Measures of Noncompactness and Condensing Operators

OT55 Operator Theory: Advances and Applications Vol. 55 Editor: I. Gobberg Tel Aviv University Ramat Aviv, Israel
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M. A. Kaashoek (Amsterdam) T. Kailath (Stanford) H. G. Kaper (Argonne) S.T.Kuroda(Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) E. Meister (Darmstadt) B. Mityagin (Columbus) J. D. Pincus (Stony Brook) M. Rosenblum (Charlottesville) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville)
M. S. Livsic (Beer Sheva) R. Phillips (Stanford) B. Sz.-Nagy (Szeged)
R. R. Akhmerov M.I. Kamenskii A.S.Potapov A.E. Rodkina B . N . Sadovskii
Measures of Noncompactness and Condensing Operators
Translated from the Russian by A. lacob
Springer Basel AG 1992
Originally published in 1986 under the title "Mery Nekompaktnosti i Uplotnyayushchie Operatory" by Nauka. For this translation the Russian text was revised by the authors.
Authors' addresses:
R.R. Akhmerov A.S. Potapov Inst. Comput. Technologies Voronezh State Teach. Training Institute Lavrentjeva 6 Faculty of Physics and Mathematics 630090 Novosibirsk ul. Lenina 86 USSR 396611 Voronezh
USSR M.I. Kamenskii B.N. Sadovskii A.E. Rodkina Voronezh State University Voronezh Institute of Department of Mathematics Civil Engineering Universitetskaja pi. 1 ul. 20 let Oktjabrija 64 394693 Voronezh 394006 Voronezh USSR USSR
ISBN 978-3-0348-5729-1 ISBN 978-3-0348-5727-7 (eBook) DOI 10.1007/978-3-0348-5727-7
Deutsche Bibliothek Cataloging-in-Publication Data
Measures of noncompactness and condensing operators / R. R. Akhmerov . . . Transi, from the Russian by A. Iacob. - Basel ; Boston ; Berlin : Birkhäuser, 1992
Einheitssacht.: Mery nekompaktnosti i uplotnjajuscie operatory <engl.> ISBN 978-3-0348-5729-1
NE: Achmerov, Rustjam R.; EST
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© Springer Basel AG 1992 Originally published by Birkhäuser Verlag Basel in 1992 Softcover reprint of the hardcover 1st edition 1992
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TABLE OF CONTENTS
1.1. The Kuratowski and Hausdorff measures of noncompactness
1.2. The general notion of measure of noncom pact ness
1.3. The measure of noncompactness f3
1.4. Sequential measures of noncompactness
1.5. Condensing operators
2.1. Fredholm operators
2.3. Fredholmness criteria for operators
2.4. The (lfl, lf2 )-norms of an operator
2.5. The measure of noncompactness of the conjugate operatm
2.6. The Fredholm spectrum of a bounded linear operator
2.7. Normal measures of noncompactness and perturbation theory
for linear operators
Vll
1
1
9
13
17
21
27
35
44
53
53
55
57
61
67
73
81
94
3.1. Definitions and properties of the index
3.2. Examples of computation of the index of a condensing operator
3.3. Linear and differentiable condensing operators
99
105
107
VI
3.4.
3.5.
3.6.
3.7.
3.S.
3.9.
Further properties of the index
Generalization of the notion of index to various classes of maps
The index of operators in locally convex spaces
The relative index
Survey of the literature
4.2. Ito stochastic equations with deviating argument
4.3. The Cauchy problem for equations of neutral type
4.4. Periodic solutions of an equation of neutral point with small delay
4.5. The averaging principle for equations of neutral type
4.6. On the stability of solutions of equations of neutral type
4.7. Floquet theory for equations of neutral type
4.S. Continuous dependence of the Floquet exponents on the delay
4.9. Measures of noncompactness and condensing operators in spaces of
integrable functions
111
lIS
128
133
137
142
151
151
159
164
175
190
203
212
216
220
233
245
INTRODUCTION
A condensing (or densifying) operator is a mapping under which the image of any set
is in a certain sense more compact than the set itself. The degree of noncompactness of a
set is measured by means of functions called measures of noncompactness.
The contractive maps and the compact maps [i.e., in this Introduction, the maps that
send any bounded set into a relatively compact one; in the main text the term "compact"
will be reserved for the operators that, in addition to having this property, are continuous,
i.e., in the authors' terminology, for the completely continuous operators] are condensing.
For contractive maps one can take as measure of noncompactness the diameter of a set,
while for compact maps can take the indicator function of a family of non-relatively com
pact sets. The operators of the form F( x) = G( x, x), where G is contractive in the first
argument and compact in the second, are also condensing with respect to some natural
measures of noncompactness. The linear condensing operators are characterized by the
fact that almost all of their spectrum is included in a disc of radius smaller than one.
The examples given above show that condensing operators are a sufficiently typical
phenomenon in various applications of functional analysis, for example, in the theory of
differential and integral equations.
As is turns out, the condensing operators have properties similar to the compact ones.
In particular, the theory of rotation of completely continuous vector fields, the Schauder
Tikhonov fixed point principle, and the Fredholm-Riesz-Schauder theory of linear equations
with compact operators admit natural generalizations to condensing operators. Therefore,
establishing that a given problem for a differential or integral equation reduces to an
equation with a condensing operator yields a considerable amount of information on the
properties of its solutions.
The first to consider a quantitative characteristic a(A) measuring the degree of non
compactness of a subset A in a metric space was K. Kuratowski in 1930, in connection
with problems of general topology. In the mid Fifties in the works of G. Darbo, L. S.
Gol'denshtein, I. Gohberg, A. S. Markus, W. V. Petryshyn, A. Furi, A. Vignoli, J. Danes,
Yu. G. Borisovich, Yu. I. Sapronov, M. A. Krasnosel'skil, P. P. Zabrelko and others various
Vlll INTRODUCTION
measures of noncompactness were applied in the fixed-point theory, the theory of linear
operators, and the theory of differential and integral equations.
This book gives a systematic exposition of the notions and facts connected with mea
sures of noncompactness and condensing operators. The main results are the characteri
zation of linear condensing operators in spectral terms and theorems on perturbations of
the spectrum (Chapter 2), and the theory of the index of fixed points of nonlinear con
densing operators, together with the ensuing fixed-point theorems (Chapter 3). Chapter
1 is devoted to the main definitions, examples, and simplest properties of measures of
noncompactness and condensing operators. In Chapter 4 we consider examples of appli
cations of the techniques developed here to problems for differential equations in Banach
spaces, stochastic differential equations with delay, functional-differential equations of neu
tral type, and integral equations.
In the treatment of the theory itself as well as of its applications we aimed at consid
ering the simplest situation, leaving the comments concerning possible generalizations for
the concluding sections or subsections. For additional information the reader is referred
to the surveys [10, 28, 160j.
The authors use this opportunity to express their gratitude to Mark Aleksandrovich
Krasnosel'skil, under whose influence many of the problems discussed here were posed and
solved.
MEASURES OF NONCOMPACTNESS
In this chapter we consider the basic notions connected with measures of noncom
pactness (MNCs for brevity) and condensing (or densifying) operators. We define and
study in detail the three main and most frequently used MNCs: the Hausdorff MNC X,
the Kuratowski MNC a, and the MNC (3. We derive a number of formulas that enable us
to compute directly the value of the Hausdorff MNC of a set in some concrete spaces. We
give the general definition of the notion of an MNC, study the so-called sequential MNCs,
and establish their connection with MNCs. We define and study the condensing operators,
and we give examples of maps that are condensing with respect to various MNCs. And
finally, we bring into consideration the ultimately compact operators and J{ -operators as
natural generalizations of the condensing maps.
1.1. THE KURATOWSKI AND HAUSDORFF MEASURES OF
NONCOMPACTNESS
In this section we define the Kuratowski and Hausdorff MNCs and study their basic
properties. The setting is that of a Banach space Ei we let n denote subsets of E, and we
use B(x, r) and B(x, r) to denote the open and respectively the closed ball in E of radius
r and center Xi B = B(O, 1).
1.1.1. Definition. The Kuratowski measure of noncompactness a(n) of the set n is
the infimum of the numbers d > 0 such that n admits a finite covering by sets of diameter
smaller than d.
As usual, by the diameter diam A of a set A one means the number sup{ Ilx - yll: x, y E
A}, which for A unbounded [empty] is taken to be infinity [resp. zero].
2 Measures of noncompactness Chap. 1
1.1.2. Definition. The Hausdorff MNC x(n) of the set n is the infimum of the
numbers 10 > 0 such that n has a finite c;-net in E.
Recall that a set SeE is called an c;-net ofn if n c S + lOB == {s + lOb: s E S, bE B}.
1.1.3. Remarks. (a) In the definition of the Kuratowski MNC one can replace
"diameter smaller than d" by "diameter no larger than d" j similarly, in the definition of
the Hausdorff MNC it is immaterial how the c;-net is defined -by closed or by open balls
of radius 10.
(b) In the definition of the Hausdorff MNC, instead of a finite c;-net one can speak of
a totally bounded one, i.e., an c;-net S that has a finite 6-net for any 6 > o.
(c) The definitions of the MNCs a and X are meaningful not only for Banach, but also
for arbitrary metric spaces.
1.1.4. Elementary properties of the Kuratowski and Hausdorff MNCs. We
list below some of the properties of the MNCs a and X that follow immediately from
the definitions. The terminology introduced in order to formulate these properties will
also be used for other MNCs. For this reason we also include in our list some properties
that are straightforward consequences of others (for example, nonsingularity follows from
regularity, monotonicity from semi-additivity, continuity from Lipschitzianity).
Thus, the MNCs a and X (denoted below by 1jJ) enjoy the following properties:
a) regularity: 1jJ(n) = 0 if and only n is totally boundedj
b) nonsingularity: 1jJ is equal to zero on everyone-element setj
c) monotonicity: n l C n 2 implies 1jJ(nt} :::; 1jJ(n2)j
d) semi-additivity: 1jJ(nl U n 2) = max{1jJ(nl ),1jJ(n2)}j
e) Lipschitzianity: 11jJ(nt} -1jJ(n2)1 :::; L",p(nl ,n2 ), where Lx = I,Lo = 2 and p denotes the Hausdorff metric (more precisely, semimetric): p(nt,n2) = inf{c; > o:nl + lOB:::> n 2,n2 + lOB :::> nl}j
f) continuity: for any neE and any 10 > 0 there is a 6> 0 such that 11jJ(n)-1jJ(nl)1 < 10 for all n l satisfying pen, n l ) < 6j
g) semi-homogeneity: 1jJ(tn) = Itl1jJ(n) for any number tj
h) algebraic semi-additivity: 1jJ(nl + n 2) :::; 1jJ(nt} + 1jJ(n2)j
i) invariance under translations: 1jJ(n + xo) = 1jJ(n) for any Xo E E.
