Journal of the Egyptian Mathematical Society (2013) xxx, xxx–xxx
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
Results on n-tupled fixed points in complete asymptotically regular metric spaces Anupam Sharma
*
Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India Received 23 July 2013; revised 29 August 2013; accepted 14 September 2013
KEYWORDS Partially ordered set; Metric space; Asymptotically regular sequence; Mixed monotone property; n-Tupled fixed point
Abstract The notion of n-tupled fixed point is introduced by Imdad, Soliman, Choudhury and Das, Jour. of Operators, Vol. 2013, Article ID 532867. In this manuscript, we prove some n-tupled fixed point theorems (for even n) for mappings having mixed monotone property in partially ordered complete asymptotically regular metric spaces. Our main theorem improves the corresponding results of Imdad, Sharma and Rao (M. Imdad, A. Sharma, K.P.R. Rao, Generalized n-tupled fixed point theorems for nonlinear contractions, preprint). 2010 MATHEMATICAL SUBJECT CLASSIFICATION:
47H10; 54H10; 54H25
ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction and preliminaries The Banach contraction principle is the most natural and significant result of fixed point theory. It has become one of the most fundamental and powerful tools of nonlinear analysis because of its wide range of applications to nonlinear equations arising of in physical and biological processes ensuring the existence and uniqueness of solutions. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics. Generalization of the above principle has been a * Tel.: +91 8909460011. E-mail address:
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heavily branch of mathematics. Existence of a fixed point for contraction type mappings in partially ordered metric space and applications have been considered by many authors. There already exists an extensive literature on this topic, but keeping in view the relevance of this paper, we merely refer to [1–13,17–31]. In [6], Bhaskar and Lakshmikantham introduced the notion of a coupled fixed point and proved some coupled fixed point theorems in partially ordered complete metric spaces under certain conditions. Afterwards Lakshmikantham and C´iric´ [17] extended these results by defining the g-monotone property, which indeed generalize the corresponding fixed point theorems contained in [6]. Since then Berinde and Borcut [8] introduced the concept of tripled fixed point and proved some related theorems. Recently Imdad et al. [14] introduced the concept of ntupled coincidence as well as n-tupled fixed point (for even n) and utilize these two definitions to obtain n-tupled coincidence
1110-256X ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2013.09.002 Please cite this article in press as: A. Sharma, Results on n-tupled fixed points in complete asymptotically regular metric spacesn-tupled fixed point theorems –>, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.09.002
2
A. Sharma
as well as n-tupled common fixed point theorems for nonlinear /-contraction mappings. For more details see [9,11,23,25]. Very recently, Soliman et al. [31] proved some n-tupled coincident point theorems for nonlinear /-contraction mappings in partially ordered complete asymptotically regular metric spaces. The purpose of this paper is to present some n-tupled coincidence and fixed point results for nonlinear contraction mappings in partially ordered complete asymptotically regular metric spaces. Our results generalize and improve the results of Imdad et al. [15]. As usual, this section is devoted to preliminaries which include some basic definitions and results related to coupled fixed point in metric spaces. From now, (X, , d) is a partially ordered complete metric space. Further, the product space X · X has the following partial order: ðu; vÞ ðx; yÞ () x u; y v 8ðx; yÞ; ðu; vÞ 2 X X: We summarize in the following the basic notions and results established in [6,8]. Definition 1 [6]. Let (X, ) be a partially ordered set and F:X · X fi X be a mapping. Then F is said to have mixed monotone property if for any x, y 2 X, F(x, y) is monotonically nondecreasing in first argument and monotonically nonincreasing in second argument, that is, for x1 ; x2 2 X; x1 x2 ) Fðx1 ; yÞ Fðx2 ; yÞ y1 ; y2 2 X; y1 y2 ) Fðx; y1 Þ Fðx; y2 Þ: Definition 2 [6]. An element (x, y) 2 X · X is called a coupled fixed point of the mapping F:X · X fi X if Fðx; yÞ ¼ x and Fðy; xÞ ¼ y: The main theoretical results of Bhaskar and Lakshmikantham [6] are the following two coupled fixed point theorems. Theorem 1. Let (X, ) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F:X · X fi X be a continuous mapping having the mixed monotone property on X. Assume that there exists k 2 [0, 1) with
If there exist x0, y0 2 X such that x0 F(x0, y0) and y0 F(y0, x0), then there exist x, y 2 X such that F(x, y) = x and F(y, x) = y. Recently, Berinde and Borcut [8] introduced the following partial order on the product space X · X · X: ðu; v; wÞ ðx; y; zÞ () x u; y v; z w 8ðx; y; zÞ; ðu; v; wÞ 2 X X X: Definition 3 [8]. Let (X, ) be a partially ordered set and F:X · X · X fi X be a mapping. Then F is said to have mixed monotone property if F is monotone nodecreasing in first and third argument and monotone noincreasing in second argument, that is, for any x, y, z 2 X x1 ; x2 2 X; x1 x2 ) Fðx1 ; y; zÞ Fðx2 ; y; zÞ y1 ; y2 2 X; y1 y2 ) Fðx; y1 ; zÞ Fðx; y2 ; zÞ z1 ; z2 2 X; z1 z2 ) Fðx; y; z1 Þ Fðx; y; z2 Þ: Definition 4 [8]. An element (x, y, z) 2 X · X · X is called a tripled fixed point of the mapping F:X · X · X fi X if F(x, y, z) = x, F(y, x, y) = y and F(z, y, x) = z. Inspired by the results on coupled fixed points, Karapinar [16] introduced the notion of quadrupled fixed point and proved some related fixed point theorems in partially ordered metric spaces. Definition 5 [16]. An element (x, y, z, w) 2 X · X · X · X is called a quadrupled fixed point of the mapping F: X · X · X · X ·fi X if F(x, y, z, w) = x, F(y, z, w, x) = y, F(z, w, x, y) = z and F(w, x, y, z) = w. The following concept of an n-fixed point was introduced by Gordji and Ramezani [12]. We suppose that the product space Xn = X · X · · X(n times) is endowed with the following partial order, where n is the positive integer (odd or even): (x1, x2, . . . , xn), (y1, y2, . . . , yn) 2 Xn nþ1 ðx1 ; x2 ; . . . ; xn Þ ðy1 ; y2 ; . . . ; yn Þ () x2i1 y2i1 8i 2 1; 2; . . . ; 2 n hnio : ðx1 ; x2 ; . . . ; xn Þ ðy1 ; y2 ; . . . ; yn Þ () x2i y2i 8i 2 1; 2; . . . ; 2
k dðFðx; yÞ; Fðu; vÞÞ 6 ½dðx; uÞ þ dðy; vÞ for each x u and y v: 2
Definition 6 [12]. An element (x1, x2, . . . , xn) 2 Xn is called an n fixed point of the mapping F:Xn fi X if
If there exist x0, y0 2 X such that x0 F(x0, y0) and y0 F(y0, x0), then there exist x, y 2 X such that F(x, y) = x and F(y, x) = y.
xi ¼ Fðxi ; xi1 ; . . . ; x2 ; x1 ; x2 ; . . . ; xniþ1 Þ 8i 2 f1; 2; . . . ; ng:
Theorem 2. Let (X, ) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Assume that X has the following property: (i) if nondecreasing sequence {xn} fi x, then xn x for all n; (ii) if nonincreasing sequence {yn} fi y, then yn y for all n. Let F:X · X fi X be a mapping having the mixed monotone property on X. Assume that there exists k 2 [0, 1) with k dðFðx; yÞ; Fðu; vÞÞ 6 ½dðx; uÞ þ dðy; vÞ for each x u and y v: 2
Remark 1. The concept of n-tupled fixed point is given by Imdad et al. [14], which is quite different from the concept of Gordji and Ramezani [12]. A detailed version on n-tupled fixed point is given in next section. Very recently, Soliman et al. [31] proved results on n-tupled coincidence point in complete asymptotically regular metric space and called this space as generalized complete metric space. Some more definitions related to our script are as follows: Definition 7. Let (X, d) be a metric space. Then a sequence {xn} in X is said to be Cauchy if limn;m!1 dðxn ; xm Þ ¼ 0 for all n; m 2 N.
