on $$\gamma $$ Γ -convergence of vector-valued mappings

$: On $$\Gamma $$ Γ -convergence of vector-valued mappings$
PositivityDOI 10.1007/s11117-013-0272-2 Positivity

On �-convergence of vector-valued mappings

Rosanna Manzo

Received: 31 July 2013 / Accepted: 30 December 2013© Springer Basel 2014

Abstract This paper deals with a new concept of limit for sequences of vector-valuedmappings in normed spaces. We generalize the well-known concept of �-convergenceto the case of vector-valued mappings and specify notion of��,μ-convergence similarto the one previously introduced in Dovzhenko et al. (Far East J Appl Math 60:1–39,2011). In particular, we show that ��,μ-convergence concept introduced in this paperpossesses a compactness property whereas this property was failed in Dovzhenko etal. (Far East J Appl Math 60:1–39, 2011). In spite of the fact this paper containsanother definition of ��,μ-limits for vector-valued mapping we prove that the ��,μ-lower limit in the new version coincides with the previous one, whereas the ��,μ-upper limit leads to a different mapping in general. Using the link between the lowersemicontinuity property of vector-valued mappings and the topological propertiesof their coepigraphs, we establish the connection between ��,μ-convergence of thesequences of mappings and K -convergence of their epigraphs and coepigraphs in thesense of Kuratowski and study the main topological properties of ��,μ-limits. Themain results are illustrated by numerous examples.

Keywords Vector-valued mapping · Partial ordered spaces · �-convergence ·Compactness result

Mathematics Subject Classification (2010) 46B40 · 49J45 · 90C29 · 49N90

R. Manzo (B)Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata,Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, SA, Italye-mail: [email protected]

R. Manzo

1 Introduction

The notion of �-convergence, that was originally introduced in the pioneering works[7,8], is an important characteristic for dealing with sequence of functionals and forunderstanding their limiting properties. See [1,3,6] for a formal and more complexanalysis. This concept plays a central role in different fields, like homogenization the-ory, phase transitions, singular perturbations, boundary value problems in perforateddomains, approximation of variational problems and many others. Many examplesand applications are scattered throughout the literature. In particular, one of the maintopics in the theory of �-convergence of functionals is concerned with the character-ization (by such convergence) of the Painlevé–Kuratowski convergence of the graphsof convex or non-convex subdifferentials. Such a characterization is recognized to becrucial for a good behaviour of a class of functions satisfying some stability properties(see, for instance, [1]).

In this paper, we present a new concept of limit for sequences of vector-valuedmappings in normed spaces. We deal with the case when the mappings take valuesin a real Banach space Y endowed with some topology τ and Y is partially orderedby a pointed convex closed cone �. Possible applications of this convergence are theset-valued analysis and variational analysis of vector optimization problems.

Note that the theory of vector optimization [13,18] has many differences withrespect to the scalar optimization. One of them is the fact that in the vectorial casethey are usually interested in the set of all solutions (of a certain type such as efficientsolutions, weakly efficient solutions and properly efficient solutions), and not only ina single one. The second difference, which motivated our efforts in this field, concernsthe following observation: if the scalar problem inf {I (x) : x ∈ X∂} has a non-emptyset of solutions, then

inf {I (x) : x ∈ X∂ } = min {I (x) : x ∈ X∂} = min[closure {I (x) : x ∈ X∂}].

However, in the case of vector optimization, the typical situation is (see [10]):

Min (S) �= ∅, Min [closure (S)] �= ∅, and Min (S) ∩ Min [closure (S)] = ∅.

Therefore, the formulations of the main results in vector optimization, such as existenceresults, approximate solutions, optimality conditions and many others differ from theirscalar counterparts. In particular, it is an urgent question of modern vector optimizationtheory to develop a vector variational principle which would inherit the fundamentalidea of the Ekland’s variational principle. As a result, we could assign to a vectoroptimization problem a “slightly perturbed” one having a nonempty set of efficientsolutions which are at the same time approximate solutions (in some sense) to theoriginal problem. Hence, the “right” interpretation of a “slightly perturbed” objectivemapping, which would be close in some sense to the original one, is a crucial pointin such analysis. So, this motivate us to study the limiting properties of vector-valuedmappings.


It is a well-known result of scalar-valued analysis that every sequence of functionals{Fk : X → R

}k∈N

can be associated with two limiting functionals—the �-lower and

�-upper limits of this sequence as k → ∞. When these functionals coincide, they saythat the corresponding sequence is �-convergent. �-convergence is closely related toG- and H -convergence (see [4,9,20,21]). One of the principle result of the theoryof �-convergence is the fact that in the scalar case there exists a close connectionbetween �-convergence of functionals and set convergence of their epigraphs in thesense of Kuratowski. However, the problem is that in the vectorial case, the closure ofepigraphs for vector-valued mappings may lead to sets which have not the structure tobe an epigraph for any mapping. So, the direct transfer of the �-convergence conceptto the vector-values case is a non-trivial matter, in general.

Thus, our prime interest in this article is to introduce some appropriate definition of�-limits for sequences of mapping from X into Y and to study their main properties. Inparticular, we specify the notion of ��,μ-convergence that was previously introducedin [12] for the sequences of locally compact mappings, and derive the conditions whichguarantee the compactness property of such convergence. It is worth to note that thispaper contains another definition of�-limits for vector-valued mapping different fromthe one proposed in [12]. Nevertheless, we show that the ��,μ-lower limit in the newversion coincides with the previous one, whereas the ��,μ-upper limit can lead toa different mapping in general. We also study the connection between the ��,μ-convergence of the sequences of mappings and K -convergence of their epigraphs andcoepigraphs in the sense of Kuratowski and give the main topological properties of��,μ-limits. In particular, we show that the hypothesis about geometrical interpretationof the ��,μ-upper limit, that was put forwarded in the previous paper [12], becomestrue. All our principle results are illustrated by numerous examples.

2 Notation and preliminary results

Let X and Y be two real Banach spaces. We assume that these spaces, as topologicalspaces, are endowed with some topologies σ = σ(X) and τ = τ(Y ), respectively. Fora subset A ⊂ Y we denote by intτ A and clτ A its interior and closure with respect tothe τ -topology, respectively. We introduce the following sets

N∞ := {N ⊆ N | N \ N finite} and N �∞ := {N ⊆ N | N infinite} .

Let {yk}k∈N be a sequence in Y . We write ykτ→ y0, if y0 is the limit of the sequence

{yk}k∈N with respect to the τ -topology of Y . Moreover, we write ykτ,N−→ y0 in the case

of convergence of a subsequence designated by an index set N ∈ N �∞ or N ∈ N∞. It isclear that every subsequence of {yk}k∈N can be expressed by {yk}k∈N , where N belongsto N �∞. In the case of N ∈ N∞, {yk}k∈N denotes a subsequence of {yk}k∈N that arisesby omitting finitely many members. For instance, a subsequence of a subsequence{yk}k∈N (N ∈ N �∞) can be expressed by some N ∈ N �∞ with N ⊆ N as {yk}k∈N .

R. Manzo

Definition 2.1 We say that a sequence of pairs {(xk, yk) ∈ X × Y }k∈N μ-converges

to (x0, y0) if xkσ→ x0 and yk

τ→ y0 as k → ∞.

