Continuity of modulus of noncompact convexity for minimalizable measures of noncompactness

Abstract

We consider the modulus of noncompact convexity ∆X,ϕ(ε) associated with the minimalizable
measure of noncompactness ϕ. We present some properties of this modulus, while the main result of
this paper is showing that ∆X,ϕ(ε) is a subhomogenous and continuous function on [0, (BX)) for an arbitrary
minimalizable measure of compactness ϕ in the case of a Banach space X with the Radon-Nikodym property.

Full text

Functional Analysis,
Approximation and
Computation
7 (3) (2015), 39–46
Published by Faculty of Sciences and Mathematics,
University of Niˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/faac
Continuity of modulus of noncompact convexity for minimalizable
measures of noncompactness
Amra Reki´c-Vukovi´ca, Nermin Okiˇci´cb, Vedad Paˇsi´cc, Ivan Arandjelovi´cd
aDepartment of Mathematics University of Tuzla Univerzitetska 4 75000 Tuzla Bosnia and Herzegovina
bDepartment of Mathematics University of Tuzla Univerzitetska 4 75000 Tuzla Bosnia and Herzegovina
cDepartment of Mathematics University of Tuzla Univerzitetska 4 75000 Tuzla Bosnia and Herzegovina
dFaculty of Mechanical Engineering University of Belgrade Kraljice Marije 16 11000 Belgrade Serbia
Abstract. We consider the modulus of noncompact convexity ∆X,ϕ(ε) associated with the minimalizable
measure of noncompactness ϕ. We present some properties of this modulus, while the main result of
this paper is showing that ∆X,ϕ(ε) is a subhomogenous and continuous function on [0,(BX)) for an arbitrary
minimalizable measure of compactness ϕin the case of a Banach space Xwith the Radon-Nikodym property.
1. Introduction
One of the basic concepts of geometry of Banach spaces is that of uniform convexity, introduced by
Clarkson [3]. Smulian [11] characterized the property dual to uniform convexity, i.e. uniform smoothness
and pointed out that uniform convexity and uniform smoothness describe geometric properties of finite
dimensional subspaces of a normed space. Most of the problems in fixed point theory, however, have global
character, which was the motivation for considering infinite dimensional counterparts of classical geometric
notions. One of these is nearly uniformconvexity introduced by Huffin [7]. Goebel and Sekowski [6] define
the modulus corresponding to nearly uniform convexity and use the concept of measure of noncompactness
to give a new classification of Banach spaces. They prove that whenever the characteristic of uniform
noncompact convexity of any Banach space is less than 1, the space is reflexive and has normal structure.
1.1. Fundamental concepts and definitions
In this paper Xdenotes the Banach space, B(x,r) an open ball centred at xof radius r,BXthe unit ball
and SXthe unit sphere in X. If A⊂Xwe denote by Athe closure of a set Aand by coA the convex envelope
of A.
Definition 1.1. Let Bbe a family of bounded subsets of X. We call the mapping ϕ:B → [0,+∞)the measure of
noncompactness defined on X if it satisfies the following :
2010 Mathematics Subject Classification. 46B20, 46B22.
Keywords. Modulus of noncompact convexity; measure of noncompactness.
Received: 29 May 2015; Accepted: 9 July 2015
Communicated by Dragan S. Djordjevi´
c
Email addresses: amra.rekic@untz.ba (Amra Reki ´
c-Vukovi´
c), nermin.okicic@untz.ba (Nermin Okiˇ
ci´
c), vedad.pasic@untz.ba
(Vedad Paˇ
si´
c), iarandjelovic@mas.bg.ac.rs (Ivan Arandjelovi´
c)

