The sets of certain classes in generalized metric spaces

THE SETS OF CERTAIN CLASSES IN GENERALIZED METRIC SPACES

Tadeusz Konik

Institute of Mathematics and Computer Science. Gzestochowa University of Technology

Abstract. In this paper some property of sets of certain classes in the generalized metric spaces are considered. In last section of this paper an example of a certain set of these classes in two-dimensional Euklidean space will be given.

1. Introduction

Let E be a certain non-empty set and let I be any non-negative real function defined on the Cartesian product EQ x EQ of the family EQ of all non-empty subsets of the set E. The pair (E, I) we shall call the generalized metric space.

Let o, 6 be arbitrary non-negative real functions defined in a certain right- -hand side neighbourhood of 0 such that

a(r) > 0 and b(r) »0 (1)

r^0+ r-^0+

By Si(p,r~)

f

\d Si(p,r)^

j we denote in this paper so-called a(r), b(r)- -neighbourhoods of the sphere Si (p, r) with the centre at the point p and the radius r in the space (E,l).

We say that the pair (A, B) of sets of the family EQ is (a, b)-clustered at the point p of the space (E,t), if 0 is the cluster point of the set of all numbers r > 0 such that A n S

(p, r)

^ 0 and B n S

(p, r)

.) ^ 0.

Let fc be any, but fixed positive real number, and let by the definition (see the paper [9]):

T/(a,6,fe,p) = {(A,B) : A,B € EQ, the pair (A,B) is (o. &)-clustered at the point p of the space (E, I) and

(2)

The set T/(a, 6, k,p) defined by the formula (2) we call the relation of (a, b)-

-tangency of order k at the point p (shortly: the tangency relation) of sets in

the generalized metric space (E,l).

If (A, B) e Tj(a, 6, &,p), then we say that the set A £ EQIS (a, &)-tangent of order k to the set B € -Eb at the point p of the space (E, I).

We say (see [3]) that the set A 6 EQ has the Darboux property at the point p of the generalized metric space (E, 1), and we shall write this as: A € D^P(E, /), if there exists a number r > 0 such that A n <Sz(p, r) 7^ 0 for r 6 (0, r).

In this paper we shall consider certain problems concerning the tangency of sets of the classes Mp^ having the Darboux property at the point p of the generalized metric spaces (E,l), for I € #/. A certain theorem for the sets of these classes will be given here.

2. On a certain theorem

Let p be an arbitrary metric of the set E. We shall denote by d^pA the diameter of the set A € EQ, and by p(A, B) the distance of sets A, B € EQ in the metric space (E,p).

Let / be any subadditive increasing real function defined in a certain right- -hand side neighbourhood of 0, such that /(O) = 0.

By $f we will denote the class of all functions / fulfilling the conditions:

1° I : EQ xE⁰ — > (0,oo),

2° f(p(A,B))<l(A,B)<f(dp(AUB)) for A,B(EE0.

It is easy to check that every function I € $/ generates in the set E the metric IQ denned by the formula:

lo(x,y) = f(p(x,y)) for x,y e E (3) Let us put by definition (see [6])

Mpjk = {A G EQ : p € A' and there exists fi> 0 such that for an arbitrary e > 0 there exists 8 > 0 such that

for every pair of points (x,y) € [A,p;/j,,k]

if p(p,x) < 6 a n d < £> ^en - - < ^£} ( 4 )

1 } P^k(p,x) p^k(p,x) * ^ '

where A! is the set of all cluster points of the set A G EQ and

[A,p- ft, k] = {(x, y):xeE,yEA and ftp(x, A) < pk(p, x) = pk(p, y)} (5) Theorem 1. If the set A e EQ is (a, 6) -tangent of order k to the set B e EQ at the point p e E for an arbitrary function I e gf omd for every point x such

that (x,y) € [A,p;fj,,k] there exists a point y € AnSi(p,r)a^ and A > 0 such that

p(x,y)<\p(x,A) (6) then A is the set of the class Mp>k-

Proof. Let (A,B) e T/(a,6,fe,p) for I G #/ and A,B € EO- From here, in particular, it follows that

(A,B)£Ti(a,b,k,p) for I e &d and A, 5 € £0 (7) where id denotes the identity function defined in a certain right-hand side neighbouhood of 0. Because every function I € -5^ generates in the set E the metric p (see definition of the class #/), then from here and from (7) follows

)

,BnSi(p,r}

) - >0 (8)

Putting I = dp, from (8) we get

^dp((A n Si(p, r)a(r)) U (5 n $(p, r)6(r))) - . 0 (9)

r — +

1"

Because

dp(A n Si(p, r)a(r)) < dp((A n Si(p,r)a(r)) U (B n Si(p,r)6(r))) then from here, from (9) we obtain

4:4(^n^(p,r)aM) > 0 (10)

rk v ' r^o+

From (10) it follows that for an arbitrary e > 0 there exists ft > 0 such that

—rdp(Ar\a(r\) < - for 0 < r < ft (11)

rK v ' 2

Now we shall prove that for every pair of points (x, y) of the set [A, p~, /J,, k]

'^x) < £ (12)

if only

r = p(p,x) < 6 and Pfcy ^ < 6 (13) Let us put n = I and 8 = min(l, ^, ^i). From here, from (6), (11) and from the triangle inequality we have

e e e

2 2 2

what means that A is the set of the class 3. On a certain set of the class Mpik

In this Section we will give an example of a certain set of the class Mp>k in two-dimensional Euklidean space, and will use Theorem 2.3 of the paper [7]

for certain subsets of this set.

