CC BY
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EXTENSIONS OF THE BANACH CONTRACTION PRINCIPLE IN MULTIPLICATIVE METRIC SPACES
Badshah-e-Romea, Muhammad Sarwarb
University of Malakand, Department of Mathematics, Chakdara Dir(L), Pakistan
a e-mail: baadeshah@yahoo.com,
ORCID iD: http://orcid.org/0000-0001-6004-5962 b e-mail: sarwarswati@gmail.com, ORCID iD: http://orcid.org/0000-0003-3904-8442
https://dx.doi.org/10.5937/vojtehg65-13342
FIELD: Mathematics.
ARTICLE TYPE: Original Scientific Paper ARTICLE LANGUAGE: English
Abstract:
In this paper, we have proven several generalizations of the Banach contraction principle for multiplicative metric spaces. We have also derived the Cantor intersection theorem in the setup of multiplicative metric spaces. Non-trivial supporting examples are also given.
Key words: Multiplicative metric, Multiplicative open ball, Multiplicative Cauchy sequence, Multiplicative contraction.
Introduction
The study of fixed points of mappings satisfying certain contractive conditions has many fruitful applications in various branches of mathematics; hence, it has extensively been investigated by many authors (Rad, et al, nd), (Radenovic, et al, nd), (Mustafa, et al, 2016, pp.110-116), (Radenovic, et al, 2016, pp.38-40). The Banach contraction principle has been the most versatile and effective tool in the fixed-point theory (Banach, 1922, pp.133-181). Generalization of the Banach contraction principle has been one of the most investigated branches of research. Matthews (1994, pp.183-197) introduced the concept of partial metric space as a part of the study of denotational semantics of dataflow networks, showing that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. Hitzler (2001) generalized the Banach contraction principle in the context of a dislocated metric space.
ACKNOWLEDGEMENT: The authors are grateful to the editor and anonymous referees for their careful reviews, valuable comments and remarks to improve this manuscript.
Zeyada (2005, pp.111-114) improved the work of Hitzler in a dislocated quasi metric space. Shatanawia & Nashine (2012, pp.37-43) studied the Banach contraction principle for nonlinear contraction ina partial metric space. Suzuki (2008, pp.1861-1869) characterized metric completeness by the generalized Banach contraction principle. Boyd and Wong (1969, pp.458- 464) showed that the constant used in the Banach contraction principle can be replaced by an upper semi-continuous function. Hadzic and Pap (2001) extended the contraction principle to probabilistic metric. Jainet al. (2012, pp.252-258) generalized the Banach contraction principle for cone metric spaces. There have been a number of generalizations of a metric space. Some examples of such generalizations are given above. One such generalization is a multiplicative metric space, where Ozavsar and Cevikel (2012) introduced the notion of multiplicative contraction mappings and derived some fixed-point results for such mappings on a complete multiplicative metric space.
Hxiaoju, et al. (2014) established some common fixed points for weak commutative mappings on a multiplicative metric space.
In the current paper, we establish an extension of the famous Banach contraction principle in multiplicative metric spaces. The Banach theorem is extended in two ways:
1. The contraction constant depends on the multiplicative distance between the points under consideration.
2. The behavior of d(x; T x) is considered instead of the comparison of d(T x, T y) and d(x, y).
The derived results carry the fixed-point results of Dugundji and Granas (1982) in a metric space to a multiplicative metric space. Furthermore, to complete the proof of the extension of the Banach theorem, we also derived the Cantors intersection theorem in multiplicative metric spaces.
Definition 1.1. (Bashirov et al, 2008) A multiplicative metric on a nonempty set X is a mapping
d: X *X ^ R satisfying the following condition:
(1) d(x, y) > 1 for all x, ye X;
(2) d(x, y) = 1 if and only if x = y;
(3) d(x, y) = d(y, x) for all x, y eX;
(4) d(x, z) < d(x, y). d(y, z) for all x, y, z e X. The pair (X,d) is called a multiplicative metric space.
