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PRACTICAL PROBLEMS AND SOLUTIONS TO THE USE OF THEORETICAL LAWS IN THE SCIENCES OF THE 21ST CENTURY
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ON SUMES OF ALMOST PERIODIC (A.P.) MULTI-VALUED FUNCTIONS
Nizomxonov Erkinxon Nizom o'g'li1 Nizomxonov Sanjarxon Erkinxon o'g'li2
lUniversity of Tashkent for Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan 2University of Tashkent for Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan [email protected], [email protected] https://doi.org/10.5281/zenodo.13364849 Abstract: This paper examines the basic properties of the sum of almost periodic multivalued functions whose values are compact sets of the space RAn. It is much more difficult to prove that the sum of two almost-periodic (a.p.) multi-valued functions is a multi- valued a.p. function. The first proof of this theorem for a single-valued function was given by G.Bohr. Subsequently, Bochner gave others a definition for unambiguous a.p. functions on which the almost-periodicity of the sum follows directly. Subsequently, it turned out that Bochner's definition is very useful in many other questions of the theory of a.p. functions. Let us now give a definition for multi- valued a.p. functions, according to Bochner, and prove the equivalence of this definition with Bohr's definition.
Keywords: Almost periodic multivalued functions, spaces RAn, equivalence, arbitrary infinite sequence, real numbers, Sum of multivalued numbers
1 INTRODUCTION
In this work considered basic properties almost periodic functions, values are compact sets space Rn. These properties generalizes famous properties unambiguous functions (see [1], [2]). results applicable to research differential inclusions with almost periodic right in parts.
In what follows, via R denoted by the number axis, through a = a(A, B) is the Hausdorff metric on comp ( R n ) and ||.4|| = a(A, B). Basic concepts theories multivalued functions are indicated in [5]. It is much more difficult to prove that the sum of two almost-periodic (a.p.) multi-valued functions is a multivalued a.p. function. The first proof of this theorem for a single-valued function was given by G.Bohr. Subsequently, Bochner gave others a definition for unambiguous a.p. functions on which the almost-periodicity of the sum follows directly. Subsequently, it turned out that Bochner's definition is very useful in many other questions of the theory of a.p. functions. Let us now give a definition for multi- valued a.p. functions, according to Bochner, and prove the equivalence of this definition with Bohr's definition.
2 MATERIAL AND METHODS
Let F ( t) = R ^ comp ( R n ) is a - continuous ambiguous function . Here are the main definitions About definition-1. A bunch of E real numbers are called relatively tight if exists such a number l > 0, which is in each interval (a, a + b)cR length l contained Although would be one number of many E.
About definition-2. The number t is called e - almost-period (e - a.p.) of the multivalued functions F (t) = R
^ comp (Rn) if
sup ( F (t +t), F (t)) < e (1)
About definition-3. (Bohr). a - continuous ambiguous
function F (t) = R ^ comp ( R n ) is called e - almost-
period (e - a.p.), if for each e > 0 there exists relatively
dense set of e - almost-periods this functions.
Each periodic ambiguous function is also a.p. functions.
In the very in fact, if w period F, then all numbers of the
form nw (n = ±1, ±2, ±3, ...) also is periods F, and
therefore - almost-periods F for any e>0. It remains to
note that a bunch of nw relatively tight.
Note elementary properties multi-valued a.p. functions
that should directly from their definitions.
and if F (t)- multi-valued a.p. function, then X F (t) (X
eR) also ambiguous a.p. functions.
If F (t)- multi-valued a.p. functions, then F (t +c) (c -
real number) also ambiguous a.p. functions.
