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19
QPI/in/lHAflHI/1 HAyMHM M^A-^I/l OPI/in/lHAflbHblE HAyMHblE CTATbll ORIGINAL SCIENTIFIC PAPERS
A NOTE ON THE MEIR-KEELER THEOREM IN THE CONTEXT OF b-METRIC SPACES
Mirjana V. Pavlovic3, Stojan N. Radenovicb
a University of Kragujevac, Faculty of Science, Department of Mathematics and Informatics, Kragujevac, Republic of Serbia, e-mail: mpavlovic@kg.ac.rs, ORCID iD: http://orcid.org/0000-0001-6257-8666, b King Saud University, College of Science, Mathematics Department, Riyadh, Saudi Arabia, e-mail: radens@beotel.rs,
ORCID iD: https://orcid.org/0000-0001-8254-6688
DOI: 10.5937/vojtehg67-19220; https://doi.org/10.5937/vojtehg67-19220
FIELD: Mathematics (Mathematics Subject Classification: primary 47H10,
secondary 54H25) ARTICLE TYPE: Original Scientific Paper ARTICLE LANGUAGE: English
Abstract:
In this note we consider the famous Meir-Keeler's theorem in the context of b-metric spaces. Our result generalizes, improves, compliments, unifies and enriches several known ones in the existing literature. Also, our proof of Meir-Keeler's theorem in the context of standard metric spaces is much shorter and nicer than the ones in (Ciric, 2003) and (Meir & Keeler, 1969, pp.326-329).
Keywords: b-metric space, b-complete, b-Cauchy, Meir-Keeler conditions, Picard sequence.
Definitions, notations and preliminaries
Let (X, d) be a standard metric space and f : X ^ X be a self-mapping. In the context of these spaces, the following (Meir-Keeler) conditions are well known: For each s> 0 there exists 8 - 8(s)> 0 such that for all x, y e X holds
ACKNOWLEDGMENT: The first author is grateful for the financial support from the Ministry of Education and Science and Technological Development of the Republic of Serbia (Metode numericke i nelinearne analize sa primenama, 174002).
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s< d(x,y)<s + 5 implies d(fx, fy)<s (1)
or
k s < d (x, y)<s + 5 implies d(fx, fy)<s (2)
o or / is contractive and
s< d(x,y)<s + 5 implies d(fx, fy)<s. (3)
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oc One says that the mapping f defined on the standard metric space
E (X, d) is contractive if d(fx, fy)< d(x, y) holds, whenever x ^ y. o For more details, see (CiriC, 2003, pp.30-33, pp.56-58).
o In 1969, Meir-Keeler proved the following:
Theorem 1 (Meir & Keeler, 1969, pp.326-329, Theorem) Let (X, d ) £ be a complete metric space and let f be a self-mapping on X satisfying
< (1). Then f has a unique fixed point, say u e X, and for each
x e Xf"x = u .
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Inspired by the above Meir-Keeler theorem, Ciric proved the following, slightly more general result:
0 Theorem 2 (Ciric, 2003, Theorem 2.5) Let (X, d) be a complete
1 metric space and let f be a self-mapping on X satisfying (2). Then f has
o a unique fixed point, say u e X, and for each x e X,limfnx = u .
The example which follows shows that Ciric's result is a proper generalization of the famous Meir-Keeler theorem:
Example 3 Let X = [0,1]u {3n -1} eN u J 3n + — \ be a subset
I 3n J neN
of real numbers with the Euclidean metric and let f be a self-mapping on X defined by
fx = 0, if 0 < x < 1 and x e {3n - l}ni
JneN
fx = 1, if x e|3n + 3nj
neN
Then one can verify that f satisfies (2) while it does not satisfy Meir-Keeler condition (1). For all details, see (Ciric, 2003, p.33).
Remark 1 Both previous theorems are true if the self-mapping f : X ^ X satisfies condition (3).