Let � be a τ -closed convex cone in Y which is supposed to be pointed, i.e. � ∩(−�) = ∅. We do not impose the typical assumption that the ordering cone � hasa non-empty interior. In many interesting and important cases, this property does nothold and it seems to be rather restrictive. Throughout this paper, we assume that Yis partially ordered with the ordering cone �. We denote with ≤� a partial orderingintroduced by the cone�, that is, given elements y, z ∈ Y , we write y ≤� z wheneverz ∈ y +� and y <� z for y, z ∈ Y , if z − y ∈ intτ �.

Definition 2.2 A sequence {yk}k∈N ⊂ Y is called �-monotone if either yk+1 ≤� yk ,∀ k ∈ N or yk+1 ≥� yk , ∀ k ∈ N.

Definition 2.3 We also say that a sequence {yk}k∈N ⊂ Y is�-bounded below (above)if there exists an element y∗ ∈ Y such that y∗ ≤� yk (y∗ ≥� yk), ∀ k ∈ N.

Definition 2.4 An element a ∈ Y is said to be the �-infimum of a subset A ⊂ Y (insymbols a = inf� A), if the assertion y ∈ Y and y ≤� a := inf�A is equivalent to thefollowing one: y ≤� z for every z ∈ A. Similarly, the�-supremum of a subset A ⊂ Y ,whenever it exists, sup�A, is defined as an element of Y such that sup�A ≤� y if andonly if b ≤� y for every b ∈ A.

It is easy to see that, in contrast to the scalar case, an element a = inf� A maynot belong to A, in general. Indeed, let Y = R

2 and let � = R2+ be the ordering

cone of positive elements in R2. We set A = {(0, 2), (2, 0)}. In this case, we have

inf� A = (0, 0) �∈ A. As the same time, the following monotonicity property isinherent: if ∅ �= A ⊆ B ⊂ Y •, then inf�B ≤� inf�A.

Following Krasnosel’skii [16], let us introduce two singular elements −∞� and+∞� in Y . We assume that these elements satisfy the following conditions:

1) − ∞� ≤� y ≤� +∞�, ∀y ∈ Y ; 2) + ∞� + (−∞�) = 0Y .

Then +∞� is the �-greatest element of Y , and the element −∞� is its �-smallestelement. Let Y • denote a semi-extended Banach space: Y • = Y ∪ {+∞�} assumingthat

‖ + ∞�‖Y = +∞ and y + λ(+∞�) = +∞, ∀ y ∈ Y and ∀ λ ∈ R+.

We suppose that inf� A = −∞� provided a subset A ⊂ Y is unbounded below in〈Y,�〉.Definition 2.5 We say that a sequence {yn}n∈N in Y τ -converges to the element{−∞�} if inf� {yn}n∈N = −∞�.

Let Lτ {yk} denote the set of all cluster points of {yk}k∈N with respect to the τ -topology of Y , that is, y ∈ Lτ {yk} if there is a subsequence {yi }i∈N with N ∈ N �∞such that yi

τ,N−→ y in Y as i → ∞. If this set is lower unbounded, i.e., inf� Lτ {yk} =


−∞�, we assume that {−∞�} ∈ Lτ {yk}. If sup� Lτ {yk} = +∞�, we assume that{+∞�} ∈ Lτ {yk}.

Let x0 ∈ X be a fixed element. In what follows for an arbitrary sequence of mappings{ fk : X → Y }k∈N we make use of the following set:

Lμ({ fk}, x0) :=⋃

{xk }k∈N∈Mσ (x0)

Lτ { fk(xk)}, (2.1)

where Mσ (x0) is the set of all sequences {xk}k∈N ⊂ X such that xkσ→ x0 in X .

Hereinafter we will associate any mapping f : X∂ → Y with its natural extensionf̂ : X → Y • to the entire space X , where

f̂ (x) ={

f (x), x ∈ X∂ ,+∞�, x /∈ X∂ .

(2.2)

Definition 2.6 We say that a mapping f : X → Y • is locally compact if for everyx0 ∈ dom f there exist a sequentially τ -compact set M ⊂ Y and a neighborhoodUσ (x0) of x0 in X such that f (x) ∈ M, ∀ x ∈ Uσ (x0) ∩ dom f .

Definition 2.7 We say that a mapping f : X → Y • is �-bounded below (resp.,above) at some point x0 ∈ dom f if for every sequence {xk}k∈N such that xk

σ→ x0,the corresponding sequence of values { f (xk)}k∈N ⊂ Y is �-bounded below (resp.,above) in the sense of Definition 2.3.

In conclusion of this section, we recall the definitions of lower semicontinuity andsequential lower semicontinuity of a vector-valued mapping introduced, respectively,in [19] and [5]. To this end, for every x ∈ X , let us denote the system of all openneighbourhoods of x in (X, σ ) by Nσ (x), and let Nτ (0Y ) be the system of all τ -openneighbourhoods of zero in Y .

Definition 2.8 A mapping f : X → Y • is said to be lower semicontinuous at x0 ∈ Xwith respect to the μ-topology of X × Y (μ-l.s.c.), if for any V ∈ Nτ (0Y ) and anyb ∈ Y satisfying b ≤� f (x0), there exists a neighborhood U ∈ Nσ (x0) such thatf (U) ⊂ b + V +� ∪ {+∞�}.Remark 2.9 If f (x0) ∈ Y (i.e. f (x0) ∈ dom f ), then Definition 2.8 means that forevery V ∈ Nτ (0Y ), there exists a neighborhood U ∈ Nσ (x0) such that f (U) ⊂f (x0)+ V +� ∪ {+∞�} (see [5,19]).

Definition 2.10 A mapping f : X → Y • is said to be sequentially lower semicontin-uous at x0 ∈ X with respect to the μ-topology of X × Y (sμ-l.s.c.), if for any b ∈ Ysatisfying b ≤� f (x0) and for any sequence {xk}k∈N of X which σ -converges to x0,there exists a sequence {bk}k∈N (in Y ) τ -converging to b and satisfying bk ≤� f (xk),for any k ∈ N.

Remark 2.11 If x0 ∈ dom f , then Definition 2.10 can be expressed as follows: foreach sequence {xk}k∈N σ -converging to x0, there exists a sequence {bk ∈ Y }k∈N τ -converging to f (x0) such that bk ≤� f (xk) for all k ∈ N.

R. Manzo

Remark 2.12 As immediately follows from Definition 2.8, the following facts areequivalent

(a) a mapping f : X → Y • is μ-l.s.c. at x ∈ dom f ;(b) f (x) = sup�U∈Nσ (x) inf�z∈U f (z).

Indeed, since the implication (b) ⇒ (a) is obvious, let us show that (a) implies(b). Indeed, if we admit assertion (a), then there exists a neighborhood U ∈ Nσ (x0)

such that the condition inf�z∈U f (z) ∈ inf�b∈V(

f (x) + b) +� holds true. Hence,

sup�U∈Nσ (x) inf�z∈U f (z) ∈ inf�b∈V

(f (x)+b

)+�. Since, this inclusion remains valid

for every V ∈ Nτ (0Y ), we can pass to the supremum over all V ∈ Nτ (0Y ) in the righthand side. Taking into account the fact that

sup�V∈Nτ (0Y )

inf�b∈V

(f (x)+ b

) = f (x),

we finally get sup�U∈Nσ (x) inf�z∈U f (z) ∈ f (x) + �, which is equivalent to theinequality

f (x) ≤� sup�U∈Nσ (x)

inf�z∈U

f (z).