Example 1.1. Let R denote the set of n-tuples of positive real numbers. And let d*: r x r ^ r be defined as
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Where x=(x1, x* ..., xn) , y=(y1, y*, ..., yn)e R ■ ned as
a if a > 1 \a I =! 1
11 - if a < 1
. a
Then, clearly, d*(x, y) is a multiplicative metric (Bashirov et al, 2008). Example 1.2. Let (X,d) be a metric space, then the mapping da defined on X as follows is a multiplicative metric,
da (x, y) = ad(x,y) where a > 1. The following definitions are given by Ozavsar and Cevikel (2012). Definition 1.2. Let (X,d) be a multiplicative metric space. If a eX and r>1, then a subset
Br (a) = B(a; r) = {x e X : d(a; x) < r}
of X is called a multiplicative open ball centered at a with the radius r. Analogously, one can define a multiplicative closed ball as
Br (a) = B(a; r) = {x e X : d(a; x) < r}
Definition 1.3. Let A be any subset of a multiplicative metric space (X,d). A point x^X is called a limit point of A if and only if
(A n Be (x)) -{x} * 0 for every £> 1 .
Definition 1.4. Let (X,d) and (Y,p) be given multiplicative metric spaces and a eX. A function f: (X, dis said to be multiplicative continuous at a, if for given s> 1 , there exists a 5 >1 such that
d(x, a)<S ^d(f (x), f (a)) < s or equivalentlyf (B{a;S)) e B(f (a); s). Where B(a;S) and B(f(a); s)
are open balls in (X,d) and (Y,p) respectively. The function f is said to be continuous on X if it is continuous at each point of X.
Definition 1.5. A sequence {xn} in a multiplicative metric space (X,d) is said to be multiplicative convergent to a point xeX if for a givens> 1 there exists a positive integer n0 such that d (xn, x) <sfor all n > n0 or equivalently, if for every multiplicative open ball B)( x) there exists a positi-
ve integer n0 such that n > n0 ^ xn e bs(x) then the sequence {x„} is said to bemultiplicative-convergent to a point x eX denoted by xn^ x(n ^
Definition 1.6. A sequence {xn} in a multiplicative metric space (X,d) is said to be multiplicative Cauchy sequence if for every s>\ there exists a positive integer n0
such that d(xn, xm) <s for all n, m> n0.
Definition 1.7. A multiplicative metric space (X,d) is said to be complete if every multiplicative Cauchy sequence in X converges in X in the multiplicative sense.
Definition 1.8. Let (X,d) be a multiplicative metric space. A mapping f: X ^ X is called a multiplicative contraction if there exists a real number a where 0<a<1 such that
d(f (xl), f (x2)) < d(xy x2f for all x2 e X.
Theorem 1.1. In a multiplicative metric space, every multiplicative convergent sequence is a multiplicative Cauchy sequence.
Lemma 1.1. A sequence {xn} in a multiplicative metric space (X,d) is a multiplicative Cauchy sequence if and only if d (xn, xm) ^ 1 (n, m ^ x)
Theorem 1.2. (Banach Contraction Principle): Let (X,d) be a multiplicative metric space and let f: X^X be a multiplicative contraction. If (X,d) is complete, then f has a unique fixed point.
Main Results
In this section, we are attempting to extend the famous Banach contraction principle into multiplicative metric spaces.
Theorem 2.1. Let (M,d) be a complete multiplicative metric space and let T: M^M. Also assume that for each a>1 there is a Y(a)>1 such that if d(x,Tx)<Y(a) then T(Ba(x))^Ba(x).
If d(Tny,Tn + ly) ^ 1 for some y^M, then the sequence {rny\ converges to a fixed point of T .
Proof. We first show that |Tnyj is a multiplicative Cauchy sequence.
Let, for the sake of brevity, define Tny = yn . Given a>1, choose a natural number n0 so that d(yn,yn+l) <yja for all n > n0.