The following two properties are isolated as separate subsections in view of their
importance.
1.1.5. Theorem. The Kuratowski and Hausdorff MNCs are invariant under passage
to the closure and to the convex hull: 1jJ(n) = 1jJ(Q) = 1jJ( co n).
Sec. 1.1 The Kuratowski and Hausdorff measures of noncompactness 3
Proof. The invariance under passage to the closure is obvious. The invariance of X
under passage to the convex hull is also quite readily established: if S is a finite c-net of
the set f!, then co S is a totally bounded c-net of the set co f!.
Let us prove that a( co f!) = a(f!). Suppose f! = U;;'=t f!k and diam f!k < d for all k.
It is readily checked that co f! is the union of all possible sums of the form E;;'=t AkCO f!k, where the vector A = (At, ... ,Am) runs through the standard simplex u (i.e., Ak ;::: 0 and
E;;'=t Ak = 1). Let c > O. The union of all such sums can be approximated, with arbitrary
accuracy 8(c) in the sense of the Hausdorff metric (8(c) -+ 0 as c -+ 0), by finite unions of
sums of the same form, in which A runs through a finite c-net U e of the simplex u. Now
from the properties of the Kuratowski MNC we obtain
m m
m m
a(cof!) $ d + 28(c),
a( co f!) $ a(f!).
The opposite inequality is obvious. QED
1.1.6. Theorem. Let B be the unit ball in E. Then a(B) finite-dimensional, and a(B) = 2, X(B) = 1 in the opposite case.
X(B) = 0 if E is
Proof. The first assertion follows from the regularity of the MNCs a and x. Turning to the second assertion, we first prove it for x. Clearly, the center of the ball
B forms a I-net for B, and so X(B) $ 1. Suppose X(B) = q < 1. Pick c > 0 such that
q + c < 1, and let {x 1, . .. ,x m} be a (q + c)-net for B:
m
From the properties of the MNC X it follows that
q = X(B) $ (q + c)X(B) = q(q + c).
But this implies q = 0 (because q + c: < 1), which is possible only if B is totally bounded.
This contradicts the infinite-dimensionality of the space E.
To prove the second assertion for a we make use of the Lyusternik-Shnirel'man-Borsuk
theorem on antipodes (see [84]): if S is a sphere in an n-dimensional normed space and Ak
(k = 1, ... , n) is a cover of S by closed subsets of that space, then at least one of the sets
Ak contains a pair of diametrically opposite points, i.e., diamA ~ diamS. Thus, suppose
that E is infinite-dimensional. Clearly, a(B) ::; 2. Suppose a(B) < 2. Then B c U;=l n k, where diam n k < 2 for all k = 1,... , n (wi th no loss of generality one can assume that
all the sets n k are closed). Now taking the section of B by an arbitrary n-dimensional
subspace En and setting Ak = nk n En, we arrive at a contradiction with the theorem on
antipodes. QED
1.1. 7. Theorem. The K uratowski and Hausdorff MNCs are related by the inequali-
ties
x(n) ::; a(n) ::; 2X(n).
In the class of all infinite-dimensional spaces these inequalities are sharp.
Proof. The inequalities themselves are consequences of the following obvious remarks:
1) if {XI, . .. , Xm} is an c-net of n, then {nn(Xk+cB)}k'=1 is a cover of n by sets of diameter
2cj 2) if {ndk'=l is a cover of n with diamnk ::; d and if Xk E n k, then {Xl, ... ,Xk} is a
d-net of n. The sharpness of the second inequality follows from Theorem 1.1.6. The following
example shows that the first inequality is also sharp. Take for E the space Co of sequences
of numbers that converge to zero, with the norm IIxll = sup Ixd, and let n = {edk:l be
the standard basis in co. Since the diameter of any set containing more than one element is
equal to 1, a(n) = 1. On the other hand, X(n) = 1 because the distance from any infinite
subset of n to any element of Co is not smaller than 1. QED
We should mention here that for some spaces E the inequality X(n) ::; a(n) can be
improved. For instance, one can show that for the space lp one has
V'2x(n) ::; a(n).
Let us prove one more important property of the MNCs a and x.
1.1.8. Theorem. The intersection of a centered system of closed subsets of a Banach
space is nonempty if this system contains sets of arbitrarily small K uratowski (or, which
in view of Theorem 1.1.7 is equivalent, Hausdorff) MNC.
Sec. 1.1 The Kuratowski and Hausdorff measures of noncompactness 5
Proof. Let 9J1 be the given centered system. Notice that if 9J1 contains a set 110
which has MNC equal to zero, and hence, thanks to the regularity of a, is compact, then
the assertion of the theorem is a trivial consequence of the definition of compactness: it
suffices to pass to the system 9J1' = {11 n no: 11 E 9J1}. In the general case, we pick a
sequence 11n E 9J1 such that O'(11n) -4 0 as n -4 00 and we show that the set 110 = n~l 11n is compact, and that after adding 110 to 9J1 the system remains centered. As above, this
will imply the assertion of the theorem.
The compactness of 110 follows from its closedness and the obvious fact that a(11o) = o. Now let us show that for any finite subsystem 9J10 C 9J1 the set A = (nf!E!lJlo 11) n 110 is
nonempty. We choose a sequence {xn} such that Xn E (nf!E!lJlo 11) n(n~=l 11k). Since
a({xn}~=l) = a({xn}~=N)::; a(11N) -4 0 as N -4 00, this sequence is relatively compact.
Consequently, the set of its limit points is nonempty. It remains to remark that every limit
point belongs to A. QED
We next prove a number of formulas that enable us to compute the Hausdorff MNC
in the spaces lp, Co, C, Lp and Loo.
1.1.9. The Hausdorff MNC in the spaces lp and Co. In the spaces lp and Co
of sequences summable in the p-th power and respectively sequences converging to zero the
MNC X can be computed by means of the formula
X(11) = lim sup 11(1 - Pn)xll, (1) n--+oo xEf!
where P n is the projection onto the linear span of the first n vectors in the standard basis.
Proof. If Q is a [X(n) + f]-net of n, then 11 C Q + [X(11) + f]B. Hence, one can
represent each x E n in the form x = q+ [X(11)+f]b, where q E Q and b E B. Consequently,
sup 11(1 - Pn)xll ::; sup 11(1 - Pn)qll + [X(11) + fl· xEf! qEQ
Since Q is finite, the first term in the right-hand side tends to zero when n -4 00, and so
lim sup 11(1 - Pn)xll ::; X(11) + f, n--+oo xEf!
which in view of the arbitrariness of f yields one of the inequalities needed to establish (1).
To prove the opposite inequality, we notice that
Using the properties of X and the total boundedness of PnD, we obtain
xeD) ~ x(PnD) + X[(1 - Pn)D] = X[(1 - Pn)D] ~ sup 11(1 - Pn)xll· xEO
Since n is arbitrary, this gives
xeD) ~ lim sup 11(1 - Pn)xll. QED n-.oo xEO
1.1.10. The Hausdorff MNC in the space era, b]. In the space C[a, b] of continu
ous real-valued functions on the segment [a, b] the value of the set-function X on a bounded
set D can be computed by means of the formula
xeD) = ~ lim sup max Ilx - xrll, 26-.0 xEO 0:-:::r:-:::6
where Xr denotes the T-translate of the function x:
{ X(t+T), ifa~t~b-T, xr(t) =
x(b), ifb- T ~ t ~ b.
(2)
Proof. Pick an arbitrary c > 0 and construct a finite [XeD) + c]-net Q of the set D. Let xED. Denote by y an element of Q such that IIx - yll :::; X(D) + c. Finally, let 6> 0
and T E [0,6]. Then
Ilx - xrll :::; Ilx - yll + Ily - yrll + IIYr - xrll :::; 211x - yll + IIY - yrll
Consequently,
sup max Ilx - xrll ~ 2x(D) + 2c + max max IIY - Yrll. xEO 0:-:::r:-:::6 yEQ 0::;r:-:::6
Letting 6 -+ 0 and taking into account that the finite family Q is equicontinuous, one
obtains
lim sup max Ilx - xrll ~ 2x(D) + 2E, 6-.0 xEO 0:-:::r:-:::6
which in view of the arbitrariness of c yields the inequality
1 . - hm sup max Ilx - xrll ~ xeD). 26-.0 xEO 0::;r:-:::6
(3)
Sec. l.1 The Kuratowski and Hausdorff measures of noncompactness 7
In proving the opposite inequality we shall assume that the functions x E n are
extended from the segment [a, b] to the whole real line by the rule: x(t) = x(a) for t s:: a, x(t) = x(b) for t ~ b. We define the operators Rh and Ph (h > 0) through the formulas
1 . (RhX)(t) = 2(max{x(s):s E [t-h,t+h]}+mm{x(s):s E [t-h,t+h]})
and respectively 1 jt+h
(PhX)(t) = 2h x(s)ds. t-h
It is not hard to see that the set PhRh(n) is relatively compact in C[a, b]. We claim that
it constitutes a (q2h/2)-net of the set n, where q2h = sUPxE!1 maxO~T~h Ilx - xTII. In fact,
1 jt+h 1 jt+h IIPhRhx - xii = max I-h (RhX)(s)ds - -h x(t)dsl
a~t9 2 t-h 2 t-h
1 jt+h s:: 2h max I(RhX)(S) - x(t)lds. a~t~b t-h
(4)
If It - sl s:: h, then obviously
min{x(r):r E [s - h,s + h]} s:: x(t) s:: max{x(r):r E [s - h,s + h]}.
Consequently, I(RhX)(S) - x(t)1 s:: 2\ maxO~T~2h Ilx - xTII, whence I(RhX)(S) - x(t)1 < q2h/2. From this and (4) it follows that X(n) s:: Q2n/2. Letting h ----. 0 we obtain
1 . X(n) ::; - hm sup max Ilx - xTII,
2 6~O xE!1 O~T:<;6
which completes the proof of equality (2). QED
(5)
1.1.11. Generalization to the space C( J{, R m). Formula (2) admits the following
genemlization to the case of the space C( J{, R m) of continuous functions on a compact
space I{ with values in Rm (see [23]), equipped with the norm Ilxllc = maXtEK Ilx(t)ll:
X(n) = sup inf supradx(V), B VEB xE!1
(6)
where B is a basis of neighborhoods of some point of J{ and rad x(V) denotes the infimum
of the milii of all balls in R m that contain x(V).