Bakhtin (Bakhtin, 1989, pp.26-37) and Czerwik (Czerwik, 1993, pp.5-11) introduced ¿»-metric spaces (as a generalization of metric spaces) and proved the contraction principle in this context. In the last period, many authors have obtained fixed point results for single-valued or set-valued functions, in the context of ¿-metric spaces. Now we give the definition of a ¿-metric space:
Definition 1.1 (Bakhtin, 1989, pp.26-37), (Czerwik, 1993, pp.5-11) Let X be a nonempty set and let s > 1 be a given real number. The function d : X x X ^ [0, ro) is said to be a ¿-metric if, and only if, for all x, y, z e X the following conditions hold:
b1) d(x, y)- 0 if, and only if, x - y;
b2) d(x, y)- d(y, x);
b3) d(x, z)< s[d(x, y) + d(y, z)].
A triplet (X, d, s > l) is called a ¿-metric space with the coefficient s.
It should be noted that the class of ¿-metric spaces is effectively larger than that of standard metric spaces, since a b-metric is a metric when s - 1. The following example shows that, in general, a ¿-metric does not necessarily need to be a metric (Chandok et al, 2017, pp.331345), (Dosenovic et al, 2017, pp.851-865), (Dubey et al, 2014), (Dung & Hang, 2018, pp.298-304), (Faraji & Nourouzi, 2017, pp.77-86), (Jovanovic et al, 2010), (Jovanovic, 2016), (Kir & Kiziltunc, 2016, pp.1316), (Kirk & Shahzad, 2014).
Example 4 Let (X,p) be a standard metric space, and d(x,y)={p(x,y))p ,p> 1 is a real number. Then d is a ¿-metric with s - 2p 1, but d is not a standard metric on X.
Otherwise, for more concepts such as ¿-convergence, ¿-completeness, ¿-Cauchy and ¿-closed set in ¿-metric spaces, we refer
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the reader to (Dosenovic et al, 2017, pp.851-865), (Dubey et al, 2014), (Dung & Hang, 2018, pp.298-304), (Faraji & Nourouzi, 2017, pp.77-86), (jovanovic et al, 2010), (Jovanovic, 2016), (Kir & Kiziltunc, 2016, pp.13-£ 16), (Kirk & Shahzad, 2014), (Koleva & Zlatanov, 2016, pp.31-34), (Chifu | & Petru§el, 2017, pp.2499-2507), (Kumar et al, 2014, pp. 19-22), (Miculescu & Mihail, 2017, pp.1-11), (Paunovic et al, 2017, pp.41624174), (Singh et al, 2008, pp.401-416), (Sintunavarat, 2016, pp.397-416), dc (Suzuki, 2017), (Zare & Arab, 2016, pp.56-67)
The following two lemmas are very significant in the theory of a fixed point in the context of 6-metric spaces. ^ Lemma 1.2 (Jovanovic et al, 2010, p.15, Lemma 3.1) Let {an }neNu{0}
y be a sequence in a 6-metric space (X,d, s > l) such that
d (an, an+i )< kd (an-l, an)
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sequence in a ö-metric space (X, d, s > l).
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Lemma 1.3 (Miculescu & Mihail, 2017, pp.1-11, Lemma 2.2) Let < {an }neWuj0} be a sequence in a 6-metric space (X,d, s > l) such that
* d (an , an+1 kd («n-^ an )
for some k e [0,l), and each n = 1,2,... Then {an} is a 6-Cauchy sequence in a ö-metric space (X, d, s > l).
Remark 2 In (Dosenovic et al, 2017, pp.851-865), it is proven that the previous lemmas are equivalent.
Since in general a ö-metric is not necessarily continuous, many papers related with ö-metric spaces used the following lemmas to prove the main results.
Lemma 1.4 (Aghajani et al, 2014, pp.941-960, Lemma 2.1) Let ((, d, s > l) be a ö-metric space. Suppose that {an} and {bn} are ö-convergent to a and b, respectively. Then
\ d (a, b)< lim inf d (an, bn )< lim sup d (an, bn )< s 2d (a, b).
s n^o.
In particular, if a=b, then we have limn^ro d(an, bn)- 0. Moreover, for each c e X, we have d.
