EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 295-311 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global n-tupled common fixed point theorems via α-series in ordered metric spaces Manju Grewal1, Ramesh Kumar Vats2, Amit Kumar3,∗ 1,2 Department of Mathematics National Institute of Technology, Hamirpur, India. 3 Department of Mathematics & Statistic, Banasthali University, Banasthali, Rajasthan, India. Abstract. The aim of this paper is to establish some results about the existence and uniqueness of n-tupled fixed point that extend the previous results, using the concept of an α-series for sequence of mappings having mixed monotone property in the framework of ordered complete metric spaces. The main result is supported with the aid of an illustrative example. 2010 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: α-series, Compatible mappings, Common fixed point, Ordered metric space 1. Introduction For standard terminology and notations in fixed point theory, not specifically men- tioned or defined we refer the reader to the standard textbook [21]. Throughout the paper, for a nonempty set X, ∏r λ=1X λ denote the product space ∏r λ=1X λ = X×X×X×· · ·×X. The existence of a fixed point for contraction type mappings in metric spaces along with applications have been taken a considerable attention. The Banach contraction principle is one of the earliest and the most important results in the area of fixed point theory. Several authors have improved, generalized, and extended this classical result in nonlinear analysis. The notion of coupled fixed point is introduced by Bhaskar and Lakshmikantham [7]. Afterwards Lakshmikantham and Ciric [18] extended this notion by defining the g- monotone property in partially ordered spaces. For a detailed study on coupled coincidence and coupled common fixed point results, we refer the reader to [5, 11, 12, 13, 18]. Berinde and Borcut [6] introduced the concept of tripled fixed point. An enough considerable work have been done in this area by several authors (see, for instance, [1, 2, 3, 4, 16, 17, 23]). In 2010, Samet and Vetro [20] extended the idea of coupled fixed point to higher dimensions by introducing the notion of fixed point of n-order (or n-tupled fixed point, ∗Corresponding author. Email addresses: amitsu48@gmail.com (Amit Kumar), ramesh−vats@rediffmail.com (Ramesh Kumar Vats),manjudahiyautku@gmail.com (Manju Grewal) http://www.ejpam.com 295 c© 2017 EJPAM All rights reserved. M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 296 where n ∈ N, n ≥ 2) and presented some n-tupled fixed point results in complete metric spaces. In 2011, Gordji and Ramezani [14] introduced and investigated the concept of an n-tupled fixed point. In 2013, Imdad et al. [15] generalized the idea of n-tupled fixed point by considering even-tupled coincidence point by initiating the idea of mixed g-monotone property on Xn and proved an even-tupled coincidence point theorem for nonlinear f - contraction mappings satisfying mixed g-monotone property. However, the concept of n-tupled fixed point given by Imdad et al. [15], which is quite