1.1.12. Generalization to the space Lex>([a, b], R m). Let Lex>([a, b], R m) be the
space of eqni1Jalence classes x of measurable, essentially bounded functions ( [a, b] ----. R m,
endowed with the norm Ilxll = vraisuPtE[a,bjll~(t)11 = infeEx SUPtE[a,bjIIW)II. Then for
mula (6) remains valid in Loo([a, b], Rm) for an appropriate interpretation of the notations
involved, namely, B stands for an arbitrary maximal filter of measurable sets in [a, b] and
xCV) stands for n ~(V), where ~ runs through x and V runs through the set of all subsets
of V of full measure.
1.1.13. The Hausdorff MNC in Lp[a, b]. In the space Lp[a, b] of equivalence classes
x of measurable functions ~: [a, b] -+ R with integrable p-th power, endowed with the norm
IIxll = U: Ix(t)IPdt)l/p, the Hausdorff MNC can be computed by means of the formula
x(12) = ~ lim sup max Ilx - xrll, 2 6-.0 rEf! 0:-:;r:-:;6
(7)
where Xr denotes the 7-translate of the function x (see 1.1.10) or, alternatively, the Steklov
function
27 t-r
(here x is extended outside [a, b] by zero).
We conclude this section by describing how the notions of Kuratowski and Hausdorff
MNC can be extended to uniform (in particular, locally convex) spaces.
1.1.14. The Kuratowski and Hausdorff MNCs in uniform spaces. Let E be
a uniform space, P a family of pseudometrics that are uniformly continuous on E X E, 9Jl
the set of all subsets of E that are bounded with respect to any pseudometric PEP, and
A the set of all functions a: P -+ [0,00), endowed with the uniform structure generated by
pointwise convergence and with the natural partial order: al :::; a2 means al (p) :::; a2 (p)
for all pEP.
1.1.15. Definition. The Kuratowski [resp. HausdorffJ measure of noncompactness
on the space E generated by the family of pseudometrics P is the function a: 9Jl -+ A [resp.
X:9Jl-+ A] defined as [a(12)](p) = inf{d > 0: 12 admits a finite partition into subsets whose
diameters with respect to the pseudometric p are no larger than d} [resp. [X(Q)](p) = inf{e > 0: 12 has a finite e-net with respect to p}].
The properties described in 1.1.14 can be reformulated in an obvious manner for the
Kuratowski and Hausdorff MNCs in uniform spaces.
1.1.16. The inner Hausdorff MNC. It is readily seen that the Kuratowski MNC
of a set 12 is an "intrinsic characteristic" of the metric space (Q, p), where p is the metric
induced by the norm on E. In contrast, the Hausdorff MNC depends on the "ambient"
Sec. 1.2 The general notion of measure of noncompactlless
space E, specifically, on how much freedom one has in the choice of the clements of an E-net.
The definition of the Hausdorff MNC can be modified so that it becomes an intrinsic char
acteristic of sets. Specifically, we define the inner H a1lsdorjj mca.811.TC of n07l.compacincss
Xi(n) of the set n to be the number
Xi(n) = inf{E > 0: n has a finite E-net in n}.
Clearly, the set-function Xi is nonsingular, Lipschitzian, semi-homogeneous, alge
braically semi-additive, and invariant under translations. It is also readily seen that Xi
is regular and invariant under passage to the closure. At the sCllne time, Xi is not invariant
under passage to the convex hull. In fact, let E = m, the space of bounded numerI
cal sequences, and let n = {Xn, -xn}~=l' where Xn = (1, ... ,1, -1, 1, 1, ... ). Clearly, '--v--'
It-I
Xi(n) ~ 2, since the distance between any two points of n is equal to~. On the other
hand, co n contains the zero element of E, and the distance from zero to any point of
con does not exceed 1. Hence, Xi(n) :::; 1. On the same example nne can check that Xi is
neither monotone, nor semi-additive.
1.2. THE GENERAL NOTION OF MEASURE OF NONCOMPACTNESS
In this section we give an axiomatic definition of the notion of a rneasure of nonCOll1-
pactness and consider a number of examples. As we already mentioncd, an important role
is played by the invariance of the Kuratowski and Hausdorff .tvINCs under passage to the
convex closure.
1.2.1. Definition. A function lj;, defined on the set of all su.lJset,\ of II. Banach spa.CC
E with values in some partially ordered set (Q, :S:), is called II. m.CIJ.8'f1.n; of n07l.c01fl.pIJ.cinc.5s
iflj;(con) = lj;(n) for a.ll neE.
As we established above, the Kuratowski and Hausdorff :tvINCs satisfy the condition
of this general definition. On the contrary, the inner Hausdorff :tvINC (see 1.1.16) is not
an MNC in the sense of the general definition.
The term "measure of noncompactness" IS not completely appropriatc for such a
general definition, because only the functions that were silid above to be regular serve
to actually measure the degree of noncompactness. However, in various problems it is
convenient to use different MNCs, including nonregular ones (and which nevertheless have
some connection with noncompactness), and it turns out that the ollly property they share
is precisely the invariance under passage to the convex closure. All the notions enumerated
in 1.1.4 carryover to general MNCs. In defining Lipschitzianity [continuity] it is natural to
require that Q be endowed with a metric [resp. topology], and in defining semi-homogeneity
and algebraic semi-additivity it is necessary that an operation of multiplication by scalars
and respectively one of addition be given on Q.
and
1/;1 (Q) = { 0,
otherwise,
are MNCs (in the sense of Definition 1.2.1). Notice that if E is infinite-dimensional then
1/;1 is not continuous (with respect to the natural topology on the set where it takes its
values, the real line ). It also clear that 1/;2 is not a regular MNC.
1.2.3. Products of MNCs. Let 1/;1, ... ,1/;n be MNCs in E with values in Q1, . .. , Qn,
respectively, and let F: Q1 x .. , X Qn --t Q be a map. Then, as is readily verified, the
function 1/;(Q) = F(1/;1(Q), ... ,1/;n(Q)) is an MNC; its properties are determined by the
properties of the measures 1/;1, ... ,1/;n and of the map F. In the following subsections we give a number of examples of MNCs in concrete spaces.
Most of them will be needed later.
1.2.4. MNCs in C([a, b], E). Let E be a Banach space with norm II . II and 1/;
be a monotone MNC on E. Let C([a, b], E) denote the space of all continuous E-valued
functions on [a, b] with the norm IIxll = maxtE[a,b] IIx(t)lI. Define a scalar function 1/;c on
the bounded subsets of C([a, b], E) by the formula
1/;o(Q) = 1/;(Q[a, b]),
where Q[a, b] = {x(t): x E Q, t E [a, b]}. 1/;c is an MNC in the sense of the general definition.
In fact,
(co Q)[a, b] C co(Q[a, b])
for any bounded subset Q C C([a, b], E), whence 1/;0( co Q) :::; 1/;o(Q). The opposite in
equality is obvious.
\Ve should remark here that if the MNC 1/; is regular and the set Q is equicontinuous,
then by the Ascoli-Arzela theorem 1/;o(Q) = 0 is equivalent to Q being relatively compact.
Sec. 1.2 The general notion of measure of noncompactness 11
In Chapter 4 we shall need two other MNCs in C([a, b], E), tPh and tPb (see 4.1.6 and
4.2.6), defined as follows. Let mfa, bj and lJl[a, bj denote the partially ordered linear space
of all scalar functions defined on [a, bj and respectively the subspace of mfa, bj consisting of
all nondecreasing functions. The MNCs tPh and tPb on C([a, b], E) with values in mfa, bj
and lJl[a, b], respectively, are defined by means of the formulas
[tP~(n)](t) = tP[n(t)j
[tPb(n))(t) = Xt(nt),
where net) = {x(t):x E n} c E, n t = {Xt = xl[a,tj:x E n} c C([a,b],E), and Xt is
the Hausdorff MNC in the space C([a, tj, E). The MNC tPb is monotone, nonsingular,
invariant under translations, semi-additive, algebraically semi-additive, and continuous. If
tP enjoys the properties enumerated, then so does the MNC tPh. If tP is continuous and the
set n is equicontinuous, then tPhCn) E C[a, bj.
1.2.5. An MNC in Cl([a, bj, E). Let Cl([a, bj, E) denote the Banach space of
the continuously differentiable functions x: [a, bj -+ E, equipped with the norm Ilxllcl = Ilxllc + Ilx'llc. The m[a,bj-valued function tPCI, defined on the bounded subsets of
Cl([a,b],E) by the formula
where n'(t) = {x'(t): x En}, is an MNC. If the set n' = {x': x E n} is equicontinuous and
the MNC tP is continuous, then tPCI (n) E C[a, bj.
1.2.6. An MNC in Cn([a, bj, E). Let Cn([a, b], E) denote the Banach space of
the n-times continuously differentiable functions x: [a, bj -+ E, endowed with the norm
IIXllcn = L:?=o Ilx(i)llc· Then each MNC tP on C([a,b],E) generates an MNC tPcn on
Cn([a, bj, E) by the rule
where n(n) = {x(n): x En}.
1.2.7. An MNC in co. Let Co be the Banach space of the numerical sequences
that converge to zero, with the norm Ilxll = max IXil. Let n(x) denote the number of
coordinates of the vector x which are larger than or equal to 1. For an arbitrary bounded
set n C Co we put
n(n) = minn(x), zEn
Clearly, n( co D) ::::: neD). On the other hand, if the first k coordinates of any vector xED are larger than or equal to 1, then the same holds true for the first k coordinates of any
vector y E co D, and so n( co D) ~ neD). This means that the function n, and together
with it the function 1/;, are MNCs. Next, it is readily seen that
and consequently
so that the MNC 1/; is semi-additive. Notice, however, that 1/; is not nonsingular, is not
invariant under translations, and is not regular.
We conclude this section by describing yet another MNC which, roughly speaking, is
different from zero on bounded noncompact sets only in nonreflexive spaces.
1.2.8. The measure of weak noncompactness. Let E be a Banach space and
B the unit ball in E. The function w:2 E ~ [0,(0), defined as w(D) = inf{e > O:D has a
weakly compact e-net in E}, is called the measure of weak noncompactness.
The measure of weak noncompactness is an MNC in the sense of the general definition
provided E is endowed with the weak topology. This assertion follows from the obvious
invariance of w under the passage to the convex hull and the invariance under the passage
to the weak closure, estahlished below:
w(D) = w(wclD), (1)
where wcl stands for weak closure.
Thus, let us prove (1). Suppose w(D) < e. By the definition of w, there exists a weakly
compact set C C E such that DeC + eB. By the Kreln-Shmul'yan theorem (see [34]),
the set co C is weakly compact, and hence weakly closed. Consequently, co C + eB is also
weakly compact. Now the inclusion D C co C + eB implies
wclD c coC + eB,
which in turn yields w(wclD)::::: e. In view of the arbitrariness of e > w(D), this gives
w(wclD) ::::: w(D).
The opposite inequality follows from the obvious monotonicity of w. This establishes
equality (1).