At the same time f (x) ≥� inf�z∈U f (z) for all U ∈ Nσ (x). As a result, combiningthe last two relations, we arrive at the equality (b).

As a consequence of this observation, we give the following result.

Lemma 2.13 Let Nσ (X) be the family of all σ -open subsets of X, and let α :Nσ (X) → Y • be an arbitrary set mapping. Then the mapping f : X → Y • definedas f (x) = sup�U∈Nσ (X) α(U) is μ-l.s.c. on X in the sense of Definition 2.8.

Proof By analogy with the proof of Lemma 6.9 in [6], we note that for every open setU ∈ Nσ (X) and every y ∈ U , we have U ∈ Nσ (y). Therefore, f (y) ≥� α(U) and,hence, inf�y∈U f (y) ≥� α(U) for every U ∈ Nσ (X). This implies

f (x) := sup�U∈Nσ (x)

α(U) ≤� sup�U∈Nσ (x)

inf�y∈U

f (y), ∀ x ∈ X.

Since the opposite inequality f (x) ≥� sup�U∈Nσ (x) inf�y∈U

f (y) is trivial, the mapping

f is μ-l.s.c. on X (see Remark 2.12). ��Since the lower semicontinuity property of vector-valued mappings is crucial in

solving many problems arising in mathematical analysis and in vector optimizationtheory, regularization techniques for extended real-valued functions are important tools[14,15,18]. These processes aim to construct, on the basis of a given mathematicalobject, another object having better properties than the initial one. In view of this, werecall the following concept.


Definition 2.14 Let f : X → Y • be a given mapping. We say that I f : X → Y • isits lower μ-semicontinuous regularization, if

(i) I f : X → Y • is a μ-l.s.c. mapping;(ii) I f (x) ≤� f (x) for every x ∈ X ;

(iii) I f (x) ≥� h(x) ∀ x ∈ X whenever h : X → Y • is a lower μ-semicontinuousmapping such that h(x) ≤� f (x) for every x ∈ X .

In what follows, by analogy with the classical variational analysis, we use thefollowing observation.

Lemma 2.15 Let f : X → Y • be a given mapping and let I f : X → Y • be itsμ-l.s.c. regularization. Then inf�z∈U f (z) = inf�z∈U I f (z).

Proof Let U be an σ -open subset of X and let g : X → Y • be the mapping defined as

g(x) = inf�z∈U

f (z), if x ∈ U , and g(x) = −∞�, otherwise.

This rule implies that g : X → Y • is μ-l.s.c. and g ≤� f on X . Hence, g ≤� I f onX by Definition 2.14 and it immediately leads us to the relation

inf�z∈U

f (z) = inf�x∈U

g(x) ≤� inf�x∈U

I f (x).

Since the opposite inequality is obvious, this concludes the proof. ��

3 Definition of ��,μ-convergence

Let X and Y be given normed spaces which are assumed to be endowed with sometopologies σ = σ(X) and τ = τ(Y ), respectively. We assume that the space (Y, τ ) ispartially ordered by a τ -closed convex pointed cone �.

With any sequence of mappings { fk : X → Y •}k∈N we associate two limit map-pings (that always exist):

1. The ��,μ-lower limit of the sequence { fk : X → Y •}k∈N with respect to the μ =σ ×τ -topology of X ×Y , denoted by ��,μ− lim infk→∞ fk , is the mapping fromX into Y • defined by

(��,μ− lim inf

k→∞ fk

)(x) = sup�

U∈Nσ (x)limτ

k→∞ inf�m≥k

inf�z∈U

{ fm(z)}. (3.1)

2. The ��,μ-upper limit of the sequence { fk : X → Y •}k∈N with respect to the μ =σ ×τ -topology of X ×Y , denoted by ��,μ− lim supk→∞ fk , is the mapping fromX into Y • defined by

(��,μ− lim sup

k→∞fk

)(x) = sup�

U∈Nσ (x)limτ

k→∞ sup�m≥k

inf�z∈U

{ fm(z)}. (3.2)

R. Manzo

Here, limτk→∞ gk stands for the limit of {gk}k∈N in Y with respect to the τ -topology.

For instance, if Y is dual to some separable real Banach space Z (Y = Z∗) and Y isendowed with the weak-∗ topology, then relation (3.1) should be understood as follows

⟨(��,μ− lim inf

k→∞ fk)(x), ϕ

⟩Y,Z

=⟨

sup�U∈Nσ (x)

g(U), ϕ⟩Y,Z, ∀ϕ ∈ Z ,

where⟨g(U), ϕ⟩Y,Z := limk→∞

⟨inf�m≥k inf�z∈U { fm(z)} , ϕ

⟩Y,Z and 〈·, ·〉Y,Z is the

duality pairing between Y and Z .

Remark 3.1 We note that in the case when Y = R,� is the cone of positive elementsin R (� = R+), and τ is the topology of pointwise convergence in R, then the notion of��,μ-limits [see (3.1)–(3.2)] can be reduced to the classical definitions of�-lower and�-upper limits for sequences of functions from X into R (see, for instance, [1,3,6,9]).Indeed, in this case Definition 2.4 implies that inf�A = inf A and sup�A = sup Afor any subset A ⊂ R. It remains to note that limk→∞ inf

m≥kam = lim infm→∞ am and

limk→∞ supm≥k am = lim supm→∞ am for any numerical sequence {am}m∈N ⊂ R.

Definition 3.2 We say that the sequence of mappings { fk : X → Y •}k∈N is ��,μ-convergent at x ∈ X if the following equality holds


k→∞ fk

)(x) =


k→∞fk

)(x). (3.3)

This common value is then denoted by(��,μ− limk→∞ fk

)(x), i.e.

(��,μ− lim

k→∞ fk

)(x) =


k→∞ fk

)(x) =


k→∞fk

)(x).

(3.4)

When equality (3.3) holds for every x ∈ X , the sequence { fk : X → Y •}k∈N is said tobe ��,μ-convergent and the limit mapping

(��,μ− limk→∞ fk

)(x) defined by (3.4)

is called the��,μ-limit of the sequence { fk}k∈N with respect to theμ = σ×τ -topologyof X × Y .

Remark 3.3 Due to Definition 2.4, we have the obvious inequality ��,μ−lim infk→∞ fk ≤� �

�,μ− lim supk→∞ fk . Hence, the sequence { fk : X → Y •}k∈N

��,μ-converges to f : X → Y • if and only if

sup�U∈Nσ (x)

limτ

k→∞ sup�m≥k

inf�z∈U

{ fm(z)} ≤� f (x) ≤� sup�U∈Nσ (x)

limτ

k→∞ inf�m≥k

inf�z∈U

{ fm(z)}

for every x ∈ X .


Remark 3.4 The ��,μ-limits have a local character. Indeed, if Bσ (x) is a base for theneighbourhood system of x in (X, σ ), then the definition of ��,μ-limits (3.1)–(3.2)can be simplified with the form


k→∞ fk

)(x) = sup�

U∈Bσ (x)limτ

k→∞ inf�m≥k

inf�z∈U

{ fm(z)},(��,μ− lim sup

k→∞fk

)(x) = sup�

U∈Bσ (x)limτ

k→∞ sup�m≥k

inf�z∈U

{ fm(z)}.