different from the concept of Gordji and Ramezani [14]. In this paper, we shall also point out some useful remarks on mappings whenever found prominent or pertinent as we proceed with this article. Our focus however will be on results that gives the guarantee about the existence and uniqueness of n-tupled fixed point that extend the previous results in the framework of ordered complete metric spaces, using the concept of an α-series for sequence of mappings having mixed monotone property in ordered complete metric spaces. In order to do so, we propose a notion of compatible mapping for mapping F : ∏r i=1X i → X and self mapping g akin to compatible mapping as introduced by Choudhary and Kundu [13] for bivariate mapping F and self mapping g. The methodology is analogous to those used in [15]. Finally, the main result of the manuscript is supported with the aid of an illustrative example. 2. Preliminaries In this section, we collect some definitions, properties and results which will be fre- quently used in this paper. As in [19] we define a metric on X, a mapping d : X × X → R such that for all x, y, z ∈ X: (i) d(x, y) = 0 if and only if, x = y; (ii) d(x, y) = d(z, x) + d(z, y). Thus, in light of the above properties one can easily deduce that d(x, y) ≥ 0 and d(y, x) = d(x, y) for all x, y ∈ X. The last requirement is called the triangle inequality. If d is a metric on X, then we say that (X, d) is a metric space. Definition 1. [10] A triple (X, d,�) is called an ordered metric space if (X, d) is a metric space and (X,�) is a partially ordered set. Imdad et al. [15] introduced the concept of mixed monotone property and g-mixed monotone property for n-tupled mapping F : ∏r λ=1X λ → X in the following way: Definition 2. Let (X,�) be a partially ordered set and F : ∏r λ=1X λ → 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, ∀ x11, x12 ∈ X,x11 � x12 ⇒ F (x11, x 2, . . . , xr) � F (x12, x 2, . . . , xr), ∀ x21, x22 ∈ X,x21 � x22 ⇒ F (x1, x21, . . . , x r) � F (x1, x22, . . . , x r), ∀ x31, x32 ∈ X,x31 � x32 ⇒ F (x1, x2, x31, . . . , x r) � F (x1, x2, x32, . . . , x r), ... ∀ xr1, xr2 ∈ X,xr1 � xr2 ⇒ F (x1, x2, x3, . . . , xr1) � F (x1, x2, x3, . . . , xr2). M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 297 Definition 3. [15] Let (X,�) be a partially ordered set. Let F : ∏r λ=1X λ → X and g : X → X be two mappings. Then the mapping F is said to have the mixed g-monotone property if F is g-nondecreasing in its odd position arguments and g-nonincreasing in its even position arguments, that is ∀ x11, x12 ∈ X, g(x11) � g(x12)⇒ F (x11, x 2, . . . , xr) � F (x12, x 2, . . . , xr), ∀ x21, x22 ∈ X, g(x21) � g(x22)⇒ F (x1, x21, . . . , x r) � F (x1, x22, . . . , x r), ∀ x31, x32 ∈ X, g(x31) � g(x32)⇒ F (x1, x2, x31, . . . , x r) � F (x1, x2, x32, . . . , x r), ... ∀ xr1, xr2 ∈ X, g(xr1) � g(xr2)⇒ F (x1, x2, x3, . . . , xr1) � F (x1, x2, x3, . . . , xr2). Now, we introduce the concept of compatible mapping for mapping F : ∏r λ=1X λ → X and self mapping g akin to compatible mapping as introduced by Choudhary and Kundu [13] for mapping F and self mapping g. Definition 4. Let F : ∏r λ=1X λ → X and g : X → X be two mappings. Then F and g are said to be compatible if lim n→+∞ d(g(F (x1n, x 2 n, . . . , x r n)), F (g(x1n), g(x2n), . . . , g(xrn))) = 0, lim n→+∞ d(g(F (x2n, x 3 n, . . . , x r n, x 1 n)), F (g(x2n), g(x3n), . . . , g(xrn), g(x1n))) = 0, lim n→+∞ d(g(F (x3n, x 4 n, . . . , x 1 n, x 2 n)), F (g(x3n), g(x4n), . . . , g(x1n), g(x2n))) = 0, ... lim n→+∞ d(g(F (xrn, x 1 n, . . . , x r−1 n )), F (g(xrn), g(x1n), . . . , g(xr−1n ))) = 0, whenever {x1n}, {x2n}, . . . , {xrn} are sequences in X, such that lim n→+∞ F (x1n, x 2 n, . . . , x r n) = lim n→+∞ g(x1n) = x1, lim n→+∞ F (x2n, x 3 n, . . . , x r n, x 1 n) = lim n→+∞ g(x2n) = x2 ... lim n→+∞ F (xrn, x 1 n, . . . , x r−1 n ) = lim n→+∞ g(xrn) = xr, (1) for all x1, x2, . . . , xr ∈ X. The following is the definition of reciprocally continuity and weakly reciprocally con- tinuity for mapping F : ∏r λ=1X λ → X and self mapping g: Definition 5. Let F : ∏r λ=1X λ → X and g : X → X be two mappings. Then F and g are said to be M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 298 (i) Reciprocally continuous if lim n→+∞ g(F (x1n, x 2 n, . . . , x r n) = g(x1) and lim n→+∞ F (g(x1n), g(x2n), . . . , g(xrn)) = F (x1, x2, . . . , xr); lim n→+∞ g(F (x2n, x 3 n, . . . , x r n, x 1 n) = g(x2) and lim n→+∞ F (g(x2n), g(x3n), . . . , g(xrn), g(x1n)) = F (x2, x3, . . . , xr, x1); ... lim n→+∞ g(F (xrn, x 1 n, . . . , x r−1 n ) = g(xr) and lim n→+∞ F (g(xrn), g(x1n), . . . , g(xr−1n )) = F (xr, x1, . . . , xr−1). whenever {x1n}, {x2n}, . . . , {xrn} are sequences in X, such that lim n→+∞ F (x1n, x 2 n, . . . , x r n) = lim n→+∞ g(x1n) = x1, lim n→+∞ F (x2n, x 3 n, . . . , x r n, x 1 n) = lim n→+∞ g(x2n) = x2 ... lim n→+∞ F (xrn, x 1 n, . . . , x r−1 n ) = lim n→+∞ g(xrn) = xr, for some x1, x2, . . . , xr ∈ X. (ii) Weakly reciprocally continuous if lim n→+∞ g(F (x1n, x 2 n, . . . , x r n) = g(x1) or lim n→+∞ F (g(x1n), g(x2n), . . . , g(xrn)) = F (x1, x2, . . . , xr); lim n→+∞ g(F (x2n, x 3 n, . . . , x r n, x 1 n) = g(x2) or lim n→+∞ F (g(x2n), g(x3n), . . . , g(xrn), g(x1n)) = F (x2, x3, . . . , xr, x1); ... lim n→+∞ g(F (xrn, x 1 n, . . . , x r−1 n ) = g(xr) or lim n→+∞ F (g(xrn), g(x1n), . . . , g(xr−1n )) = F (xr, x1, . . . , xr−1). (2) Whenever {x1n}, {x2n}, . . . , {xrn} are sequences in X, such that lim n→+∞ F (x1n, x 2 n, . . . , x r n) = lim n→+∞ g(x1n) = x1, lim n→+∞ F (x2n, x 3 n, . . . , x r n, x 1 n) = lim n→+∞ g(x2n) = x2 ... lim n→+∞ F (xrn, x 1 n, . . . , x r−1 n ) = lim n→+∞ g(xrn) = xr, for some x1, x2, . . . , xr ∈ X. M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 299 Remark 1. Every pair of reciprocally continuous mapping (F, g) is weakly reciprocally continuous but not conversely. Definition 6. Let (X, d,�) be an ordered metric space. We say that X is regular if the following conditions hold: (i) if a non-decreasing sequence {xn} is such that xn → x, then xn � x for all n ≥ 0, (ii) if a non-increasing sequence {yn} is such that yn → y, then yn � y for all n ≥ 0. Definition 7. [22] Let {sn} be a sequence of non-negative real numbers. We say that a series ∑+∞ n=1 sn is an α-series, if