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CONTRACTIVE CONDITIONS IN jb-METRIC SPACES
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Tatjana M. Dosenovic3, Mirjana V. Pavlovicb, Stojan N. Radenovicc
a University of Novi Sad, Faculty of Technology, Novi Sad, Republic of Serbia, e-mail: [email protected], ORCID iD: http://orcid.org/0000-0002-3236-4410, b University of Kragujevac, Faculty of Sciences, Department of §
Mathematics and Informatics, Kragujevac, Republic of Serbia, e-mail: [email protected], ORCID iD: http://orcid.org/0000-0001-6257-8666, c University of Belgrade, Faculty of Mechanical Engineering, o
Belgrade, Republic of Serbia, _-
e-mail: [email protected],
ORCID iD: © http://orcid.org/0000-0002-7417-1342
http://dx.doi.org/10.5937/vojtehg65-14817
FIELD: Mathematics (Mathematics Subject Classification: primary 47H10,
secondary 54H25) o
ARTICLE TYPE: Original Scientific Paper ARTICLE LANGUAGE: English
Abstract:
The purpose of this paper is to consider various contractive conditions in b-metric spaces which have been recently published. Our results improve and complement many recent results from this field. Using the recently obtained result by R. Miculescu and A. Mihail (Miculescu & Mihail, 2017, pp.1-11) the authors of this article show that the proofs of the majority of known results in the context of b-metric spaces can be shortened. Keywords: metric space, common fixed point, altering distance function, point of coincidence, weak compatibility.
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 (Matematicki modeli nelinearnosti, neodredenosti i odlucivanja, 174009) and from the Provincial Secretariat for Higher Education and Scientific Research, Province of Vojvodina, Republic of Serbia, Project no. 142-451-2838/2017-01. The second 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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Introduction
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| It is well known that the Banach Contraction Principle (Banach, 1922,
pp.133-181) states that, if a self-mapping T of a complete metric space (M, d) is a contraction mapping, then T has a unique fixed point (say u) and for each v e M the corresponding Picard sequence |T" (v)j converges w to this fixed point u. In general, this principle has been generalized in two di-§ rections. On the one hand, the usual contractive condition is replaced by a ° weakly contractive condition. On the other hand, the action spaces are re-_i placed by metric spaces endowed with an ordered or partially ordered struc-o ture or with some kind of generalized metric space (like cone metric space, ^ G-metric space, partial metric space, fuzzy metric space, etc.). o In 1989 I. A. Bakhtin (Bakhtin, 1989, pp.26-37) and in 1993 S. Czerwik
i- (Czerwik, 1993, pp.5-11) introduced a new distance on a non-empty set £ which is called a ¿»-metric. A ¿»-metric space is an attempt to generalize the ¡i metric space by replacing only the triangle inequality introducing one real constant. Their definition of this new kind of generalized metric space is the following.
Definition 1 (Bakhtin, 1989, pp.26-37), (Czerwik, 1993, pp.5-11) Let w M be a (non-empty) set and K > 1 a given real number. A function ej d1: M xM ^ [0, <x>) is called a ¿-metric on M if, for all p, q, r e M , the following conditions hold:
(b1) d1 (p, q) = 0 if and only if p = q;
(b2) di (p, q) = di (q, p); (b3) di(p,r)<K(di(p,q) + di(q,r)). > In this case, (M, di, K) is called a ¿-metric space.
If (M, <) is still a partially ordered set, then (M, < ,di,K) is called an
ordered ¿-metric space.
Otherwise, for all other definitions of the notions in ¿-metric spaces such as ¿-convergence, ¿-Cauchy sequence, ¿-completeness, see (Abbas et al, 2016, pp.1413-1429), (Ansari et al, 2017, pp.315-329), (Bakhtin, 1989, pp.26-37), (Huang et al, 2015), (Jovanovic, 2016), (Radenovic et al, 2017a, 2017b), (Roshan et al, 2013), (Zhang et al, 2017, pp.1334-1344) and the reference therein.
