Critical remarks on “Existence of the solution to second order differential equation through fixed point results for nonlinear F-contractions involving w0-distance” Текст научной статьи по специальности «Математика»

<D

O >

CO C\l o (M

Critical remarks on "Existence of the solution to second order differential equation through fixed point results for nonlinear F-contractions involving w0-distance"

Zoran Kadelburga, Nicola Fabianob, E Milica Savatovicc, Stojan Radenovicd

O a University of Belgrade, Faculty of Mathematics,

Belgrade, Republic of Serbia, e-mail: [email protected], ORCID iD: © https://orcid.org/0000-0001-9103-713X

o b University of Belgrade, "Vinca" Institute of Nuclear Sciences - National

m Institute of the Republic of Serbia, Belgrade, Republic of Serbia,

> e-mail: [email protected], corresponding author

<t ORCID iD: ©https://orcid.org/0000-0003-1645-2071

University of Belgrade, School of Electrical Engineering, Belgrade, Republic of Serbia, e-mail: [email protected], ORCID iD: © https://orcid.org/0000-0003-0439-1451 d University of Belgrade, Faculty of Mechanical Engineering, ^ Belgrade, Republic of Serbia,

^ e-mail: [email protected],

ORCID iD: © https://orcid.org/0000-0001-8254-6688

x

h DOI: 10.5937/vojtehg71-46505;https://doi.org/10.5937/vojtehg71-46505

FIELD: mathematics ^ ARTICLE TYPE: original scientific paper

Abstract:

Introduction/purpose: In this paper, several critical remarks are presented concerning the paper of Iqbal & Rizwan: Existence of the solution to second order differential equation through fixed point results for nonlinear F-contractions involving w0-distance from 2020. Methods: Conventional theoretical methods of functional analysis. Results: It is shown that their use of the non-decreasing "control" function F instead of a strictly increasing one in Wardowski-type results usually produces contradictions.

Conclusion: It is shown that such results can be obtained in a more general class of metric-like spaces, where strict monotonicity is the only as-

ACKNOWLEDGMENT: The research of Milica Savatovic was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, Project No 174032.

sumption that has to be imposed on the function F. An example is pre- ¡5

sented showing that the obtained results are stronger than the classic ^

ones. oo

Key words: F-contraction, fixed point, metric-like space, strictly increas- £ ing function.

Introduction and preliminaries

o

CP

~o

<u

X

o

o

D. Wardowski's result from (Wardowski, 2012) can be considered as one of the significant generalizations of S. Banach's basic result from 1922. In this generalization, Wardowski used a function F : (0, ^ R satisfying the following three properties:

(F1) F is strictly increasing, i.e., ti < t2 implies F(ti) < F(t2);

(F2) for any sequence {an} c (0, an = 0 if and only if

F(an) = -to (i.e., limt^o+ F(t) = -to);

(F3) there exists k e (0,1) such that lim^0+ tkF(t) = 0.

He proved that a self-map T on a complete metric space (X, d) has a unique fixed point if there exists a positive number t and a function F satisfying the previously mentioned conditions, such that, for all x,y e X 8 with Tx = Ty,

t + F(d(Tx,Ty)) < F(d(x,y)) (1)

holds.

Later on, in another work (Wardowski, 2018), he generalized the mentioned result, replacing the condition (1) by the following one

<u o c <u

CO

x

p(d(x, y)) + F(d(Tx, Ty)) < F(d(x, y)), (2)

where p is some function from (0, to itself satisfying lim infp(s) > 0 for each t > 0 (he called such mappings (p, F)-contractions).

In a multitude of subsequent papers, dozens of authors used Wardowski's approach for mappings acting in various spaces (such as 6-metric | spaces, partial metric spaces, metric-like spaces, cone metric spaces, G-metric spaces, rectangular metric spaces, as well as in spaces endowed with a w-distance) - a review of these results until 2022 can be found in (Fabiano et al, 2022). We recall here the definitions of w-distance p of O. Kada, T. Suzuki and W. Takahashi and metric-like n of A. Amini-Harandi. n

ro

Definition 1. (Kada et al, 1996) Let (X,d) be a metric space and let a mapping p : X x X ^ [0, satisfy:

865

(p1) p(x, z) < p(x, y) + p(y, z) for all x,y,z G X; (p2) for any x g X, the function p (x, ) : X ^ [0, +rc>) is d-lower semi-continuous;

o (p3) for any e > 0, there exists 5 > 0 such that p (z, x) <5 and p (z, y) <

co 5 imply d (x,y) <e.