there exist 0 < α < 1 and nα ∈ N such that ∑k i=1 si ≤ αk for each k ≥ nα. Example 1. The series ∑+∞ n=1 1 n2 is an α-series. Remark 2. [22] It is bring here to notice that each convergent series of non-negative real terms is an α-series. However, there are also divergent series that are α-series. As for instance; the series ∑+∞ n=1 1 n , is an α-series. Definition 8. Let X be a nonempty set. An element (x1, x2, . . . , xr) ∈ ∏r λ=1X λ is called r-tupled fixed point of the mapping F : ∏r λ=1X λ → X if F (x1, x2, . . . , xr) = x1, F (x2, x3, . . . , x1) = x2, F (x3, x4, . . . , x2) = x3, ... F (xr, x1, . . . , xr−1) = xr. (3) Example 2. Let (X, d,�) be an ordered metric space with � as natural ordering and let F : ∏r λ=1X λ → X be a mapping defined by F (x1, x2, . . . , xr) = (x1 · x2 · · ·xr)2, for any x1, x2, . . . , xr ∈ X. Then (0, 0, . . . , 0) and (1, 1, . . . , 1) are both r-tupled fixed points of F . Definition 9. Let X be a nonempty set. An element (x1, x2, . . . , xr) ∈ ∏r λ=1X λ is called r-tupled coincidence point of the mappings F : ∏r λ=1X λ → X and g : X → X if F (x1, x2, . . . , xr) = g(x1), F (x2, x3, . . . , x1) = g(x2), F (x3, x4, . . . , x2) = g(x3), ... F (xr, x1, . . . , xr−1) = g(xr). (4) Example 3. Let (X, d,�) be an ordered metric space with � as natural ordering and let F : ∏r λ=1X λ → X be a mapping defined by F (x1, x2, . . . , xr) = sin(x1 · x2 · · ·xr), for any x1, x2, . . . , xr ∈ X and g : X → X be mapping defined by g(x) = x2. Then (0, 0, . . . , 0) is r-tupled coincidence point of F and g. M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 300 Proposition 1. Let (X, d,�) be an ordered metric space. Let g be a self-mapping on X and {Ti}i∈N be a sequence of mappings from ∏r λ=1X λ → X. Then the pair of mappings (Ti, g) is said to have property (A) if for x1, x2, . . . , xr, y1, y2, . . . , yr ∈ X d(Ti(x 1, x2, . . . , xr), Tj(y 1, y2, . . . , yr)) ≤ βi,j [d(g(x1), Ti(x 1, x2, . . . , xr)) +d(g(y1), Tj(y 1, y2, . . . , yr))] +γi,jd(g(y1), g(x1)) (5) with g(x1) � g(y1), g(x2) � g(y2), g(x3) � g(y3), ... g(xr) � g(yr), where 0 ≤ βi,j , γi,j < 1 for i, j ∈ N and lim n→+∞ supβi,n < 1. Proposition 2. Let (X,�) be a partially ordered set and {Ti}i∈N be a sequence of mappings from ∏r λ=1X λ → X. Then {Ti}i∈N is said to have property (B) if for x1, x2, . . . , xr, y1, y2, . . . , yr ∈ X Ti(x 1, x2, . . . , xr) � Ti+1(y 1, y2, . . . , yr), Ti+1(y 2, y3, . . . , yr, y1) � Ti(x2, x3, . . . , xr, x1), ... Ti+1(x r, x1, x2 · · · , xr−1) � Ti(yr, y1, y2, . . . , yr−1). (6) 3. Main results Theorem 1. Let (X, d,�) be an ordered metric space. Let g be a continious self-mapping on X and {Ti}i∈N be a sequence of mappings from ∏r λ=1X λ → X such that i) Ti( ∏r λ=1X λ) ⊆ g(X), g(X) is regular and complete subset of X; ii) {Ti}i∈N have g-mixed monotone property, the pair {Ti}i∈N and g are compatible, weakly reciprocally continuous and satisfy property (A) and (B). iii) There exists x10, x 2 0, . . . , x r 0 ∈ X such that g(x10) � T0(x10, x20, . . . , xr0); g(x20) � T0(x20, x30, . . . , xr0, x10); g(x30) � T0(x30, x40, . . . , x10, x20); ... g(xr0) � T0(xr0, x10, . . . , x