Please cite this article in press as: A. Sharma, Results on n-tupled fixed points in complete asymptotically regular metric spacesn-tupled fixed point theorems –>, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.09.002
Results on n-tupled fixed points in complete asymptotically
3
A metric space (X, d) is said to be complete if every Cauchy sequence converges.
a mapping such that F has the mixed monotone property on X. Assume that for all x1, x2, . . . , xn, y1, y2, . . . , yn 2 X for which x1 y1, x2 y2, x3 y3, . . . , xn yn,
Definition 8. A sequence {xn} in a metric space (X, d) is said to be asymptotically regular if limnfi1d(xn+1, xn) = 0.
wðdðFðx1 ; x2 ; . . . ; xn Þ; Fðy1 ; y2 ; . . . ; yn ÞÞÞ
Every Cauchy sequence is asymptotically regular but the converse may not be true in metric spaces. Example 1. Let X ¼ R and d(x, y) =PŒx yŒ for all x, y 2 X. Then {xn} in X defined by xn ¼ nk¼1 k1 is asymptotically regular but not Cauchy. Definition 9. A metric space is called complete asymptotically regular if every asymptotically regular sequence {xn} in X converges to some point in X. Let U denote the set of all functions /:[0, 1) fi [0, 1) which satisfy that limtfir/(t) > 0 for all r > 0 and limt!0þ /ðtÞ ¼ 0. Let W denote the set of all functions w:[0, 1) fi [0, 1) which satisfy (i) w(t) = 0 if and only if t = 0, (ii) w is continuous and nondecreasing, (iii) w(s + t) 6 w(s) + w(t) for all s, t 2 [0, 1). Examples of typical functions / and w are given in [19].
2. Main results In this paper we used the new definitions of n-tupled fixed point and mixed monotone property given by Imdad et al. [14]. Throughout the paper n stands for a general even natural number. In the sequel we have the following definitions: Definition 10 [14]. Let (X, ) be a partially ordered set and F: Xn fi X be a mapping. The mapping F is said to have the mixed monotone property if F is nondecreasing in its odd position arguments and nonincreasing in its even position arguments, that is, 8 for all x11 ;x12 2 X; x11 x12 ) F x11 ;x2 ;x3 ;...;xn Fðx12 ;x2 ;x3 ;...;xn Þ > > > > > for all x21 ;x22 2 X; x21 x22 ) F x1 ;x21 ;x3 ;...;xn Fðx1 ;x22 ;x3 ;...;xn Þ > > < for all x31 ;x32 2 X; x31 x32 ) F x1 ;x2 ;x31 ;...;xn Fðx1 ;x2 ;x32 ;...;xn Þ > >. > > .. > > > : for all xn1 ;xn2 2 X; xn1 xn2 ) F x1 ;x2 ;x3 ;...;xn1 Fðx1 ;x2 ;x3 ;...;xn2 Þ: ð2:1Þ
Definition 11 [14]. An element (x1, x2, . . . , xn) 2 Xn is called an n-tupled fixed point of the mapping F:Xn fi X if 8 Fðx1 ; x2 ; x3 ; . . . ; xn Þ ¼ x1 > > > > > Fðx2 ; x3 ; . . . ; xn ; x1 Þ ¼ x2 > > < Fðx3 ; . . . ; xn ; x1 ; x2 Þ ¼ x3 > >. > .. > > > > : Fðxn ; x1 ; x2 ; . . . ; xn1 Þ ¼ xn :
ð2:2Þ
Now we are equipped to prove our main results as fo Theorem 3. Let (X, ) be a partially ordered set and (X, d) be a complete asymptotically regular metric space. Let F:Xn fi X be
