satisfy p(x0, ¯x) ≤ λ

Theorem 3.4.5 (Ekeland’s variational principle) Impose, additionally to the sumptions of Theorem 3.4.4, that F (x) have the strict domination property, for every

as-x ∈ domF Then, there eas-xists (¯x, ¯y) ∈ grF satisfying

(i’) ¯y + ε

λp(x0, ¯x)k0 ≤K y0, with ¯y ∈ SMinKF (¯x);

λp(¯x, x)k0 6≤K y for all (x, y) ∈ grF , with (x, y) 6= (¯¯ x, ¯y)

If (x0, y0) is an εk0-minimizer of F , then ¯x can be chosen such that p(x0, ¯x) ≤ λ.Note that (ii’) of Theorem 3.4.5 is strictly stronger than (ii) of Theorem 3.4.4.Our results improve or recover recent ones in the literature when applied to theirparticular cases

For a brief of Chapter 4, first note that, dealing with the EVP for a vectorfunction f : X → Y , there have been three approaches The first one is applying

scalarization techniques to convert the vector problem to a scalar case The secondmethod is introducing a vector metric d : X × X → Y to write versions of vectorEVP similar to scalar ones The third way, which has recently been most applied,

partial-ordering cone of Y , and discussing the EVP in this direction Very recently,Bednarczuk and Zagrodny (2009) used a convex set D ⊆ K and set perturbations

clearly more general, including directional perturbations as the special case with

(2009, 2010) Notice that there have been also studies of EVP for bifunctions f (., ).But, in fact, using such functions brings no new information, see e.g., discussions

in Bao and Khanh (2003) In Chapter 4, we develop the idea of Bednarczuk andZagrodny (2009) to establish versions of EVP for set-valued maps First we dealwith a kind of Kuroiwa’s minimizers

F (x0) + ε

λd(x, x

(a) F (x0) ⊆ F (¯x) + ε

λd(x0, ¯x)D + K;

(b) F (¯x) 6⊆ F (x) + ε

λd(¯x, x)D + K for all x ∈ X \ {¯x}.

all x ∈ domF ), then ¯x can be chosen to satisfy d(x0, ¯x) ≤ λ

hold

We explain relations between the conclusion (b) of Corollary 4.3.4 using setperturbations and the conclusion of the EVP using directional perturbations:

(¯b) F (¯x) 6⊆ F (x) + d(¯x, x)k0+ K, for all ¯x ∈ X \ {¯x}.

First, since the right side of (b) is larger than that of (¯b) (if k0 ∈ D), (b) is strongerthan (¯b) To have a ”quantitative” relation, let UD (Uk0) stand for the set of all ¯xsatisfying (b) ((¯b), respectively) Then it is not hard to see that

= 1, a ≥ 0, b ≥ 0}, d((a1, a2), (b1, b2)) = p(b1− a1)2+ (b2− a2)2 and F be defined

Therefore, the set limε→0+Uε.k0 is not compact for any direction k0 ∈ K \ {0}, i.e.,

minima

Now we deal with Pareto minimizers We prove also minimal element theoremsunder set perturbations These types of results are closely related to EVP for Paretominimizers and can be regarded as a background for EVP

(x0, y0) ∈ grF Impose the following assumptions

λd(¯x, x)D + K for all (x, y) ∈ grF with (x, y) 6= (¯x, ¯y).

then ¯x can be chosen to satisfy d(x0, ¯x) ≤ λ

The thesis includes many examples to assist understanding the results and theirlimitations, and especially to illustrate relations to existing results in the literature,including advantages of what obtained here Hence they show also that the resultsimprove known ones to different extents In Chapter 4, we focus on using set per-turbations with the original metric of the underlying space X However, one canuse generalized distances, defined additionally on X, similarly as for Chapters 2 and

3 to get corresponding extensions

The aim of Chapter 5 is twofold First we introduce several new kinds of continuity of a set-valued map to give a better insight of its behavior around agiven point It turns out that these concepts help to understand in more detail andcompleteness the changes of a map when its variable varies Next we consider vari-ous semicontinuity properties of solution maps of a quasivariational inclusion prob-lem Alongside with the tremendously rapid development of use of set-valued maps

semi-in analysis, optimization and other areas, their semicontsemi-inuity properties proved

to be among the most important topics The basic semicontinuity concepts for a

liminfx→¯xG(x) ⊇ G(¯x), and outer continuous at ¯x if limsupx→¯xG(x) ⊆ G(¯x) Here

liminf and limsup are the Painlev´e-Kuratowski inferior and superior limits Close toouter continuity is the (Berge) upper semicontinuity: G is called upper semicontinu-ous (usc) at ¯x iff for each open set U ⊇ G(¯x), there is a neighborhood N of ¯x such that