Definition 2 (Khan et al, 1984, pp.1-9) A function (p:[0, <x>) ^ [0, <x>) is called an altering distance function if the following properties hold: (1) p is continuous and nondecreasing;
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(2) (t) = 0 if and only if t = 0. g
First, a very known (important) result from a ¿»-metric space is the jo following:
Theorem 1 (Czerwik, 1993, pp.5-11, Theorem 1) Let (M, dl, K) be a b-complete ¿-metric space and let T: M ^M satisfy
di(((p),T(q)) < (p{di(p,q)\p,q eM, (1)
where <(:[0, ro) ^ [0, ro) is an increasing function such that | limn^ro(n (() =0 for each fixed t >0. Then T has an exactly one fixed point v and
lim di (Tn (p), v) =0 (2)
n^ro
for each p e M.
Lemma 1 (Miculescu & Mihail, 2017, Lemma 2.2.) Let {tn} be a sequence in a ¿-metric space (M, d1, K) such that °
di (tn, tn+i) <u-di (tn_i, tn) (3)
for some ¿ue [0,1), and each n = 1,2, . Then \tn} is a ¿-Cauchy
sequence in (M, d1, K).
Remark 1 In several published papers based on the ¿-metric
1 ° concept, the authors assume that u e [0,—) instead of u e [0,1), which is Q
K
obviously weaker. Then under this weaker condition they show that the Picard sequence \tn = T(tn_1 )}n=12 , t0 e M is a b-Cauchy. For the proof, the authors used the following clear inequality:
d1(tm , tn )< Kdx (tm , tn +1)+ K 2d1 (tm +1, tn+2 )+ ... + Kn-(tn_ 2, tn _1) +
/ \ (4)
Kn_md1 (tn_1, tn ),
where n, m e N and n > m.
However, putting (p(r) = u r, r e [0, ro), ue(0,1) in (1), the proof of Theorem 1 from (Czervik, 1993, pp.5-11) follows that Picard sequence
{K = T(tn_1 )}n=1,2,..., t0 e M is a b-Cauchy.
Now, we can show that Lemma 2.2. from (Miculescu & Mihail, 2017) is an immediate consequence of the one part of Theorem 1 from (Czerwik, 1993, pp.5-11).
First of all, we give the next result:
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Lemma 2 If {tn }neN is an arbitrary sequence in the nonempy set M, then there exists at least one mapping T:M ^M such that it is Picard «3 sequence of T with t1 as the beginning point.
Proof. We define T:M ^M as T(tk) = tk+1 for k = 1,2,3,... as well as T(t) = v0 in case t eM\ {t1,t2,...,tn,...} and v0 g{tj, t2,...,tn,...}. The last oc one is possible if {,t2,...,tn,...}M and ft,t2,...,tn,...}*M.
S Proposition 1 Lemma 2.2. from (Miculescu & Mihail, 2017) is an
o immediate consequence of (Czerwik, 1993, pp.5-11, Theorem 1).
Proof. Indeed, the {tn} is a Picard sequence of the mapping defined in Lemma 2. It is obvious that the mapping T satisfies the condition (1) where p(r) = j- r, r e [0, <»), ue(0,i). Further (3) becomes di(T(tn_l),T(tn))<jud(tn_i,tn),n = 2,3,4,.. i.e. the sequence {tn} is a < ¿-Cauchy according to the proof of (Czerwik, 1993, pp.5-11, Theorem 1).
Now, by (Czerwik, 1993, pp.5-11, Theorem 1) that is, by (Miculescu, Mihail, 2017, Lemma 2.2.), the majority of already known results can be improved. Also, by using the same argument some known results can be ^ made significantly shorter and nicer. LA The first such result is the following:
Proposition 2 Let T be a self-map on a b-complete ¿-metric space y (M, d1, K) satisfying
di (T(p), T2 (p)) < jdi (p, T(p)) for some u e (0,i), (5)
§ either (i) for all p e M, or (ii) for all p eM,p ^ T(p), and suppose that o T has a fixed point. Then T has a property P.