1 6 wðdðx1 ; y1 Þ þ dðx2 ; y2 Þ þ . . . þ dðxn ; yn ÞÞ n /ðdðx1 ; y1 Þ þ dðx2 ; y2 Þ þ þ dðxn ; yn ÞÞ; where / 2 U and w 2 W. x10 ; x20 ; x30 ; . . . ; xn0 2 X such that 8 1 x0 F x10 ; x20 ; x30 ; . . . ; xn0 > > > > x2 F x2 ; x3 ; . . . ; xn ; x1 > > 0 0 > 0 0 0 < x30 F x30 ; . . . ; xn0 ; x10 ; x20 > > .. > > > . > > : n x0 F xn0 ; x10 ; x20 ; . . . ; xn1 : 0
Suppose
that
ð2:3Þ there
exist
Suppose that either (a) F is continuous or (b) X has the following property: (i) if nondecreasing sequence {xm} fi x, then xm x for all m P 0; (ii) if nonincreasing sequence {xm} fi x, then xm x for all m P 0, then there exist x1, x2, . . . , xn 2 X such that Fðx1 ; x2 ; . . . ; xn Þ ¼ x1 ; Fðx2 ; . . . ; xn ; x1 Þ ¼ x2 ; . . . ; Fðxn ; x1 ; . . . ; xn1 Þ ¼ xn : Proof. Let x10 ; x20 ; x30 ; . . . ; xn0 in X such that 8 1 x0 > > > > 2 > x > > < 03 x0 > > > > ... > > > : n x0
F x10 ; x20 ; x30 ; . . . ; xn0 2 3 F x0 ; x0 ; . . . ; xn0 ; x10 F x30 ; . . . ; xn0 ; x10 ; x20
Fðxn0 ; x10 ; x20 ; . . . ; xn1 0 Þ:
We construct sequences x1m ; x2m ; x3m ; . . . ; xnm in X as follows: 8 1 x > > > m > 2 > x > > < m x3m > >. > .. > > > > : n xm
¼ F x1m1 ; x2m1 ; x3m1 ; . . . ; xnm1 ¼ F x2m1 ; x3m1 ; . . . ; xnm1 ; x1m1 3 ¼ F xm1 ; . . . ; xnm1 ; x1m1 ; x2m1
ð2:4Þ
¼ F xnm1 ; x1m1 ; x2m1 ; . . . ; xn1 m1 for m P 1:
By the mixed monotone property of F, it is easy to show that 8 1 1 1 1 1 x > 0 x1 x2 xm1 xm > > > x2 x2 x2 x2 x2 > > 1 2 m1 m > < 03 x0 x31 x32 x3m1 x3m ð2:5Þ > > > .. > > . > > : n x0 xn1 xn2 xnm1 xnm :
Please cite this article in press as: A. Sharma, Results on n-tupled fixed points in complete asymptotically regular metric spacesn-tupled fixed point theorems –>, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.09.002
4
A. Sharma d 6 d n/ðdÞ < d; Due to (2.3)–(2.5), we have
which is a contradiction. Thus d = 0, that is,
w d x1mþ1 ;x1mþ2 ¼ w d F x1m ;x2m ;...;xnm ;Fðx1mþ1 ;x2mþ1 ;...;xnmþ1 Þ 1 6 w d x1m ;x1mþ1 þ d x2m ;x2mþ1 þ þ d xnm ;xnmþ1 n / d x1m ;x1mþ1 þ d x2m ;x2mþ1 þ þ d xnm ;xnmþ1 ; w d x2mþ1 ;x2mþ2 ¼ w d F x2m ;...;xnm ;x1m ;Fðx2mþ1 ;. ..;xnmþ1 ;x1mþ1 Þ 1 6 w d x2m ;x2mþ1 þ þ d xnm ;xnmþ1 þ d x1m ;x1mþ1 n / d x2m ;x2mþ1 þ þ d xnm ;xnmþ1 þ d x1m ;x1mþ1 ;
.. . n ;F xmþ1 ; x1mþ1 ;...; xn1 w d xnmþ1 ;xnmþ2 ¼ w d F xnm ;x1m ;.. .;xn1 m mþ1 1 n1 6 w d xnm ;xnmþ1 þ d x1m ;x1mþ1 þ þ d xn1 m ;xmþ1 n n1 / d xnm ;xnmþ1 þ d x1m þ x1mþ1 þ þ d xn1 m ;xmþ1 :
Due to above inequalities, we conclude that w d x1mþ1 ; x1mþ2 þ w d x2mþ1 ; x2mþ2 þ þ w d xnmþ1 ; xnmþ2 6 w d xm1 ; x1mþ1 þ d xm2 ; x2mþ1 þ þ d xmn ; xnmþ1 n/ d xm1 ; x1mþ1 þ d xm2 ; x2mþ1 þ þ d xmn ; xnmþ1 : From property (iii) of w, we have
ð2:6Þ
lim dm ¼ lim d xm1 ;x1mþ1 þ d xm2 ; x2mþ1 þ þ d xmn ; xnmþ1 ¼ 0;
m!1
m!1
ð2:13Þ fx1m g; fx2m g; . . . ; fxnm g
which implies that are asymptotically regular sequences in (X, d). Since (X, d) is a complete asymptotically regular metric space, there exist x1, x2, . . . , xn 2 X such that lim x1 m!1 m