Example 1. Let E = R2 be the two-dimensional Euclidean space. Let <p be a increasing function of the class C\s function together with

1st derivative) defined in a certain right-hand side neighbourhood of 0 such that 9?(0) = 0. Using the de L'Hospital's theorem and mathematical induction for k € N we can easily prove that

i/w

» 0 (14) tk

From this it follows immediately

Let us put

C = {(x,y): x>0, Q<y<pk+l(x) and k € N} (16) We shall prove that C defined by the formula (16) is the set of the class Mptk-, where p = (0,0). For this purpose let us denote

A = {(t,0) : t > 0 } and B = {(t,/+1(*)) : * > 0, fc e N} (17) Let yi, 7/2 be a points of the set C such that

yi € A n Sp(p, r), 2/2 e B n 5p(p, r) for r > 0 (18) If according to (17) and (18) we put yz = (i,<^fc+1(t)), then

(19) Hence it follows that yi = (^/i2 + </?2fc+2(t),0). From (19) and from the properties of the function ^> it results also that r —> 0+ if and only if t —> 0+. Hence and from the conditions (14), (15), (19) for r > 0 we have

_

\ 2

what means that

-^dp(C n Sp(p,r)) > 0 (20)

T^K r->0+

From here it follows that for an arbitrary e > 0 there exists 81 > 0 such that

^dp(C n Sp(p,r)) < | for Q<r<61 (21)

/ Zi

Now we shall prove that for an arbitrary e > 0 there exists 62 > 0 such that for every pair of points (x,y\) e [A,p;fi,, k]

when

)<62 and - < 6, (23) Let T/J be a projection of the point x 6 E at the set A, i.e., such point of the set A that p(x,y[) = p(x,A). Because x — (t, ±\A"2 — *2) for 0 < t < r, then

=r-t = that is to say

(24) Let /u. = 2, #2 = niin(^,|). Hence, from (23), (24) and from the triangle

p(y'i,y] 2p(x, A) e

pk(p,x) ~~ pk(p,x) ~~ pk(p,x) 2 which yields the inequality (22) .

Lastly we shall prove that for an arbitrary e > 0 there exists £3 > 0 such that for every pair of points (#,3/2) G [B,p\ k]

if only

r = p(p,x)«5

and ^± < 6

(26)

From the properties of the function 99 it follows that

(/+1(*))'|t=o = 0 (27)

what means that the set B is tangent to the axis x at the point p. From here it follows that in a certain right-hand side neighbourhood of 0 the function y = ipk+l(t) is a convex function. Let y'% be a projection of the point x € E at the set B, i.e., such point of the set B that p(x,yl2) = p(x,B). Let L be a tangent line to the set B at the point y'2 , and let y e L n Sp(p,r], where Sp(p, r) denotes the sphere with the centre at the point p € E and the radius r > 0 in the metric space (E,p). From here, on the base of the inequality (24), it follows that

)<p(x,y'2)<p(x,B} (28) Hence and from the triangle inequality we get

p(x, yi) < p(x, y) < p(x, y'

) + p(y'

,y} < 2p(x, B} (29)

pk(p,x)

which yields the inequality (25).

Let fj, = 2, 8 — min(Si, 62,63) and let (x,y) be an arbitrary pair of points belonging to the set [(7,p;/^, k]. In this example: p(x,C) = p(x,A), or p(x,C) = p(x,B), or x € C.

Let us suppose that p(x, C) = p(x, A). From here, from the triangle inequal- ity, from (21) and (22) it follows that for an arbitrary e > 0 there exists 6 > 0 such that for every pair of points (x, y) € [C, p; /x, k], if

, , c , p(x,C) c

r = p(pjx)<8 and ~-^—^ < ^ pK(p,x) then

p(x,y) ^ p(x,yi) | p(y,yi) ^ p(x,yl) | 1 rf g < ^ pk(p,x) pk(p,x) pk(p,x) ~ pk(p,x) rk p p

Similarly, if p(x, C} = p(x, B) then from here, from the triangle inequality, from (21) and (25) it follows that for an arbitrary e > 0 there exists 6 > 0 such that for every pair of points (x,y) €E [C,p; /x, k], if

r = p(p, x) < 6 and —-^—r < 8 then

pk(p,x) ~ pk(p,x) pk(p,x) ~ pfc(p, a;) rk

If x € C, then from (21) it follows immediately that for an arbitrary e > 0 there exists 6 > 0 such that for every pair of points (x,y) e [C,p; /^, k]

when

n(f. C\) < 8 and

Hence, from (30) and (31) it follows that the set C defined by the formula (16) belongs to the class Mpik-

Evidently, the set C of the form (16) has the Darboux property at the point p of the metric space (E, p). From the above it follows that C € Mp^k n DP(E, p).

Because the sets A, B defined by the formula (17) have the Darboux property at the point p of the space (E, I). and are subsets of the set C e Mp^,

then from here and from Theorem 2.3 of the paper [7] it follows that the set A is (a, fe)-tangent of order k (k e N) to the set B at the point p of the space ( E , l ) , when I G §j, and the functions 0,6 fulfil the condition

^ ,0 and M ,0 (33)

'0

References

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