NEW ZEALAND JOURNAL OF MATHEMATICS Volume 27 (1998), 177-182

ON THE MEASURES OF NONCOMPACTNESS IN SOME METRIC SPACES

D a r i u s z B u g a j e w s k i a n d E w a G r z e l a c z V k

(Received March 1997)

Abstract. In this paper we calculate the Hausdorff and the Kuratowski mea­sure of noncompactness for any bounded subsets of the space R2 with the “river” metric or with the radial metric.

1. Introduction

The Hausdorff and the Kuratowski measure of noncompactness have very im­portant applications in nonlinear functional analysis and to differential and integral equations in abstract spaces. By using these indexes one can formulate conditions for many fixed-point results and in existence theorems for equations in Banach spaces (see [1], [4], [8]).

In [2] we investigated some properties of Kuratowski’s measure of noncompact­ness in vector spaces with a translation invariant metric or with a homogeneous metric. In particular, we calculated Kuratowski’s measure of noncompactness of some subsets of R2 with the “river” metric or with the radial metric.

The aim of this paper is to calculate both the Kuratowski and the Hausdorff measure of noncompactness of any subsets of R2 with the above mentioned metrics. For this purpose, we formulate some simple conditions. To calculate the measures of noncompactness in these spaces it is enough to verify if a given set satisfies these conditions.

2. R2 with the “River” Metric

For convenience we recall the definitions of the Kuratowski and the Hausdorff measure of noncompactness. Let (X , d) be a metric space and D be a bounded subset of X . The Kuratowski measure of noncompactness of D — a ( D ) — is defined as the infimum of positive numbers e such that D can be covered by a finite number of sets of diameter not greater than e. The Hausdorff measure of noncompactness of D — (3(D)— is defined as the infimum of positive numbers e such that D can be covered by a finite number of balls with centres in D and radius not greater than £.

For the basic properties of a and (3 we refer [1]. In what follows the following inequalities between a and (3 will be very useful:

(3(D) < a(D) < 2(3(D) for each founded subset D e l (1)

(cf. [1]).

1991 A M S Mathematics Subject Classification: 54E50.

178 DARIUSZ BUGAJEWSKI AND EWA GRZELACZYK

Now, consider the space R2 with the “river” metric defined as follows

\ y i - V 2 \ , if x i = x 2, \yi\ + I2/2I + \xi - x 2\ , if X l / ^ 2 ,

d ( v i , v 2 ) =

where v\ = (a?i,2/1), v2 = (x2,y2) G R2.We introduce the following:

Definition 1. Let D be any bounded subset of R2. We say that y' £ R satisfies:

(a) the condition A*(D), if for every y < y' there exists at least countable (but not finite) number of points vn = (xn, yn) £ D such that

xn / xm for n ^ m and y < yn < y' for n £ N;

(b) the condition A*(D), if for every y > y' there exists at least countable (but not finite) number of points vn = (xn, yn) £ D such that

xn 7 xm for n ^ m and y > yn > y' for n £ N.

Put y*(D) = supy/ \y'\, where y' satisfies A*(D) or A*(D).If exists any number y' satisfying neither A*(D) nor A*(D), put y*{D) = 0.

Our first result is given by the following:

Theorem 2. For any bounded subset D e l 2 with the “river” metric, we have

a (D ) = 2 y*(D) and (3(D) = y*{D).

Proof. 1. It is clear that if exists a number y' satisfying neither A*(D) nor A*(D), then D consists of sets containing in a finite number of segments, each of them is generated by points (Xi,yi), where Xi is fixed. Hence 01(D) — /3(D) = 0.

Consider now any bounded set D such that there exists y' satisfying A*(D) or A.{D).

2. First, we prove that

a(D) > 2y-(D). (2)

Let (Dj) j=i,...,™ be a cover of D such that maxi<j<m d(Dj) < e for some e > 0, where d(-) denotes the diameter of the corresponding set. Consider the sets An = {v = (x,y) £ D : \y\ > y*(D) — n £ N. Then in view of Defini­tion 1, for every n £ N there exist j n £ { 1 , . . . , m} and u” , v £ D such that x \ 7 #2 and v™, V2 £ Djn fl An.Since

d(v?,v%) > \y?\ + \y%\ > 2y*(D) - for every n £ N,

£ > 2y*(D). Hence (2) is satisfied.

3. In this point we show that a(I) = 2, (3(1) — 1, where I = {(x,y) € I 2 : \y\ <1, |a:| < 1}. Let A* be the closed ball of centre , 0) and radius r(Ak) = 1+^r and A~k- the closed ball of centre ( — 0) and radius r(A~k) = 1 + ^r, where n £ N and k = 0 , . . . , 2n — 1. It can be easily verified that balls At , A ~k, k = 0 , . . . , 2n — 1 form a finite cover of D for every n £ N. Thus (3(1) < 1. In view of the inequality(2) we infer that a(I) > 2y*(I), and therefore by (1), we obtain

1 / t \ ^ i T\

ON THE MEASURES OF NONCOMPACTNESS IN SOME METRIC SPACES 179

Hence a(I) = 2 and (3(1) — 1.