Then, p is called a w-distance on X.

cm o cm

0£ yy

0£ ZD

o o

-J

<

o

x

o

LU

I—

^

Q1 <

Definition 2. (Amini-Harandi, 2012) A metric-like on a nonempty set X is a function n : X x X ^ [0, +rc>) if the following conditions hold for all

x,y,z G X:

(u1) n(x,y) = 0 implies x = y; (p2) n(x,y) = n(y,x); (ju3) fi(x, y) < fi(x, z) + p(z, y). Then (Xis called a metric-like space.

Note that, see, e.g. (Iqbal & Rizwan, 2020) or (Kadelburg & Radenovic, 2024), for a w-distance p on a set X, the mapping

| n(x, y) = max{p(x, y), p(y, x)}

CD S2

y The notions of convergent and Cauchy sequences, and continuous functions, were introduced in metric-like spaces as follows.

X LU I—

o

is a metric-like on X.

Definition 3. (Amini-Harandi, 2012) Let (X, d) be a metric-like space and

o {xn} be a sequence in X.

1. The sequence {xn} is said to converge to x g X if

d(x, xn) = d(x, x).

2. {xn} is a Cauchy sequence if d(xm,xn) exists and is finite.

3. The space (X,d) is said to be complete if every Cauchy sequence {xn} in X converges to some x g X such that

limm,n^+^ d(xm, xn) = d(xj x) = limn^+^ d(xn, x).

4. A mapping T : X ^ X is continuous at a point x g X if xn = x implies Txn = Tx.

On the other hand, some authors considered different conditions on the "control" function F, culminating in O. Popescu's and G. Stan's proof, see

(Popescu & Stan, 2020), Theorem 5, that in fact just condition (F1) is suffi- £ cient for obtaining the basic result from (Wardowski, 2012), see also (Fabi- 1 ano et al, 2022), Theorem 2.3 and Remark 2.4. °°

It is natural to ask whether the property (F1) of the function F can be ^ replaced by a weaker property that F is non-decreasing, but not strictly increasing. Of course, there are a lot of such functions. a.

In the recent paper (Iqbal & Rizwan, 2020), the authors tried to generalize the results of papers (Wardowski, 2012) and (Wardowski, 2018) in two ways - firstly, instead of metric d they used w-distance p or metric-like ¡j,. On the other hand, instead of the assumption (F1), the authors of (Iqbal & Rizwan, 2020) used the weaker assumption

(F1') the function F is non-decreasing on (0, i.e. ti < t2 implies F(ti) < F(t2), together with the assumptions (F2) and (F3).

Unfortunately, all their results obtained under the assumption (F1') may be incompatible with a Wardowski-type contractive condition. To show that, the following observation can be useful.

Now, it is easy to construct examples like the following one.

Example. Consider X = [0, with the standard metric d(x, y) = \x - y\ and the mapping T : X ^ X given by Tx = 4x. The function

satisfies conditions (F1'), (F2) and (F3) (but not (F1)!). However, if 4 < \x - y\ < 2, then the condition (1) reduces to t + 0 < 0, i.e., t < 0, which is incompatible with the basic assumption t > 0.

Lemma 4. If F : (0, +œ) ^ R is a non-decreasing, but not strictly increas- g ing function, then there exists an interval (a, b) c (0, +rc>) such that the restriction of F to this interval is constant.