¼ x1 ; lim x2m ¼ x2 ; . . . ; lim xnm ¼ xn : m!1
ð2:7Þ
w d x1mþ1 ; x1mþ2 þ d x2mþ1 ; x2mþ2 þ . . . þ d xnmþ1 ; xnmþ2 1 1 2 2 n n 6 w d xm ; xmþ1 þ d xm ; xmþ1 þ þ d xm ; xmþ1 n/ d x1m ; x1mþ1 þ d x2m ; x2mþ1 þ þ d xnm ; xnmþ1 : ð2:8Þ 1 1 2 2 n n Set dm ¼ d xm ; xmþ1 þ d xm ; xmþ1 þ þ d xm ; xmþ1 , then we have ð2:9Þ
which yields that wðdmþ1 Þ 6 wðdm Þ;
for all m:
ð2:10Þ
Since w is nondecreasing, we get that dm+1 6 dm for all m. Hence {dm} is a nonincreasing sequence. Since it is bounded below, there is some d P 0 such that lim dm ¼ d:
m!1
ð2:14Þ
x1 ¼ lim x1m ¼ lim x1m1 ; x2m1 ; . . . ; xnm1 m!1 m!1
1 ¼ F lim xm1 ; lim x2m1 ; . . . ; lim xnm1 ¼ Fðx1 ; x2 ; . . . ; xn Þ: m!1
m!1
m!1
Analogously, we also observe that x2 ¼ lim x2m ¼ lim x2m1 ; . .. ;xnm1 ; x1m1 m!1 m!1
¼ F lim x2m1 ;. .. ; lim xnm1 ; lim x1m1 ¼ Fðx2 ; . .. ;xn ; x1 Þ: m!1
m!1
m!1
.. .
xn ¼ lim xnm ¼ lim xnm1 ; x1m1 ;. . .; xn1 m1 m!1 m!1
n 1 n1 Þ: ¼ F lim xnm1 ; lim x1m1 ; .. .; lim xn1 m1 ¼ Fðx ; x ; . .. ; x m!1
m!1
Thus we have
Combining (2.6) and (2.7), we get that
for all m;
m!1
Suppose that assumption (a) holds. Then by (2.4) and (2.14), we have
m!1
w d x1mþ1 ; x1mþ2 þ d x2mþ1 ; x2mþ2 þ þ d xnmþ1 ; xnmþ2 6 w d x1mþ1 ; x1mþ2 þ w d x2mþ1 ; x2mþ2 þ n þ w d xmþ1 ; xnmþ2 :
wðdmþ1 Þ 6 wðdm Þ n/ðdm Þ;
ð2:12Þ
ð2:11Þ
We shall show that d = 0. Suppose on contrary that d > 0. Letting m fi 1 in (2.9) and having in mind that we suppose limtfir/(t) > 0 for all r > 0 and limt!0þ /ðtÞ ¼ 0, we have
Fðx1 ; x2 ; . . . ; xn Þ ¼ x1 ; Fðx2 ; . . . ; xn ; x1 Þ ¼ x2 ; . . . ; Fðxn ; x1 ; . . . ; xn1 Þ ¼ xn1 : Let us assume that assumption (b) holds. Since x1m ; x3m ; n1 . . . ; xm are nondecreasing and x1m ! x1 ; x3m ! x3 ; . . . ; n1 n1 xm ! x and also x2m ; x4m ; . . . ; xnm are nonincreasing and x2m ! x2 ; x4m ! x4 ; . . . ; xnm ! xn , we have x1m x1 ; x2m x2 ; x3m x3 ; . . . ; xnm xn 8m; by assumption ðbÞ: Consider now dðx1 ; Fðx1 ; x2 ; . . . ; xn ÞÞ 6 dðx1 ; x1mþ1 Þ þ dðx1mþ1 ; Fðx1 ; x2 ; . . . ; xn ÞÞ ¼ d x1 ; x1mþ1 þ d F x1m ; x2m ; . . . ; xnm ; Fðx1 ; x2 ; . . . ; xn Þ 1 < d x1 ; x1mþ1 þ w d x1m ; x1 þ d x2m ; x2 n n n þ þ d xm ; x / d x1m ; x1 þ d x2m ; x2 þ þ d xnm ; xn : ð2:15Þ Taking m fi 1 in (2.15) and using (2.14), we get that d x1 ; F x1m ; x2m ; . . . ; xnm ¼ 0 ) x1 ¼ F x1m ; x2m ; . . . ; xnm :
Please cite this article in press as: A. Sharma, Results on n-tupled fixed points in complete asymptotically regular metric spacesn-tupled fixed point theorems –>, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.09.002
Results on n-tupled fixed points in complete asymptotically Analogously, we get that
Proof. The set of n-tupled fixed point is non empty due to Theorem 3. Assume that (x1, x2, x3, . . . , xn) and (y1, y2, y3, . . . , yn) are two n-tupled fixed points of F, that is,
: x2 ¼ F x2m ; . . . ; xnm ; x1m ; . . . ; xn ¼ F xnm ; x1m ; . . . ; xn1 m Thus we proved that F has an n-tupled fixed point.