A. Reki´c-Vukovi´c et al. /FAAC 7 (3) (2015), 39–46 40
1. ϕ(B)=0⇔B is a relatively compact set;
2. ϕ(B)=ϕ(B),∀B∈ B
3. ϕ(B1∪B2)=max{ϕ(B1), ϕ(B2)},∀B1,B2∈ B.
Some of well known measures of noncompactness are Kuratowski measure (α), Hausdorffmeasure (χ) and Istratescu
measure (β), see e.g. [5], [8] and [12]. We call a measure of noncompactness ϕaminimalizable measure of
noncompactness if for every infinite bounded set A and for every ε > 0there exists its subset B ⊂A which is
ϕ-minimal and such that ϕ(B)≥ϕ(A)−ε.
A measure ϕis a strictly minimalizable measure of noncompactness if for every infinite, bounded set A there
exists its subset B ⊂A which is ϕ-minimal and such that ϕ(B)=ϕ(A).
Remark 1.2. Clearly every strictly minimalizable measure is a minimalizable measure of noncompactness. See e.g.
[8] and [5] for more on measures of noncompactness.
Definition 1.3. A modulus of noncompact convexity associated with an arbitrary measure of noncompactness ϕis
a function ∆X,ϕ : [0, ϕ(BX)] →[0,1] given by
∆X,ϕ(ε)=inf{1−d(0,A) : A⊆BX,A=coA =A, ϕ(A)> ε}.
Banas [4] considered a modulus ∆X,ϕ(ε) for ϕ=χ, where χis the Hausdorffmeasure of noncompactness.
For ϕ=α, where αis a Kuratowski measure of noncompactness, ∆X,α(ε) represents the Goebel-Sekowski
modulus of noncompact convexity, see [6].
For ϕ=β, where βis the separation measure of noncompactness, ∆X,β(ε) represents the Dominguez-Lopez
modulus of noncompact convexity, see [10].
Definition 1.4. We define the characteristic of noncompact convexity of X associated with a measure of noncompact-
ness ϕby
εϕ(X)=sup{ε≥0 : ∆X,ϕ(ε)=0}.
For moduli ∆X,ϕ(ε) with respect to ϕ=α, χ, β we have the following inequalities:
∆X,α(ε)≤∆X,β(ε)≤∆X,χ(ε),
and hence
εα(X)≥εβ(X)≥εχ(X).
It is known that Xis nearly uniformly convex (NUC) if and only if εϕ(X)=0, for ϕ=α, χ, β, see e.g. [5].
Banas [4] showed that the modulus ∆X,χ(ε) in the case of a reflexive space Xis a subhomogenous
function continuous on [0,1). Prus [9] gave a result which links the continuity of the modulus ∆X,ϕ(ε)
with the uniform Opial condition which implies the normal structure of the space. In [1] we demonstrated
certain properties of the modulus ∆X,ϕ(ε) as well as its continuity for a strictly minimalizable measure of
noncompactness ϕon Banach spaces with the Radon-Nikodym property.
Definition 1.5. We say that the Banach space X is with the Radon-Nikodym property if and only if every nonempty
bounded subset A ⊂X is dentable, i.e. if and only if for all ε > 0, exists x ∈A, such that x <co(A\B(x, ε)), see e.g.
[13].
The aim of this paper is to show that the modulus ∆X,ϕ(ε) is a subhomogenous and continuous function
on [0, ϕ(BX)) for an arbitrary minimalizable measure of compactness ϕin the case of a Banach space Xwith
the Radon-Nikodym property.
Structure of paper. This paper has the following structure. Section 1 provides the background infor-
mation and the fundamental concepts and definitions. Section 2 contains several results which provide the
properties of the ∆X,ϕ modulus. Section 3 contains the main result of the paper, namely the result on the
continuity of the modulus ∆X,ϕ. Section 4 contains the discussion of the obtained results.