Proof.Since F is non-decreasing but not strictly increasing, then there are -§ a,b e (0, such that a < b and F(a) = F(b). But then F is a constant function on (a, b). □

<u o c <u -t—f CO

'x

CO

log t, 0 <t< 1, E

F(t) = <0, 1 < t < 2,

\og(t - 1), t> 2,

o m

CD

N

<D CO

"J One possible additional assumption when we use the function F satis-

fying just the condition (F1') could be that

{ d(x,y): x,y e X }n(U Ic) = to, > c

co

o where Ic is the interval (a, b) such that F(t) = C for each t e (a, b). of In order to improve and generalize the results from (Iqbal & Rizwan, ^ 2020), we first state the following two known lemmas that are of interest o in themselves and can be used in proving the Cauchyness of a Picard se° quence {xn} = {Tnx0} in both metric and metric-like spaces, see some o references on these lemmas in (Fabiano et al, 2022).

o Lemma 5. Let (X, d) be a metric-like space and {xn} be a Picard sequence in it. If

d(xn+i,xn) < d(x

n, xn i ) ,

<

for all n e N, then xn = xm whenever n = m.

Lemma 6. Let (X, d) be a metric-like space and {xn} be a sequence in X So such that {d(xn+i,xn)} is a non-increasing sequence and that Tn limn^+^ d(xn+i,xn) = 0. If {xn} is not a Cauchy sequence, then for some

CD

S2 e > 0 there exist two sequences {mk} and {nk} of positive integers with nk > mk > k, such that the following sequences tend to e+ as k ^

x

LU I—

o o

d (x2mk ,x2nk ) , d (x2mk , x2nk-i) , d (x2mk+i,x2nk )

d (x2mk-i,x2nk + i ) , d (x2mk+i,x2nfc + i) , •••

Remark 7. Lemma 6 is true without the hypothesis that the sequence {d(xn+i,xn)} is non-increasing. In that case one can get that the following sequences tend to e+ as k ^

d (xmk ,xnk ) , d (xmk ,xnk- i) , d (xmk+i,xnk ) , d (xmk-i,xnk + i ) , d (xmk+i,xnk+i) , •••

In this paper, besides already mentioned problems with using non-decreasing functions in Wardowski-type results, we show how generalizations of such results can be derived in the framework of metric-like spaces, using just strict monotonicity of the "control" function F. Modifying the original Wardowski's example, we show that the obtained results are stronger than Banach-type ones.

Results

869

LT)

oo I

In this part of the work, p will be a function that maps (0, +<x>) to itself and for which lim infp(s) > 0 is fulfilled for each t > 0, while F will be £ a strictly increasing function that maps (0, +rc>) to R. We aim to generalize and improve all three results from (Iqbal & Rizwan, 2020) by replacing the | metric d and the w-distance w with a metric-like j. Concerning the function F, just its strict monotonicity will be assumed. This will also extend D. Wardowski's result from (Wardowski, 2018).

~o

<u

X

o

Theorem 8. Let (X, j) be a complete metric-like space and T : X — X. If p and F are functions with the properties stated above, and such that, for |

all x,y G X,

x = y and j(Tx, Ty) > 0 implies p(j(x, y)) + F(j(Tx, Ty)) < F(j(x, y)), S

(3)

then T has a unique fixed point in X.

Proof. We first prove the uniqueness of a possible fixed point. Indeed, if x and y be two distinct fixed points of the mapping T, then both conditions | would be met and still it would hold that p(j(x, y))+F(j(x, y)) < F(j(x, y)), which is a contradiction with p(j(x,y)) > 0.

Similar as in the case of metric spaces, the continuity of mapping T follows from the contractive condition (2); however, due to the definition of limits in metric-like spaces (see Definition 3), the proof is a bit different. Namely, we have to prove that, for every sequence {xn} in X, j(xn, x) — j(x, x) = 0 as n — implies that j(Txn, Tx) ^ 0 as n ^

But it follows from the contractive condition (2) that j(Txn,Tx) < j(xn, x), implying that j(Txn, Tx) ^ 0 as n ^ It remains to prove that j(Txn, Tx) j(Tx, Tx), which will follow if we show that j(Tx,Tx) = 0. However, j(Tx,Tx) < 2j(Tx,Txn) according to the triangle relation. ^ Thus, we have proved that the continuity follows from the contractive con- | dition. ro