U ⊇ G(N ) When Y is a topological vector space, G is said to be Hausdorff uppersemicontinuous (H-usc in short; Hausdorff lower semicontinuous, H-lsc, respectively)

be continuous at ¯x iff it is both lsc and usc at ¯x and to be H-continuous at ¯x iff it is

were proposed to highlight geometrically the nature of semicontinuity A new andweaker semicontinuity notion was defined by Morgan-Scalzo (2006) and applied towell-posedness considerations Recently, in Khanh-Luc (2008) other semi-limits ofset-valued maps were determined, called inferior open and superior open limits Thecorresponding kinds of semicontinuity were defined in the same way as for inner andouter continuity These new concepts help to link semicontinuities of a set-valued

the map by its graph In Chapter 5, we go further in this direction by introducingsome new kinds of semi-limits and corresponding semicontinuities to obtain a morecomplete picture of local behaviors of a set-valued map Let X and Y be topological

set-valued maps

y ∈ intG(x) for all x ∈ U }

limsupo∗x→¯xG(x) := {y ∈ Y : there is a net {xα} ⊆ X converging to ¯x such

correspond-ingly to semi-limits, we propose the following new kinds of semicontinuity.

is of, for each λ ∈ Λ,

(QVIPλ) : finding ¯x ∈ K1(¯x, λ) such that, for each y ∈ K2(¯x, λ),

0 ∈ F (¯x, y, λ)

We choose to study this model since, though simple and relatively little mentioned

in the literature, it is equivalent to other models, which are frequently discussedrecently and englobe most of optimization-related problems To motivate our choice

of this model, we state the following other two general settings, recently dealt with in

inclusion problem was investigated

(QVIP1λ) : find ¯x ∈ K1(¯x, λ) such that, for each y ∈ K2(¯x, λ),

P (¯x, y, λ) ⊆ Q(¯x, y, λ)

below), the main factor is the inclusion With other constraints, the methods of

vary while using analogous study methods

Let R(x, y, λ) be a relation linking x, y ∈ X and λ ∈ Λ Note that R can

be identified as the subset M = {(x, y, λ) ∈ X × X × Λ : R(x, y, λ) holds} of theproduct space X ×X ×Λ A model with only constraints different from the followingquasivariational relation problem was studied

(QVRPλ) : find ¯x ∈ K1(¯x, λ) such that, for each y ∈ K2(¯x, λ),

R(¯x, y, λ) holds

of optimization-related problems as special cases Now we show the equivalence of

we simply set Z = X × X × Λ and F (x, y, λ) = (x, y, λ) − M Then R(x, y, λ) holds

λ) with F (x, y, λ) ≡ Q(x, y, λ) and P (x, y, λ) ≡ {0} Finally,

simpler form

Since our quasivariational inclusion problem contains many problems as specialcases, including equilibrium problems, variational inequalities, optimization prob-lems, Ekeland’s variational principle, fixed point and coincidence point problems,complementarity problems, Nash equilibrium problems, etc, from our results we canderive consequences for such special cases In this last chapter, we consider onlyseveral quasiequilibrium problems as illustrative examples In particular, we inves-tigate a very specific scalar equilibrium problem to see that Ekeland’s variationalprinciple can be applied to get good semicontinuity results, which cannot be derived

space and f : X × X × Λ → R For λ ∈ Λ, we are concerned with the followingscalar (parametric unconstrained) equilibrium problem

f (¯x, y, λ) + d(¯x, y) ≥ 0

Assume that the solution set Σ(λ) of (EPλ) is nonempty for all λ in a hood of ¯λ.

neighbor-To develop other conditions for lower semicontinuity of Σ, which are more able in some cases, we need the following auxiliary problem called a parametricEkeland’s variational problem, for λ ∈ Λ,

(para-metric) Ekeland’s variational principle We obtain the following sufficient condition

(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

Then ˆΣ(λ) 6= ∅

Moreover, assume further that X is compact and

(a) f (x, , ) is lsc for all x ∈ ˆΣ(¯λ)