Hence, if two sequences { fk : X → Y •}k∈N and {gk : X → Y •}k∈N coincide on anopen subset U of X , then their ��,μ-lower limits, as well as their ��,μ-upper limits,coincide on U .

Our next intension is to show that the definition of ��,μ-lower limit can be reducedto the form recently proposed in [12]. Namely, we have the following result.

Proposition 3.5 Let { fk : X → Y •}k∈N be a given sequence and let j ∈ N be a fixedindex value. Then for every x ∈ X and for every neighbourhood U ∈ Nσ (x), thefollowing relation holds true

inf�m≥k

inf�z∈U

{ fm(z)} = inf�z∈U

inf�m≥k

{ fm(z)}. (3.5)

Proof Having fixed x ∈ X and k ∈ N, for every open set U ∈ Nσ (x) we have

inf�z∈U

inf�m≥k

{ fm(z)} ≤� inf�z∈U

{ fm(z)}, ∀ m ≥ k.

Hence, inf�z∈U inf�m≥k { fm(z)} ≤� inf�m≥k

inf�z∈U

{ fm(z)}. On the other hand, the rela-

tion

inf�m≥k

inf�z∈U

{ fm(z)} ≤� inf�m≥k

{ fm(z)}, ∀ z ∈ U

leads us to the opposite inequality inf�m≥k inf�z∈U { fm(z)} ≤� inf�z∈U

inf�m≥k

{ fm(z)}.This concludes the proof. ��

As an obvious consequence of this proposition, we have the following result.

Corollary 3.6 The ��,μ-lower limit of the sequence { fk : X → Y •}k∈N with respectto the μ = σ × τ -topology of X × Y can be represented in the form


k→∞ fk

)(x) = sup�

U∈Nσ (x)limτ

k→∞ inf�z∈U

inf�m≥k

{ fm(z)}, ∀ x ∈ X. (3.6)

Further we give the main properties of��,μ-limits which seem to be rather obviousand their fulfilment immediately follows from relations (3.1)–(3.2). To begin with, we

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note that the following propositions follow immediately from relations (3.1)–(3.2) andLemmas 2.13 and 2.15.

Proposition 3.7 The ��,μ-limits do not change if we replace the mappings fk :X → Y • by their lower μ-semicontinuous regularizations I fk : X → Y • introducedin Definition 2.14. Namely, the following relations hold:

��,μ− lim infk→∞ fk = ��,μ− lim inf

k→∞ I fk , ��,μ− lim supk→∞

fk = ��,μ− lim supk→∞

I fk .

(3.7)

Proposition 3.8 The mappings ��,μ− lim infk→∞ fk : X → Y • and ��,μ−lim supk→∞ fk : X → Y • are μ-l.s.c. on X in the sense of Definition 2.8.

Proof Indeed, it is enough to apply Lemma 2.13 to the set mappings

α(U) := limτ

k→∞ inf�m≥k

inf�z∈U

{ fm(z)} and β(U) := limτ

k→∞ sup�m≥k

inf�z∈U

{ fm(z)} ,

defined for every σ -open subset U of X . ��Proposition 3.9 If

{fkn : X → Y •}

n∈Nis a subsequence of { fk : X → Y •}k∈N, then

��,μ− lim infk→∞ fk ≤� �

�,μ− lim infn→∞ fkn , ��,μ− lim sup

k→∞fk ≥� �

�,μ− lim supn→∞

fkn .

In particular, if { fk : X → Y •}k∈N ��,μ-converges to f : X → Y •, then{fkn : X → Y •}

n∈N��,μ-converges to f : X → Y • as well.

Proof The proposition follows immediately from the definition of ��,μ-limits [see(3.1)–(3.2)] and from the properties of the the�-infimum and�-supremum in (Y,�)(Definition 2.4). ��Proposition 3.10 If μ1 and μ2 are two topologies on X × Y such that μi = σi × τ ,where σ1 is weaker than σ2, then

��,μ1 − lim infk→∞ fk ≤� �

�,μ2− lim infk→∞ fk, ��,μ1− lim sup

k→∞fk ≤� �

�,μ2

− lim supk→∞

fk .

Proof For simplicity, we will prove only the first inequality. For every x ∈ X , let usdenote by Nσ1(x) and Nσ2(x) the set of all open neighbourhoods of x in the topologiesσ1 and σ2 respectively. Since Nσ1(x) ⊂ Nσ2(x), we obtain

sup�U∈Nσ1 (x)

(limτ

k→∞ inf�m≥k

inf�z∈U

{ fm(z)})

≤� sup�U∈Nσ2 (x)

(limτ

k→∞ inf�m≥k

inf�z∈U

{ fm(z)}),

which is the inequality to be proved. ��


Proposition 3.11 If {gk}k∈N is another sequence of mappings from X into Y • andfk ≥� gk on X for every k ∈ N, then

��,μ− lim infk→∞ gk ≤ ��

�,μ− lim infk→∞ fk and ��,μ− lim sup

k→∞gk

≤ ��,μ− lim sup

k→∞fk . (3.8)

Proof We will prove only the first inequality. Let x be any element of X , let U ∈ Nσ (x)be some its neighbourhood, and let k0 ∈ N be an arbitrary index value. Since gk ≤� fk

it follows that inf�z∈U {gm(z)} ≤� inf�z∈U { fm(z)} for every m ≥ k0. Hence,

inf�m≥k0

inf�z∈U

{gm(z)} ≤� inf�m≥k0

inf�z∈U

{ fm(z)}, ∀U ∈ Nσ (x), ∀ k0 ∈ N.

Since this inequality holds for every k0 ∈ N, we get

limτ

k→∞ inf�m≥k

inf�z∈U

{gm(z)} ≤� limτ

k→∞ inf�m≥k

inf�z∈U

{ fm(z)} , ∀U ∈ Nσ (x).

Passing to the supremum over all U ∈ Nσ (x), we conclude

sup�U∈Nσ (x)

limτ

k→∞ inf�m≥k

inf�z∈U

{gm(z)} ≤� sup�U∈Nσ (x)

limτ

k→∞ inf�m≥k

inf�z∈U

{ fm(z)} .

The proof is complete. ��We show how definitions of ��,μ-limits work in some examples.

Example 3.12 Let X = R, Y = R2, let σ and τ be the topologies of pointwise

convergence in R and R2 respectively, and let the ordering cone� be defined by� ={

y ∈ R2 : y2 ≤ 0 & y1 + y2 ≥ 0

}. Let us define the sequence

{fk : R → R

2}

k∈Nas

follows

fk(x) = (sin(kx), cos(kx))t , ∀ k ∈ N. (3.9)

It is easy to see that this sequence has no τ -cluster points in R2. Moreover, since

| sin(ky)| ≤ 1 and | cos(ky)| ≤ 1 on R, and

inf� A =[

1−3

], where A =

{y ∈ R

2 : |yi | ≤ 1, i = 1, 2},

we have

[sin(ky)cos(ky)

]≥�

[1

−3

]for all y ∈ R and all k ∈ N. As the same time, for

every x ∈ R and every neighbourhood U ∈ Nσ (x) there exists a number k0 ∈ N suchthat

inf�z∈U

{ fk(z)} = inf�x∈U

{[sin(kx)cos(kx)

]}=

[1

−3

], ∀ k ≥ k0.