Since d(yn,yn+l) = d(yn,Tyn)we get T(B^(yn))cB^(yn). This gives y+i = Tyn eB4-a(yt) and TJyn = Tyn+J eB^(yn) by induction for all j > 0.
Then d(yk ,y )<d(yk ,y ).d(y ,y )<Va.Va=a for all j,k>n
/v / /v n n / 0
J 0 0 J
It means {yn} = {T"y) is a multiplicative Cauchy sequence and, due
to the completeness of M, converges to some point z eM . Now we claim that z=Tz. Suppose by way of contradiction that z* Tz then d(z, Tz)=fi>1.
Choosezn eB^(z) such that d(zn,zp <r^P . Then, by the
hypothesis of the theorem T(B (z ))eB (z ). Therefore, TzeB3p(zn).
VP n vP n
2
But since d(Tz,z) < d(Tz,zn). d(z , z) ^ d(Tz, zn) > ^^ > L= p3.
n n n d(zn, z) 3L
2
As L3 >4LforL >1.
Therefore, Tz^B3^(zn) gives a contradiction. Hence, Tz=z. This completes the proof.
Theorem 2.2. Let (M,d) be a complete multiplicative metric space, and let T: M^M be a mapping satisfying
d (Tx, Ty) <S[d(x, y)]. (2.1)
Where S:[1, <») ^ [l, <») is any non-decreasing (not necessarily continuous) mapping such that sn (t) ^ l for each fixed t>l.
Then the sequence {Tnx } converges to a fixed point of T in M.
Proof. We claim that S(t) < t for each t>1; because if t <S(t) for some t>1, then by monotonicityS(?) <S[S(t)], which by induction implies that t < Sn (t) for all n>0, implying that t< 1, which is a contradiction. Now, by equation (2.1), we have d(Tnx,Tn+1 x) <Snd(x,Tx). Hence
d (Tnx,Tn + jx) ^ 1 for each x e M.
Let a be given, and r(a) = u e B(x,a), using multiplicative triangular inequality we have
Let a be given, and r(a) = > L If d(x,Tx) <r(a) for any
a
d (Tu, x ) < d (Tu, Tx ).d (Tx, x ) < S[ d (u, x )]./ < S (a).-= a. 7
8(a)
^ i^m + p ^m +1 + p x , T/^m ^m + 1 w , r n
d(T ^x,T ^x)<d(T x,T x) <(s +1) for all p>0. As k< 1, so letting p ^ œ, we have s < 1, which is a contradiction.
Now let a > 1, n = p(a,a) and y = min{Va, a1 . Let d(x,Tx) <y and ze B(x,a). Using multiplicative triangular inequality, we have d(Tz, x) < d(Tz, Tx).d(Tx, x) . We distinguish the following two cases:
Case-1 : d(x, z) <ja , then
d(Tz,x) < d(Tz, Tx)d(Tx,x)<d(z,x)d(Tx,x)<\/a.\/a=a
Case-2: 4a <d(x,z) <a, then
It means Tm e B(x,a). The rest of the proof is followed by Theorem 2.1.
Theorem 2.3. Let (M,d) be a complete multiplicative metric space and T: M ^M be a map satisfying
d(Tx,Ty) < d(x, y)S(xy). (2.2)
Where S:M x M ^ [1,<») has the property that for any closed interval [a, b] e[1, sup{S(x,y) | a < d(x,y) < b} = fa,b) < 1, here fa,b) denotes a maximum value of S( x, y) for all x, y e[ a, b]. Then T has a unique fixed point veM and Tnx ^v for each x eM.