Otherwise, if T is a map which has a fixed point v, then v is a also a fixed point of Tn for every natural number n. However, the converse is false. For, consider M = [0,1], T is defined by T(p) = 1 - p. Then T has a
unique fixed point at 1 but Tn = I for each n >1, which has every point
2'
of [0,i] as a fixed point. On the other hand, if M = [0,^], T(p) = cosp, then T is nonexpansive and every iterate of T has the same fixed point as T. Involutions are also examples where F(t)^ F(t"). See, e.g. (Jeong & Rhoades, 2005, pp.71-105) and the references therein.
We shall say that a map T has a property P if F(t) = F(t") for every n e N.
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Proof (of Proposition 2). The statement for n = 1 is trivial. Therefore, g
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we shall assume that n >1 is a given (fixed natural number). It is clear that
F (T ) c F (Tn) Let v e F (Tn).
Case 1. Suppose that T satisfies (i). Then, using (5), d1 (v, T (v )) = dx (Tn (v), Tn+1 (v)) = d1 (T (Tn1 (v)) T2 (Tn1 (v))) < ud1 (Tn1 (v), T (Tn1 (v))) = ud1 (T (Tn_2 (v)),T2 (Tn_2 (v))) < uUdx (Tn_2 (v), T(Tn_2 (v))) < ... <und1 (v, T(v)), which implies that v = T (v).
Case 2. Let now T satisfy (ii).
If v = T(v), then there is nothing to prove. Suppose, if possible, that s> v ^ T (v). Then a repetition of the argument for Case 1 again leads to d1 (v, T (v)) < und1 (v, T (v)), which implies that v = T (v) and F (Tn) = F (t).
Remark 2 Proposition 1.8. obviously generalize the corresponding result, Theorem 1.1. from (Jeong & Rhoades, 2005, pp.71-105), for standard metric spaces.
Corollary 1 Let T be a selfmap of a b-complete ¿-metric space (M, d13 K) satisfying d1 (TT(p), T(q)) < ud1 (p, q) for all p, q e M and for some u e (0,1). (6) a
Then T has a property P.
Proof.Indeed, condition (6) implies (5). Also, by (Czerwik, 1993, pp.5-11, Theorem 1) F(t) ^0. Then the result follows according to Proposition 2.
The next is also generalization of one result from a metric to a ¿-metric space.
Proposition 3 Let T be a selfmap of a b-complete ¿-metric space (M, dj satisfying
d1 (T(p), T2 (p)) < ud1 (p, T(p)) for all p e M and some u e (0,1). (7)
Then F(T) ^0, if T is a b _ continuous.
Proof.Let p0 e M be an arbitrary point and let {pn} be a corresponding Picard sequence. For each n e {0}^ N we have d (pn+1, pn+ 2 ) = d (t (pn), T2 (pn)) < udx (pn, T (pn)) = ¿1 (pn, pn+1). (8)
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Further, according to (Miculescu & Mihail, 2017, Lemma 2.2.) (see also (6)) follows that {pn} is a b-Cauchy sequence. Since (M, di) is a
S ¿-complete ¿-metric space there is v e M such that pn ^ v as n
The continuity of T implies that T(v) = v, i.e., F(t)^0. Jungck's result in the concept of ¿-metric spaces: Theorem 2 Let ((, di, K) be a ¿-metric space and T, S: M ^ M, T(M) e S(M) be self mappings such that for all p, q e M. o di (T(p),T(q))< jdi (S(p), S(q)), where ue(0,i) (9)
AL Also, assume that, at least one of the following conditions hold:
° (i) (d(M),di) or (S(M),di) is ¿-complete;
^ (ii) ((,di,K) is ¿-complete, S is ¿-continuous and T and S are
commuting.
< Then T and S have a unique point of coincidence. Moreover, if T
and S are weakly compatible (for case (i)) then they have a unique common fixed point in M.
Proof. First, we notice that if a point of coincidence of T and S
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< exists, then it is unique. Indeed, if wi and w2 are two distinct points of u coincidence of T and S, then there exist two points ui,u2 eM,ui ^ u2, such that T(ui) = S(ui) = wi ^ w2 = S(u2) = T(u2). Now, by (9) we have
d1 (wi, w2 ) = d (d (vi \ T (v2 S < Jdi (S (vi \ S (v2 ^ = jdi (wi, w2 ) < di (wi, w2 \
NO which is a contradiction.