r−1 0 ). (7) . M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 301 If ∑+∞ i=1 ( βi,i+1+γi,i+1 1−βi,i+1 ) is an α-series, then {Ti}i∈N and g have a r-tupled coincidence point. Proof. Let (X,�) be a partially ordered set, g be a self-mapping on X and {Ti}i∈N be a sequence of mappings from ∏r λ=1X λ → X. Since Ti( ∏r λ=1X λ) ⊆ g(X), one can always have x11, x 2 1, . . . , x r 1 ∈ X such that g(x11) = T1(x 1 0, x 2 0, . . . , x r 0), g(x21) = T1(x 2 0, x 3 0, . . . , x r 0, x 1 0), ... g(xr1) = T1(x r 0, x 1 0, . . . , x r−1 0 ). Again we can choose x12, x 2 2, . . . , x r 2 ∈ X such that g(x12) = T2(x 1 1, x 2 1, . . . , x r 1), g(x22) = T2(x 2 1, x 3 1, . . . , x r 1, x 1 1), ... g(xr2) = T2(x r 1, x 1 1, · · · , x r−1 1 ). Continuing in this way, we can construct the {x1m}, {x2m}, . . . , {xrm} sequences as follows : g(x1m+1) = Tn(x1m, x 2 m, . . . , x r m), g(x2m+1) = Tn(x2m, x 3 m, . . . , x r m, x 1 m), ... g(xrm+1) = Tn(xrm, x 1 m, . . . , x r−1 m ). (8) Now our claim is that for all m ≥ 0. g(x1m) � g(x1m+1), g(x2m) � g(x2m+1), . . . , g(xrm) � g(xrm+1). (9) We shall our claim by the principle of Mathematical induction. Since  g(x10) � T0(x10, x20, . . . , xr0) = x11, g(x20) � T0(x20, x30, . . . , xr0, x10) = x21, g(x30) � T0(x30, x40, . . . , x10, x20) = x31, ... g(xr0) � T0(xr0, x10, . . . , x r−1 0 ) = xr1. and  g(x11) = T0(x 1 0, x 2 0, . . . , x r 0), g(x21) = T0(x 2 0, x 3 0, . . . , x r 0, x 1 0), ... g(xr1) = T0(x r 0, x 1 0, . . . , x r−1 0 ). M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 302 Thus we get  g(x10) � g(x11), g(x20) � g(x21), g(x30) � g(x31), ... g(xr0) � g(xr1). Which shows that (9) holds for m = 0. Now assume that (9) holds for some m > 0. From (8) and (9), one can deduce that g(x1m+1) = Tm(x1m, x 2 m, . . . , x r m) � Tm+1(x 1 m+1, x 2 m, . . . , x r m) � Tm+1(x 1 m+1, x 2 m+1, . . . , x r m) ... � Tm+1(x 1 m+1, x 2 m+1, . . . , x r m+1) = g(x1m+2). g(x2m+1) = Tm+1(x 2 m, x 3 m, . . . , x r m, x 1 m) � Tm+1(x 2 m+1, x 3 m, . . . , x r m), x1m � Tm+1(x 2 m+1, x 3 m+1, . . . , x r m, x 1 m) ... � Tm(x2m+1, x 3 m+1, . . . , x r m+1, x 1 m+1) = g(x2m+2). g(x3m+1) = Tm(x3m, x 4 m, . . . , x r m, x 1 m, x 2 m) � Tm+1(x 3 m+1, x 4 m, . . . , x r m, x 1 m, x 2 m) ... � Tm+1(x 3 m+1, x 4 m+1, . . . , x r m+1, x 1 m+1, x 2 m+1) = g(x3m+2). Continuing in this way g(xrm+1) = Tm(xrm, x 1 m, x 2 m, . . . , x r−1 m ) � Tm+1(x r m+1, x 1 m, x 2 m, . . . , x r−1 m ) � Tm+1(x r m+1, x 1 m+1, x 2 m, . . . , x r−1 m ) M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 303 ... � Tm+1(x r m+1, x 1 m+1, x 2 m+1, · · · , xr−1m+1) = g(xrm+2). Thus by the principle of Mathematical induction, we conclude that (9) holds for all n ≥ 0. Therefore, g(x1m) � g(x1m+1) g(x2m) � g(x2m+1) g(x3m) � g(x3m+1) ... g(xrm) � g(xrm+1). We consider the sequences {x1m}, {x2m}, . . . , {xrm} in X constructed in (8) and represent them by δm such that δm = d(g(x1m), g(x1m+1)) + d(g(x2m), g(x2m+1)),+ . . . ,+d(g(xrm), g(xrm+1)). Now from the property (A) of {Ti}i∈N and g in (5) we get d(g(x11), g(x12)) = d(T0(x 1 0, x 2 0, . . . x r 0)), T1(x 1 1, x 2 1, . . . x r 