(b) Σ(¯λ) ⊆ cl ˆΣ(¯λ)

Then Σ is lsc at (¯λ)

In each chapter of this thesis, comparisons between our results and recent knownones, including even comparisons when applied to particular cases, are provided.Numerous corollaries and examples are also given to illustrate the main results

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Contents lists available at ScienceDirectNonlinear Analysis

journal homepage: www.elsevier.com/locate/na

A generalized distance and enhanced Ekeland’s variational principle for vector functions

aDepartment of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Viet Nam

bDepartment of Mathematics, Cantho University, Cantho, Viet Nam

of existing versions of Ekeland’s variational principle (EVP) 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 Our results are proved to be stronger than the classical EVP and many generalizations in the literature, even in the usual one-dimensional case, by numerous corollaries and examples We include equivalent formulations of our enhanced EVP as well.

© 2010 Elsevier Ltd All rights reserved.

1 Introduction

The celebrated Ekeland variational principle (EVP, in short) [ 1 ] has shown to be a powerful tool in nonlinear analysis and optimization In the past three decades, a great deal of efforts have been made to generalize this principle and its equivalent formulations In this paper 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 ( Theorem 3.2 ).

To see the generality of this theorem we show that it improves, as an example, the main result of a recent paper [ 2 ] Then

we focus on our second contribution, which is using Theorem 3.2 as a basic tool to prove generalized EVP in various settings.

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 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

∗Corresponding author Tel.: +84 8 5122796; fax: +84 8 7242195.

E-mail addresses:pqkhanh@hcmiu.edu.vn (P.Q Khanh), dnquy@ctu.edu.vn (D.N Quy).

0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved.

doi:10.1016/j.na.2010.06.005

are far from having a point as desired, since they are unbounded from below However, in this paper 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 This idea motivates our main commitment in this paper, which is using generalized distances to weaken this assumption w -distance was introduced in [ 3 ] and used also in [ 4 , 5 ] Another generalized distance was proposed in [ 6 τ -distance was defined in [ 7 ], which is more general than the two above-mentioned distances In [ 8 ]

τ -function, which is incomparable with τ -distance, was introduced All these distances are used to weaken the mentioned lower semicontinuity assumption In this paper we propose the notion of weak τ -function, which is more general than all the aforementioned generalized distances Examples 2.5 and 4.2 show that, even for simple one-dimensional cases, our enhanced EVP using this definition holds while the classical principle fails due to the lack of lower semicontinuity.

We weaken the lower semicontinuity assumption by using weak τ -functions and also impose some conditions only in a fixed direction under consideration (see Definition 2.8 ) Moreover, we prove also enhanced versions of the EVP by using a proposed general lower closed transitive relation ( Theorems 3.2 , 4.1 , 4.6 , 4.10and 4.60) We derive general equivalent forms ( Theorem 6.1) of our enhanced EVP as well We prove the existence of p-strict minimizers of p-perturbations of the vector function f in question, i.e perturbations in terms of weakτ-functions instead of the metric We investigate also p-sharp minimizers of f and strong minimizers of its p-perturbations too Moreover, we point out that the EVP stated under kinds

of (relaxed) semicontinuity conditions for functions of two variables is equivalent to that stated in terms of functions of one variable (we observe in the literature that several authors argue that the former may be stronger than the latter), but under lower closedness of suitable transitive relations, the former is in fact stronger (see Remark 4.7 , Theorem 4.10and

Example 4.8 ) We restrict ourselves to the case of single-valued mappings The EVP for multivalued mappings is the aim of another work [ 9 ] Note that all above-encountered generalized distances, including that we use in this paper, are scalar and single-valued Vector metrics have been also used to develop the EVP for vector functions, see e.g [ 10 ] Recently set-valued vector metric was proposed for the study of the EVP in [ 2

The organization of the paper is as follows In the remainder of this section we recall some needed definitions In Section 2

we propose definitions of weak τ -function, generalized lower semicontinuities of a vector function Section 3 defines lower closedness of a transitive relation and establishes a theorem on minimal elements of lower closed relations The generality

of this theorem is shown by a consequence which includes properly the main result of the recent paper [ 2 ] Section 4 is

devoted to versions of the EVP for functions on metric spaces with values in a Hausdorff locally convex space Y Some results for the case of Y being a normed space, which is more specific but more important in practice, are developed in