9 Further, the condition T (M) c S (M) implies that there exists
Jungck's sequence jn = T(vn) = S(vn+1), where {vn} is a sequence in M, v0 e M is an arbitrary point. We shall prove that the sequence {jn} is a ¿-Cauchy. Indeed, for each n e{0}^N we have that
di(d n+1, jn+2 d1 (T (vn+1), T (vn+2)) < jdi (S (vn+1), S (vn+2)) = jdi (.jn , j'n+J
i .e., for all n e{0}o> N the sequence {jn} satisfies condition (3). This means that it is ¿-Cauchy.
Now, let (i) holds. Therefore, since (S(M),di) is a ¿-complete ¿-metric space, it follows that there exists ve M such that T(vn) = S(vn+1) = jn ^ Sv as n We will prove that T(v) = S(v). In order to prove this equality, we have
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K d(T(v),S(v)) < dx (T(v),T())+dx (T(vn),S(v))<pdx (S(v),S()) + dx((,S(v)) K
= udx (S (v), jn_1)+d1 ((, S (v)) ^ u • 0 + 0 = 0.
Hence, T(v) = S(v) = w is a point of coincidence (unique) of the pair <6 (T, S).
If ((M), dj is a ¿-complete the proof is very similar. If (ii) holds, then since (M, dx) is ¿-complete, there exists ve M such that T(vn) = S(vn+1) = jn ^ v, as n ^ro. Since both self-mappings T and S are ¿-continuous, we have when n ^ ro:
S(T(vn)) ^ S(v) and T(S(vn)) ^ T(v) when n ^ ro. 5
Since T and S are commuting, we again obtain that T(v) = S(v) = is a point of coincidence (unique) of the pair (T, S).
For both cases (i) and (ii), according to the known Jungck's result, it <3 follows that w is a unique common fixed point of T and S.
The next is a common fixed point theorem of the Zamfirescu type in b _ metric spaces.
Theorem 3 (Jovanovic, 2016), (Khan et al, 1984, pp.1-9), (Rhoades, | 1977, pp.257-290, Theorem 4.3.) Let (M, d1, K) be a ¿-complete ¿-metric space and let T: M ^M be a mapping and let there exist nonnegative Q numbers a, b, c such that for all p, q e M at least one of the following conditions:
1 0 dx (T (p), T (q) )< adx (p, q); 20 d1 (T(p), T(q)) < b[d (p, T(p)) + d1 (q, T(q))J 30 d1 (T (p) T (q)) < c [d1 (p, T (q)) + dx (q, T (p))]
holds.
If a < —,b c then T has a unique fixed point.
K 2K2 2 K2
Remark 3 By using (Miculescu & Mihail, 2017, Lemma 2.2) the conditions for a,b,c can be relaxing, that is., we get a < 1,b <1 i c < 1
o <1J
W > S ro
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2 2K (for details see Theorem 2.2. below).
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Main results
£ In this section, we shall consider several important as well as
significant contractive conditions announced in the existing literature. Readers can compare all these conditions to the corresponding ones in the context of standard metric spaces, for more details see (Rhoades, 1977, pp.257-290).
w Let Yx be the family of all nondecreasing functions
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< It is well known that if W\ then Wi(f) < t if t >0 as well as
y/x : [0, œ) ^ [0, œ) such that lim (0 = 0, for all t >0.
Yi (0) = 0.
ft Our first result is the improvement of the proof in (Abbas et al, 2016,
pp.1413-1429, Theorem 2.2.)