h
Corollary 1. Let (X, ) be a partially ordered set and (X, d) be a complete asymptotically regular metric space. Let F:Xn fi X be a mapping such that F has the mixed monotone property on X. Assume that there exists k 2 [0, 1) such that for all x1, x2, . . . , xn, y1, y2, . . . , yn 2 X for which x1 y1, x2 y2, x3 y3, . . . , xn yn, dðFðx1 ; x2 ; . . . ; xn Þ; Fðy1 ; y2 ; . . . ; yn ÞÞ k 6 ½dðx1 ; y1 Þ þ dðx2 ; y2 Þ þ þ dðxn ; yn Þ: n
ð2:16Þ
Suppose that there exist x10 ; x20 ; x30 ; . . . ; xn0 2 X such that 8 1 x0 F x10 ; x20 ; x30 ; . . . ; xn0 > > > > x2 F x2 ; x3 ; . . . ; xn ; x1 > > 0 0 > 0 0 < 03 x0 F x30 ; . . . ; xn0 ; x10 ; x20 > > .. > > > . > > : n x0 F xn0 ; x10 ; x20 ; . . . ; xn1 : 0
Fðx1 ; x2 ; x3 ; . . . ; xn Þ ¼ x1 ; Fðy1 ; y2 ; y3 ; . . . ; yn Þ ¼ y1 Fðx2 ; x3 ; . . . ; xn ; x1 Þ ¼ x2 ; Fðy2 ; y3 ; . . . ; yn ; y1 Þ ¼ y2 .. . Fðxn ; x1 ; x2 ; . . . ; xn1 Þ ¼ xn ; Fðyn ; y1 ; y2 ; . . . ; yn1 Þ ¼ yn : We shall show that (x1, x2, x3, . . . , xn) and (y1, y2, y3, . . . , yn) are equal. By the assumption of the theorem, there exists (z1, z2, z3, . . . , zn) 2 Xn that is comparable to (x1, x2, x3, . . . , xn) and (y1, y2, y3, . . . , yn). Define sequences z1m ; z2m ; . . . ; n zm such that z10 ¼ z1 ; z20 ¼ z2 ; . . . ; zn0 ¼ zn and 8 1 zm ¼ Fz1m1 ; z2m1 ; z3m1 ; . . . ; znm1 > > > < z2m ¼ F z2m1 ; z3m1 ; . . . ; znm1 ; z1m1 > ... > > : n zm ¼ Fðznm1 ; z1m1 ; z2m1 ; . . . ; zn1 m1 Þ8m:
ð2:18Þ
Since (x1, x2, . . . , xn) is comparable with ðz10 ; z20 ; . . . ; zn0 Þ, we may assume that ðx1 ; x2 ; . . . ; xn Þ ðz1 ; z2 ; . . . ; zn Þ ¼ ðz10 ; z20 ; . . . ; zn0 Þ:
Suppose that either (a) (b)
5
F is continuous or X has the following property: (i) if nondecreasing sequence {xm} fi x, then xm x for all m P 0; (ii) if nonincreasing sequence {xm} fi x, then xm x for all m P 0,
then there exist x1, x2, . . . , xn 2 X such that
Recursively, we get that ðx1 ; x2 ; . . . ; xn Þ z1m ; z2m ; . . . ; znm
8 m:
ð2:19Þ
By (2.3) and (2.19), we have w d x1 ; z1mþ1 ¼ w d Fðx1 ; x2 ; . . . ; xn Þ; F z1m ; z2m ; . . . ; znm 1 6 w d x1 ; z1m þ d x2 ; z2m þ . . . þ d xn ; znm n / d x1 ; z1m þ d x2 ; z2m þ . . . þ d xn ; znm ; w d x2 ; z2mþ1 ¼ w d Fðx2 ; . . . ; xn ; x1 Þ; F z2m ; . . . ; znm ; z1m 1 6 w d x2 ; z2m þ þ d xn ; znm þ d x1 ; z1m n / d x2 ; z2m þ þ d xn ; znm þ d x1 ; z1m ;