Sec. 1.3 The measure of noncompactness (3 13
As one can readily verify, the MNC w is nonsingular, semi-additive, algebraically
semi-additive, invariant under translations, and satisfies the inequality w(n) ~ X(n). The
regularity property for w has the following meaning: w(n) = 0 if and only if wcl (n) is
weakly compact (see [31]).
1.2.9. Theorem. The measure of weak noncompactness of the unit ball in E is equal
to zero if E is reflexive, and to one in the opposite case.
Proof. Recall that the unit ball B is weakly compact if and only if E is reflexive. The
first assertion of the theorem is obvious. Now suppose E is not reflexive. The inclusion
B C {O} + 1 . B gives w(B) ~ 1. Suppose w(B) < 1. Then there exist an c: E (0,1) and a
weakly compact set C such that B C C + c:B. But then B C co C + c:B, and consequently
(1 - c:)B + c:B c coC + c:B.
Now we use the following assertion [139]: if n1 + c:B c n2 + c:B, and if O2 is convex and
closed, then n1 C n2 . In our case this yields
(1- c:)B C coCo
Therefore, (1- c: )B is a weakly closed subset of the set co C, which is itself weakly compact
by the Krein-Shmul'yan theorem. Consequently, (1 - c:)B is weakly compact, and then so
is B, which contradicts the nonreflexivity of E. QED
1.3. THE MEASURE OF NONCOMPACTNESS {3
In this section we describe and study yet another MNC which is useful in applications.
1.3.1. Deflnition. The measure of noncompactness (3(n) of the subset n of the
Banach space E is the infimum of the numbers r > 0 for which n does not have an infinite
r-Iattice or, equivalently, the supremum of those r > 0 for which n has an infinite r-Iattice.
We remind the reader that a set 0 1 is called an r-lattice, or a lattice with parameter
r, if Ilx - yll ;::: r for all x,y E 0 1 • A set 0 1 cO with this property is called an r-Iattice
of o. Every r-Iattice n1 C 0 that is maximal (i.e., such that it cannot be enlarged to an
r-Iattice inside 0) is obviously an r-net of o.
1.3.2. Remarks. It is not hard to see that the MNC (3 is regular, monotone, semi
additive, semi-homogeneous, and invariant under translations and under passage to the
closure of the set. Further, it is a straightforward matter to check that the MNCs a, X and
(3 are related by the inequalities
xeD) ~ (3(D) ~ a(D).
Also, a simple argument establishes the Lipschitzianity of 13:
Less obvious properties of the MNC 13 are established in the next subsections.
1.3.3. Theorem. The MNC (3 is algebraically semi-additive.
Proof. Let Dl and D2 be arbitrary bounded subsets of E. Fix some f > O. By
the definition of (3, the set Dl + D2 has a countable [f3(Dl + D2) + f]-lattice {zd. Write
Zi = Xi + Yi, where Xi E Dl and Yi E D2. Then for all i i- j,
(1)
Let us show that from the sequence {Xi} one can extract a subsequence {u;} such that
(2)
Then in view of (1) the corresponding subsequence {v;} of {yd will be a [f3(D l + D2) (3(Dd - 2f]-lattice of D2, and consequently
In view of the arbitrariness of f, this implies the needed inequality f3(D l + D2) ~ f3(D]) + (3(D2)'
To extract the subsequence {ud consider some maximal [f3(Dd + f]-lattice of the set
{Xl,X2""}' It is finite and, as remarked above, it forms a [f3(Dd +f]-net of {X],X2,"'}' Hence, there is a term of the sequence {xd such that its closed [(3(Dd + f]-neighborhood
contains an infinite subsequence of {x;}. Denote the first term with this property x~ and
then take the terms of the subsequence described above that lie after x~ in the original
sequence and relabel them as x~, i = 2,3, .... Thus, we extracted from {xd an infinite
subsequence, all of whose terms lie at distance ~ (3( D]) + f from its first term. In exactly
the same manner, from the sequence {x~: i ~ 2} we can extract a subsequence {x;: i ~ I}
with the same property, and then continue to produce subsequences {xt}, {xt}, and so on.
Sec. 1.3 The measure of noncompactness (3 15
Now define the sought-for sequence to be Ui = x:, i = 1,2, .... The recipe used guarantees
that inequality (2) holds for all i and j. QED
1.3.4. Theorem. The function (3 is invariant under passage to the convex hull, and
hence it is an MNC in the sense of the general definition.
Proof. In view of the monotonicity of (3 it suffices to prove the inequality (3( co n) ~
(3(n). For an unbounded set it is obvious, so we shall assume that n is bounded. Suppose
that the needed inequality does not hold for some n, and pick numbers b and c such that
(3(n) c) and fix in
it an arbitrary element i11. Now i11 is a finite convex combination of elements of n. As is
readily verifiedl, among the latter one can find x suc!: that Ilx - YII ~ rl for Y belonging to
an infinite set Y I C YI. Setting YI = x, Y1 = {YI} U Y I, we get the first two objects in our
construction.
Further, consider the sets n~ = n n B(YI , b), n~ = n" n~ , and notice that any Y E co n
can be represented as Y = (1- /-LI)U~ + /-LluL with u~ E coni, i = 0,1 and /-LI E [0,11. For
each Y fix such a representation, defining in this manner two functions: /-LI = /-LI (y) and
u~ = ui (y). Denote the set {ui: Y E Yd by ut and define a binary indicator al as follows:
al = 1 if (3(UI) > c and al = 0 otherwise.
In the second case, when al = 0, one necessarily has (3(Un > c. This follows from
the inclusion Y1 C c02(Uf U UI) (here CONn stands for the set of all convex combinations
of at most N elements of n), the inequality
(3)
which is established in the next subsection, the monotonicity and semi-additivity of the
function (3, and the fact that (3(YJ) > c. Therefore, in either of the two cases, (3(Uf') > c.
Proceeding in analogous manner we construct objects Yn, Yn, n~, /-Ln, u~, U~ (i = 0,1),
an for each positive integer n, taking care at each new step to single-out a set Yn in
U~n. In more detail, Yn+l is an rn+rlattice (rn+! > c) in con~n, Yn+! E Yn+l n n~n, Yn+l " {Yn+!} C U~n, n~+l = n~n n B(Yn+! , b), n~+! = n~n "n~+!, the functions /-Ln+! = /-Ln+I(U~n), U~+l = u~+!(u~n), are defined for u~n E con~n by some fixed decompositions
u~n = (1- /-Ln+!)U~+1 = /-Ln+IU~+I' U~+l = {U~+l: u~n E Yn+d, an+! = 1 if (3(U~+I) > c,
and an+! = 0 in the opposite case. Throughout the construction (3(U:~.1') > c.
Notice that if an = 1 and m ~ n + 1, then IIYm - Ynll > b. In fact, by construction,
Ym E n~m C n~n = n~, i.e., Y ¢ B(Yn,b). This immediately implies that the set {Yn:an =
I} is a b-Iattice in n. Since b > !3(n), the set {n:an = I} is finite.
Thus, there is a k such that an = 0 for n ~ k, and hence Yn " {Yn} C U~_l for
n > k. Let m > k. Since U~+l is a function of u~n and an = 0 for n ~ k, u?" is a function
of u~. Consider the set Y{t-l = {u~ E Yk+l:U?" E Ym}, which, like Yk+l, is an infinite
rHrlattice; also, !3(Y{t-l) ~ rHl > c.
We claim that every element u~ E Yk+1 is representable in the form
m-k
u~ = L 6j Uk+j + 6u?", (4) j=l
where 6j ~ 0, 6 = rrj=~k(l - f.lk+j), L:j=~k 6j + 6 = 1. Indeed, for m = k + 1 this is
precisely the representation u~ = (1 - f.lk+l)U~+l + f.lk+luk+l' and the step from m to
m + 1 is made by substituting in (4) the analogous representation for u?,,: m-k
u% = L 6j Uk+l + 6[(1 - f.lm+dU~+l + f.lm+lU!,,+l] j=l
m-k
= L 6j Uk+j + 6f.lm+l U!,,+1 + 6(1 - f.lm+dU~+l· j=l
This is precisely a representation of the needed form. It is convenient to recast (4) in the
form m-k-l
u~ = L 6j Uk+j + (6m- k + 6)u!" + 6(u~ - u!,,). j=l
From this equality it follows that
m-k
Yk+1 C cOm-k ( U Uf+j) + 6B(O, d), j=l
where d is the diameter of co n. We next show that the f.li admit the bound f.li ~ P > 0,
and consequently 6 can be made arbitrarily small by taking m sufficiently large. Then
with the aid of (3) and (4) we conclude that for some j ~ 1,
which contradicts the equality aj+k = O.
The bound f.li ~ P = (c - b) / (d - b) follows from the relation
Sec. 1.4 Sequential measures of noncompactness 17
and the inequalities Ilu~~l' - y;ll ~ ri > C (u~~l' E Y;), Ilu? - Yill ::; b and Ilu} - Yill ::; d. Notice that the denominator d - b is strictly positive because d ~ (3( co n) > b.
To complete the proof it remains to establish inequality (3).
1.3.5. Lemma. For any nonnegative integer N,
where CONn denotes the set of all convex combinations of at most N elements of n.
Proof. We use the representation
CONn = U (Aln + A2 n + ... + ANn), AEtT
where A = (Al,A2,'" ,AN) runs through the standard simplex a. As in the proof of
Theorem 1.1.5, the above union can be approximated, with arbitrary accuracy 8(e:) in the
Hausdorff metric, by a finite union of the same form, where now A runs through a finite
e:-net a, of a. Using the Lipschitzianity, semi-additivity, algebraic semi-additivity, and
semi-homogeneity of the function {3, we get
(3(CONn) ::; {3 [ U (Aln + A2 n + ... + ANn)] + 28(e:) AEtT,
N
;=1
Since e: > 0 is arbitrary and tire:) --+ 0 as e: --+ 0, this yields the needed inequality (3). QED
1.4. SEQUENTIAL MEASURES OF NONCOMPACTNESS
To this point we took as the domain of definition of an MNC a colection of sets which,
together with any of its members n, contains the closure of its convex hull. Now let us
consider functions of countable sets. Of course, a collection of countable sets does not
satisfy the aforementioned requirement; nevertheless, the functions studied here are in
many respects analogous to the MNCs. Below E continues to denote a Banach space.
1.4.1. Definition. Let SE be the collection of all bounded and at most countable
subsets of the space E. A function t/J: SE -t [0,00) is called a sequential measure of
noncompactness if it satisfies the following condition: 111,112 E SE and 111 C co 112 implies
t/J(11d :::; t/J(112 ).
Lipschitzianity, algebraic semi-additivity, semi-homogeneity, and invariance under transla
tions for sequential MNCs are defined exactly in the same way as for ordinary MNCs.