4. Arguing analogously as in point 3 one can verify that

a(Ia,b) = 26, /3{Ia,b) = b,where Ia^ = {(x, y) e R2 : |y| < b, \ x - a |< b}, a E R, b > 0.

5. Now, we prove that

0(D) < y'(D). (3)

If t/*(D) = sup(l,B)eD |y|, then D can be covered by a finite number of rectangles Iai,y*(D)- In view of the point 4 of our proof we know that /?(/ai,y*(D)) = V*(D)- Hence, by the property of (3 we obtain /3(D) < y*(D).

Further, if y*(D) < sup^x y eD |y|, then in view of Definition 1, for every e > 0 there exists a finite number of segments which are parallel to the axe OY and contain points (x, y) £ D such that | y |> y*(D) + £ . Hence fi({(x,y) e D : |y| > y*(D) + £}) = 0. Moreover, the finite number of rectangles Ia-,y*(D)+e covers all points (x,y) € D such that |y| < y*(D) + s. By the point 4, /3(laj,y'(D)+e) = V*(D) + e, since e > 0 is arbitrarity we obtain /3(D) < y*(D).

6. To end our proof let us remark that in view of (1), (2), (3) we obtain

y'(D) < ia (D ) < 0(D) < y*(D).

Hence a(D) = 2y*(D), /3(D) = y*(D). □

3. R2 with the Radial Metric

In this section we shall consider R2 with the radial metric defined as follows:

{ q(vi,i>2) , if 0 ,v\,v-2 are collinear,

£>(t>i,0) + q(v2, 0) , otherwise,

where g denotes the usual Euclidean metric and v\ = (x\,y\), V2 — (#2^ 2) € R2. Analogously, as in Section 2 we introduce the following:

Definition 3. Let D be any bounded subset of R2. We say that w' € R+ satisfies:

(a) the condition W*(D) if for every w < w' there exists at least countable (but not finite) number of points vn = (xn,yn) £ D such that vn, vm, (0,0) are not collinear for n ^ m and w' > g( (xn, yn), (0,0)) > w for n G N.

(b) the condition W*(D) if for every w > w' there exists at least countable (but not finite) number of points vn — (xn,yn) € D such that vn, vm, (0,0) are not collinear for n ^ m and w' < g((xn, yn), (0,0)) < w for n G N.

Put w*(D) — sup^, w', where w' satisfies W*(D) or W*(D). If there exists any nonnegative number satisfying neither W*(D) nor W*(D), put w*(D) = 0.

The following theorem states, roughly speaking, that in the space R2 with the radial metric the situation concerning the measures of noncompactness is similar as in the case of R2 with the “river” metric.

Theorem 4. For any bounded subset D of R2 with the radial metric, we have a(D) = 2 w*(D) and 0(D) = w*(D).

180 DARIUSZ BUGAJEWSKI AND EWA GRZELACZYK

Proof. 1. Obviously, if exists a nonnegative number w' satisfying neither W *(D ) nor W*(D), then D consists of sets containing in a finite number of seg­ments such that any two different points belonging to one of them are collinear with (0,0). Hence a (D ) = /3(D) — 0.

Now, let D be any bounded subset of M2 such that there exists w' satisfying W*(D) or W*(D).

2. We prove that

a(D) > 2w*(D). (4)

Let ( D j ) j - 1,... ,m be a cover of D such that maxi<j<m d(Dj) < £ for some e > 0, and let An = {v = (x,y) G D : x2 + y2 = r2, where r > w (D) — ^ }, n G N. Then for every n G N there exist j n G { 1 , . . . , m) and v™, v% G D such that v™ ^ v%, v™, vg, (0,0) are not collinear and v™, G An fl Djn. Since

d(vi ,v2 ) = f?(v™,0) + q(v%,0) > 2w*(D) — ^ for every n G N,

e > 2w*(D). Thus a(D) > 2w*(D) and (4) is proved.

3. In this point we prove that

/3(D) < w*(D). (5)

Obviously, if w*(D) = swp(x ^ eD g((x, y), (0,0)), then D is contained in the closed ball of centre (0,0) and radius w*(D). So in this case (5) is satisfied.

Let w*(D) < sup(x ^ eD g((x,y), (0,0)). Then in view of the Definition 3, for every £ > 0 there exist at most the finite number of segments such that any two diffrent points belonging to one of them are collinear with (0,0) and they contain all points (x,y) G D with the property \y\ > w*(D) + £. Hence /3({(x,y) G D : \y\ > w*(D) + £}) = 0. Moreover, (3({(x,y) G D : \y\ < w*(D) + e}) < w*(D) + £, so in view of the arbitrarity of e > 0 we obtain (3(D) < w*(D).