R. Manzo

Hence,

limτ

k→∞ inf�m≥k

inf�z∈U

{ fm(z)} =[

1−3

], limτ

k→∞ sup�m≥k

inf�z∈U

{ fm(z)} =[

1−3

].

Since the last relations hold true for every U ∈ Nσ (x), this leads us to the conclusion

��,μ− lim infk→∞ fk = ��,μ− lim sup

k→∞fk =

[1

−3

]on R.

Thus,

[1

−3

]is the ��,μ-limit of the sequence (3.9).

It is worth to note that��,μ-limit essentially depends on the choice of ordering cone�. Indeed, if we let� to be the natural cone of positive elements in R

2 (� = R2+), then

by analogy with the previous reasoning it can be shown that ��,μ− limk→∞ fk =[−1−1

]on R.

Example 3.13 Let � be a bounded open domain in RN . We set X = L2(�)

and Y = L1(�). We assume that these spaces are endowed with the weaktopologies σ and τ , respectively. Let � be the natural ordering cone of posi-tive elements in L1(�), i.e. � = {

g ∈ L1(�) : g(x) ≥ 0 a. e. in �}. Let �∗ ={

ψ ∈ L∞(�) : ∫�ψ(x)y(x) dx ≥ 0 ∀ y ∈ �}

be the dual cone. Let {χk}k∈N be asequence in L∞(�) such that c1 ≤ χk(x) ≤ c2 almost everywhere in � for everyk ∈ N, where the constants c1, c2 satisfy condition 0 < c1 ≤ c2. We define thesequence of mappings

{fk : L2(�) → L1(�)

}k∈N

as follows

fk(u) = χk u2, ∀ k ∈ N, ∀ u ∈ L2(�).

Let us assume that the following hypothesis holds: there exist a sequence {βk}k∈N ⊂L∞ and an element ζ ∈ L∞(�) satisfying the properties:

(i) βk(x) > 0 a.e. in �∀ k ∈ N; (i i) βk∗⇀ 1 in L∞(�);

(i i i) βkχk∗⇀ ζ in L∞(�); (iv) β2

kχk∗⇀ ζ in L∞(�);

(v) the implications[β2

kχk y ⇀ ζ y in L1(�)]

⇒[

sup�m≥k

(β2

mχm y)⇀ ζ y in L1(�)

]

and[βkχk y ⇀ ζ y in L1(�)

]⇒

[inf�m≥k

(βmχm y) ⇀ ζ y in L1(�)]

hold for every y ∈ L1(�).

Let us fix u ∈ L2(�) and let us define the elements uk = βku, k ∈ N. Then, dueto the initial assumptions, it is easy to check that uk ∈ L2(�) for every k ∈ N anduk ⇀ u in L2(�) as k → ∞. Moreover, in this case we have


limk→∞ 〈ϕ, fk(uk)〉L∞(�);L1(�) = lim

k→∞

∫

�

ϕ(x)u2(x)︸︷︷︸L1(�)-function

χk(x)β2k (x) dx

=∫

�

ϕ(x)ζ(x)u2(x) dx =⟨ϕ, ζu2

⟩L∞(�);L1(�)

, ∀ϕ ∈ L∞(�),

i.e. the sequence { fk(uk)}k∈N converges to ζu2 weakly in L1(�). Note that, the prop-erties of boundedness in L1(�) and equi-integrability for

{fk(uk) = χkβ

2k u2

}k∈N

areobviously true. Since for every neighbourhood U of u in the weak topology of L2(�),we have uk ∈ U for k large enough, it follows that

⟨ϕ, limτ

k→∞ sup�m≥k

inf�v∈U

fm(v)

⟩

L∞(�);L1(�)

:= limk→∞

⟨ϕ, sup�

m≥kinf�v∈U

fm(v)

⟩

L∞(�);L1(�)

≤ limk→∞

⟨ϕ, sup�

m≥k

(β2

mχmu2)⟩

L∞(�);L1(�)

by (v)1= limk→∞ 〈ϕ, fk(uk)〉L∞(�);L1(�)

=⟨ϕ, ζu2

⟩L∞(�);L1(�)

, ∀ϕ ∈ �∗.

Hence, taking the supremum over all weak neighbourhoods U of u, we obtain

⟨ϕ, ��,μ− lim sup

k→∞fk(u)

⟩

L∞(�);L1(�)

≤⟨ϕ, ζu2

⟩L∞(�);L1(�)

, ∀ϕ ∈ �∗,

(3.10)

that is, ��,μ− lim supk→∞ fk(u) ≤� ζu2 (here, μ stands for the product of the weaktopologies of L2(�) and L1(�)).

Our next step is to prove the opposite inequality for the ��,μ-lower limit of theoriginal sequence. To this end, we use the following obvious inequality

χkv2 ≥ −χku2

k + 2χkukv = −χku2k + 2χkβkuv a.e. in �, ∀ v ∈ L2(�).

Hence, by definition of the cone �, we have

fk(v) ≥� − fk(uk)+ 2χkβkuv for every v ∈ L2(�). (3.11)

Taking into account the definition of the weak neighbourhood in L2(�) and property(iii), it becomes evident the following assertion: for a given �-positive element ψ ∈L1(�), there exists a neighbourhood V ∈ Nσ (u) such that

〈ϕ, χkβkuv〉L∞(�);L1(�) :=∫

�

χkβkuvϕ dx >∫

�

χkβku2ϕ dx −∫

�

ψϕ dx

=⟨ϕ, χkβku2 − ψ

⟩L∞(�);L1(�)

, ∀ v ∈ V, ∀ψ ∈ �∗, ∀ k ∈ N,

R. Manzo

which implies χkβkuv ≥� χkβku2 − ψ for every v ∈ V ∈ Nσ (u) and every k ∈ N.Therefore, combining this fact with (3.11), we come to the relation

inf�v∈V

fk(v) ≥� − fk(uk)+ 2χkβku2 − 2ψ = −χkβ2k u2 + 2χkβku2 − 2ψ.

Taking into account that

inf�m≥k

inf�v∈V

fm(v) ≥� − sup�m≥k

(χmβ

2mu2

)+ sup�

m≥k

(2χmβmu2

)− 2ψ,

(iii)–(v)-properties lead us to the relation

⟨ϕ, limτ

k→∞ inf�m≥k

inf�v∈V

fm(v)

⟩

L∞(�);L1(�)

:= limk→∞

⟨ϕ, inf�

m≥kinf�v∈V

fm(v)

⟩

L∞(�);L1(�)

≥ − limk→∞

⟨ϕ, χkβ

2k u2

⟩L∞(�);L1(�)

+ 2 limk→∞

⟨ϕ, χkβku2

⟩L∞(�);L1(�)

−2 〈ϕ,ψ〉L∞(�);L1(�)

=⟨ϕ, ζu2 − 2ψ

⟩L∞(�);L1(�)

, ∀ψ ∈ �∗.