A. Reki´c-Vukovi´c et al. /FAAC 7 (3) (2015), 39–46 41
2. ∆X,ϕ modulus properties
Theorem 2.1. Let X be a Banach space with Radon-Nikodym property and let ϕbe a minimalizable measure of
noncompactness. The modulus ∆X,ϕ(ε)is a subhomogenous function, i.e. we have that for all k ∈[0,1] and for all
ε∈[0, ϕ(BX)]
∆X,ϕ(kε)≤k∆X,ϕ(ε).
Proof. Let η > 0 be arbitrary and ε∈[0, ϕ(BX)]. By definition 1.3 there exists a convex closed subset A⊂BX,
ϕ(A)> ε such that
1−d(0,A)<∆X,ϕ(ε)+η .
For arbitrary k∈[0,1] the set kA ⊂BXis convex and closed. Hence we have that ϕ(kA)=kϕ(A)>kε.
As ϕis a minimalizable measure, for δ=ϕ(kA)−kε
2>0 there exists an infinite ϕ-minimal set B⊂kA,
such that
ϕ(B)≥ϕ(kA)−δ ,
i.e. ϕ(B)>kε. From the Radon-Nikodym property of the space X, for ε > 0 there exists x0∈Bsuch that
x0<co B\Bx0,ε
2 .
If we define the set B∗=C+1−k
∥x0∥x0, where C=co B\Bx0,ε
2, then
1−d(0,B∗)<k(∆X,ϕ(ε)+η).(1)
The set B∗⊂BXis closed and convex and we have that ϕ(B∗)=ϕ(C)=ϕ(B)>kε. If we take the infimum
over all the sets B∗,ϕ(B∗)>kε, in inequality (1) we get that
∆X,ϕ(kε)≤k(∆X,ϕ(ε)+η),
which proves the theorem due to η > 0 being arbitrary.
Several additional properties of the modulus ∆X,ϕ arise from Theorem 2.1.
Lemma 2.2. Let ϕbe a minimalizable measure of noncompactness defined on a Banach space X with the Radon-
Nikodym property. Then the modulus ∆X,ϕ(ε)is a strictly increasing function on [εϕ(X), ϕ(BX)].
Proof. Let ε1,ε2∈[εϕ(X), ϕ(BX)] and ε1< ε2. If we put k=ε1
ε2
<1, using Theorem 2.1 we have that
∆X,ϕ(ε1)= ∆X,ϕ(kε2)≤k∆X,ϕ(ε2)<∆X,ϕ(ε2).
Lemma 2.3. Let ϕbe a minimalizable measure of noncompactness defined on a Banach space X with the Radon-
Nikodym property. For every ε∈[0, ϕ(BX)] we have that
∆X,ϕ(ε)≤ε .

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Proof. If ε∈[0,1], then using Theorem 2.1, switching the roles of kand εaround and putting ε=1, we have
that
∆X,ϕ(ε)≤ε∆X,ϕ(1) ≤ε .
If ε∈(1, ϕ(BX)], then due to monotonicity of the modulus ∆X,ϕ(ε) we have that
∆X,ϕ(ε)<∆X,ϕ(ϕ(BX)) =1< ε .
Lemma 2.4. Let ϕbe a minimalizable measure of noncompactness defined on a Banach space X with the Radon-
Nikodym property. Then the function f (ε)=
∆X,ϕ(ε)
εis nondecreasing on [0, ϕ(BX)] and for ε1+ε2≤ϕ(BX)we
have that
∆X,ϕ(ε1+ε2)≥∆X,ϕ(ε1)+ ∆X,ϕ(ε2).(2)
Proof. Let ε1,ε2∈[0, ϕ(BX)] be such that ε1≤ε2. Putting k=ε1
ε2
, we get that
f(ε1)=
∆X,ϕ(ε1)
ε1
=
∆X,ϕ(kε2)
ε1
.
Using the subhomogeneity of the function ∆X,ϕ(ε) we have that
f(ε1)≤
∆X,ϕ(ε2)
ε2
=f(ε2),
which shows that f(ε) is a nondecreasing function on [0, ϕ(BX)]. We also have that
∆X,ϕ(ε1)+ ∆X,ϕ(ε2)≤k∆X,ϕ(ε2)+ ∆X,ϕ(ε2)
=ε1+ε2
ε2
∆X,ϕ(ε2)
≤(ε1+ε2)
∆X,ϕ(ε1+ε2)
ε1+ε2
= ∆X,ϕ(ε1+ε2),
which shows inequality (2).
Lemma 2.5. Let ϕbe a minimalizable measure of noncompactness on a Banach space X with the Radon-Nikodym
property. Then for all ε1, ε2∈(ε1(X), ϕ(BX)], such that ε1≤ε2, we have that
∆X,ϕ(ε2)−∆X,ϕ(ε1)
ε2−ε1≥
∆X,ϕ(ε2)
ε2
.(3)
Proof. Let k=ε1
ε2≤1. Using Theorem 2.1 we get that
∆X,ϕ(ε2)−∆X,ϕ(ε1)= ∆X,ϕ(ε2)−∆X,ϕ(kε2)
≥∆X,ϕ(ε2)−k∆X,ϕ(ε2)
=ε2−ε1
ε2
∆X,ϕ(ε2),
which shows the inequality (3).