About definition-4. a- a continuous multivalued
function F(t): R ^ comp(Rn) is called a normal
function if the family of multivalued functions [F(t +
h)} (—m < h < m) is compact in the sense of uniform
convergence in a the -metric on the entire R, that is, if
an infinite sequence of functions
F(t + hi), F(t + h2), F(t + h3), ... a subsequence uniformly converging on the whole R. 1-theorem. In order for a- a continuous multi-valued function F(t): R ^ comp(Rn) to be a.p. function, it is necessary and sufficient for it to be a normal function. Proof. Necessity. Let F(t):R ^ comp(Rn) a multivalued a.p. function and h1, h2, h3,... an arbitrary
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infinite sequence of real numbers. It should be shown that from
F(t + h-), F(t + h2), F(t + h3), ... sequence, we can choose a uniformly convergent subsequence. Let's use the diagonal method. Let us denote by
t-, t2, t3, .countable, everywhere dense set of real numbers. Since F(t)- is bounded (see Theorem 1 [6]), then from the sequence
F(t- + h-), F(t- + h2), F(t- + h3), ... we can choose a convergent subsequence:
F(t- + h-i), F(t- + h-2), F(t1 + h13), ... Likewise for the number sequence
F(t2 + h-J, F(t2 + h-2), F(t2 + h13), ... we can choose a convergent subsequence:
F(t2 + h2-), F(t2 + h22), F(t2 + h23), ... Let's continue this process and then consider the diagonal sequence of functions F(t + hu), F(t + h22), F(t + h33), ... (2) Let us show that the diagonal sequence (2) converges at all points of a countable, everywhere dense set. In fact, let be tk an arbitrary point on a countable set of points. According to our construction, the sequence F(tk + hkl), F(tk + hk2), F(tk + hk3),
converges, and for n> k (t = tk) all terms of sequence (2) also converge into sequence (3). 3 RESULTS AND DISCUSSION Therefore, sequence (2) converges at any point tk. Let us now show that sequence (2) converges on the entire R.
Let t0 i be an arbitrary real number and £ > 0. Let us choose a number I = l(^) according to Definition 1 and
a number 5 = 5 Q) according to Theorem 2 (see [6]). We cover pthe interval [0, Z] with intervals 5 and in each of the intervals of length 5 we select a point from a countable everywhere dense set. Let us denote the selected points by s^ s2, s3,..., sp. At fixed £ p is also fixed, therefore, from the convergence of sequence (1) at points s^ s2, s3,..., sp it follows that it is possible to specify such a sufficiently large integer N = n(e) such that the following inequalities r, i > N = N(e) are satisfied p:
a (F(sj + hrr), F(sj + hu)) < (j = lTp) (4)
Let t almost-period of a multi-valued function F(t) enclosed in (-t0, -t0 + I). Then the numbers s0 = t0 + t lie in (0, I) and therefore, for some j |s7- — s0| < 5. Hence,
a (f(sj + hkk), F(so + hkk)) <£~, (k = 1, 2, ...)
(5)
From inequalities ( 4 ) and ( 5 ) it follows
a(F(to + hrr), F(to + ha))
< a(F(So + hrr), F(So + hrr)) + a (f(s0 + hrr), F(sj + hrr)) + +a (f(sj + hrr), F(sj + hu))
+ a (f(sj + hu), F(so + hu)) + a(F(so + hu), F(to + hu)) < £ And since N does not depend on the number fo, but to were £ chosen arbitrarily, then (1) converges uniformly on R. The need has been proven. Sufficiency. LetF(t): R ^ comp(Rn) a is a continuous multivalued function on R and the family [F(t + h)} (—&> < h < x>) is compact. It should be shown that F(t) is a multi- valued a.p. function. Let us assume the opposite, that is, suppose that for some £ = £0 it is impossible to specify the corresponding length l(£0). This means that there is a sequence of intervals whose L-, L2, ... lengths r-i^, r2, ... grow indefinitely so that none of the intervals Ln contain £0 almost-periods multivalued functions F(t).