Section 5 , in particular those containing and improving results of [ 11 ] Equivalent formulations of the EVP are discussed in Section 6 In the final Section 7 we show that our results improve – or include as special cases - those in [ 12–15 , 8 , 5 , 16 , 4

Let X be a metric space, Y be a topological vector space and K⊆Y be a convex cone (containing zero) Let f :X→Y be a

vector function f is said to be K -lower semicontinuous (K -lsc) at x iff∀e∈Y and∀{x n} →x such that f(x n)+e≤K0, one has

f(x) +e≤K 0, where y≤K z means that z∈y+K f is called K -lower semicontinuous from above (K -lsca) at x (see [17 ]) iff, for each convergent sequence {x n} →x such that f(x n+1) ≤K f(x n), ∀n, one has f(x) ≤K f(x n), ∀n We always say that f has

a property on A⊆X iff f has this property at every point of A We omit ‘‘on A’’ if A=domf := {x∈X: ∃y∈Y,y=f(x)}

A subset B⊆Y is called quasibounded from below if there is a bounded subset M⊆Y such that B⊆M+K and is called

bounded from below if there is y∈Y such that B⊆y+K Note that boundedness from below implies quasiboundedness

from below but not vice versa In what follows, instead of saying that F(X) is quasibounded or bounded from below we

simply say that F is so For a transitive relation<in X , a subset A⊆X is termed<-complete iff any Cauchy sequence in A,

which is <-decreasing, converges to a point in A For a subset B⊆Y the notations int B, cl B, co B and cone B stand for the

interior, closure, convex hull and conical hull (called also cone generated by B) of B.

2 Weakτ-functions and generalized lower semicontinuity of a function

Let, throughout the paper if not otherwise specified, (X,d) be a metric space.

Definition 2.1 (See [ 8]) A function p:X×X→R+ is called a τ-function iff the following four conditions hold, for x,y,z∈X ,

( τ1) (triangle inequality) p(x,z) ≤p(x,y) +p(y,z) ;

( τ 2) (lower semicontinuity) ∀x∈X , p(x, )is R+ -lsc;

( τ3) if x n,y n∈X satisfy lim n→∞ p(x n,y n) = 0 and limn→∞sup {p(x n,x m) :m>n} = 0, then limn→∞ d(x n,y n) = 0; ( τ4) p(x,y) =0 and p(x,z) =0 imply that y=z.

We propose a weaker notion as follows.

Definition 2.2 A function p : X×X → R+ is said to be a weak τ -function iff it satisfies three ( τ 1), ( τ 3) and ( τ 4) of

Definition 2.1

Definition 2.3 (See [ 7]) A function p:X×X→R+ is called a τ-distance on X iff there is a functionη :X×R+ →R+ such

that the following conditions are satisfied, for x,y,z∈X and t∈R+ ,

( τ1) (triangle inequality) p(x,z) ≤p(x,y) +p(y,z) ;

( τ 0

2) (weak lower semicontinuity) if x n → x and lim n→∞sup { η(z n,p(z n,x m)) : m ≥ n} = 0 for some z n ∈ X , then

p(w,x) ≤ lim infn→∞ p(w,x n) for all w ∈X ;

Lemma 2.4 Anyτ-distance is a weakτ-function.

Proof If p is aτ-distance then p satisfies (τ 3) by Lemma 3 of [ 7 ] and also ( τ 4) by Lemma 2 of [ 7 

The following example shows that a weak τ -function may help to consider the EVP.

Example 2.5 Let X=R+ ,Y =R∪ {+∞} , γ >0 and p:R×R→ [ 0 , +∞) be defined by

each case of x,y,z: the three points are different, one pair coincide and the three points coincide, are easily carried out So

p is a weakτ -function Let

where a is the unique solution of e a−a− γ − 1 =0 So perturbation f(x)+d(x,y) does not have strict minimizers (recall that,

if f is lsc, the EVP says that this perturbation has a strict minimizer) But we have seen that, for a suitable weakτ -function

p, the perturbation f(x) +p(x,y) admits strict minimizers.

In what follows we need also the following facts.

Lemma 2.6 (See [ 8, Lemma 2.1]) Let p be a weakτ-function If a sequence{x n}satisfies the condition lim n→∞sup {p(x n,x m) :

m>n} =0, then{x n}is a Cauchy sequence.