— d(a, c)< lim inf d(an, c)< lim sup d(an, c)< sd(a, c).
Lemma 1.5 (Paunovic et al, 2017, pp.4162-4174, Lemma 2.3) Let (X, d, s > l)be a ¿-metric space and } a sequence in X such that
lim nnw d (an, an+i) = 0.
If } is not 6-Cauchy, then there exist s> 0 and two sequences {m(k)} and \n(k)} of positive integers such that the following items hold:
SS < lim jf d(am(k> an(kM ) < lim SUUP d(am(k)> an(kM ) < SS2 ,
s
s < lim jf d(am(k' an(k)) < lim d(am(kYl, an(k)) < S2 ,
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s < lim ifd(am(k)' an(k)) < lim SUP d(am(k), an(k)) < S , ^
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S < lim inf d(m(k^, an(k ),—) < lim SUP d(m(k >1, an(k Yl) < ss3 . 5
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In particular, if s - 1 and {an} is not a ¿-Cauchy sequence, then _
to
there exists s > 0 as well as two sequences {m(k)} and \n(k)} of positive integers such that the sequences
d (am(k > an (k)), d (a m(k), an(k )+1 )d(am(k)+i,an(k)) andd(am(k)+i,aMk)+i) (4) J
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tend to s as k ^ ro. Q-
Main result
Now, according to the last Lemma (the condition s -1), we formulate and prove the following result:
Theorem 5 Let (X, d) be a complete metric space and let f be a contractive self-mapping on X satisfying the next condition: Given s > 0, there exists 8 > 0 such that for all x, y e X
s < d (x, y) < s + 8 implies d(fx, fy) < s . (5)
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Then f has a unique fixed point, say u e X, and for each
x e X, limn^ fnx = u .
35 Proof. Let x0 in X be arbitrary. Consider the sequence of iterates
{/% } If d(fnx0, fn+1x0 )= d(fnx0, ffnx0 )= 0 for some n e N , then fnx0 is a fixed point off. Assume now that d(fnx0, fn+1 x0)> 0 for
V -0 n=0- " " v/"x0'fn +x0 ) = d fX fTx0 ) = 0 TOr s0me n>
cm
oc an = f "x0 is a Tixea point otf. Assume now tnat d\f "x0,f "*x0 )>
=5 all n g N. Since f is contractive, the sequence fd (fnx0, fn+1 x0 is u strictly decreasing. Therefore, there exists the limit of this sequence, say s, and d(fnx0, fn+1x0)>s Tor all n g N. Assume that s> 0. In this I case, by hypothesis, there exists a suitable 5 = 5(s)> 0 such that (5) holds. From the definition of s, it follows that there is n g N such that
DC s< d (fnx0, fn+1 x0 )<s + 5. (6)
<
According to (5), we get that
d (ffnx,, ffn)= d (fn+1 x0, fn+2 x0 )< s , a contradiction. Therefore limn^œ d (f nx0, fn+1 x0 )= 0.
Now we show that f nx0 }0 is a Cauchy sequence. IT this is not the
o case, by applying Lemma 1.5 to the sequence {fnx0 , we get that
J there exist s > 0 and two sequences of positive integers {m(k)} and
o f(k)} such that n(k)> m(k)> k, and sequences (4) tend to s+ as
o k . Using the condition (5) with x = am\k), y = an\k) and the
5 = 5(s)> 0, ones obtains that there exists a positive integer l such that Tor each k > l, we have
s < d (am (k ), an\k ))= d\fam(k )-1, fan(k )-1 )< s + 5 impUeS d\fam \k ), fan\k )) < s .
This contradicts the Tact that
d\fam(k) fan\k))= d(am\k)+1,an\k)+1 s+ ask ^ œ.
Hence, f nx0 } is a Cauchy sequence.
The proof is further as in (Ciric, 2003) and (Meir & Keeler, 1969, pp.326-329).
To our knowledge, it is not known whether Meir-Keeler's and Ciric's theorems hold in the context of a b-metric space. Also, there is no known example that confirms that conditions (1) or (2) or (3) holds in the context of ¿-metric spaces but that f does not have a fixed point.