In order to prove the existence of at least one fixed mapping point of T, starting from an arbitrary point xo g X, form the corresponding Picard sequence {xn}. If, for some k, xk = xk-\ holds, then according to the first part of the proof, xk -i is a unique fixed point of T. Hence, assume that xn = xn-l for each n g N. Then, putting x = xn-l and y = xn into the contractive condition (3), it directly follows that the sequence j(xn,xn+l)

o

O "ro <u N OT

<D

O <

X Lll

O >

is non-increasing and so it has a limit as n ^ If we denote it by H*, using the property of strict monotonicity of the function F, we get a contradiction with p(p*) > 0. Then, according to Lemma 5, assuming that the constructed sequence {xn} is not a Cauchy sequence, the conditions for applying of Lemma 6 are fulfilled. Namely, by putting x = xnk, y = xmk we get

o >

co CM o CM

0£ yy

0£

o Passing to the limit as k ^ we get a contradiction with

p(^(xnk ,xmk )) + F{p(xnk+1 ,xnk+l)) < F(p(xnk ,xmk )) ■

liminfp(s) > 0.

^ Since {xn} is a Cauchy sequence in the complete metric-like space

(X, f), then there exists a (unique) point x* e X such that

x

o

jJU

>- lim f(xn,xm) = lim f(xn,x*) = f(x*,x*) = 0.

<t

Since T is continuous, Tx* = x* holds. □

We now state several consequences of our Theorem 8. « Putting p(t) = t for each t > 0 in the contractive condition (3) of The-

u orem 8, where t is a positive constant, we get the main result of D. War-g dowski from (Wardowski, 2012), but in the framework of metric-like spaces

and with a weaker assumption on the function F :

o Corollary 9. Let (X, f) be a complete metric-like space and T : X ^ X. If F : (0, +rc>) ^ R is a strictly increasing function, such that, for all x,y e

X,

x = y and f(Tx, Ty) > 0 implies t + F (f(Tx, Ty)) < F fx, y)), then T has a unique fixed point in X.

The following example (which is adapted from (Wardowski, 2012), Example 2.5) shows that our Corollary 9 is stronger than the Banach-type fixed point result in metric-like spaces.

n(n + 1)

Example. Consider the set X = {Sn : n e N}, where Sn = —^—-, and let f(x,y) = max{x,y} for x,y e X. Then, (X,f) is a complete metric-like space (of course, it is not a metric space). Let a mapping T : X ^ X be given by TSi = Si and TSn = Sn -1 for n > 1.

We show first that T is not a Banach-type contraction in (X, ¡). Indeed, it is

¡(TSn,TSi) max{S„_i, 1} Sn_1

lim -TE<—— = lim -ro = lim ——

n^+œ fx(Sn,S1) n^+œ max{Sn, 1} n^+œ Sn

lim 1 (n - 1)n = 1,

2° Similarly, in this case it is

max{TSm, TSn} emax{TSm,TSn}_max{Sm,Sn} = Sm_1 &Sm-1_Sm < &_m < g_1_

maX{Sm, Sn} Sm

Hence, all conditions of Corollary 9 are fulfilled and the conclusion follows.

Remark 11. Since convergence, Cauchyness, completeness and continuity are defined in the same way for all following classes of spaces: partial,

LT)

oo I

^f

CD oo £±

O

n^+œ 2n(n + 1) ' 73

which means that ¡(TSn, TS1 ) < A ¡(Sn, S1) cannot hold for any A < 1 and all n e N.

Now, we show that the contractive condition of Corollary 9 holds if we take F(t) = t + log t and t = 1. In this case, this condition can be rewritten is as

1(Tx, Ty) n(Tx,Ty)_v(x,y) < e_1

¡(x,y) ~

for all x,y e X with x = y and ¡(Tx, Ty) > 0, i.e., in our example, as |

o

o

<u

max{TSm,TSn} emax{TSm,TSn}_max{Sm,Sn} < e_1 P

maX{STO, Sn}

"D C

o o <u

CO

Note that TSm = TSn holds (for m > n) if and only if one of the following holds: 1° m > 2, n = 1 or 2° m > n > 1. Consider separately these two cases.

1° In this case we have °

max{TSm,TSi} emax{TSm,TS1}-max{Sm,S1} = Sm-1 pSm-1-Sm < p-m < £-1

max{STO,Si} Sm cu

o c <u co

x

ro E <u

ro o

Remark 10. Since every partial metric space, in the sense of (Matthews, ° 1994), is also a metric-like space, Theorem 8 and Corollary 9 are also true in the class of partial metric spaces.

ro <u N ci

<u

"O

ro

<u

o >

CO

metric-like, partial b-metric (Shukla, 2014) and b-metric-like spaces (Al-ghamdi et al, 2013), then Theorem 8 can most likely be formulated and proved for all these classes of spaces, including the most general one—b-metric like spaces.