Fðx1 ; x2 ; . . . ; xn Þ ¼ x1 ; Fðx2 ; . . . ; xn ; x1 Þ ¼ x2 ; . . . ; Fðxn ; x1 ; . . . ; xn1 Þ ¼ xn : Proof. It is sufficient to take w(t) = t and /ðtÞ ¼ 1k t in the n previous theorem. h
.. . Now we shall prove the uniqueness of n-tupled fixed point. For a product Xn of a partial ordered set (X, ), we define a partial ordering in the following way: 8ðx1 ; x2 ; . . . ; xn Þ; ðy1 ; y2 ; . . . ; yn Þ 2 Xn ðx1 ; x2 ; . . . ; xn Þ ðy1 ; y2 ; . . . ; yn Þ () x1 y1 ; x2 y2 ; x3 y3 ; . . . ; xn yn :
ð2:17Þ
We say that (x1, x2, . . . , xn) is equal to (y1, y2, . . . , yn) if and only if x1 = y1, x2 = y2, . . . , xn = yn. Theorem 4. In addition to the hypotheses of Theorem 3, suppose that for real (x1, x2, . . . , xn), (y1, y2, . . . , yn) 2 Xn there exists, (z1, z2, . . . , zn) 2 Xn that is comparable to (x1, x2, . . . , xn) and (y1, y2, . . . , yn), then F has a unique n-tupled fixed point.
w d xn ; znmþ1 ¼ w d F xn ; x1 ; .. . ;xn1 ; F znm ;z1m ; . .. ; zn1 m 1 6 w d xn ; znm þ d x1 ; z1m þ þ d xn1 ; zn1 m n : / d xn ; znm þ d x1 ; z1m þ þ d xn1 ; zn1 m Set dm ¼ d x1 ; z1m þ d x2 ; z2m þ . . . þ d xn ; znm . Then due to above inequalities, we have wðdmþ1 Þ 6 wðdm Þ n/ðdm Þ for all m; ð2:20Þ which implies dm+1 6 dm. Hence the sequence {dm} is decreasing and bounded below. Thus there exists d P 0 such that limmfi1dm = d. Now, we shall show that d = 0. Suppose to the contrary that d > 0. Letting m fi 1 in (2.20), we obtain wðdÞ 6 wðdÞ n lim /ðdm Þ < wðdÞ; m!1
Please cite this article in press as: A. Sharma, Results on n-tupled fixed points in complete asymptotically regular metric spacesn-tupled fixed point theorems –>, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.09.002
6
A. Sharma
which is a contradiction. Therefore d = 0. That is, limmfi1dm = 0. Consequently, we have lim d x1 ; z1m ¼ 0; lim dðx2 ; z2m Þ ¼ 0; . . . ; lim dðxn ; znm Þ ¼ 0:
m!1
m!1
m!1
ð2:21Þ Similarly, we show that lim d y1 ; z1m ¼ 0; lim d y2 ; z2m ¼ 0; . . . ; lim d yn ; znm ¼ 0:
m!1
m!1
m!1
ð2:22Þ Combining (2.21) and (2.22) yields that (x1, x2, x3, . . . , xn) and (y1, y2, y3, . . . , yn) are equal. h
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Please cite this article in press as: A. Sharma, Results on n-tupled fixed points in complete asymptotically regular metric spacesn-tupled fixed point theorems –>, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.09.002