However, with a stronger condition than (1), we have the positive result. Hence, our main result is the following:
Proof. It is clear that for all x, y e X we obtain
d (fx, fy )< kd (x, y ), (8)
where k = e [0,l).
Let a0 e X be an arbitrary point. Define the sequence {an} by an+1 = fan for all n > 0. If an = an+1 for some n, then an is a fixed point (unique) of f and the results follows.
So, suppose that an ^ an+1 for all n > 0. From the condition (8), we obtain
d (an, an+i )< kd (an an). (9)
Further, according to (Miculescu & Mihail, 2017, pp.1-11, Lemma 2.2.) we obtain that {an} is a b-Cauchy sequence in a b-metric space
(X, d). By the b-completeness of (X, d), there exists u e x such that lim an = u . (10)
Finaly, (8) and (10) imply that fu = u, i.e. u is a unique fixed point of f in X.
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Theorem 6 Let (f, d, s > l) be a b-complete b-metric space and let -q ; self-mapping on X satisfy the following condition: Given s > 0, there exists 5 > 0 such that
s < d (x, y ) < s + 5 implies sad (fx, fy ) < s , (7)
where a > 0 is given. Then f has a unique fixed point, say u e X, and for each §
x e X, limnfnx = u.
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For the following facts and definitions, we refer to (Aghajani et al, 2014, pp.941-960), (Jovanovic, 2016) and (Kirk & Shahzad, 2014) and the references therein.
Definition 2.1 Let f and g be self-mappings of a nonempty set X such that f (X)<z g(X). Let x0 e X be an arbitrary point. Then
fx0 e g(X), so we can assume that fx0 = gx1 = y0 (say) for some x1 e X. Again, fx1 e g(X), so we can choose x2 e X such that o fxi = gx2 = y1 (say). Similarly, we can construct two sequences {xn} and ^ {yn} such that yn = jxn = gxn+1 for all n > 0. Here the sequence {yn} is called a corresponding Jungck sequence for the point x0 e X.
Definition 2.2 Let f and g be the self-mappings of a nonempty set < X. If z = fx = gx for some x in X, then x is called a coincidence point of f and g, and z is called a point of coincidence off and g. The mappings f and g are called weakly compatible if they commute at their coincidence points.
ëj Lemma 2.3 Let f and g be the weakly compatible self-maps of a
nonempty set X. IT f and g have a unique point of coincidence z = fx = gx, then z is the unique common fixed point off and g.
o Now, we announce the following result which generalizes Theorem 5
¡3 in several directions:
Theorem 7 Let ((, d, s > l) be a ¿-complete ¿-metric space and let
f,g: X ^ X be two self-maps such that f (X)e g(X), one of these
two subsets of X being b-complete. Suppose the following conditions hold:
for each s > 0 there exists 5 > 0 such that s < d(gx, gy) < s + 5 implies sad((•, fy) < s and fx = fy whenever gx = gy, where a > 0 is given.
Then f and g have a unique point of coincidence, say z e X. Moreover, for each x0 e X, the corresponding Jungck sequence {yn} ^
can be chosen such that limnyn = z. In addition, if f and g are weakly
compatible, then they have a unique common fixed point. Finally, we have an open question:
Prove or disprove the following:
References
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X <u
■ Let (X,d,s> l) be a ¿-complete b-metric space and
f, g: X ^ X be two given mappings such that f (Xg(X), one of
these two subsets of X being ¿»-complete. Assume that the following conditions hold:
for each s> 0, there exists 5 = 5(s)> 0 such that
s< d (gx, gy)<s + S implies d (fx, fy)<s and fx = fy, whenever
gx = gy.
Then f and g have a unique point of coincidence, say z e X. ^ Moreover, if f and g are weakly compatible, then they have a unique common fixed point.
o <u o
Aghajani, A., Abbas, M., & Roshan, J. 2014. Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces. Mathematics Slovaca, 64(4), pp.941-960. Available at: https://doi.org/10.2478/s12175-014-0250-6. ,o
Bakhtin, I.A. 1989. The contraction principle in quasimetric spaces. Funct. g Anal, 30, pp.26-37.