Proof. Using condition (2.2), for any positive integer n andxeM, we have
d (Tn + 1x, Tn + 2x) < d (Tnx, Tn + 1x)S(Tnx'Tn + 1x)
< d (Tnx, Tn + TU + 1x) < d (Tnx, Tn + 1x)
That is {d(Tnx,T"+1x)}n is a decreasing sequence and therefore converges to some e> 1. We claim that e=1. Suppose on the contrary that e> 1. Obviously, there will be some positive integer n 0 such that d(Tnx,Tn+1x)e[s,s+1] for all n > n0. We can choose an integer m > n0 and let k e f (s,s + 1), we get by inductio
in
<u E o IT
d(Tz, x) < d(Tz, Tx) .d(Tx, x) < d(z, x)^(zx) .d(Tx, x) .a1 ^ = a. In both cases, T(BJ(xj) e Ba(x). Consequently, the existence of the
fixed point of T follows from Theorem 2.1.
For the uniqueness of the fixed point T, consider Tz = z ^ w = Tw
where x, w e M. Using (2.2), we have, d(z, w)=d(Tz, Tw) <d(z, w/(z'w). Which for S(z, w) < 1, gives a contradiction. Hence z = w.
Definition 2.1. The sequence of non-empty sets {S^} in a multiplicative metric space M is said to be a nested sequence of sets, if
1) S 3S ,, n = 1,2,... ' n — n + 1
2) The diameter S( S ) of S tends to 1 as n ^-<x>.
n n
Theorem 2.4. (Cantor's Intersection Theorem)A multiplicative metric space (M,d) is complete if and only if every nested sequence of closed sets has a non-empty intersection.
Proof. Suppose the multiplicative metric space M is complete and let S13S2 3S3 3... 3Sn 3... be a nested sequence of closed sets. Select a
point xn in Sn, n = 1,2,... We show that {xn} is a Cauchy sequence. Let
s>1 be given. As S( Sn) for n ^1. There will be a positive integer
nQ such that S( Sn) = sup d(a, b) <s for n > nQ.
a, b e S
n
Now x is in S 3 S for m>n. Therefore, x , x e S for all
m n m m n n
0
m,n>n0 and d(x , x )<s. Hence, {x } is a Cauchy sequence in M: Due
^^ n n
to the completeness of M x ^x eM Next, we are going to show that
n
«
xe n S . For any integer n>n ,the elements x ,x ,,...are all inS .
« 0 n n + 1 n
n = 1
As x is the limit point of the set of these points of S^, so x is the limit point of S as well. Also, as S is closed, therefore xeS for n >n0. Hence,
x e n S Conversely, suppose every nested sequence of closed
n
n = 1
sets has a non-empty intersection. We shall show that M is complete. Let {xn} be a Cauchy sequence in M Then for every s>1 there will be a positive integer n such that d(x , x ) <s V m, n>n0.
0 m n 0
1
Take s=22 and let n be a positive integer such that
1
d (x , x ) < 2 2 n, > n0.
n n, 1 0
Let S1 = B(xn ,2), take s=24 and let «2 be a positive integer such
1
that d (x , x ) < 24 n2 > n1.
n1 n2 21
1
_ 1 -Let S2 = B(x« ,22). Again take s=28 and let n3 be a positive inte-
n2
gersuch that d(x ,x )<28 n > n .
a v n ' n ' 3 2
2 3
_ 1
Let S3 = B(x« ,24). Clearly SfS2 and S3 are closed and
S13S2 3S3. Continuing in the same way, choose n <n <n3 <... <n <...
1
and closed sets S 3S 3X 3... 3S 3... with S = B(x 22^ *). As
12 3 k k n ' 7
k
S(S, ) when k ^^ ,therefore these sets form a nested sequence of
k
tx> tx>
closed sets. By our assumption n Sk Let x e n sk , then for some
k = 1 k = 1
1
integer k x e S^ for all k > k0 . That is d(xn^, x) < 2 2 , k > k0. It means ^ x. But {xn } is a subsequence of a Cauchy sequence {xn}, therefore, xn ^ x e M . This completes the proof.