Let, h- be an arbitrary number and h2 be chosen so that h2 — h- it lies in the interval LVl = L-. Choose LV2 what rV2 > lh2 — h-l and number h3 so that both numbers h3 — h- and h3 — h2 lie in the interval LV2. We select the interval LV3 from the condition,
rV3 > maxih — h-l, ^ — h2l, W3 — h-D and a number h4 such that both numbers h4 — h-, h4 — h2, h4 — h3 lie in the intervals LV3. In general, LVn, we choose so that
rn > ^max {lh^ — hv\] and hn+- so that the numbers
(hn+- — hm) £ LVn, (m = 1, n). Let us now show that from the sequence of functions
F(t + h-), F(t + h2), F(t + h3), ... You cannot choose a uniformly convergent subsequence supa(F(t + hnl), F(t + hn2)) =
t£R
= supa(F(t + hm — hn2), F(t))
t£R
< £0
because by construction all numbers hn- —
h„
(n1 > n2) are in an interval Lv
and none of
the numbers in this interval is £0 an almost-period of a multi-valued function F(t).
The resulting contradiction completes the proof of the theorem. Note that in [3] va [4] this theorem was proved by a different method.
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Let us show that for sets on comp(Rn) the following inequality holds:
a((A, B), (A', B')) < a(A, A') + a(B, B') (6)
Indeed, let it r = a(A, A'), p = a(B, BA) be £ > 0 any. Then A c Ur+E(A'), B c Up+£(B') where Ua(M) a is a neighborhood of the set M. For any, x E A, y E B there exist x' E A', y' E B' such that Hx — x'H <r + £, Hy — y'H < p + £. Then
p(x + y, A'+ B') < H(x + y) — (x'+ y')H <
Hy — y'H < r + p + 2£
Hence,
A + B cUr+p+2E(A' +B'). Similarly, it is proved that A' +B' cUr+p+2E(A+B). That's why
a((A, B), (A', B')) <r + p + 2£ 4 CONCLUSIONS Since £ > 0any, then inequalities (6) are proven. 2-theorem. Sum of multi- valued a.p. functions is also a multi-valued a.p. function.
Proof. multi F(t), Q(t): R ^ comp(Rn)- valued a.p. functions. According to Theorem 1, the families [F(t + h)}, [Q(t + h)} (—m<h<m) are compact. It is easy to see that the family of functions [F(t + h) + Q(t + h)} (—m <h<m) is also compact. In fact, let, for example, be given a sequence of functions F(t + hn) + Q(t + hn). First, we choose a subsequence h'n, for which the sequence of functions F(t + h'n) converges uniformly, and then from the subsequence h'n we select a subsequence h" for which Q(t + h") it converges uniformly. Because a(F(t + h) + Q(t + h), F(t) + Q(t))< a(F(t + h), F(t)) + a{Q(t + h), Q(t)), then the subsequence F(t + h'^) + Q(t + hJOconverges uniformly to the function F(t) + Q(t). Therefore, the family [F(t + h) + Q(t + h)} (—m < h < m) is compact and, therefore, by Theorem 1, it is F(t) + Q(t) a multivalued a.p. function. Let us show that for sets from comp(Rn) the following inequality is true:
a((A, B), (A', B')) < max[a(A, A') + a(B, B')} (7)
Indeed, let r = a(A, A'), p = a(B, B'), V£ > 0
any and for certainty p>r. Then
A c Ur+E(A'), B c Up+E(B'),
AUB c Ur+£(A') U Up+£(B') c Up+£(A' U B').
In a similar way it is verified that
A' UB' c Ur+£(A) U Up+£(B) c Up+£(A U B).
Where follows (7).
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[3]. Banzaru T. Aplicatii multi voce aproape-periodioe, Bul.Sti.Pehnic Inat. Polytechnic Fimisoava . Mat. fiz.- mec . 19(33). fabc , 1 (1974), p 25-26.
[4]. Banzaru T., Cvivat N. Asupva applicate multi voce apvoape-periodice cu volovi in spatii uniforme. Bul. Sti Pehnis Inst. Polytechnic Fimisoava, Mat.- fiz., 1981, 26 (40) fasc (2) p.47-51.
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