Since this inequality holds for every�-positive element ψ ∈ L1(�), by the definitionof ��,μ-lower limit, this implies that ��,μ− lim infk→∞ fk(u) ≥� ζu2. Combiningthis relation with inequality (3.10), we arrive at the conclusion: under assumptions (i)–(v) the sequence

{fk = χk u2 : L2(�) → L1(�)

}k∈N

��,μ-converges to the mappingF(u) = ζu2 as k → ∞. It is worth to note that the conditions (i)–(v) can be omittedprovided the sequence {χk}k∈N possesses the property: the implication

[ζ 2 y

χk⇀ ζ y in L1(�)

]⇒

[sup�m≥k

(ζ 2 y

χm

)⇀ ζ y in L1(�)

]

holds for every y ∈ L1(�), where the element ζ ∈ L∞(�) is defined (by Banach-

Alaoglu theorem) as follows 1/χk∗⇀ 1/ζ in L∞(�) as k → ∞. Indeed, having set

βk = ζ/χk for every k ∈ N, we immediately come to the fulfilment of the conditions(i)–(v).

In conclusion of this section, we give sufficient conditions of the compactnessfor the ��,μ-convergence in the space X × Y . In order to do this, we introduce thefollowing concept.

Definition 3.14 Let (Y, τ ) be a real linear topological space with an ordering cone�.We say that the cone� possesses the strong D-property (or it is called strongly Daniell)if � is convex pointed τ -closed and for every �-bounded τ -convergent sequence{yk}k∈N in Y the following equality holds

limτ

k→∞ inf�m≥k

ym = limτ

k→∞ yk = limτ

k→∞ sup�m≥k

ym .


Remark 3.15 Usually, a convex cone� is called Daniell, if every decreasing sequence,(i.e. i < j ⇒ y j ≤� yi ) which has a lower bound, τ -converges to its �-infimum[13]. So, in this sense Definition 3.14 provides the ordering cone with some strongerproperties. We note also that this property is obviously true if τ is the norm topologyof Y and � is a convex pointed τ -closed cone with a non-empty topological interior[16].

Theorem 3.16 Let (Y, τ ) be a real linear topological space with an ordering stronglyDaniell cone �. Let { fk : X → Y •}k∈N be a sequence of locally compact mappingswhich are assumed to be essentially �-bounded below, i.e. there exists an elementb ∈ Y such that

fk(x) ≥� b, ∀ x ∈ X and ∀ k ∈ N. (3.12)

Assume that X satisfies the second axiom of countability. Then the original sequence{ fk : X → Y •}k∈N has a ��,μ-convergent subsequence.

Proof Let Bσ = {U j}

j∈Nbe a countable base for the σ -topology of X . To begin

with, we show that for every j ∈ N there exist an element g j ∈ Y • and an index set

N j ∈ N �∞ such that the subsequence

limτ

k→∞k∈N j

inf�x∈U j

{ fk(x)} (3.13)

has a τ -limit in Y •. Indeed, in view of the condition (3.12), for every fixed j ∈ N, the

sequence{

inf�x∈U j fk(x)}

k∈Nis �-bounded below (see Definition 2.3). Hence, if

this sequence is bounded with respect to the norm of Y , then by the locally compactness

property of { fk : X → Y •}k∈N, we can extract a subsequence{


k∈N j

with N j ∈ N �∞ such that inf�x∈U j fk(x)τ,N j→ g j for some element g j ∈ Y . As for

the case when the sequence{


k∈Nis unbounded with respect to the

norm of Y , this sequence is still �-bounded below. Hence, there exists an index set

N j ∈ N �∞ such that inf�x∈U j fk(x)τ,N j→ +∞�. Therefore, we can take g j = +∞�.

As a result, following a classical diagonalization process, we can extract an indexset N∗ ∈ N �∞ such that

limτ

k→∞k∈N∗

inf�x∈U j

{ fk(x)} exists in Y • for every j = 1, 2, . . . . (3.14)

Hence, for every x ∈ X we can define a vector-valued mapping f : X → Y • asfollows

R. Manzo

f (x) := sup�U∈Bσ (x)

limτ

k→∞k∈N∗

inf�y∈U

{ fk(y)}, where Bσ (x) = {U j ∈ Bσ : x ∈ U j}.

Further, we note that if the sequence

{yk := inf�

x∈U j

{ fk(x)}k∈N∗

}

k∈N∗τ -converges to

some element q ∈ Y , then condition (3.14) and Definition 3.14 lead us to the relation

limτ

k→∞k∈N∗

inf�m≥k

inf�x∈U

{ fm(x)}= limτ

k→∞k∈N∗

inf�x∈U

{ fk(x)}= limτ

k→∞k∈N∗

sup�m≥k

inf�x∈U

{ fm(x)} (3.15)

which holds true for every U ∈ Bσ (x). Hence,

sup�U∈Bσ (x)

limτ

k→∞k∈N∗

inf�m≥k

inf�y∈U

{ fm(y)} = f (x) = sup�U∈Bσ (x)

limτ

k→∞k∈N∗

sup�m≥k

inf�y∈U

{ fm(y)},

that is, ��,μ− lim infk→∞ fk(x) = f (x) = ��,μ− lim supk→∞ fk(x).

It remains to consider the case when the sequence{

yk := inf�x∈U j { fk(x)}k∈N∗}

k∈N∗is unbounded with respect to the norm of Y . Then, as has been shown earlier, +∞�

is its τ -limit. Hence,

limτ

k→∞k∈N∗

inf�x∈U

{ fk(x)} = limτ

k→∞k∈N∗

sup�m≥k

inf�x∈U

{ fm(x)} = +∞, ∀U ∈ Bσ (x). (3.16)

As for the sequence{inf�m≥k inf�x∈U { fm(x)}

}k∈N∗ , it is�-bounded below by (3.12)

and monotonically increasing, i.e. inf�m≥k inf�x∈U { fm(x)} ≤� inf�m≥k1 inf�x∈U{ fm(x)} for every k1 > k, (k, k1 ∈ N ). Hence, there exists an index set N∗∗ ⊂ N∗such that the corresponding subsequence

{inf�m≥k inf�x∈U { fm(x)}

}k∈N∗∗ is strictly

monotone and, therefore, is unbounded with respect to the norm of Y . As a result,limτ

k→∞k∈N∗ inf�m≥k inf�x∈U { fm(x)} = +∞�, and combining this fact with(3.16), we arrive at a relation like (3.15) with N∗∗ instead of N∗. Thus, the sequence{ fk : X → Y •}k∈N∗∗ ��,μ-converges to f by Remark 3.4. ��

4 Geometric interpretation of ��,μ-convergence

One of the most famous result of the classical �-convergence theory deal with thefact that there is strong connection between �-convergence for the sequences ofextended real valued functions and the set convergence (in Kuratowski sense) of theirepigraphs. Namely, �-convergence is nothing but the set convergence of the cor-responding epigraphs. However, in the case of ��,μ-convergence for vector-valuedmappings, the direct transfer of this property is a non-trivial matter and, moreover,this is impossible in general. The problem is that in the vectorial case, the closure ofepigraphs for vector-valued mappings may lead to sets which have not the structureof an epigraph for any mapping. Indeed, let X = R, Y = R

2, and let � = R2+ be the


ordering cone in Y . We define a vector-valued mapping f : [0, 1] → R2 as follows:

f (x) =[

01

]if x ≤ 1

2, and f (x) =

[10

]if x >

1

2. (4.1)

Then

cl (epi f ) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

(x, y) ∈ [0, 1] × R2, y ≥�

[10

]if x < 1

2 ,

(x, y) ∈ [0, 1] × R2, y ≥�

[01

]if x > 1

2 ,

( 12 , y

)such that

(y ≥�

[10

])∨

(y ≥�

[01

])= “truth”.