Notice that in Lemma 2.1 of [ 8] it is assumed that p is aτ -function, but (τ 2 ) is not used in the proof (In fact this proof

in [ 8 ] is incomplete, since only limn→∞ d(x n,x n+1) = 0 is verified However, one can show that limn, m→∞ d(x n,x m) = 0.)

Lemma 2.7 (See [ 18, Lemma 3.4]) Let p be a weakτ-function,{x n} →x andΓ :X → 2X be a set-valued mapping such that the following conditions hold

(i) x n+1∈ Γ (x n)andΓ (x n+1) ⊆ Γ (x n), for all n∈N;

(ii) limn→∞sup {p(x n,u) :u∈ Γ (x n)} =0;

(iii) x∈ Γ (x n), for all n∈N.

ThenT

n∈N Γ (x n) = {x}.

If, in addition,

(iv) Γ (x) 6= ∅andΓ (x) ⊆ Γ (x n), for all n∈N,

then x is an invariant point ofΓ(i.e.Γ (x) = {x}) Conversely, if p(x,x) =0 for all x∈X , and x is an invariant point ofΓ, then there is a sequence{x n}, which converges to x and satisfies all conditions( i ) – ( iv ).

Now we pass to generalizing lower semicontinuity.

Definition 2.8 Let Y be a vector space, K ⊆Y be a convex cone, f :X→Y be a vector function and e∈Y

(i) f is said to be(e,K) -lower semicontinuous ( (e,K)-lsc, in short) at x iff for each r ∈ R, each sequence{x n} →x with

f(x n) +re≤K 0, one has f(x) +re≤K0.

(ii) f is called(e,K) -lower semicontinuous from above ( (e,K)-lsca, in short) at x iff for each r ∈ R, each sequence{x n}

converging to x, from f(x0) +re≤K 0 and from f(x n+1) +t n e≤K f(x n), for all n∈N and for some nonnegative sequence{t n}, it follows that f(x) +re≤K0.

Note that these generalized semicontinuities are defined for a vector space Y without any topological structure.

Definition 2.8(i) is clear and shows that if f is K -lsc at x then f is (e,K )-lsc at x for every e∈K To see clearer the relation

between K -lower semicontinuity from above and (e,K )-lower semicontinuity from above, we observe that f is K -lsca at

x if and only if for each e ∈ Y , each sequence{x n}converging to x, from f(x0) +e≤K 0 and f(x n+1) ≤K f(x n) , ∀n, one has

f(x) +e≤K 0 Indeed, for the ‘‘if’’ assume that f(x n+1) ≤K f(x n) , ∀n For each fixed n, the sequence{x n+p}p satisfies the

conditions f(x n+p+1) ≤K f(x n+p) Then by the assumption with e= −f(x n), one has f(x) −f(x n) ≤K 0, i.e f(x) ≤K f(x n) , ∀n.

The ‘‘only if’’ is obvious.

From this observation, it is evident that if f is K -lsca at x then f is (e,K )-lsca at x for every e∈K Furthermore, (e,K )-lower

semicontinuity implies (e,K )-lower semicontinuity from above.

3 Lower closed transitive relations and minimal elements

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 section we propose the notion of lower closedness of a general transitive relation on a metric space endowed with a weak τ-function p and use it to establish the existence of minimal elements of such relations.

Note that we avoid here the usage of the Zorn lemma.

Definition 3.1 A transitive relation<on a metric space X is called lower closed iff for any< -decreasing (i.e <x n

< <x2<x1) sequence converging to x one has x<x n, ∀n∈N.

Theorem 3.2 (Minimal Elements for Lower Closed Relations) Let<be a lower closed transitive relation and p be a weakτ 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{x n}in X is asymptotic by p (i.e lim n→∞ p(x n,x n+1) =0) Then,

there exists x∈S< (x0)such that

S< (x) = ∅ or S< (x) = {x}

Moreover, if <is reflexive, then S< (x) = {x}.

Proof Starting by x0we construct a sequence {x n}in S< (x0)as follows: having x n∈S< (x n−1), we choose x n+1∈S< (x n) by the following rule

(a) if p n:= sup {p(x n,x) :x∈S< (x n)} < +∞, x n+1is taken so that

p(x n,x n+1) ≥ 1

(b) if p n= +∞, x n+1is taken so that p(x n,x n+1) ≥ 1.