Remark 12. Just like Theorem 2.1, the other two Theorems 2.2 and 2.3 from (Iqbal & Rizwan, 2020) can be formulated and proved within the class w of metric-like spaces. And then only strict growth of the function F has to be assumed. One can use some function p or a given constant t as in our § Theorem 8 or Corollary 9.

0 Remark 13. Finally, note that the authors of (Iqbal & Rizwan, 2020), in

1 the examples and applications at the end of the paper, used just functions w with strict growth, in contrast with the theoretical results in the paper which ^ they claim to hold when a non-decreasing function F is used. Moreover, ft in all these examples and applications, the only function F that is used is

F(t) = ln t, which is trivially known to produce only very well-known results which can be treated in a classical way, without using the ideas from the papers (Wardowski, 2012) and (Wardowski, 2018). That was also one of <5 our motivations to consider and discuss the work (Iqbal & Rizwan, 2020).

CD S2

| References

i

^ Alghamdi, M.A., Hussain, N. & Salimi, P. 2013. Fixed point and coupled fixed

o point theorems on b-metric-like spaces. Journal of Inequalities and Applications, ¡3 art.number:402. Available at: https://doi.org/10.1186/1029-242X-2013-402.

Amini-Harandi, A. 2012. Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory and Applications, art.number:204. Available at: https://doi.org/10.1186/1687-1812-2012-204.

Fabiano, N., Kadelburg, Z., Mirkov, N., Sesum Cavic, V. & Radenovic, S. 2022. On F-contractions: A survey. Contemporary Mathematics, 3(3), pp.327-342. Available at: https://doi.org/10.37256/cm.3320221517.

Iqbal, I. & Rizwan, M. 2020. Existence of the Solution to Second Order Differential Equation Through Fixed Point Results for Nonlinear F-Contractions Involving wo-Distance. Filomat, 34(12), pp.4079-4094. Available at: https://doi.org/10.2298/FIL2012079I.

Kada, O. Suzuki, T. & Takahashi, W. 1996. Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Mathematica Japonic^, 44(2), pp.381-391. Available at: https://cir.nii.ac.jp/crid/1570009749812799360.

Kadelburg, Z. & Radenovic, S. 2024. Some new observations on w-distance

cd

Matthews, S.G. 1994. Partial Metric Topology. Annals of the New York Academy

and ^-contractions. Matematicki Vesnik, 76(1), in press.

of Sciences, 728(1), pp.183-197. Available at: https://doi.org/10.1111/j.1749-6632.1994.tb44144.x.

Popescu, O. & Stan, G. 2020. Two Fixed Point Theorems Concerning F- ° Contraction in Complete Metric Spaces. Symmetry, 12(1), art.number:58. Available at: https://doi.org/10.3390/sym12010058.

Shukla, S. 2014. Partial b-Metric Spaces and Fixed Point Theorems. Mediterranean Journal of Mathematics, 11, pp.703-711. Available at: https://doi.org/10.1007/s00009-013-0327-4.

Wardowski, D. 2012. Fixed points of a new type of contractive mappings in complete metricspaces. Fixed Point Theory and Applications, art.number:94. Available at: https://doi.org/10.1186/1687-1812-2012-94.

Wardowski, D. 2018. Solving existence problems via F-contractions. Proceedings of the American Mathematical Society, 146(4), pp.1585-1598. Available at: https://doi.org/10.1090/proc/13808.