A. Reki´c-Vukovi´c et al. /FAAC 7 (3) (2015), 39–46 43
3. The continuity of the modulus of noncompact convexity
The main result of this paper is the following
Theorem 3.1. Let ϕbe a minimalizable measure of noncompactness and let X be a Banach space with the Radon-
Nikodym property. Then the modulus of noncompact convexity ∆X,ϕ(ε)is a continuous function on [0, ϕ(BX)).
The proof of Theorem 3.1 follows from two lemmas that deal with the continuity of the modulus ∆X,ϕ(ε)
from below and above respectively, i.e. Lemma 3.2 and Lemma 3.3.
Lemma 3.2. Let ϕbe a minimalizable measure of compactness and let X be a Banach space with the Radon-Nikodym
property. Then the modulus of noncompact convexity ∆X,ϕ(ε)is a continuous function from below on [0, ϕ(BX)).
Proof. Let ε0∈[0, ϕ(BX)) be arbitrary and let ε < ε0. By Definition 1.3, for arbitrary η > 0 there exists a
convex and closed subset A⊂BX,ϕ(A)> ε such that
1−d(0,A)<∆X,ϕ(ε)+η .
If ϕ(A)≥ε0> ε, then clearly
inf{1−d(0,A) : A=coA,A⊂BX, ϕ(A)≥ε0} ≤ ∆X,ϕ(ε)+η ,
i.e.
∆X,ϕ(ε0)≤∆X,ϕ(ε)+η ,
whence, due to η > 0 being arbitrary, we complete the proof. Therefore, let ε < ϕ(A)< ε0and γ=ε0−ϕ(A)>
0. Since the measure of noncompactness ϕis minimalizable, for arbitrary γ > 0, there exists an infinite
ϕ-minimal subset B⊂Asuch that
ϕ(B)≥ϕ(A)−γ=2ϕ(A)−ε0.
Let n0∈Nbe such that 2ϕ(A)−ε0
n0
<diamB
2. Since B⊂BXis a bounded subset of the space Xwith the
Radon-Nikodym property, for r=2ϕ(A)−ε0
n0
there exists x0∈Bsuch that
x0<co B\B(x0,r).
Let C=co B\B(x0,r). The set C⊂Bis convex and closed and we have that 1 −d(0,C)≤1−d(0,B)≤
1−d(0,A)<∆X,ϕ(ε)+η. Besides, since Bis a ϕ-minimal set, we have that
ϕ(C)=ϕ(B)≥2ϕ(A)−ε0.
Let k=1+1−d(0,C)
2and let us look at the set A∗=kC ∩BX.A∗is closed and convex and we have that
A∗⊆kC ⊂kB and hence
1−d(0,A∗)≤1−d(0,kC)<1−d(0,C)<∆X,ϕ(ε)+η .
Since Bis ϕ-minimal so is kB and hence
ϕ(A∗)=ϕ(kB)=kϕ(B)≥k(2ϕ(A)−ε0).
Put δ=ε0
21−1
k. Then for arbitrary ε∈(ε0−δ, ε0) we have that
ϕ(A∗)≥k(2ε−ε0)>k(2(ε0−δ)−ε0)=ε0.