(4.2)

However, as follows from (4.2), for the cross-section of cl (epi f ) at the pointx0 = 1

2 , we have cl (epi f )|x=x0�= y +�. Hence, the set cl (epi f ) does not have the

structure of epigraph for any vector-valued mapping.For this reason, we begin this section with some notion.

Definition 4.1 The coepigraph of a mapping f : X → Y • is the complement of thestrict hypograph of f , i.e., coepi f = {(x, y) ∈ X × Y | y ≮� f (x)}.

Definition 4.2 Let {Sk}k∈N be a sequence of subsets of the topological space (X, σ ).The K -lower limit of the sequence {Sk}k∈N, denoted by K− lim infk→∞ Sk , is the setof all points x ∈ X with the following property: for every U ∈ Nσ (x) there exists anindex set N ∈ N∞ such that U ∩ Sk �= ∅ for every k ∈ N . The K -upper limit, denotedby K− lim supk→∞ Sk , is the set of all points x ∈ X with the following property: forevery U ∈ Nσ (x) there exists an index set N ∈ N �∞ such that U ∩ Sk �= ∅ for everyk ∈ N .

Remark 4.3 It follows immediately from this definition that ∅⊆ K− lim infk→∞ Sk ⊆K− lim supk→∞ Sk and K− lim supk→∞ Sk = ⋂

m∈Nclσ

⋃k≥m Sk . Moreover, the

K -lower limit and K -upper limit of any sequence {Sk}k∈N are closed (possibly empty)subsets of X , and the K -limit of a constant sequence {Sk = S}k∈N coincide with theclosure of S. For more details, we refer to [17].

Definition 4.4 Let f : X → Y • be a given mapping. We say that I cf : X → Y • is its

sequential lower semicompact regularization, if

(i) I cf : X → Y • is a locally compact and sμ-l.s.c. mapping;

(ii) I cf (x) ≤� f (x) for every x ∈ X ;

(iii) I cf (x) ≥� h(x), ∀ x ∈ X whenever h : X → Y • is a sequentially lower semi-

continuous and locally compact mapping such that h(x) ≤� f (x) for everyx ∈ X .

We also need a few technical results.

R. Manzo

Proposition 4.5 ([12]) Let Y be a normed space partially ordered by a τ -closedconvex pointed cone � with nonempty topological interior. Let I be an index set andlet { fi : X → Y •}i∈I be a collection of mappings. Then

⋃i∈I

coepi ( fi ) = coepi

(inf�i∈I

{ fi }), (4.3)

where coepi ( fi ) stands for the coepigraph of fi : X → Y •.

Proposition 4.6 ([12]) Let Z be a nonempty subset of X. Let Y be a normed spacepartially ordered by a τ -closed convex pointed cone �. Let { fk}k∈N and f be givenmapping from Z into Y •. Assume the sequence { fk}k∈N is monotonically increasingand for every x ∈ X there exists an element y ∈ Y such that y ≥� fk(x) for all k ∈ N.Then the following equality holds true

limτ

k→∞ sup�z∈Z

fk(z) = sup�z∈Z

limτ

k→∞ fk(z). (4.4)

Proposition 4.7 ([11]) Let Y be a normed space partially ordered by a τ -closedconvex pointed cone � with nonempty topological interior. Let f : X → Y • bea locally compact mapping. Then its sequential lower semicompact regularizationf̂ : X → Y • can be represented as follows

f̂ (x) := inf�Lμ( f, x) = sup�U∈Nσ (x)

inf�z∈U

f (z). (4.5)

We are now in a position to state the first result of this section concerning the con-nection between ��,μ-convergence of vector-valued mappings and K -convergence oftheir coepigraphs, defined by Definition 4.1.

Theorem 4.8 Let Y be a normed space partially ordered by a τ -closed convex pointedcone �. Let { fk : X → Y •}k∈N be sequence of locally compact mappings such thatfor every x ∈ X there exists an element y ∈ Y satisfying y ≥� fk(x) for all k ∈ N.Assume that the μ-topology on X × Y satisfies the first axiom of countability. Then

K− lim supk→∞

(coepi fk

)= coepi


k→∞ fk

). (4.6)

Proof Due to Remark 4.3, we have the following representation for the K -upper limit

K − lim supk→∞

coepi fk =⋂

m∈N

clμ

⎛⎝⋃

k≥m

coepi fk

⎞⎠. (4.7)

Then by the initial assumptions and Proposition 4.5, we obtain

⋂m∈N

clμ

⎛⎝⋃

k≥m

coepi fk

⎞⎠ =

⋂m∈N

clsμ

(coepi

(inf�k≥m

{ fk})). (4.8)


Taking into account the fact that the sequential closure of coepigraph is also thecoepigraph of some mapping (see Proposition 3.10 in [12]), we conclude

⋂m∈N

clsμ

(coepi

(inf�k≥m

{ fk}))

=⋂

m∈N

coepi gm, (4.9)

where gm is a sequential lower semicompact regularization of inf�k≥m { fk} in the senseof Definition 4.4 (see Theorem 3.8 in [12]). Hence, {gm}m∈N forms a monotonicallynon-decreasing sequence of locally compact mappings with sequentially μ-closedcoepigraphs. Moreover, using the fact that gm(x) ≤� fm(x) for all x ∈ X , we deduce:the sequence {gm}m∈N is essentially�-bounded above on X , i.e. there exists an elementy ∈ Y such that y ≥� gk(x) for all k ∈ N. As a result, having applied Proposition 4.7,it leads us to the following representation for the mappings gm : X → Y • [see (4.5)]

gm(x) = sup�U∈Nσ (x)

inf�z∈U

inf�k≥m

{ fk(x)}, ∀ x ∈ X, ∀ m ∈ N. (4.10)

Further, we note that by monotonicity of {gm}m∈N and definition of the coepigraph,the following representation is valid

⋂m∈N

coepi gm = coepi(

limm→∞ gm

). (4.11)

As a result, summing up the representations (4.7)–(4.11), we obtain

K− lim supk→∞

(coepi fk

)= coepi

(limτ

m→∞ sup�U∈Nσ (x)

inf�z∈U

inf�k≥m

{ fk(z)})

by Proposition 4.6= coepi

(sup�

U∈Nσ (x)limτ

m→∞ inf�z∈U

inf�k≥m

{ fk(z)})

by Proposition 3.5= coepi

(sup�

U∈Nσ (x)limτ

m→∞ inf�k≥m

inf�z∈U

{ fk(z)})

= coepi


k→∞ fk

).

The proof is complete. ��As an obvious consequence of this theorem we can establish the following properties

of ��,μ-lower limit mapping ��,μ− lim infk→∞ fk : X → Y • (see [12]).

Corollary 4.9 Let Y be a normed space partially ordered by a τ -closed convex pointedcone�. Let { fk : X → Y •}k∈N be a sequence of locally compact mappings such thatfor every x ∈ X there exists an element y ∈ Y such that y ≥� fk(x) for all k ∈ N.Then

(i) The coepigraph of ��,μ-lower limit mapping is sequentially μ-closed.

R. Manzo

(ii) The ��,μ-lower limit of { fk : X → Y •}k∈N is a sμ-l.s.c. mapping.(iii) If fk = f for every k ∈ N, where f : X → Y • is a locally compact mapping,

then ��,μ− lim infk→∞ f is the sequential lower semicompact regularizationof f in the sense of Definition 4.4.