< Theorem 4 Let (M, ^ , K > 1) be a partially ordered b-complete
b-metric space and let T: M — M be an increasing mapping with respect to
< such that there exists an element p0 e M with po ^ T(v0). Assume that
K • 1 \K •d( q) • d(T(p), T(q)) < Wl (M, (p, q)) + L, • N, (p, q)
CD 1 + _di (p, T(p))
:*c 2
o
for all comparable elements p,q e M, where L, > 0, Í Mi(p, q) = max jd, (p, q), )} (11)
and
N,(p, q) = mm{dl(p,T(p)\ d,(p,T (q)), d, (q,T(p)), d, (q,T (q)) } (12) If T is continuous, then T has a fixed point.
Proof. If po *T(po) then p0 ^T(p0) Further, for the Picard sequence we can assume that d,(pn,pn+,)>0 for all n e{o}o>N. Now, we will prove that
d, ((n, pn+,) < v d, ((n-,, pn ^ for all n e N. (13)
K
Indeed, since
(10)
1 + Kd1 ^pn) = 1 + Kd1(pn mp„) >_1±d1jp^11p„)_ >1 1 11 1+^ d1 (Pn-l, ^„-J 1+2 d1 p„) 1+2d1 p„)
<U
then by using (10) with p = pn-1, q = pn, we obtain
Kd1 (n+1, p„ ) = Kd1 (T(p„ ), T(p„-1 )) < ^ (d1 (p„ , p„-1 ))+L • N1 (p„ , p„-1 ).
Because M1 (p„ , p„-J = d1 (p„-1, p„ (d1 (p„ , p„-1 )) < d1 (p„ , p„-J and £
N1 (p„, pn-1) = 0 we obtain that (13) holds.
This means that the sequence {p„} is a ¿-Cauchy, according to Lemma 2.2. from (Miculescu & Mihail, 2017) which then converges to some u e M. The continuity of T implies that u is a fixed point of T. 8
Remark 4 All that shows that our approach gives a much shorter and J nicer proof than the ones in (Ansari et al, 2017, pp.315-329). Also, ¿y the § same method, the proofs of all results in (Ansari et al, 2017, pp.315-329) can ¿e improved.
In fact, the main (important) question is the following: Does some given contractive condition in the framework of any class of generalized metric spaces imply (give) that the corresponding Picard sequence is a Cauchy (in this class)? The previously contractive condition is such. We proved that for it holds d1 (p„,p„+)< jud1 (pn-1,p„) for all neN and some
u e (0,1). Since K >1 and u = — then the result follows by (Miculescu &
K
Mihail, 2017, Lemma 2.2.).
In the framework of b - metric spaces, the following two results are specific.
Theorem 5 Let ((, d1, K) ¿e a ¿-complete ¿-metric space and let T: M ^ M ¿e a ¿-continuous mapping. Also let
d1 (Tp, Tq) < ad1 ((, Tp) + bd1 (q, Tq), for all p, q e M, a, b > 0, a + b <1 that is
d1 (Tp,Tq) < ad1 (p,Tq) + bd1 (Tp, q),for all p, q e M, a, b > 0, a + b < —
K
In each of the given cases, T has a unique fixed point (say v) and for any u e M the sequence {t„ (u)v as n ^ro.
o
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Proof. In the first (Kannan) case, we obtain that
_ dJ (pn+J, pn) • dj (pn, pn-J), while in the second one (Chatterjea), we
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P + b)K
have dj(Pn+j,Pn)( ^ • dj(Pn,Pn-j) According to ^cutec^ &
2 - (a + b)K
Mihail, 2017, Lemma 2.2.) it follows that in both cases that the Picard sequence {Tnp0 }nej0juW, p0 g M is a b-Cauchy. Since T is b-continuous, g the result follows.
O Conclusion
z
X
uj Based on the previous discussion, we can conclude that the proofs of
>- the majority results in the existing literature for the concept of b-metric < spaces can be significantly shortened by using (Miculescu & Mihail, 2017, Lemma 2.2.).