There is then n0 ∈ N such that,∀n ≥ n0, p n < +∞ Indeed, otherwise we would have an < -decreasing sequence

{x n}with p(x n,x n+1) 6→ 0, a contradiction Now, we check the assumptions of Lemma 2.7 for Γ =S< By the transitivity

of < , (i) holds By (1) and the asymptoticity of the sequence {x n} , (ii) is satisfied Lemma 2.6 implies that {x n} is a Cauchy

sequence and hence converges to some x Since<is lower closed, x∈S< (x n), ∀n, i.e (iii) is fulfilled.Lemma 2.7 yields that

T

n∈NS< (x n) = {x}, i.e S< (x) = ∅or S< (x) = {x}

If <is reflexive, then (iv) is also satisfied and x is an invariant point of S< 

We will apply Theorem 3.2 in the sections to come for establishing enhanced EVP in various settings This theorem is rather general and seems to contain a large part of existing versions of the EVP in the literature Here we have no attempt to prove this, but as an example, we will now derive the main result, Theorem 3.8, of a very recent paper [ 2 ] For this we recall

some preliminaries From now on, unless otherwise specified, let Y be a Hausdorff locally convex space, K ⊆Y be a convex

cone containing zero, k0∈K\ −clK , Y∗be the topological dual of Y , K+be the positive polar of K , i.e.

K+:= {y∗∈Y∗: hy∗,ki ≥ 0 , ∀k∈K} ,

and z∗ ∈ K+such that z∗ (k0) = 1 (the existence of z∗is guaranteed by the separation theorem) We extend Y similarly

as for the one-dimensional case by an additional element, denoted by +∞ , with the usual rules for addition of elements

and multiplication with reals We adopt that y≤K+∞ , ∀y ∈ Y , and avoid indeterminate expressions like 0.(+∞) K is

called based (well based) iff there exists a (bounded) convex set B⊆K , called a base of K , such that 06∈clB and R+B=K

K is normal iff for all nets{x i} and {y i}in Y such that 0≤K x i≤K y i , for all i, and{y i} → 0, then one has {x i} → 0 K is

called w -normal iff the above convergence is in the weak topology (and so → is changed by the weak convergence → w).

It is known [ 15] that K isw-normal if and only if Y∗=K+−K+ A subset M ⊆Y is said to be K -bounded from below by

We denote the lower sector of x with respect to the relation≤D, i.e {x0∈X:x0≤D x}, by S D(x) Let C D:= coneco ∪ {D(x,y) :

∀x,y∈X}and K D:= (C D\ { 0 } +K) ∪ { 0 } Then, K Dis a convex cone.

Lemma 3.4 (See [ 2, Lemma 3.7(b)]) Let x0 ∈ X and the set (f(X) −f(x0)) ∩ (−K D)be K -bounded by scalarizations Then

y∗(f(.))is bounded from below on S D(x0)for all y∗∈K+.

Some sufficient conditions for the relation ≤Dto be lower closed are given in the following lemma.

Lemma 3.5 Let X,Y,K, α,D,f and≤D be as above Then the transitive relation≤D is lower closed if either of the following conditions holds

(i) S D(x)is closed for each x∈X ;

(ii) for z∈X if {x n} ⊆S D(z),{x n} → ¯x and f(x n) ≤K f(x m), ∀n>m, then x¯ ∈S D(z).

Proof (i) It is obvious, as≤Dis a transitive relation.

(ii) Assume that {x n} → ¯x and x n+1≤D x n, ∀n ∈ N Fix n and apply assumption (ii) to z = x n We have x n+p≤D x n+q,

∀p,q∈N:p>q By the definition of≤D , there is d pq∈D(x n+p,x n+q) such that

f(x n+p) −f(x n+q) + αd pq≤K0

Hence, f(x n+p) ≤K f(x n+q) As {x n+p} → ¯x, when p→ +∞ , by (ii), we have ¯x∈S D(x n) , i.e ≤Dis lower closed 

Corollary 3.6 Let X be a complete metric space, x0∈X ,α >0 and D be an sv-K -metric Assume that

( A 1) K isw-normal and K D is based;

( A 0 2) 0 6∈clw∪p(x,y)≥δ D(x,y), ∀δ >0;

( A03) ≤D is lower closed;

( A 4) the set(f(X) −f(x0)) ∩ (−K D)is K -bounded by scalarizations.