Chandok, S., Jovanovic, M., & Radenovic, S. 2017. Ordered b-metric spaces and Geraghty type contractive mappings. Vojnotehnicki glasnik/Military Technical Courier, 65(2), pp.331-345. Available at: https://doi.org/10.5937/vojtehg65-13266.
Chifu, C., & Petru§el, G. 2017. Fixed point results for multivalued hardy-rogers contractions in b-metric spaces. Filomat, 31(8), pp.2499-2507. Available at: https://doi.org/10.2298/fil1708499c.
Czerwik, S. 1993. Contraction mappings in b-metric spaces. Acta Math. Inform., Univ. Ostrav, 1(1), pp.5—11. Available at: https://dml.cz/handle/10338.dmlcz/120469. Accessed: 10.10.2018.
Ciric, Lj. 2003. Fixed Point Theory: Contraction Mapping Principle.Belgrade: Faculty of Mechanical Engineering.
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Dosenovic, T., Pavlovic, M., & Radenovic, S. 2017. Contractive conditions in b-metric spaces. Vojnotehnicki glasnik/Military Technical Courier, 65(4), pp.851-865. Available at: nttps://aol.org/10.5937/vojteng65-14817.
Dubey, A.K., Shukla, R., & Dubey, R.P. 2014. Some fixed point results in b-metric spaces. Asian J. Math. Appl., article ID ama0147.
Dung, N.V., & Hang, V.T.L. 2018. On The Completion OT b-Metric Spaces. Bulletin of the Australian Mathematical Society, 98(2), pp.298-304. Available at: https://doi.org/10.1017/s0004972718000394.
Faraji, H., & Nourouzi, K. 2017. A generalization of Kannan and Chatterjea fixed point theorem on complete b-metric spaces. Sahand Communications in Mathematical Analysis (SCMA), 6(1), pp.77-86. Available at: https://doi.org/10.22130/SCMA.2017.23831.
Jovanovic, M., Kadelburg, Z., & Radenovic, S. 2010. Common Fixed Point Results in Metric-Type Spaces. Fixed Point Theory and Applications, 2010, Article ID:978121. Available at: https://doi.org/10.1155/2010/978121.
Jovanovic, M. 2016. Contribution to the theory of abstract metric spaces. Belgrade. Available at: http://nardus.mpn.gov.rs/handle/123456789/7975. Accessed: 10.10.2018.
Kir, M., & Kiziltunc, H. 2016. On Some Well Known Fixed Point Theorems in b-Metric Spaces. Turkish Journal of Analysis and Number Theory, 1(1), pp.1316. Available at: https://doi.org/10.12691/tjant-1-1-4.
Kirk, W., & Shahzad, N. 2014. Fixed Point Theory in Distance Spaces. Switzerland: Springer International Publishing.
Koleva, R., & Zlatanov, B. 2016. On fixed points Tor Chatterjea's maps in b-metric spaces. Turkish Journal of Analysis and Number Theory, 4(2), pp.31-34. Available at: http://www.sciepub.com/TJANT/abstract/6009. Accessed: 10.10.2018.
Kumar, M.P., Sachdeva, S., & K. Banerjee, S. 2014. Some Fixed Point Theorems in b-metric Space. Turkish Journal of Analysis and Number Theory, 2(1), pp.19-22. Available at: https://doi.org/10.12691/tjant-2-1-5.
Meir, A., & Keeler, E. 1969. A theorem on contraction mappings. Journal of Mathematical Analysis and Applications, 28(2), pp.326-329. Available at: https://doi.org/10.1016/0022-247x(69)90031-6.
Miculescu, R., & Mihail, A. 2017. New fixed point theorems Tor set-valued contractions in b-metric spaces. Journal of Fixed Point Theory and Applications, 19(3), pp.2153-2163. Available at: https://doi.org/10.1007/s11784-016-0400-2.