A. Reki´c-Vukovi´c et al. /FAAC 7 (3) (2015), 39–46 44
Therefore
inf{1−d(0,A∗) : A∗⊂BX,A∗=coA∗, ϕ(A∗)> ε0} ≤ ∆X,ϕ(ε)+η
and hence
∆X,ϕ(ε0)≤∆X,ϕ(ε)+η ,
whence due to η > 0 being arbitrary, we have that
lim
ε→ε0−
∆X,ϕ(ε)= ∆X,ϕ(ε0).
Lemma 3.3. Let ϕbe a minimalizable measure of noncompactness and let X be a Banach space with the Radon-
Nikodym property. Then the modulus of noncompact convexity ∆X,ϕ(ε)is a function continuous from above on
[0, ϕ(BX)).
Proof. Let η > 0 and ε0∈[0, ϕ(BX)). By definition 1.3 there exists a convex and closed subset A⊂BX,
ϕ(A)> ε0, such that
1−d(0,A)<∆X,ϕ(ε0)+η .
Since the measure of noncompactness ϕis minimalizable, for arbitrary γ > 0, there exists an infinite ϕ-
minimal subset B⊂A, such that ϕ(B)≥ϕ(A)−γ. Let n0∈Nbe such that ϕ(A)−γ
n0
<diamB
2. Using the
Radon-Nikodym property of Xfor the set Band r=ϕ(A)−γ
n0
, there exists x0∈Bsuch that
x0<co B\B(x0,r).
In this case the set C=co B\B(x0,r)is a convex and closed subset of the set Bfor which we have
1−d(0,C)<∆X,ϕ(ε0)+η
and
ϕ(C)=ϕ(B)≥ϕ(A)−γ .
Let k=1+1−d(0,C)
2. The set A∗=kC ∩BXis a convex and closed set such that
1−d(0,A∗)<∆X,ϕ(ε0)+η ,
and ϕ(A∗)=ϕ(kB)=kϕ(B)≥k(ϕ(A)−γ).
Let ε′∈2ε0
3−d(0,C), ε0be arbitrary and let γ=ε0−ε′. Using the above, there exists a subset B⊂Asuch
that
ϕ(B)≥ϕ(A)−ε0+ε′> ε′.
Hence for the set A∗we have that
ϕ(A∗)=kϕ(B)>kε′.

A. Reki´c-Vukovi´c et al. /FAAC 7 (3) (2015), 39–46 45
If we choose δ=kε′−ε0, then for an arbitrary ε∈(ε0, ε0+δ) we have that
ϕ(A∗)>kε′> ε .
Therefore
inf{1−d(0,A∗) : A∗⊂BX,A∗=coA∗, ϕ(A∗)> ε} ≤ ∆X,ϕ(ε0)+η ,
i.e.
lim
ε→ε0+
∆X,ϕ(ε)= ∆X,ϕ(ε0).
4. Discussion
It is known that the separation measure of noncompactness βis a minimalizable measure on a complete
metric space. Therefore it is minimalizable on Banach spaces with the Radon-Nikodym property. Using
Theorem 3.1 we can conclude that the modulus ∆X,β(ε) with respect to the separation measure βis a
continuous function on [0, ϕ(BX)) for a Banach space Xwith the Radon-Nikodym property. For example,
since the spaces lpand Lp(1 <p<+∞) are reflexive, they are with Radon-Nikodym property. Therefore
∆lp,β(ε) and ∆Lp,β (ε) are continuous functions on 0, ϕ Blpand 0, ϕ BLprespectively.
The Kuratowski measure of noncompactness αis not minimalizable on spaces lp(1 <p<+∞). Indeed,
if αwas minimalizable on these spaces, then for ε∈0,2n
√2−1
n
√2(n∈N) and for Blp, there would exist an
α-minimal set A⊂Blpsuch that α(A)≥α(Blp)−ε=2−ε. Since every α-minimal set is also β-minimal and
because α(A)=β(A) (see [5], Lemma III. 2.9), we have that
2−ε≤β(A)≤β(Blp)=21
p.
Because of the choice of εthis would mean that p≤1, which is a contradiction with this choice of p.
Therefore, the measure of noncompactness αis strictly minimalizable on the space l1. Namely, since
β(A)=2χ(A) for an arbitrary bounded set A⊂l1, see ([5], Corollary X.4.7), and the general property
χ(A)≤β(A)≤α(A)≤2χ(A)
holds, we conclude that α(A)=β(A)=2χ(A). Since l1is weakly compactly determined, see [2], the
Hausdorffmeasure of noncompactness χis strictly minimalizable on l1, which means that the measure α
is also strictly minimalizable on l1. Hence l1is the separable dual of the space c0, so it is a space with the
Radon-Nikodym property. Using Theorem 3.1 we conclude that ∆l1,α(ε) is a continuous function on [0,2).
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