All these results are in the following papers (Aghajani et al. 2014, pp. 941-960), (Allahyar et al, 2014), (Chandok et al, 2017a), (Chandok et al, 2017b, pp.331-345), (Demma & Vetro, 2015), (Ding et al, 2016, pp. 151164), (Dung & Hang, 2016, pp. 267-284), (Harandi, 2014, pp. 351-358), (Kaushik et al, 2017), (Khamsi & Husain, 2010, pp. 3123-3129), (Kir &
>o Kiziltunc, 2013, pp. 13-16), (Kumam et al, 2015), (Latif et al, 20915, pp. 1 363-377), (Liu & Gu, 2016, pp. 5909-5930), (Ozturk & Ansari, 2017, pp. m 45-52), (Parvaneh et al, 2013), (Petrusel et al, 2017, pp. 199-215), (Piri & Kumam, 2016), (Roshan et al, 2014a, pp. 725-737), (Roshan, et al, 2015), o (Roshan et al, 2014b, pp. 613-624), (Sarwar et al, 2017, pp. 3719-3731), (Sarwar & Rahman, 2015, pp. 70-78), (Sintunavarat, 2016, pp. 397-416), (Zabihi & Razani, 2014).
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Татьяна М. Дошеновича, Мирьяна В. Павлович6, Стоян Н. Раденовичв ю- а Университет в г. Нови-Сад, Технологический факультет, г. Нови-Сад,
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ОБЛАСТЬ: математика (математическая тематическая классификация: з первичная 47Н10, вторичная 54Н25)
О ВИД СТАТЬИ: оригинальная научная статья
ЯЗЫК СТАТЬИ, английский
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о В данной работе представлен анализ различных условий сжатия в
ш Ь-метрических пространствах, которые недавно были
£ опубликованы. На основании исследований настоящих
< результатов мы дополнили и откорректировали многие аспекты результатов в данной области. Так, например, исследовав недавние результаты, полученные Р. Микулеску и А. Михаилом (МсЫевси & М'эИ, 2017, рр.1-11), авторы настоящей статьи доказали, что многие известные результаты в контексте
< Ь-метрических пространств могут быть значительно сокращены. ^ Ключевые слова: метрическое пространство, общая >о фиксированная точка, функция изменения расстояния, точка
совпадения, низкая совместимость.
° КОНТРАКТИВНИ УСЛОВИ У Ь-МЕТРИЧКИМ ПРОСТОРИМА
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О Тат]ана М. Дошенови^3, Мир]ана В. Павлович , Сто]ан Н. Раденови^8
Универзитет у Новом Саду, Технолошки факултет, Нови Сад, Република Срби]а,
б Универзитет у Крагу]евцу, Природно-математички факултет,
Институт за математику и информатику, Крагу]евац, Република Срби]а, в Универзитет у Београду, Машински факултет, Београд, Република Срби]а
ОБЛАСТ: математика (математичка тематска класификаци]а: примарна
47Н10, секундарна 54Н25) ВРСТА ЧЛАНКА. оригинални научни чланак JЕЗИК ЧЛАНКА: енглески
Сажетак:
Цил> овог рада }есте да размотри разне контрактивне услове у Ь-метричким просторима ко}и су недавно об]авл>ени. Наши резултати поправл>а]у и допук>у}у многе недавне резултате из
овог контекста. КористеПи недавно доби}ени резултат ю
Р. Микулескуа и А. Михаила, (Miculescu & Mihail, 2017, pp.1-11) ""
аутори овог чланка показ^у да докази многих познатих резултата ю у контексту Ь-метричких простора могу бити доста скраПени.
Къучне речи: метрички простор, за}едничка фиксна тачка, функци]а ф
промене раздаъине, тачка коинциденци]е, слаба компатибилност. го
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Paper received on / Дата получения работы / Датум приема чланка: 17.08.2017. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 20.09.2017.
Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 22.09.2017.
© 2017 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). This article is an open access article distributed under the о
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© 2017 Авторы. Опубликовано в «Военно-техническии вестник / Vojnotehnicki glasnik / Military Technical Courier» (www.vtg.mod.gov.rs, втг.мо.упр.срб). Данная статья в открытом доступе и о
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© 2017 Аутори. Обjавио Воjнотехнички гласник / Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). Ово jе чланак отвореног приступа и дистрибуира се у складу са Creative Commons licencom (http://creativecommons.org/licenses/by/3.0/rs/).
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