The Banach theorem can also be extended in another way, where the behavior of d(x,Tx) is considered instead of comparing d(Tx,Ty) and d(x, y). Many of such generalizations relay on the following general principle involving minimizing sequences for suitable real valued functions:
Theorem 2.5. Let (M,d) be a complete multiplicative metric space and y:M ^ [1, <») be an arbitrary (not necessarily continuous) function. Assume that inf{y(x)y(y) | d(x,y) >p} = X(p) > 1 for all p > 1 (2.3).
Then each sequence {x^} in M such that w(xn) converges to
one and the same point zeM.
Proof. Let Sn = {x | y(x) < y(x )}. Any finite family of these nonempty
sets has a nonempty intersection. We shall show that S(Sn) ^ 1. As
y(xn) ^ 1; so, for any given s>1, there will be a positive integer n0 such
that y(x ) <Jx(s) for all n > n0.
n y
For any x, y e Sn with n > n0 we have y(x)y(y) <A(s). Condition (2.3) gives d(x, y) <s, so S(Sn) <s. But s>1 is arbitrary, so S(Sn) ^ 1.
Moreover, as S(Sn) = S(Sn) ^ 1, so, using Cantor's Intersection Theorem
« _
2.4, we conclude that there is unique z e n S . Since xn e S for each n,
n n = 1
therefore x^ ^z. Now let {yn} be another sequence with y(yn) ^1, therefore ¥(xn ^)y/(yn) , arguing as before and using relation (2.3), it follows that d(x , y^) and therefore yn ^z.
The following theorem is an obvious consequence of the above result. Theorem 2.6. Let (M,d) be a complete multiplicative metric space and F1: M ^ [1,«) be a continuous mapping. Assume the function
y (x) = d(x, F, (x)) satisfying condition (2.3) and inf d(x, F1(x)) = 1. Then F
1 xeM
has a unique fixed point.
Proof. Notice that the Banach fixed-point theorem in a multiplicative metric space follows from theorem 2.6. If
d(F1(x), F1(y)) < d(x, y )a where ae (0,1). Then condition (2.3) is
valid for \y(x) = d(x, F1(x)), because
d(x, y)1—a = d(x' y )a <-^^-< d(x, F1 (x)).d(y, F1 (y)).
d(x, y)"a d(Fi(x),Fi(y)) ^
Using inf d (x, F1( x)) = 1, we get that d(F1 (x), F1n+11(x)) for each
xeM 1 1
xeM. This completes the proof.
The following corollary is readily derivable from the Banach Contraction principle
Corollary 2.1. Let (M,d) be a complete multiplicative metric space
and B=B(x0, r)={x| d(x, x0)<r} where r > 1.
A
Let T:B ^ M be a mapping such that d(Tx,Ty) < d(x,y) for all
1—A
x,y e B where A e (0,1). If d(Tx0,x0) <r , then T has a unique fixed point.
Proof. Choose s < r such that d(Tx0,x0) <s1~A <r1—A. Next, we
show that T maps the closed ball C={x | d(x, x0) < s} into itself. If x e C, then
using the contractive condition of T and multiplicative triangular inequality we
have d(Tx, x0) < d(Tx, Tx0 ).d( Tx0, x0) < d(x, x0 )A. s1—A < sA .s1—A = s.
As C is closed, so the Banach Contraction Principle completes the proof. We conclude with the following example which supports Theorem 2.3. Example 2.1. Let M = [0.01,1]. Consider the multiplicative metric
d : M x M ^ [1, <») defined by d(x y) = e y1 ■ Then (M,d) is a complete multiplicative metric space. The mapping T: M ^ M defined by
3
T(x) =-, satisfies the following multiplicative contractive condition
5 + x
d(Tx,Ty)<d(x,y)5(x,y), where MxM^[0,m) defined S{x,y) = ^,
has the property that for any closed interval
[a,b] c [1, m), sup{S(x,y) | a < d(x,y) < b} = ¿u(a,b) < 1.
Obviously, T has a unique fixed point 0.5413812651 e M .