It is an immediate consequence of the definition that every sequential MNC enjoys
the monotonicity property.
The next theorem shows that every sequential MNC t/J generates in E an ordinary
MNC ;fi, which "inherits" the properties of t/J.
1.4.2. Theorem. Let t/J be a sequential MNC in E. Then the rule
;fi(11) = sup{tP(C): C ESE, C C 11}, (1)
yields a monotone MN C in E, defined (and finite) on all bounded s ets. Moreover, if t/J has anyone of the properties enumerated in the preceding subsection, then ;fi also has that
property.
Proof. We first show that the function ;fi takes finite values on the bounded subsets
neE. Let en en (n = 1,2, ... ,) be countable sets such that
Let C = U::'=1 Cn; then clearly
Consequently, ;fi(11) = t/J(C) < 00.
En route we showed that in (1) one can always replace sup by max. The monotonicity
of ;fi is plain.
Now let us show that ;fi is an MNC. Suppose 11 is a bounded subset of E. The
inequality ;fi(11) :::; ;fi( co 11) follows from the monotonicity of;fi. To prove the opposite
inequality consider an arbitrary countable set C C co 11 and arrange its elements in a
sequence {Yn}. The inclusion Yn E co 11 is equivalent to Yn having a representation
r(m)
k=1
Sec. 1.4 Sequential measures of noncompactness 19
with Xnmk E n, CXnmk ~ 0, L:~~~) CXnmk = 1. Denote the set of all elements Xnmk by Cl.
Clearly, Cl is countable and C c co Cl , so that 1jJ( C) :::; 1jJ( Cl ). Thus, for any countable
set C c co n there exists a countable set Cl C n whose sequential MNC is at least equal
to that of C. This yields the needed inequality ~(con) :::; ~(n), and thus the first part of
the theorem is proved.
The proof of the second part is tedious, but trivial, and we omit it. QED
1.4.3. The MNC 1jJo. Any MNC 1jJ, defined on the bounded subsets of E, generates
in a natural manner a sequential MNC 1jJo (the restriction of the original measure to
the collection of all at-most-countable subsets of E). One is naturally led to asking:
under which conditions does the MNC ~o, constructed by means of formula (1) from the
sequential MNC 1jJo, coincide with the original MNC 1jJ? A complete answer to this question
is not know. A partial answer for the case of the Hausdorff MNC X is given in the following
two subsections.
holds for any ball B in the Banach space E.
Proof. From the properties of the Hausdorff MNC and Theorem 1.4.2 it follows
that the MNC XO is monotone, invariant under translations, semi-homogeneous, and semi
additive. Hence, by Theorem 1.1.6, it suffices to verify that XO(B), where B is the unit
ball, is equal to 0 or 1 according to whether the dimension of E is finite or infinite. If
dimE < 00, then obviously XO(B) = o. Now suppose dimE = 00. Since
XO(B) = max{x(C): C ESE, C C B} :::; X(B) :::; 1,
it suffices to show that XO(B) ~ 1. Suppose this is not the case: XO(B) = q < 1. Fix
an c: E (0,1 - q). Let C E SE,C C B, and let A be a finite (q + c:)-net of C in E, i.e.,
C C UXEA[x + (q + c:)BJ. Then XO(C) :::; (q + C:)XO(B) (here we used the aforementioned
properties of the MNC XO). Consequently,
XO(B) = max{x(C): C ESE, C C B}
= max{xO(C): C ESE, C c B} :::; (q + C:)XO(B),
whence XO(B) = 0, because q+c: < 1. We therefore conclude that any countable set C C B is totally bounded, which of course is not the case if the space is infinite-dimensional. QED
1.4.5. Theorem. The function Xo does not necessarily coincide with x. However,
the inequalities
hold for any bounded set 11 c E.
Proof. First let us provide an example where the MNCs Xo and X are distinct. Let
A be the set of ordinals of countable power. Let E denote the set of all bounded functions
x: A -+ R that satisfy the following condition: for any x E E there exists an ax E A such
that x(a) = 0 for all a 2: ax. Clearly, E is a linear space (with respect to the natural linear
operations). It is readily verified that endowed with the norm
Ilxll = sup{la(x)l: a E A}
E is a Banach space. Now let
11 = {x E E: 0::; x(a)::; 1 for alIa E A}.
Let us show that
X(11) = 1. (4)
This will establish the first assertion of the theorem as well the as the fact that in the
inequalities (2) (if they hold) the constant 1/2 is sharp.
Let C = {xX}~=I be an arbitrary countable subset of 11. Clearly, one can find an
element a* E A such that a* 2: ax for all n (indeed, A is not countable). Then one can
readily check that the element x* E E defined by the formula
x*(a) = {1/2, ~f a::; a:, 0, If a> a ,
provides an 1/2-net for the set 11. Consequently, XO(11) ::; 1/2. The opposite inequality
follows from (4) and inequalities (2), which will be established below.
Let us prove (4). First of all, X(11) ::; 1, since 11 is a subset of the unit ball. Next, let
{Yi }f=I be an arbitrary finite collection of elements of E. Pick ao E A such that a y; < ao
for all i = 1, ... ,k, and define Xo E 11 by the rule
xo(a) = {O, ~f a:f ao, 1, If a = ao.
Sec. 1.5 Condensing operators 21
Obviously, IIYi - Xo II = 1 for all i = 1, ... ,k. Hence, no finite collection of elements can
form an c-net of the set !l for c < 1, i.e., X(!l) 2:: 1.
Finally, let us prove (2). The inequality XO(!l) ~ X(!l) is plain. It remains to verify
that
(5)
The case X(!l) = 0 is trivial. Suppose X(!l) > O. Then for any given c > 0, the set !l has
no finite [X(!l) - c)-net in E. Therefore, one can produce a countable set C C E such that
Ilx - YII 2:: X(!l) - c
for all x, Y E C, x f y. But in this case, as it is easily seen,
and so _ 1 XO(!l) 2:: 2 [x(!l) - c).
Since c is arbitrary, this yields inequality (5) and completes the proof of the theorem.
QED
1.5. CONDENSING OPERATORS
In this section we introduce the condensing operators and study some of their prop
erties.
1.5.1. Definitions. Let El and E2 be Banach spaces and let c/> and t/J be MNCs
in El and E2, respectively, with values in some partially ordered set (Q, ~). A continuos
operator f: DU) C El -+ E2 is said to be (c/>, t/J )-condensing (or (c/>, t/J )-densifying) if
!l C DU), t/J[f(!l)) 2:: c/>(!l) implies !l is relatively compact. The operator f is said to
be (c/>,'l/J)-condensing (or (c/>,t/J)-densifying) in the proper sense if 'l/J[J(!l)) < c/>(!l) for any
set !l c DU) with compact closure (in a partially ordered set (Q,~) the strict inequality
a < b means that a ~ b and a f b). If the set Q is linearly ordered, then clearly the two
notions of condensing operator coincide.
Suppose that on Q there is defined an operation of multiplication by nonnegative
scalars. A continuous operator f is said to be (q,c/>,t/J)-bounded if
for any set n C D(f). Whenever El = E2 and 4> = tP we shall simply say "tP-condensing"
and "( q, tP )- boundell'. In the case q < 1, (q, tP )-bounded operators are sometimes referred
to as tP-condensing with constant q.
1.5.2. Elementary examples. (a) Any compact operator defined on a bounded
subset of a Banach space is obviously tPl-condensing, where tPl is the MNC introduced in
1.2.2 [Translator's note: throughout this book the Russian "completely continuous" will
be translated as "compact"; thus, here a "compact" operator will be one that is continuous
and compact in the sense that it maps bounded subsets of its domain into compact sets;
operators that have only the second property will be explicitly said to do so]; similarly, any
contractive operator on a bounded subset is tP2-condensing (with tP2 as defined in 1.2.2).
However, the MNCs tPl and tP2 often turn to be not so convenient to work with, as they
do not possess sufficiently nice properties; for instance, as we remarked earlier, tP2 is not
regular.
(b) A more meaningful example is provided by the compact and the contractive oper
ators, which, as one can readily see, are condensing with respect to the Kuratowski MNC
ll'.
(c) Obviously, any compact operator on a bounded set is also condensing with respect
to the Hausdorff MNC x.
1.5.3. Remark. Contractive operators are not necessarily x-condensing. In fact, let
{Pn}, {qn}, {Tn}, {Sn} be sequences of numbers in the interval (0,1) satisfying
Sn+l < Pn < qn < Tn < Sn ~ 0 (n ~ 00),
and let {an} and {bn} be sequences of piecewise-linear functions, defined on [0,1], whose
values at 0,1, and the break points Pn, qn, Tn, Sn are shown in the following table:
t= 0 a(t) = 1 b(t) = 0
Pn qn Tn Sn 1 1 1 -1 -1 -1 o 1 -1 0 O.
Let n = {an} C C[O, 1] and define the operator f: n ~ C[O, 1] by the formula f(a n ) = bn.
Clearly, Ilan - amll = 2 and Ilbn - bmll = 1 (m ¥ n), so that f is contractive (with
contractivity constant 1/2). At the same time it is easy to show (using, say, formula (2)
in 1.1.10) that X(n) = X[J(n)](= X({bn }) = 1.
1.5.4. Elementary properties of condensing operators. In the assertions given
below it is assumed that Q is a closed cone in a Banach space and "~" is the partial order
relation defined by Q (see [85]).
(a) If the MNC 'l/Jl is regular, then any (q,'l/Jt,'l/J2)-bounded operator with q < 1 is
('l/Jt, 'l/J2)-condensing in the proper sense.
(b) The composition ft 0 h of a (ql,'l/Jl,'l/J2)-bounded operator ft and a (q2,'l/J2,'l/J3)
bounded operator h is a (ql q2, 'l/Jl, 'l/J3 )-bounded operator.
(c) If the MNC 'l/J2 is algebraically semi-additive and monotone, then the sum ft + h of a (qt,'l/Jl,'l/J2)-bounded operator ft and a (q2,'l/Jl,'l/J2)-bounded operator h is a (ql + Q2, 'l/Jl, 'l/J2)-bounded operator.
( d) If ft is a ('l/Jl, 'l/J2) -condensing operator and h is a ('l/J2, 'l/J3) -condensing operator that maps totally bounded sets into totally bounded ones, 'l/Jl and 'l/J3 are regular MNCs,
and Q = [0,00), then the composition h 0 ft is a ('l/Jl, 'l/J3 )-condensing operator.
(e) If Q = [0,00) and 'l/J2 is semi-additive, then the set of all ('l/Jl, 'l/J2 )-condensing
operators is convex.
