A measure of weak noncompactness in L

              Re v . A e a_d . Can ar . Cien e . , IX ( N ú m . 1 ) , 9 - 1 3 ( 1 9 9 7)
A MEASURE OF WEAK NONCOMPACTNESS IN L'(Q,:E,A)
I.J. Cabrera and K.B. Sadarangani
Departamento de Matemáticas, Universidad de Las Palmas de G.C.
Campus de Tafira Baja, 35017 LAS PALMAS.
We introduce a measure of weak noncompactness in the space L1(Q , :E,A)and we prove
that this measure is less or egua! to the classical De Blasi measure of weak noncompactness of
L' (Q , :E, A) . Moreover, we give a sufficient condition related with A and :E for thar both meas u res
are egua!.
l. INTRODUCTION
The theory of measures of weak noncompactness was initi ated by De Blasi in the paper [5] .
This measure was app lied successfully to nonlinear functional anal ysis, to operator theory and to the
theory of diffcrential and integral eguations (see [ l ,2,3,7,8], for axample).
In the paper 14], a formula is given to express the De Blasi measure of weak noncompact­ness
in the Lebesgue space L' (0,1). Following the same ideas of [4], in this paper we give a
meas ure of weak noncompactness in the space L.; (Q , :E, A), where A is a nonnegative measure on a
set Q with :E a cr-field of subsets of Q , which is less or egua! to the c lassical De Blasi measure of
weak 11011-compactness of L' (Q , L,A). Moreover, we give a suffic ient condition for that both
measure are egua!.
2. NOTATION ANO DEFINITIONS
Lel E be a n infinite dimensional real Ba11ach ,htCe wi th norm 11 · II and zero e lement 0.
Denote by B1, the c losed unit ball of the spacc E.
Nex t, Jet ME be the family of ali 11onemp11 and bounded subsets of E and Jet N ;; be its
subfamily consisting of a li re lati vely weakly compact s,·&
The so-called De Blasi measurc ol weak 11,,1t<:ompactness ~: ME ---t [0, oo) is defined in the
fo llowing way:
~(X) = inl{r > 0: 3Y <= f\ , , X c Y +E B1 }
1)
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Let us observe that in the case when E is a reflexive Banach space then P(X) = O, for every
X E ME . Hence it is only of interest to consider the function P in a nonreflexi ve Banach space.
Recall that P(BE) = 1 when E is nonreflexive. For other properties of the De Blasi measure of
noncompactness we refer to [5 ,8], for example.
In what follows, suppose Q is a set , L is a cr-field of n and "A a nonnegative measure on
(Q,L). By L\Q, I:, "A) we denote the following set:
L1 (Q, I:, "A) = { f: Q ~ C, measurable with l lfl d"A < 00}
Now, based on the paper [4], we introduce the function H defined on the family ML'wi:,, by the
formula:
H(X) = lim HA(X)
A( A )--;O
where A E L and HA (X) = su/ f IFld"AJ .
t eX lA
The properties of the function H will be investigated in the next section. In particular, we
show that H provides a lower estímate for PL'w.u,
ness.
3. RESULTS
We start with the following result which proves that H is a measure of weak noncompact-
THEOREM 1. The function H satisfies the fol lowing conditions:
(a) H(X) = O{::} X E N;'.
(b) X c Y ~ H(X) $ H(Y)
(e) H(cX) = icl H(X), Ve E R
(d) H(X + Y)$ H(X) + H(Y)
(e) H(convX) = H(X)
( f) H(BL'm.u, ) $ 1
\()
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Proof.
(a) is consequence of a result of [6, pag 93], while (b) follows inmediately from the defini­tion
of the function H.
In order to prove (c) !et us take a set X E M L' <n.u.i and A E :E , and assume that c E R and c 1' O,
then we have
HA (cX) = su/ JjcfjdAJ = su/¡c¡ J jrjdA) = jcjsu/ J jfjdAJ = jcj HA (X)
feX lA feX l A feX lA
and, consequently, H(cX) = jcj H(X).
In the case when c = O, we have that cX is zero function and, thus, HA (cX) = O and in this case the
equality (c) is also satisfied.
In order to prove (d), let us take a set A E :E, f E X and g E Y. Then we have
J¡r + gjdA s:; J¡r¡ctA + JjgjdA s:; HA (X) + HA (Y)
A A A
and, thus
consequently, H(X + Y) s:; H(X) + H(Y)
For the proof of (e) it is sufficient Lo prove the inequality H(convX) s:; H(X). In fact. we take
r = L,A, f, with L, A; = 1, A;~ O, f'; E X and, moreover, A E :E .
Then we can obtain
consequently, HA(convX)s:;H"(X) and, thus, H(convX) .;:; H(X).
HA (B ' ¡ ) s:; IIH 1 ' 11 s:; 1 l ,.1.u::. ,) I.CU .Lr,.)
and, thus, H(B,.,m.u ., ) s:; 1.
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This following result proves that the function H is a lower estímate for the classical De
Blasi measure of weak noncompactness pin L'(Q,L,A).
THEOREM 2. H(X) ~ PCX) for any X E M,,inu,
Proof.
Suppose that P(X) = r. Then for any E> O we can find a set Y EN:,<" °'' such that
X e Y+ (r + E)B , ,. . (see section 2). Hence, using theorem 1. we obtain
L ( il ._.1.l
H(X) ~ H( Y)+ (r +E) H(B,,«u: ,, ) ~ r + E
in virtue of the arbitrariness of E, this implies H(X):,; P(X).
Our last result will be to give a sufficient condition related with the CT-fie ld L and the
measure A for that both measure H and p are equal. This condition has been taken following the
same ideas of the paper [4].
THEOREM 3. Suppose that exists a sequence (A,,) of e lements of L such that A(A,,)
tends to zero when n---') oo, and, moreover, for any bounded subset X of L' (Q, :E, A) ,
Xn- A,, = { fXn- A,,: f E X} is relatively weakly compact in L' (Q, L,A) for every n. In these
conditions H = P .
Proof.
lt's sufficient prove that p ~ H, in virtue of theorem 2. In fact, every function
f E L1 (Q, :E, A) may be written in the form:
where XA denotes the characteristic function of the set A.
Using the representation ( 1) it may be easily show that
X e Xn-A,, + H A,, (X) BL'«u:.,J
Hence, in view of the properties of the De Blasi measure of weak noncompactness p , we get
P(X) ~ P(Xn-A,,) + HA,, (X) P(BL' tn.UJ )
In virtue of the hipothesis, ~(Xn-A) = O and as we consider the case of l.; (Q, L , A) a nonreflexive
space, because the other case is trivial, we obtain
~(X)~ HA,, (X) ~(BL' cnu) = HA,, (X)
and thus ~(X)~ H(X).
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REFERENCES
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