(iv) If{

f̂k : X → Y •}k∈N

is the sequence of sequential lower semicontinuous regu-larizations of the mappings { fk : X → Y •}k∈N, then

��,μ− lim infk→∞ fk(x) = ��,μ− lim inf

k→∞ f̂k(x), ∀ x ∈ X.

It should be emphasized that the result given by Theorem 4.8 cannot be extendedto the case of ��,μ-upper limit and K -lower set limit. Indeed, let us consider thefollowing example.

Example 4.10 Let X = R, Y = R2, and let� be the natural cone of positive elements

in R2. We define the sequence of mappings

{fk : R −→ R

2}

k∈Nas follows

fk(x)={(0, 1)t , k = 2m, m ∈N ,

(1, 0)t , k = 2m + 1, m ∈N .(4.12)

Then the K -lower limit of the corresponding sequence of coepigraphs, in view ofDefinition 4.2, takes the form:

K− lim infk→∞ (coepi fk)=

{(x, y) ∈ R × R

2 : y ≥�

[01

]∨ y ≥�

[10

]}. (4.13)

As obviously follows from (4.13), this set does not possess the structure of coepi-

graph for any vector-valued mapping. So, the relation K− lim infk→∞(

coepi fk

)=

coepi(��,μ− lim supk→∞ fk

)fails in the vectorial case.

Before we give some results concerning the limiting properties of epigraphs forsequences of locally compact mappings { fk : X → Y •}k∈N, we recall some basicnotation of Convex Analysis.

Definition 4.11 ([16]) A cone � in a normed space (Y, τ ) is called minihedral if theequality (x +�) ∩ (y +�) = (

sup� {x, y} +�)

is valid for every x, y ∈ Y .

For instance, in the case when Y = L p(�) or Y = C(�), where � is an openbounded subset of R

N , p ∈ [1,+∞), the natural cone of non-negative elements of Yis minihedral and normal for the norm topology.

Definition 4.12 ([2]) The set cone�(A) is called the �-conic hull of a set A ⊂ Y ifcone�(A) is the intersection of all sets of the form x +� in Y containing A, i.e.

cone�(A) =⋂

x∈Y : A⊆(x+�)(x +�).


It is well known that if � is a minihedral cone in Y , then the cone�(A), as thesmallest conic set in Y containing A, has the representation (see [2,16])

cone�(A) = inf�A +�. (4.14)

As a result, we have the following representation for the �-conic hull of a set A ⊂X × Y •, provided � is a minihedral cone in Y :

cone�(A) = {(x, y) ∈ X × Y • ∣∣ y ≥� inf� A|x

}.

In view of this it is worth to recall the following result, which shows that theclassical rule of the lower semicontinues regularization epi I f = clμ epi f for real-valued function f : X → R does not hold in the vectorial case.

Theorem 4.13 ([11]) Let� be a τ -closed convex pointed minihedral cone in Y with anonempty topological interior, and let f : X → Y • be a locally compact vector-valuedmapping. Then the �-cone hull of the sequential μ-closure of epi f is an epigraph ofthe sequential lower semicompact regularization of f in the sense of Definition 4.4,i.e.

cone�(clμ epi f

) = epi f̂ ,

where f̂ (x) = inf� Lμ( f, x), ∀ x ∈ X.

Further we need the following technical lemma.

Lemma 4.14 Let � be a τ -closed convex pointed cone in Y . Let f : X → Y • be agiven mapping and let A ⊂ X × Y • be any subset of the epigraph epi f . Then

epi f ⊇ cone�A.

Proof Let us fix a point x0 ∈ X and consider the set Ax0 := {(x, y) ∈ A : x = x0}.By definition of the epigraph, we have

(x0, inf�

(x,y)∈Ax0

{y} ) ∈ epi f . Hence, the set

cone�A|x=x0 := {(x, y) ∈ cone�A : x = x0} belongs to the epigraph epi f as well.Since x0 is an arbitrary point of X , this concludes the proof. ��

The following theorem shows that the connection between ��,μ-convergence ofvector-valued mappings and K -convergence of their epigraphs may have a more com-plicate form than in the scalar case.

Theorem 4.15 Let (Y, τ ) be a real linear topological space with an ordering stronglyDaniell cone�. Let { fk : X → Y •}k∈N be a given sequence of�-bounded mappings.Then the following inclusion holds

cone�

[K− lim inf

k→∞ (epi fk)

]⊆ epi


k→∞fk

).

R. Manzo

Proof Let (x, y)be an arbitrary pair of K− lim infk→∞ (epi fk). Then, as follows fromDefinition 4.2, for every U ∈ Nσ (x) there exist a sequence {(xk, yk)}k∈N ⊂ X × Yand an index set N ∈ N∞ such that

(xk, yk)μ→ (x, y) in X × Y and yk ≥� fk(xk) ≥� inf�

z∈U{ fk(z)} , ∀ k ∈ N .

Since xkσ→ x in X , it follows that

sup�k≥m

yk ≥� sup�k≥m

fk(xk) ≥� sup�k≥m

inf�z∈U

{ fk(z)} , ∀U ∈ Nσ (x), ∀ m ∈ N .

Hence,

limτ

m→∞ sup�k≥m

yk ≥� limτ

m→∞ sup�k≥m

fk(xk) ≥� limτ

m→∞ sup�k≥m

inf�z∈U

{ fk(z)}, ∀U ∈ Nσ (x).

Taking into account the strong Daniell property of the cone � and the fact that theneighbourhood U ∈ Nσ (x) is arbitrary, we finally obtain

y = limτ

m→∞ sup�k≥m

yk ≥� sup�U∈Nσ (x)

limτ

m→∞ sup�k≥m

inf�z∈U

{ fk(z)} = ��,μ− lim supk→∞

fk(x),

that is, (x, y) ∈ epi(��,μ− lim supk→∞ fk

). Thus,

K− lim infk→∞ (epi fk) ⊆ epi


k→∞fk

)(4.15)

and it remains to apply Lemma 4.14. The proof is complete. ��

Remark 4.16 It is worth to note that the inclusion (4.15) can be strict, in general.Indeed, let X = R, Y = R

2, and let � = R2+ be the ordering cone in Y . Let the

sequence { fk : X → Y }k∈N be defined as follows fk ≡ f for every k ∈ N, where themapping f : X → Y • has the representation (4.1). Then, by Remark 4.3 we have

K − lim infk→∞ (epi fk)

= cl (epi f ) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

(x, y) ∈ [0, 1] × R2, y ≥�

[10

]if x < 1

2 ,

(x, y) ∈ [0, 1] × R2, y ≥�

[01

]if x > 1

2 ,

( 12 , y

)such that

(y ≥�

[10

])∨

(y ≥�

[01

])= “truth”,


where as Proposition 3.7 leads us to the equality ��,μ− lim supk→∞ fk = I f , whereI fk : X → Y • is the lower μ-semicontinuous regularizations of f in the sense ofDefinition 2.14. Hence,

I f (x) =[

01

]if x <

1

2, I f (x) =

[10

]if x >

1

2, and I f

(1

2

)=

[00

]

end, therefore, K− lim infk→∞ (epi fk) ⊂⊂ epi(��,μ− lim supk→∞ fk

).

Acknowledgments The author wishes to thank Proff. Ciro D’Apice and P. I. Kogut for the useful discus-sions and suggestions.

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