Paunovic, Lj., Kaushik, P., & Kumar, S. 2017. Some applications with new admissibility contractions in b-metric spaces. The Journal of Nonlinear Sciences and Applications, 10(08), pp.4162-4174. Available at: https://doi.org/10.22436/jnsa.010.08.12.
Singh, S.L., Czerwik, S., Krol, K., & Singh, A. 2008. Coincidences and fixed points of hybrid contractions. Tamsui Oxf. J. Math. Sci., 24, pp.401-416.
Sintunavarat, W. 2016. Nonlinear integral equations with new admissibility types in b-metric spaces. J. Fixed Point Theory Appl., 18(2), pp.397-416. Available at: https://doi.org/10.1007/s11784-015-0276-6.
Suzuki, T. 2017. Basic inequality on a b-metric space and its applications. Journal of Inequalities and Applications, 2017:256. Available at: https://doi.org/10.1186/s13660-017-1528-3.
Zare, K., & Arab, R. 2016. Common fixed point results for infinite families in partially ordered b-metric spaces and applications. Electronic Journal of Mathematical Analysis and Applications, 4(2), pp.56-67. Available at: http://math-frac.org/Journals/EJMAA/Vol4(2)_July_2016/Vol4(2)_Papers/06_EJMAA_Vol4(2)_J uly_2016_pp_56-67.pdf. Accessed: 15.10.2018.
ЗАМЕТКА О ТЕОРЕМЕ МЕИРА-КИЛЕРА В КОНТЕКСТЕ b-МЕТРИЧЕСКИХ ПРОСТРАНСТВ
Мирьяна В. Павлович3, Стоян Н. Раденович6
а Университет в г. Крагуевац, Естественно-математический факультет,
г. Крагуевац, Республика Сербия, 6 Университет короля Сауда, Естественно-математический факультет, Департамент математики, Рияд, Саудовская Аравия
ОБЛАСТЬ: математика (математическая тематическая классификация:
первичная 47H10, вторичная 54H25) ВИД СТАТЬИ: оригинальная научная статья ЯЗЫК СТАТЬИ: английский
Резюме:
В данной работе рассматривается знаменитая теорема Меира-Килера в контексте b-метрических пространств. Наш результат обобщает, улучшает, дополняет и объединяет ранее полученные результаты, которые были опубликованы в научной литературе. Наше доказательство намного короче и лучше, чем доказательства, представленные в иных работах (ЪириЬ, 2003) и (Meir & Keeler, 1969, pp.326-329).
Ключевые слова: b-метрическое пространство, b-полная система функций, b-Коши, условия Меира-Килера, последовательности Пикарда.
БЕЛЕШКА О MEIR-KEELER-ОВОJ ТЕОРЕМИ У КОНТЕКСТУ b-МЕТРИЧКИХ ПРОСТОРА
Мир]ана В. Павлович8, Сто^ан Н. Раденови^6 а Универзитет у Крагу]евцу, Природно-математички факултет,
Крагу]евац, Република Срби]а,
6 Универзитет кра^а Сауда, Природно-математички факултет,
Департман математике, Ри]ад, Сауди]ска Арабка
O CM
О
X
ОБЛАСТ: математика (математичка тематска класификаци]а:
примарна 47H10, секундарна 54H25) ВРСТА ЧЛАНКА: оригинални научни чланак
® JЕЗИК ЧЛАНКА: енглески
о
> Сажетак:
У овом раду разматрана jе позната Мв1г-Кее1вг-ова теорема у контексту Ь-метричких простора. Наш резултат генерализ^е, ш поболшава, даjе допринос, уедите и обога^е познате
Щ резултате у научноj литератури. Тако^е, наш доказ Ме1г-Кее!ег-
о ове теореме у контексту стандардних метричких простора jе
° много краПи и прикладн^и него у радовима ЪириПа, (2003) и Мег &
< Кее!ег-а (1969, рр.326-329).
Къучне речи: b-метрички простор, b-комплетан, b-Cauchy-jee,
о Meir-Keeler-ови услови, Picard-ов низ.
>- Paper received on / Дата получения работы / Датум приема чланка: 18.10.2018.
Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 26.11.2018.
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