On the measures of non-compactness

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXI (1979)

L.

(Lublin)

On the measures of non-compactness

In [1] К. Kuratowski defined a measure of non-compactness and proved the theorem of non-empty intersection for it.

We give here a shorter proof of this theorem.

Th e o r e m

1. Let (M , d ) be a complete metric space and let b : 2M -> R be a measure of non-compactness satisfying the following conditions:

1° b(E) = 0 iff Ё is compact, 2° if E a F, then b{E) ^ b{F), 3° b{E) = b ( E u { x } ) for x e M .

Then, if for a chain , C) o f closed non-empty subsets of M we have inf (b(F): F e

F) = 0, then f] F Ф 0 .

Fe#

P roof. Let us consider a subsequence (Fn)neN such that, for all n e N , F„ + l c Fn and lim b(F„) = 0. We can choose elements x„ e Fn. Let us con-

П

• 00

sider the set D = (J {x„}. By 3° for all k e N we have n=

00 00

b(D) = b( U {*„}) = b( U {X„})

b(Fk) n=

n

and therefore b(D) = 0. All the limits of the convergent subsequences of (х„)ПбЛГ

00

are contained in F: = f) F„, which is closed and compact. Now we may n=

consider only a chain consisting of compact sets, which, as it is known, have a non-empty intersection. Q.E.D.

Let (X, T) be a linear topological space.

De f in it io n

1. We say that for Y с X a mapping / : Y -+ Y is generalized condensing if for A c Y and such that f ( A) c A from the compactness of A — c ô f ( A) it follows that A is compact (cf. [2]).

We present below a modification of the result obtained in [2].

204 L. Pasi cki

Theorem

2. If A = со A

X and / : A -*■ A is a generalized condensing mapping, rheu there exists a compact convex set C such that сô /(C ) = C .

Le m m a.

I f A = œ A , f : A - + A and there exists a compact set В a A such that f(B) a B, then there exists a С — со С c A such that œ f ( C) = C.

P roof. Let us consider a family Ж = (F = c ô F

A: cof ( F)

F).

It is non-empty, as X e . f . By the Hausdorff theorem there exists a maximal chain Ж cz Ж containing a set that has a non-empty intersection with B.

Let У

W consist of all sets G such that G n B

0 . The set С: = П G is non-empty because, by the compactness of B, f) G n B Ф 0 . Moreover, С = со С, с о /(C ) c C and c ô /( C ) e ^ , as f ( B r \ C ) c= B. By virtue of the definition of C it must be со /(С ) = C.

P r o o f o f th e th eo rem . In view of the lemma and the definition of the generalized condensing mapping, the theorem will be proved if we find a compact set В such that f ( B) с= B.

Let D be any compact subset of A, and В = В the minimal set containing D such that /(В ) с B. We see that В —/(В ) c D because B - J W ) n { X — D) is relatively open in В and would be rejected when non-empty. Now, by the definition of / , В is compact.

Corolla ry.

If (X , T ) is a topological space and for a mapping / : X -> X there exists a compact set В such that /(B ) c= B, then we can find a compact set С а X for which f ( C) = C.

P roof. We may follow the procedure that was used in the proof of our lemma, with substituting the operation “со” by “ ” (closure), and considering a maximal chain containing B.

Definition

2. We say that a mapping f : X ^ X is b-condensing (b : 2A -*■ R) if for all A

X such that b(f(A)) < oo and b ( A ) ^ b ( f ( A ) ) , f ( A ) is compact.

We see that if b satisfies 1°, 2° and

3°' b(E) = b ( œ E u Z) for all Z

A and b(Z) = 0,

then for Y

X and b(Y) <

any b-condensing mapping / : У ->У is generalized condensing.

Now, let (B, || II) be a Banach space and let s: 2B -> R denote the Hausdorff measure of non-compactness (for A

В : s(/l) = inf (re R: there exists a finite r-net in A)). It is known that for E, F cz B s satisfies 1° and the following conditions:

4° s { E u F ) = max (s(£), s(F)), 5° s ( œ E ) = s(E),

6

° if N e(E): = (

B : dist (x, E) ^ e), then there exists a function k\

R + -> R such that lim k{e) = 0, s ( Ne(E)) ^ s(E) + k(e).

Measures o f non-compactness 205

T

3. The measure of non-compactness b : 2B-+ R satisfies 1°, 4°-6°

iff there exists a non-decreasing continuous function g: R -»• R such that b = g o s . P roof, (i) For x e B , r e R + and the balls К с В we have lim b ( K( x , r + e))

= b( K( x, r) ) .

Let 0 < e; then in view of 4° and, on the other hand, by

°, b( K( x, r) ) ^ b ( K ( x , r + e ) ) ^ b ( K( x , r)) + k(e),

and similarly, for — r ^ e <

,

b( K( r — e)) ^ b( K( x, r) ) ^ b ( K( x , r — e)) + k(e)

which proves (i).

(ii) For all x e B , r c R + b( K( x, r ) ) = b(X (0,r)).

For each 0 < e < r we can choose t e R + such that K ( x , r — e) cz со [K(0, г) и {fjc}} and therefore by 4° and 5°, b ( K ( x , r — e)) ^ b(K(0, r)) and similarly b (K (0 ,r — e)) ^ b(K (x,r)). Now (ii) is a consequence of (i).

(iii) Let E с B, s(E) = r. From the definition of s and by 4°, (ii) and

° we have:

b(E) = b ( [ j K ( x ± , r + e)) = b ( K ( 0, r + e)) ^ b(E) + k(e),

i e l e

where E a ( (J К ( х ^ г + е)) and Ie is finite. Therefore

i e / e

b{E) = b ( K (0, r)) = : g (r) = g o s ( K { 0

r)) = gos { E)

which with (i) and 4° gives the thesis of our theorem.

Coroll ar y.

I f f : B —>B is a b-condensing mapping and b satisfies

4°-6°, then f is s-condensing.

Proof.. Obviously, if s (/(^ )) < s(X), then b(f(A)) ^ b(A) and equva- lently, because of the properties of g, from b( A) ^ b(f(A)) it follows that s(+ ) ^ s(f(A)).

References

[1] K. K u ra to w s k i, Sur les espaces completes, Fund. Math. 15 (1930), p. 301-309.

[2] E. A. L ifsch ic, B. N. S a d o v sk ij, Theory of fixed point for generalized condensing operators (in Russian). D.A.N. SSSR 183.2 (1968), p. 278-279.