4. In view of (1), (4), (5) we have

w*(D) < i a(D) < (3(D) < w*(D).

Hence ot(D) — 2w*(D) and (3(D) — w*(D). □

4. Application

In this section we show an application of our results to fixed point theory in hyperconvex metric spaces. Recently Espinola-Garcia [5] and Kirk, and Song Sik Shin [6] have proved the following Darbo-Sadovski-type fixed point result for these spaces. Namely -

Theorem 5. Let X be a bounded hyperconvex space and let f : X —> X be a continuous mapping. If

a ( f (A ) ) < ot(A) for any bounded subset A C X such that a(A) > 0,

then f has a fixed point.

In [3] we have proved the following generalization of the above result.

Theorem 6. Let X be a hyperconvex metric space, xq 6 X and let f be a contin­uous mapping of X into itself. If the following implication

(V is isometric to e f (V ) or V = f (V ) U {x 0}) =>■ (a(^) = 0),

where e f (V ) denotes the hyperconvex hull of f (V ) , holds for every subset V C X , then f has a fixed point.

The significance of our generalization can be illustrated by the following example. Let be given the space R2 with the radial metric. Consider the mapping / : R2 —* R2 defined as follows

f ( x , y ) = (hx,hy), (x,y) € R2,

for some h > 1. It can be proved that this mapping does not satisfy the main assumption of Theorem 5. To show this one needs to calculate the Kuratowski measure of noncompactness in R2 with the radial metric, namely Theorem 4 (see[3] for details). But this mapping satisfies all assumptions of Theorem 6.

5. Appendix

Consider the space R2 with the translation invariant metric defined as follows

ON THE MEASURES OF NONCOMPACTNESS IN SOME METRIC SPACES 181

\ >- if y\ ^ 2/2,d(vi,v2) = < \ , if yi = y2 and x\ ± x 2,

1 o , if xi = x 2 and y1 = y2,

where vi = (xi,yi ), v2 = (x2,y2) G R2. The above defined metric is a useful toolto investigate some properties of the index a in vector spaces with a metric (cf.[2]). The measures of noncompactness in this space are described by the following

Theorem 7. Let D be any bounded subset of R2 with the metric defined above.

(a) If D contains at least countable (but not finite) number of points (xn,yn) € D such that yn ^ ym for n ^ m, then

a(D) = 0(D) = 1.

(b) If D does not satisfy the assumption of the point (a), but there exists at least a countable (but not finite) number of points (xnryn) 6 D such that xn ^ xm for n ^ m, then

a(D) = fi(D) =

(c) If D does not satisfy the conditions of the point (a) and (b), then

a(D) = /3(D) = 0.

182 DARIUSZ BUGAJEWSKI AND EWA GRZELACZYK

Proof. (a) It is clear that in this case a ( D ) = 1. Assume that /3(D) < 1. Then there exists a finite cover of D by closed balls B \,. . . , Bm of radius r < 1 and centre (Xi, yi), i = 1 , . . . , m, respectively. Every ball Bi is the line y = yi, i = 1 , . . . , m, but the finite number of such lines can not cover considered set D. Thus /3(D) > 1 and, therefore /3(D) = 1.

(b) If the set D C M? satisfies the assumptions of this point, then it is clear that a(D) = Assume now that /3(D) < \. Then there exists a finite cover of D by closed balls B\,.. . ,B n of radius r < \ and centre (xi, yi), i = 1 , . . . , n, respectively. Every ball Bi consists of one point (xt, yt) only, i = 1 , . . . ,n, but a finite number of points can not cover considered set D. Thus (3(D) > \ and, therefore (3(D) —

(c) It is clear that in this case the set D consists of a finite number of points and thus a(D) = (3(D) = 0. □

References

1. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math. 60, Marcel Dekker, New York and Basel, 1980.

2. D. Bugajewski, Some remarks on Kuratowski’s measure of noncompactness in vector spaces with a metric, Comment. Math. 32 (1992), 5-9.

3. D. Bugajewski and E. Grzelaczyk, A note on a fixed point theorem in hyperconvex spaces, submitted.

4. K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Mathematics 596, Springer-Verlag, Berlin - Heidelberg - New York, 1977.

5. R. Espinola-Garcia, Darbo-Sadovski’s theorem in hyperconvex metric spaces, in the proceedings of the Workshop “Functional Analysis: Methods and Applica­tions” , Camigliatello Silano, May 29 - June 2 1995, Rend. Circ. Mat. Palermo, Serrie II, Suppl. 40 (1996), 129-137.

6. W .A. Kirk and Sang Sik Shin, Fixed point theorem in hyperconvex spaces, Hous­ton J. Math. 23 (1) (1997), 175-188.

7. K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), 301-309.8. V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract

Spaces, Pergamon Press, New York, 1981.

Dariusz Bugajewski and Ewa GrzelaczykFaculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityMatejki 48\4960-769 PoznanPOLAN Dddbb@math.amu.edu.pl