Proof. (a) In fact, the inequalities 'l/Jl (n) ~ 'l/J2 [f(n)] ~ Q'I/Jl (n), Q < 1, imply
'l/Jl cn) = 0, which in view of the regularity of 'l/Jl guarantees the relative compactness of n. Cb) Obviously,
(c) In view of the monotonicity and algebraic semi-additivity of 'l/J2,
(d) Suppose n is noncompact. Then 'l/J2[ft(n)] < 'l/Jl(n). If ft(n) is also noncom
pact, then 'l/J3{h[ftcnm < 'l/J2[ftcn)], which in conjunction with the preceding inequality
gives 'l/J3{h[ftcnm < 'l/Jlcn). If, however, ftcn) is totally bounded, then, by hypothesis,
h [ft cn)] is also totally bounded. Consequently, 'l/J3 {h [ft cnm = 0, since 'l/J3 is regular.
On the other hand, 'l/Jl (n) > 0, because n is not totally bounded and 'l/Jl is regular. Thus,
in this case, too,
(e) Let ft and h be ('l/Jl, 'l/J2 )-condensing operators and A E [0,1]. Consider the
operator f>.. = Aft + (1 - A)h and suppose that for some n,
(1)
24 Measures of noncompactness
Clearly, 1>..(0.) c co[h(n) u 12 (n)J. Using the semi-additivity of tP2, we get
tP2 [1>.(n)J ~ max{t/J2 [h (0.)], tP2 [hen)]}.
Chap. 1
(2)
Since the set where tP2 takes its values is linearly ordered, the right-hand side in (2) is equal
to either tP2[h(n)], or tP2[h(n)J. Suppose, for definiteness, that the first case occurs. Then
(1) and (2) imply the inequality
which in view of the fact that h is (tPl, tP2 )-condensing means that 0. is relatively compact.
QED
1.5.5. Condensing families. The definitions of the notions of (</>, tP )-condensing
and (q, </>, tP )-bounded operators admit natural extensions to families of operators f =
{f~:'\ E A}; in this case fen) is understood as U.xEA f.x(n). Often a family of operators
f = {f.x:'\ E A} is regarded as an operator of two variables, f:A X El -+ E2 (f(,\,x) = f.x(x)). Then instead of speaking of a "( </>, tP )-condensing" or a "(q, </>, tP )-bounded" family
one speaks of a ''jointly (</>, tP )-condensing" or a ''jointly (q, </>, tP )-bounded" operator.
1.5.6. An example of a condensing family of operators (condensing homo
topy). Suppose the operators fo, h: M C El -+ E2 are (</>, tP )-condensing, the set where
the MNCs </> and tP take their values is linearly ordered (as a consequence of which 10 and
hare (</>, tP )-condensing in the proper sense), and tP is semi-additive. Then the family of
operators f = {f.x:'\ E [0, I]}, where f.x(x) = (1- '\)fo(x) + ,\h(x), is (</>,tP)-condensing.
The proof is essentialy the same as for assertion (e) in 1.5.4. It suffices to remark
that fen) c co[Jo(n) u h(n)J.
The next theorem gives what apparently is the most common test for an operator to
be condensing; we state and prove it for families of operators.
1.5.7. Theorem. Suppose the operators in the family 1 = {f.x:'\ E A} are continuous
and admit a diagonal representation hex) = <p(,\,x,x) through an operator <p:A X M X
El -+ E2 (here El and E2 are Banach spaces, M eEl, and A is an arbitrary set).
Suppose further that for any y E El the set clI(A X M X {y}) is totally bounded, and that for any ,\ E A and any x E M the operator clI('\, x, .) satisfies the Lipschitz condition with
a constant q < 1 that does not depend on ,\ and x. Then the family f is (q, x)-bounded.
Proof. We have to show that the inequality
(3)
holds for any set n c M, where in the left- [resp. right-) hand side X denotes the Hausdorff
MNC in the space E2 [resp. Ed. If n is not bounded, then (3) is obvious, since X(n) = +00.
Now suppose n is bounded and let S be a finite [X(n) + c)-net of n in El (where c > 0 is
arbitrary). Consider the set SI = (A x n x S). It is totally bounded, being the union of
the finite collection of sets (A x n x {y}) (y E S). We claim that SI is a q[X(n) + c]-net
of the set fen) in E2. In fact, let z E fen), i.e., z = (A, x, x) for some A E A and x E n,
and let yEn be such that Ilx - yll:::; x(n) +c. Then ZI = (A,x,y) belongs to SI and
Ilzl - zll = 11(A,x,y) - (A,x,x)11 :::; qllx - yll:::; q[x(n) +4
Thus, X[J(n)) :::; q[X(n) + c], which in view of the arbitrariness of c yields (3). QED
1.5.8. Corollaries. (a) Under the hypotheses of Theorem 1.5.7, if the set M is bounded and q < 1, then the family f is x-condensing.
(b) The sum f + g of a compact operator f: El ---+ E2 and a contractive operator g: El ---+ E2 is a x-condensing operator on any bounded set M eEl.
The following result on the derivative of a (q, x)-operator is often useful.
1.5.9. Theorem. The Frechet derivative f'(xo) of any (q, x)-bounded operator f is itself (q, x)-bounded.
Proof. Let f: D(f) C El ---+ E2, Xo be an interior point of D(f), and A = f'(xo). Then, for sufficiently small hEEl,
Ah = f(xo + h) - f(xo) + w(h),
where w(h)/llhll ---+ 0 when h ---+ O. Consequently, for any bounded set n c El and c > 0
sufficiently small,
1 1 A(n) = -A (cn) c -[J(xo +cn) - f(xo) +w(cn)).
c c
U sing properties of the Hausdorff MN C and the (q, X)-boundedness of f we obtain
X[A(n)) :::; qX(n) + X[w(d"2)/c).
Letting c ---+ 0, we obviously get
X[A(Q)) :::; qX(Q). QED
26 Measures of noncom pact ness Chap. 1
1.5.10. Remark. It is readily verified that the preceding theorem remains valid if
X is replaced by an arbitrary monotone, semi-homogeneous, algebraically semi-additive,
translation-invariant MNC.
To demonstrate how the notion of a condensing operator is used we prove a fixed-point
theorem for such operators. Here we give it in a particular formulation, while in the next
subsection and in Chapter 3 we describe various generalizations.
1.5.11. Theorem. Suppose the x-condensing operator f maps a nonempty convex
closed subset M of the Banach space E into itself. Then f has at least one fixed point in
M.
Proof. We construct a transfinite sequence of closed convex sets nO' as follows:
no = M, and for 0 > 0,
if 0 - 1 exists,
Clearly, this nonincreasing (with respect to inclusion) sequence stabilizes starting with
some index 0 = 8: n", = nh for 0 ~ 8. Since n H1 = n h = cof(nh ) and f is condensing,
n h is compact. If we show that n h ::j:. 0, then the assertion of the theorem reduces to the
Schauder principle.
Let Xo E M and Xn = fn(xo) (n = 1,2, ... ). The sequence {xn} is relatively compact,
since X[J{ Xn}] = X( {Xn: n ~ 2}) = X( {xn}) and f is condensing. Hence, its limit point set
I< is not empty. It remains to observe that f( I<) = I<, and consequently n", :J I< for all
o. QED
1.5.12. Generalizations. In the preceding proof we used only the following two
properties of the MNC 1/J:
(a) X(con) = X(n). According to our general definition, this property is enjoyed by
any MNC.
(b) X({xn:n ~ I}) = X({Xn:n ~ 2}. This equality holds for any semi-additive (i.e.,
such that X( A U B) = max{x( A), X( B)}) nonsingular (i.e., such that X( {x}) = 0) MNC,
and also for any additively-nonsingular (i.e., such that X(A U {x}) = X(A)) MNC.
Thus, in Theorem 1.5.11 X may stand for an arbitrary (not necessarily real-valued)
MNC that satisfies condition (b) for any sequence {xn}. This condition, too, can be
discarded if it is known beforehand that there is a nonempty set I( C M such that
co fU() :J K.
1.6. ULTIMATELY COMPACT OPERATORS
The classes of condensing operators defined relative to distinct MNCs are of course
distinct in general, but they nevertheless share a number of general properties. In this
section we examine one of them, namely, the ultimate compactness property, which can
be formulated without resorting to MNCs. In the next section we shall study a chain of
strenghtened versions of this property, each characterizing a certain class of condensing
type operators.
1.6.1. Definition of the sequence TO/. Let M be a subset of a Banach space E. Given an operator f: M -4 E, we construct a transfinite sequence of sets {TO/} by the
following rule:
To = cof(M)j
(la)
(lb)
TO/ = n Tf3 , if 0: - 1 does not exist. (lc) f3<0/
(1)
Recall that co n denotes the closed convex hull of the set n. The main properties of
the sequence {TO/} are established in the following lemma.
1.6.2. Lemma. (a) Each TO/ is closed and convex.
(b) f(M n TO/) c TO/+1 .
(c) IfTJ < 0:, then TO/ c T1/. (d) Each TO/ is invariant under f in the sense that f(M n TO/) c TO/. (e) There is an ordinal number 0 such that TO/ = T6 for all 0: 2: o.
Proof. Assertion (a) is an obvious consequence of formulas (1), while assertion (b) is a
consequence of (lb). Next, it is readily seen that (d) is a straightforward consequence of (b)
and (c). However, for us it will be more convenient to establish (c) and (d) simultaneously
by induction on 0:. For 0: = 0 assertion (c) is trivial, while (d) follows from the inclusions:
To = co f(M) :J f(M) :J f(M n To).
Now suppose both assertions hold true for all 0: < 0:0, and let us show that they remain
true for 0: = 0:0. We have to examine two possible cases.
1. The ordinal number 0:0 - 1 exists. If TJ < 0:0, then TJ ~ 0:0 - 1, and so, by the
inductive hypothesis, TO/o- 1 C T.". Using formula (lb), assertion (d) for 0: = 0:0 -1, and
28 Measures of noncom pact ness Chap. 1
the fact that Too - 1 is closed and convex, we obtain
so that assertion (c) holds for a = ao. Next, from the inclusion Too C Too-I it follows
that M n Too eM n Too-I, whence f(M n Too) C cof(M n Too-I) = Too, i.e., (d) also
holds.
2. The ordinal number ao -1 does not exist. Then, according to (lc), Too = np<oo Tp, so that the validity of assertion (c) for a = ao is plain. From the same formula it follows
that f(M n Too) C co f(M n Tp) for any /3 < ao. Using assertion (d) for a = /3, we obtain
f(M n Too) C Tp. Consequently,
f(M n To:o) C n Tp = Too· P<O:o
Thus, assertions (c) and (d) are proved. To prove assertion (e) it suffices to remark
that if the power of the ordinal number {j is larger than the power of the set of all subsets
of the space E, then in the transfinite sequence {To:: 0 :::: a :::: {j} there must be some
repetitions; but for a nonincreasing sequence this is equivalent to stabilization. QED
1.6.3. Definition. The limit set To of the transfinite sequence (1) is called the
ultimate range (or limit range) of the operator f on the set M and is denoted by J<'°(M). The operator f: M ~ E is said to be ultimately compact (or limit compact) if the set
f[M n fOO(M)] is relatively compact. In particular, an operator f is ultimately compact
on M whenever fOO(M) = 0. The next lemma lists elementary properties of the ultimate range and of ultimately
compact operators.