Then there existsv ∈X such that

(a) [f(v) −f(x0) + αD(x0, v)] ∩ (−K) 6= ∅;

(b) [f(x) −f(v) + αD(v,x)] ∩ (−K) = ∅,∀x∈X\ { v}.

Proof For all x∈ X , as x ∈ S D(x), one has S D(x) 6= ∅ To apply Theorem 3.2 it suffices to check that any ≤D-decreasing sequence {x n}in X is asymptotic by p Suppose to the contrary the existence ofδ > 0 such that, ∀n∈N, p(x n,x n+1) ≥ δ By the definition of ≤D -decreasing, there exists d n∈D(x n,x n+1) such that

f(x n+1) ≤K f(x n) − αd n.

Then, f(x n+1) ∈f(x n) −K D for all n ByLemma 3.4 and (A4) we deduce that the real-valued sequence {y∗(f(x n))} is bounded

from below for all y∗ ∈ K+ As for all y∗ ∈ K+, y∗(.)is nonnegative on C D and hence on K D, {y∗(f(x n))} is nonincreasing Thus {y∗ (f(x n))} has a limit It follows that

Note that assumption (A02) collapses to (A2) in [ 2] if p=d and, byLemma 3.5 , (A03) is weaker than the following assumption

in [ 2

(A3) For z∈X if{x n} ⊆S D(z) , {x n} → ¯x and f(x n) ≤K f(x m), ∀n>m, then x¯ ∈S D(z)

Therefore, Theorem 3.2 generalizes and improves the main Theorem 3.8 in [ 2 ], where ( A 1) – ( A 4) are imposed The next example supplies a case where Theorem 3.8 of [ 2 ] fails to be applied but Corollary 3.6 can be.

Then K D = C D = R+and S D(x) = {y ∈X :f(y) −f(x) + |y−x| ≤ 0 } It is easily seen that S D(x) is not closed for each

x∈ [− 2 , 2 ] \ { 0 } Assumption (A3) is not satisfied Indeed, let x n= 1

0 is lower closed because the only ≤∗k

0 -decreasing sequence {x n} → 0 is {0}) Thus, Corollary 3.6 yields a point

v ∈X satisfying (a) and (b) By direct computations we see that every pointv ∈ [− 1

2 , 1

2 ] \ { 0 } satisfies (a) and (b).

We consider now two particular transitive relations related to functions perturbed by a weak τ-function p The first one

is defined via a vector function Ψ :X×X→Y∪ {+∞} satisfying the condition

Note that the relation ≤k0is transitive Indeed, if z≤k0y and y≤k0x, then(3) implies that Ψ (x,y) ∈ −K andΨ (y,z) ∈

−K By condition (H) and (τ 1) one obtains the transitivity, since

Ψ (x,z) +p(x,z)k0 ∈ ( Ψ (x,y) + Ψ (y,z)) + (p(x,y) +p(y,z))k0−K

∈ −K.

We denote the lower sector of x with respect to the relation≤k

0by S k0(x) Some sufficient conditions for the relation ≤k

0 to

be lower closed are given in the following lemma.

Lemma 3.8 Let X,Y,K,p,k0andΨ be as above.

(i) If S k0(x)is closed for each x∈X , then≤k

0is lower closed.

(ii) If K is closed, p satisfies (τ2) andΨ (x, )is (k0,K )-lsc, then S k0(x)is closed Hence, if this is satisfied for all x∈X then≤k0

is lower closed.

(iii) If K is closed, p satisfies (τ2) andΨ (x, )is (k0,K )-lsca for all x∈X , then≤k0is lower closed.

Proof (i) It is clear that each transitive relation< has this property.

(ii) For a fixed x∈X , assume that{x n}is in S k0(x) and {x n} →x We show that x∈S k0(x) By the lower semicontinuity of

By the closedness of K , passing i→ ∞one sees that x≤k0x, i.e x∈S k0(x)

(iii) Assume that x n+1≤k

0x n, ∀n∈N, and{x n} →x Fix n For each sufficiently large i∈N, by the lower semicontinuity of

p(x n, )one has Q(i) ∈ N such that,∀q>Q(i) ,

Ψ (x n,x n+q) + (p(x n,x) − 1 /i)k0∈ Ψ (x n,x n+q) +p(x n,x n+q)k0−K.