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References
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Bashirov, A.E., Kurpnar, E.M., & Ozyapc, A., 2008. Multiplicative calculus and its applications. J. Math. Analy. App, 337, pp.36-48. Available at: http://dx.doi.org/10.1016/j.jmaa.2007.03.081.
Boyd, D.W., & Wong, J.S.W., 1969. On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), pp.458-466. Available at: http://dx.doi.org/10.2307/2035677.
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Hitzler, P., 2001. Generalized Metrics and Topology in Logic Programming Semantics. National University of Ireland - University College Cork. Ph.D. Thesis.
Hxiaoju, H., Songmand, M., & Chen, D., 2014. Common fixed points for weak commutative mappings on a multiplicative metric space. Fixed Point Theory and Applications, pp.20-48. Available at: http://dx.doi.org/10.1186/1687-1812-2014-48.
Jain, Shobha, Jain, Shishir, & Jain, L.B., 2012. On Banach contraction principle in a cone metric space. J. Nonlinear Sci. Appl., 5, pp.252-258.
Matthews, S.G., 1994. Partial metric topology, 183-197. In: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci., 728.
Mustafa, Z., Huang, H., Radenovic, S., 2016. Some remarks on the paper "Some fixed point generalizations are not real generalizations". J. Adv. Math. Stud. (9), pp.110-116.
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Radenovic,S., Chandok, S., Shatanawi, W., 2016. Some cyclic fixed point results for contractive mappings. University Though, Publication in Nature Sciences, 6(2), pp.38-40. Available at: http://dx.doi.org/10.5937/univtho6-11813.
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Suzuki, T., 2008. A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society, 136(5), pp.1861-1869.
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РАСШИРЕНИЕ БАНАХОВЫХ ПРИНЦИПОВ СЖАТИЯ В МУЛЬТИПЛИКАТИВНОМ МЕТРИЧЕСКОМ ПРОСТРАНСТВЕ
Бадшах-е-Роме, Мухаммед Сарвар
Университет г. Малаканд, Математический факультет, г. Чакдара, Пакистан
ОБЛАСТЬ: математика
ВИД СТАТЬИ: оригинальная научная статья
ЯЗЫК СТАТЬИ: английский
Резюме:
В данной статье мы доказали несколько обобщений Банаховых принципов сжатия в мультипликативном метрическом пространстве. Мы также развили применяемуюКанторовутеорему подмножеств при образовании мультипликативных метрических пространств,подтвердив ее нетривиальными примерами.
Ключевые слова: мультипликативная метрика, мультипликативный открытый шар, последовательность Коши, мультипликативное сжатие.
ПРОШИРИВА^Е БАНАХОВОГ ПРИНЦИПА КОНТРАКЦШЕ НА МУЛТИПЛИКАТИВНЕ МЕТРИЧКЕ ПРОСТОРЕ
Badshah-e-Rome, Muhammad Sarwar
University of Malakand, Department of Mathematics, Chakdara Dir(L), Pakistan
ОБЛАСТ: математика
ВРСТАЧЛАНКА: оригинални научни чланак иЕЗИКЧЛАНКА: енглески
Сажетак:
У овом раду je доказано неколико генерализац^а Банаховог принципа контракц^е за мултипликативне метричке просторе. Тако^е, разв^ена ]е Канторова теорема интерсекц^е при
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образовав мултипликативних метричких простора, подржана нетриви]алним примерима.
Къучне речи: мултипликативна метрика, мултипликативна отворена кугла, мултипликативни Коши]ев низ, мултипликативна контракци}а.
Paper received on / Дата получения работы / Датум приема чланка: 22.01.2017. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 17.03.2017.
Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 19.03.2017.
© 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 terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).
© 2017 Авторы. Опубликовано в «Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier» (www.vtg.mod.gov.rs, втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией «CreativeCommons» (http://creativecommons.org/licenses/by/3.0/rs/).
© 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 лиценцом (http://creativecommons.org/licenses/by/3.0/rs/).