(b) If MI eM, then fOO(MI) C fOO(M).
(c) If f is ultimately compact on M and MI C M, then f is ultimately compact on
MI.
(d) The operator f: M ~ E is ultimately compact if and only if its ultimate range fOO(M) is compact.
(e) If the range f( M) of the operator f is relatively compact, then f is ultimately compact.
(f) The operator f: M ~ E is ultimately compact if and only if the equality co f(M n n) = n implies that n is compact.
Sec. 1.6 Ultimately compact operators 29
Proof. Since fOO(M) = T6 and T6+1 = T6, assertion (a) follows from (lb). If MI eM, then using (1) it is easily seen that T~ C TOI for all a, where {T~} is the sequence (1) for
f on MI. This yields (b), and then also (c), since J[MI n fOO(Md] C J[M n fOO(M)]. To
establish (d) it suffices to remark (see (a)) that f[MnfOO(M)] C fOO(M), and consequently
f[M n fOO(M)] is relatively compact if fOO(M) is compact. Conversely, if J[M n fOO(M)]
is relatively compact, then fOO(M) = coJ[M n fOO(M)] is compact. Next, assertion (e)
follows from the obvious inclusion J[M n fOO(M)] C f(M). Finally, in assertion (f) the
"if" part follows immediately from (a) and (d), and the "only if" part-from (c) and (d).
QED Assertion (d) provides a trivial example of ultimately compact operator. The basic
examples are the condensing operators.
1.6.5. Theorem. Let M be a closed set,1/; a monotone MNC in E, and f: M -+ E
a 1/;-condensing operator. Then f is ultimately compact on M.
Proof. By assertion (a) of Lemma 1.6.4,
co f[M n fOO(M)] = fOO(M). (2)
Hence,
Taking into account the monotonicity of 1/;, we conclude that
(3)
But then from the definition of a condensing operator it follows that the set M n fOO(M)
is relatively compact, and since M and fOO(M) are closed, it is compact. Together with it
the set f[MnfOO(M)] is also compact (since f is continuous), i.e., f is ultimately compact.
QED
1.6.6. Theorem. Suppose the operator f is condensing with respect to an arbitrary
MNC 1/; and maps the closed convex set M into itself. Then f is ultimately compact on
M.
Proof. It suffices to remark that, under the assumptions of the theorem, fOO(M) C
M. Therefore, (2) immediately implies (3) (with the equality sign) with no need of assum
ing that 1/; is monotone. QED
1.6.7. Remark. The definition of an ultimately compact operator extends in the
usual manner to families f = {f>.: A E A}j in this case the transfinite sequence {Ta} is con
structed following the same rules (1) as above, where now by f(n) one means U>'EA f>.(n). If the family f = {f>.: A E A} is given by a function of two variables f: A x M -t E, then
one often says that f is a jointly ultimately compact operator.
We wish to emphasize right away the following circumstance: if fo, It: M -t E are
ultimately compact operators, then the family f = {f>.: A E [0, I]}, f>.(x) = (1 - A)fo + Ait (x), which effects a homotopy from fo to It, is not necessarily ultimately compact.
1.6.8. Example (a linear homotopy that is not ultimately compact). Let
E be the Banach space Co of the sequences that converge to zero, with the norm Ilxll = max { I x i I: i = 1, 2, ... }, and let B be the closed unit ball in E. Define the operators fo and
It by the formulas
and
It is readily verified that fo and It are ultimately compact on B. In fact, fff'(B) = To = {O}
is a compact set. For the operator It the sets Ta with 0: < w (where w is the first transfinite
number) can be described as
TOt = {(I, ... ,I,x1,x2, ... ) E co:lxil $1 (i = I,2, ... )}. '--v--'
Ot+1
Consequently, fro(B) = Tw = nOt<w TOt = 0 (indeed, the sequence (1, ... ,1, ... ) does not
belong to co), which means that It is ultimately compact.
Next, for the family f = {(I - A)fo + Afd the sets TOt with 0: < w are described as
TOt = {(YO, ... ,YOt,X1,X2, ... ) E co:O $ Yj $1 (j = 0,1, ... ,0:), IXil $1 (i = I,2, ... )}.
Hence, the set Tw for the family f, where, obviously, the stabilization of the sequence {Ta} obviously begins, has the form
Tw = {(X1,X2, ... ) E co:O $ Xi $1 (i = I,2, ... )}.
Therefore, fOO(B) = Tw is not compact, and so the family f is not ultimately compact on
B. In connection with the definition of the ultimate range we notice the following two
interesting facts. First, to reach the ultimate range we may have indeed to construct a
genuine transfinite sequence-an example is given below in 1.6.9. On the other hand, if
the operator in question is condensing with respect to a sufficiently nice MNC and if one
is interested in the occurrence of the first compact set in the sequence {Ta} rather than in
its stabilization, then one can always restrict oneselves to the first transfinite number-the
proof of this fact is given in 1.6.10-1.6.13.
1.6.9. Example. Let A denote the set of all ordinals smaller than or equal to a
fixed ordinal number 8 > w of second kind, i.e., such that 8 - 1 does not exist. Let
meA) be the Banach space of all bounded functions x: A -+ R, endowed with the norm
Ilxll = sup{lx(a)l:a E A}, and let B be the closed unit ball in meA) centered at zero. If
x E meA) and a E A, let Sax denote the element of meA) defined by the rule:
(in particular, Sox = 0). Define the operator f: B -+ meA) by the rule
(fx)(a) = IISaxllx(a) (a E A).
Notice that feB) C B. Next, if x, y E B and a E A, then
l(fx)(a)-(fy)(a)1 = IIISaxllx(a)-IISaylly(a) 1$ IISaxlllx(a)-y(a)I+ly(a)IIISax-SaYII
:::; Ilxllllx - yll + Ilyllllx - yll :::; 211x - yll·
Thus, the operator f satisfies the Lipschitz condition with constant 2, and so it is contin-
uous.
Now let us consider the sequence {Ta} constructed for f and prove by induction on
a that { Ra+IB, if a < w,
~= W RaB, if a ~ w,
where Ra = I - Sa. In fact, for a = 0 (4) becomes the equality To = RIB, which is readily
checked. Now assume that (4) holds for all a < ao, and let us check that it remains
valid for a = ao. Suppose first that ao - 1 exists and ao < w. Then, by hypothesis,
Tao - 1 = RcroB, and so
If ao - 1 exists and ao > w (the case ao = w cannot occur), then ao - 1 ~ w, and an
analogous argument shows that Tao = RaoB.
Now suppose that ao - 1 does not exist. Then
TOIO = n Tfj = n RfjB = ROIoB, fj<OIo fj<OIo
as needed. From (4) it immediately follows that for any a < 8 the set f(TOI ) is not relatively
compact. At the same time the operator f is ultimately compact, since foo(B) = T6+1 =
{a}. Similar examples can be constructed in spaces with "sufficiently nice" properties.
Indeed, let 8 be an ordinal number of countable power, 8 =I w, and, as above, set A
= {a: ° :::; a :::; 8}. Fix some bijection 4>: N -t A, where N denotes the natural numbers,
and define an operator f in the Hilbert space h by the rule
() { a, if4>(n) =0, fx n =
sup{lxt/>(m)I: 4>(m) < 4>(n)}Xn, if 4>(n) > 0.
Then it is readily verified that the properties of f are analogous to those of the operator
constructed above.
In the proofs, given below, of Lemmas 1.6.10 and 1.6.11 and of Theorem 1.6.12 we
shall consider, for the sake of simplicity, the case of the Hausdorff MNC, and in 1.6.13 we
shall indicate which of its properties were actually used.
1.6.10. Lemma. Let Rn (n = 1,2, ... ) be subsets of the bounded set Q in the metric
space E, and let U denote the family of all sets A that are representable in the form
00
(5)
where An is finite and An eRn for all n. Then there is an A* E U such that
X(A*) = sup{X(A):A E U}(= s). (6)
Proof. Let Ak (k = 1,2, ... ) be sets in U such that limk--+oo X(Ak) = s, and let the
representation (5) for Ak be Ak = U:::l A~. Consider the set Ak = U:::k A~. Since Ak differs from Ak by only finitely many terms, we have
Now set A * = U~l A k. The set A * belongs to the family U, because A * = U~l U:'=k A~ = U:'=l U~=l A~ and U~=l A~ C Rn (n = 1,2, ... ). We claim that (6) holds. In fact, the
inequality x( A *) :::; s follows from the inclusion A * E U established above. On the other
hand, X(A*) ~ X(A k ) for any k, since A* :) Ak. Consequently,
X(A*) ~ lim X(Ak) = s. QED k-HXJ
1.6.11. Lemma. Let (E,p) be a complete metric space, M a closed bounded set in
E, f: M -... E a x-condensing operator, and ReM. Then
lim X[r(R)] = O. n-->oo
(7)
(Notice that the operator fn is not necessarily defined on the whole set R. In that case
by r(R) we mean the image under r of the part of R on which r is defined.)
Proof. Denote Rn = r(R) (n = 1,2, ... ). Clearly, Rn C f(M), and since f is
x-condensing, the set Q = f(M) is bounded. Thus, all conditions of Lemma 1.6.10 are
satisfied, and consequently there is a set A* E U such that (6) holds. Let the representation
(5) of A* be A* = U::"=l A~. Next, for each x E A~ pick an element y E r-I(R) = Rn- l
such that fey) = x, and then denote the set of all elements y constructed in this manner
by Bn- l . Now set B = U::"=l En. Clearly, B E U, and so X(B) :::; X(A*). Furthermore,
00
and so
x[f(B)] = x(A*) 2 x(B). (8)
Since f is x-condensing, from this inequality it follows that the set B is totally bounded
and hence, thanks to the completeness of E, that B is compact. But then feB) is also
compact, since X[J(B)] = o. Finally, (8) implies X(A*) = O. Thus, we showed that any set
A E U is totally bounded (X(A) :::; X(A*) = 0).