о

о

ф "о

о -а с

Критические замечания о статье «О существовании 8

решения дифференциального уравнения второго порядка <"

через результаты о неподвижной точке для нелинейных F-сжатий с использованием т0-дистанцией»

о

Зоран Кадельбург3, Никола Фабиано6, ™

Милица Саватовичв, Стоян Раденовичг

3 Белградский университет, факультет математики

г. Белград, Республика Сербия

б ф

б Белградский университет, Институт ядерных исследований %

«Винча» - Институт государственного значения для Республики -

Сербия, г Белград, Республика Сербия, корреспондент

Белградский университет, факультет электротехники, г. Белград, Республика Сербия г Белградский университет, Машиностроительный факультет, г. Белград, Республика Сербия го

РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА, О

27.25.17 Метрическая теория функций, го

27.39.15 Линейные пространства, ф

снабженные топологией, порядком N и другими структурами ВИД СТАТЬИ: оригинальная научная статья

873

<u о

ф

ф "О

го

Резюме:

Введение/цель: В этой статье представлено несколько критических замечаний относительно статьи, написанной в 2020 году Iqbal & Rizwan: Existence of the solution to > second order differential equation through fixed point results for

nonlinear F-contractions involving w0-distance.

Методы: Общепринятые теоретические методы функци-

ей

ш онального анализа.

з Результаты: Доказано, что использование ими неубываю-

§ щей "управляющей" функции F вместо строго возрастаю-

щей в результатах типа Вардовского обычно приводит к

0 противоречиям.

1 Выводы: Показано, что такие результаты могут быть ш получены в более общем классе пространств подобных >_ метрическим, где строгая монотонность является единственным условием, которое необходимо наложить на функцию Е Приведен пример, показывающий, что полученные результаты сильнее классических.

Ключевые слова: Е-сжатие, неподвижная точка, метриче-от ское пространство, строго возрастающая функция.

^ Критичке напомене о чланку „Посща^е решена

диференци]алне ]едначине другог реда помогу резултата о

ш непокретно] тачки F-контракциja користе^ем ад0-дистанцу" н

О Зоран Каделбурга, Никола Фабиано6,

^ Милица Саватови^6, Сто]ан Раденови^г

Универзитету Београду, Математички факултет, Београд, Република Ср6и]а

6 Универзитет у Београду, Институт за нуклеарне науке "Винча" - Национални институт Републике Срби]е, Београд, Република Срби]а, ауторза преписку в Универзитету Београду, Електротехнички факултет,

Београд, Република Срби]а г Универзитету Београду, Машински факултет, Београд, Република Срби]а

ОБЛАСТ: математика

КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад

Сажетак:

оо

Увод/цил>: У овом раду изнето je неколико критичких напо-мена у вези са радом Iqbal & Rizwan: Existence of the solution to second order differential equation through fixed point results for nonlinear F-contractions involving w0-distance, из 2020. године. 5

Œ TD Ф

X

Методе: Конвенционалне теор^ске методе функционалне анализе.

Резултати: Показано je да шихова употреба нeопадаjуhe „контролне" функц^е F уместо стриктно растуПе у резул-татима типа Вардовског обично производи контрадикци-

е

Закъучак: Такви резултати могу се добити у општиjоj класи метричких простора, где je строга монотоност je-дина претпоставка ща се мора наметнути функции F. Приказан je пример ко\и показуje да су доби/ени резултати jачи од класичних.

Къучне речи: F-контракц^а, фиксна тачка, простор сли-чан метрици, строго растуЬа функц^а.

со ОО Œ

о

о

го

ф

2 ф

¡t=

тз ф

"О

о

"О с о

о ф

(Я

ф

Paper received on / Дата получения работы / Датум приема чланка: 12.08.2023. Manuscript corrections submitted on / Дата получения исправленной версии работы / _ Датум достав^а^а исправки рукописа: 21.11.2023. 8

Paper accepted for publishing on / Дата окончательного согласования работы / Датум Ф коначног прихвата^а чланка за об]ав^ива^е: 22.11.2023.

ф

© 2023 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier ^

(http://vtg.mod.gov.rs, http://втг.мо.упр.срб). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ^

(http://creativecommons.org/licenses/by/3.0/rs/).

© 2023 Авторы. Опубликовано в "Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier" (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией "Creative Commons" ^

(http://creativecommons.org/licenses/by/3.0/rs/). ф

© 2023 Аутори. Об]авио Во]нотехнички гласник/ Vojnotehnicki glasnik / Military Technical Courier ^ (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Ово ]е чланак отвореног приступа и дистрибуира се у складу са Creative Commons лиценцом (http://creativecommons.org/licenses/by/3.0/rs/). О

e

N

Е? ur

lb el

d a

875