Suppose now that (7) does not hold. This means that there are a positive eo and an
infinite increasing sequence of positive integers nk (k = 1,2, ... ) such that
(9)
for any k. Fix an arbitrary element Xl E r'(R) and choose X2 E r2(R) such that
p( Xl, X2) ~ eo. The existence of such an element X2 follows from the fact that inequality
(9) means, in particular, that the set fn 2 (R) has no finite eo-net in E. For the same reason,
{xI,xd cannot be an eo-net for f n3 (R), hence, there is an X3 E fn3(R) whose distance to
{Xl, X2} is not smaller than co. Continuing in this manner we produce a sequence {x d which, on the one hand, is necessarily totally bounded (because it can be completed to a
set A that belongs to U) and, on the other hand, is not totally bounded since the distance
between any of its elements is not smaller than co. We thus arrived at a contradiction,
which completes the proof of the lemma. QED
1.6.12. Theorem. Let E be a Banach space, M a closed bounded set in E, and
f: M ---+ E a x-condensing operator. Then the sets T ex , constructed according to the rules
(1), are compact for 0: ~ W.
Proof. It suffices to show that the set
00
n=O
is totally bounded. In turn, to establish the total boundedness of Tw it suffices to establish
the total boundedness of any of its countable subsets A, which obviously can be written
as a union of one-element sets An C Tn. Notice that the conditions of Lemma 1.6.10 are
fulfilled here (with Tn in the role of Rn and To in that of Q), and so there is an A* E U with
maximal MNC. Thus, our task reduces to proving that X(A*) = O. Write A* = U::"=l A~, where A~ C Tn and is finite. If X E A *, then in its (lin )-neighborhood there is an element
of the form m m
(10) i=l i=l
Let A~ denote the collection of elements Ii;, taken one for each x E A~. It is readily seen
that the MNC of A* = U::"=l A~ is equal to the MNC of A*: in fact, the elements of these
sets can be arranged in sequences {Yn}, {Zn} such that llYn -znll---+ 0 as n ---+ 00. Let B n- 1 denote the collection of elements Xi, constructed for all elements X E A~ in accordance
with formula (10). Then B = U::"=o Bn E U, and so X(B) ::; X(A*). On the other hand,
since cof(B):J A* and X[cof(B)] = X[J(B)J, we get
X[J(B)] = X[cof(B)] ~ X(A*) = X(A*).
Thus, X(B) ::; X(A*) ::; X[J(B)J, which in view of the fact that f is x-condensing implies
X(A*) = O. QED
1.6.13. Remark. Let us isolate those properties of the MNC that were used to
prove assertions 1.6.10-1.6.12. In Lemma 1.6.10 we used: a) real-valuedness; b) additive
Sec. 1.7 K -operators 35
nonsingularity, i.e., the invariance of the MNC with respect to the adjunction of a single
element to a (nonempty) set; c) monotonicity. In Lemma 1.6.11, in addition to a)-c), the
following properties were required: d) nonnegativity; e) subordination to the Hausdorff
MNC, i.e., existence of a constant c > 0 such that ¢(n) ~ cX(n) for any bounded set n; f) regularity. Finally, in the proof of Theorem 1.6.12 we used properties a)-c), f) and also:
g) continuity, and h) invariance under passage to the convex hull. In particular, assertions
1.6.10-1.6.12 are valid for the Kuratowski MNC and the MNC f3 (see 1.3.1).
1.7. K-OPERATORS
In ~his section we continue our study of those general properties of condensing oper
ators, the formulation of which does not involve MNCs. The starting point will be the
definition of the notion of an ultimately compact operator in the form used in subsection
1.6.4 (f): co f(n n M) = n must imply the compactness of n. It turns out that if one
requires that the compactness of n be implied by a less restrictive equality, then the prop
erties of the operator improve. This situation corresponds roughly to considering operators
that are condensing with respect to nicer MNCs. For example, if the equality in question
is replaced by co [T U f(n n M)] = n, where T is a one-element set, then one obtains the
K1-operators, for which an analogue of the Schauder principle already holds (see 1.7.3,
1. 7.4), in contrast to the case of ultimately compact operators. For K 2-operators (i.e.,
in the case where the set T in the above equality consists of two elements) it is possi
ble to construct a fixed-point index theory that is richer compared with the cases of the
ultimately compact and the K1-operators (see Chapter 3).
1. 7.1. Definition. Let A be a family of subsets of a Banach space E, and let M C E. A continuous operator f: M ~ E is called an A-operator if for any TEA and any set
neE one has the implication:
co [T U f(n n M)] = n => n is compact.
Let ](n denote the set of all n-element subsets of E (in particular, Ko = {0}), and let
](00 and ](e designate the set of all finite and respectively compact subsets of E. We let
Kn [resp. Koo , Kel denote the set of all ](n- [resp. ](00-' ](e-l operators. This section is
devoted precisely to the study of the operators in the classes K n , Koo , and K e , for which
we shall use the common term of "](-operators" and the notation K = {Kn};:"=o uKoo UKe •
1.7.2. Remarks. (a) Clearly,
(1)
(b) Earlier we proved (see Lemma 1.6.4, assertion (f)) that the class of Ko-operators
coincides with that of ultimately compact operators.
Next we prove a generalization of Schauder's theorem to the class of Kl-operators (cf.
Theorem 1.5.11).
1. 7 .3. Theorem. Suppose the KI -operator f maps the nonempty closed convex
subset M of the Banach space E into itself. Then f has at least one fixed point in M.
Proof. Fix some point Xo E M and denote by wt the family of all sets n c M with
the property that
co [{xo} U f(n)] en.
Clearly, wt is nonempty, since M E wt. Set noo = nflEm n. It is readily verified that Xo E noo, and so noo f 0. Moreover,
noo is contained in M and is invariant under f. The assertion of our theorem will follow
from Schauder's principle provided we can show that noo is convex and compact. To this
end we observe that
In fact, if Xl E noo and Xl rt co [{xo}uf(noo )], then, as is readily checked, n oo ' {xd E wt, which contradicts the definition of noo. Since f is a Kl-operator, the equality established
above implies the compactness of noo. The convexity of noo is an obvious consequence of
the same equality. QED
The next example shows that the theorem just proved fails for ultimately compact
operators; this means, in particular, that the classes Ko and KI are distinct.
1. 7 .4. Example. In the Banach space Co of sequences that converge to zero consider
the operator f defined by the formula
(we met f before in 1.6.8, where it was shown that f is ultimately compact on the unit
ball B in co). Clearly, f maps B into itself. However, f has no fixed point in B: in fact, it
is readily verified that the only possible fixed point for f is the sequence (1,1, ... ), which
does not lie in co.
Sec. 1.7 J{ -operators 37
1. 7 .5. Theorem. len =f len+1 lor any nonnegative integer n.
Proof. We need to exhibit an operator I E len ,len+1. We take for E the space 11 of
absolutely summable sequences, and take for the domain M of the operator I the closed
ball in 11 of radius 3 and center zero. We first define I on the vectors el, e2, . .. of the
canonical basis: if k = 1, ... ,n + 1,
if k ~ n + 2;
here (k) denotes (the reminder of the division of k - 2 by n)+2. Further, let Bk (k = 1,2, ... ) denote the closed ball of radius c and center ek. The
positive number E: < 1 will be chosen later. For an arbitrary x E M we put
Notice that the balls Bk are disjoint, since c < 1. It is not hard to check that the operator I thus defined maps M into itself and is continuous. We claim that I is not a Kn+1-operator.
In fact, if in the equality
co [T U f(n n M)] = n (2)
we put
<Xl
f(n) = U leI, 2ek+l - e(k)], k=n+2
where [a, b] denotes the segment with endpoints a, bEE. Hence, if one takes for T the
(n + I)-element set {e2' e3, ... ,en+d, then (2) will hold; at the same time, n is not
compact. Thus, f ff- le n +1 .
Now let us show that if (2) holds for a noncompact set n and a finite set T, then T contains at least n + 1 points. This obviously will mean that f E len.
Thus, suppose n is noncom pact and satisfies (2) for some finite set T. Then n inter
sects infinitely many balls Bk: otherwise, the left-hand side of equality (2) would be the
convex hull of a finite set, and hence compact. Let C denote the set of all k ~ n + 2 for
which Bk n n =f 0. Then
f(n n M) c U [el,2ek+l - e(k)] C L(D), kEG
where D is the set consisting of the vectors el and 2ek+l - e(k) (k E C) and L(D) denotes
the closed linear span of D. Suppose that we can produce a set T with the following
properties:
TCnnMj
T is linearly independent (3)
p(T, L(D)) > 0
(here p(A, B) denotes the Hausdorff distance between the sets A and B). Then from (2) it
will follow that T consists of at least n + 1 points. Indeed, (2) and (3) imply the inclusion
T C L(T U D) C L(T) + L(D).
Let us label the points of the set T: T = {Xl, ... ,xn+d. Then for any i = 1, ... ,n + 1,
Xi E L(T) + Zi, (4)
where Zi E L(D). From (3) it follows, in particular, that for any Z E L(D) the set TU {z} is linearly independent, and then so is the set {Xi - Zi: i = 1, ... ,n + I}. From (4) one
derives the inclusion
{Xi -zi:i = 1, ... ,n+1} C L(T),
which means that T consists of no less than n + 1 points.
Therefore, in order to prove the theorem it suffices to exhibit a set T with properties
(3).
To this end, we first note that the following alternative holds for the set C: either
C contains all positive integers starting with some m, and m - 1 fI. Cj or there is no m
such that the last assertion is true. In the first case we take as an (n + 1 )-element set
Tl = {ek:k = m, ... ,m + n}. In the second case, we take for Tl an arbitrary (n + 1)
element subset of the (infinite) set {ek: k E C, k -1 fI. C}. From here on the proofs for the
two cases are analogous.
From the coordinate representation of the elements of Tl and D and the definition of
the norm in 11 it readily follows that the distance from any element ek of Tl to the linear
span of the set D U Tl " {ed is not smaller than 1/2 (in the second case it is even not
smaller than 1).
Now to define the sought-for set T proceed as follows: chose one representative Xk E
Bk n n for each k such that ek E T l , and set T = {xd. Thus, the elements of Tare
Sec. 1.7 K -operators 39
"c-perturbations" of the elements of T1 : Ilxk - ekll :::; c. Since the number of elements of
Tl is known beforehand (it is n + 1), we can choose c > 0 in the definition of the operator
f so that T will possess properties (3). QED
1.7.6. Remarks. (a) The class K= is different from all classes Kn (n = 0,1, ... ). In
fact, suppose that Kn = K= for some n; then in view of the inclusions Kn :J Kn +1 :J K= (see (1)) we would have Kn = Kn+l' which is impossible. It is now clear that the class Kc, too, is different from any of the classes Kn.
(b) The question of whether the classes K= and Kc coincide or not is open.
We next turn to the investigation of the relationships between condensing and K operators.
1.7.7. Theorem. Let M be a subset of a Banach space E and let f:M

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