1) ≤ β0,1[d(g(x10), T0(x 1 0, x 2 0, . . . x r 0)) + d(g(x11), T1(x 1 1, x 2 1, . . . x r 1))] +γ0,1[d(g(x10), g(x11))] = β0,1[d(g(x10), g(x11)) + d(g(x11), g(x12))] +γ0,1[d(g(x10), g(x11))]. Consequently, (1− β0,1)d(g(x11), g(x12)) ≤ (β0,1 + γ0,1)d(g(x10), g(x11)), or equivalently, d(g(x11), g(x12)) ≤ ( β0,1+γ0,1 1−β0,1 ) d(g(x10), g(x11)). Now d(g(x12), g(x13)) = d(T1(x 1 1, x 2 1, . . . x r 1), T2(x 1 2, x 2 2, . . . x r 2)) ≤ ( β1,2 + γ1,2 1− β1,2 ) d(g(x11), g(x12)) ≤ ( β1,2 + γ1,2 1− β1,2 )( β0,1 + γ0,1 1− β0,1 ) d(g(x10), g(x11)). Continuing in this way, we get d(g(x1m), g(x1m+1)) ≤ d(Tm−1(x 1 m−1, x 2 m−1, . . . x r m−1), Tm(x1m, x 2 m, . . . x r m)) M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 304 ≤ m−1∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) d(g(x10), g(x11)). (10) d(g(x2m), g(x2m+1)) ≤ d(Tm−1(x 2 m−1, x 3 m−1, . . . , x r m−1, x 1 m−1), Tm(x2m, x 3 m, . . . x r m, x 1 m)) ≤ m−1∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) d(g(x20), g(x21)). Similarly, one can inductively write d(g(x2m), g(x2m+1)) ≤ ∏m−1 i=0 ( βi,i+1+γi,i+1 1−βi,i+1 ) d(g(x20), g(x21)). d(g(x3m), g(x3m+1)) ≤ ∏m−1 i=0 ( βi,i+1+γi,i+1 1−βi,i+1 ) d(g(x30), g(x31)). ... d(g(xrm), g(xrm+1)) ≤ ∏m−1 i=0 ( βi,i+1+γi,i+1 1−βi,i+1 ) d(g(xr0), g(xr1)). (11) Adding (10) and (11), we have δm = d(g(x1m), g(x1m+1)) + d(g(x2m), g(x2m+1) + · · ·+ d(g(xrm), g(xrm+1) ≤ m−1∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) [d(g(x10), g(x11)) + d(g(x20), g(x21)) + · · ·+ d(g(xr0), g(xr1))] = m−1∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) δ0. Moreover, for p > 0 and by repeated use of the triangle inequality, ones obtain d(g(x1m), g(x1m+p)) + d(g(x2m), g(x2m+p)) + · · ·+ d(g(xrm), g(xrm+p)) ≤ [(d(g(x1m), g(x1m+1)) + d(g(x2m), g(x2m+1)) + · · ·+ d(g(xrm), g(xrm+1))) +(d(g(x1m), g(x1m+2)) + d(g(x2m), g(x2m+2)) + · · ·+ d(g(xrm), g(xrm+2))) · · ·+ (d(g(x1m+p−1), g(x1m+p)) + d(g(x2m+p−1), g(x2m+p)) + · · ·+ d(g(xrm+p−1), g(xrm+p)))] ≤ m−1∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) δ0 + m∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) δ0 + · · · + m+p−2∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) δ0 = p−1∑ k=0 m+k−1∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) δ0 = m+p−1∑ k=m k−1∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) δ0. M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 305 Now using the fact that the Geometric mean of non-negative numbers is always less or equal to the Arithmetic mean, one have d(g(x1m), g(x1m+p)) + d(g(x2m), g(x2m+p)) + · · ·+ d(g(xrm), g(xrm+p)) ≤ m+p−1∑ k=m [ 1 k k−1∏ i=0 ( βi,i+1 + γi,i+1 1− βi,i+1 ) ]kδ0. ≤ ( m+p−1∑ k=m αk)δ0. ≤ αm 1− α δ0 Now, proceeding the limit as m→ +∞, ones deduce that lim m→+∞ δm = lim m→+∞ αm 1− α δ0 = 0 as α < 1. i.e., lim m→+∞ [d(g(x1m), g(x1m+p)) + d(g(x2m), g(x2m+p)) + · · ·+ d(g(xrm), g(xrm+p))] = 0. Which further implies that lim m→+∞ d(g(x1m), g(x1m+p)) = lim n→+∞ d(g(x2m), g(x2m+p)) = · · · = lim n→+∞ d(g(xrm), g(xrm+p)) = 0. {g(x1m)}, {g(x2m)}, . . . , {g(x1m)} are all Cauchy sequences in (X, d,�). Since g(X) is complete subspace ofX, and hence there exists (x10, x 2 0, . . . , x r 0) ∈ ∏r λ=1X λ such that  g(x10) = x1, g(x20) = x2, ... g(xr0) = xr. (12) By the continuity of g and (12), we have g(g(x10)) = g(x1), g(g(x20)) = g(x2), ... g(g(xr0)) = g(xr). (13) Now using the above equations we have lim m→+∞ g(x1m+1) = lim m→+∞ Tm(x1m, x 2 m, . . . , x r m) = x1, lim m→+∞ g(x2m+1) = lim m→+∞ Tm(x2m, x 3 m, . . . , x r m, x 1 m) = x2, ... and lim m→+∞ g(xrm+1) = lim m→+∞ Tm(xrm, x 1 m, . . . , x r−1 m ) = xr. M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 306 Since {Ti}i∈N and g are weakly reciprocally continuous, thus lim m→+∞ g(Tm(x1m, x 2 m, . . . , x r m)) = g(x1), lim m→+∞ g(Tm(x2m, x 3 m, . . . , x r m, x 1 m) = g(x2), ... and lim m→+∞ g(Tm(xrm, x 1 m, . . . , x r−1 m )) = g(xr). On the other hand, the compatibility of {Ti}i∈N and g yields.  lim m→+∞ d(g(Tm(x1m, x 2 m, . . . , x r m)), Tm(g(x1m), g(x2m), . . . , g(xrm))) = 0, lim m→+∞ d(g(Tm(x2m, x 3 m, . . . , x r m, x 1 m)), Tm(g(x2m), g(x3m), . . . , g(xrm), g(x1m))) = 0, ... and lim m→+∞ d(g(Tm(xrm, x 1 m, . . . , x r−1 m )), Tm(g(xrm), g(x1m), . . . , g(xr−1m ))) = 0. Thus the above expression turns out to be lim m→+∞ Tm(g(x1m), g(x2m), . . . , g(xrm)) = g(x1) lim m→+∞ Tm(g(x2m), g(x3m), . . . , g(xrm, x 1 m)) = g(x2) ... and lim m→+∞ Tm(g(xrm), g(x1m), . . . , g(xr−1m )) = g(xr). Since {g(xim)} are non-decreasing or non-increasing according as i is odd or even, respectively. Using the regularity of g(X), we have g(xim) � xi, when i is odd; and g(xim) � xi, when i is even. Therefore g(g(xim)) � g(xi),when i is odd; and g(g(xim)) � g(xi),when i is even. Then by (5), ones obtain d(Ti(x 1, x2, . . . , xr), Tm(g(x1m), g(x2m), . . . , g(xrm))) ≤ βi,m[d(g(x1), Ti(x 1, x2, . . . , xr) +d(g(g(x1m)), Tm(g(x1m), g(x2m), . . . , g(xrm)] +γi,m[d(g(g(x1m)), g(x1))]. Proceeding limit as m → +∞ in the above inequality, using (13) with the fact that βi,m < 1, we get Ti(x 1, x2, . . . , xr = g(x1). Similarly, it can also be shown that g(x2) = Ti(x 2, x3, . . . , xr, x1) M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 307 g(x3) = Ti(x 3, x4, . . . , xr, x1, x2) ... g(xr) = Ti(x r, x1, . . . , xr−1). This shows that (x1, x2, . . . , xr) ∈ ∏r λ=1X λ is a r-tupled coincidence point of {Ti}i∈N and g. � Theorem 2. Let X be regular and (X, d,�) be a complete ordered metric space. Let {Ti}i∈N be a sequence of mappings from ∏r λ=1X λ → X such that for all x1, x2, . . . , xr, y1, y2, . . . , yr ∈ X with x1 � y1, x2 � y2, x3 � y3, . . . , xr � yr, {Ti}i∈N satisfy the following conditions: i) Tm(x1, x2, . . . , xr) � Tm+1(y 1, y2, . . . , yr); ii) d(Ti(x 1, x2, . . . , xr), Tj(y 1, y2, . . . , yr)) ≤ βi,j [d(x, Ti(x 1, x2, . . . , xr)) + d(y1, Tj(y 1, y2, . . . , yr))] + γi,jd(y1, x1). where 0 ≤ βi,j , γi,j < 1 ∀ i, j ∈ N iii) There exists (x10, x 2 0, . . . , x r 0) ∈ ∏r λ=1X λ such that x10 � T0(x10, x20, . . . , xr0), x20 � T0(x20, x30, . . . , xr0, x10), x30 � T0(x30, x40, . . . , x10, x20), ... xr0 � T0(xr0, x10, . . . , x r−1 0 ). If ∑+∞ i=1 ( βi,i+1+γi,i+1 1−βi,i+1 ) is an α-series, then {Ti}i∈N has r-tupled fixed point. Proof. The proof easily follows from the proof of Theorem 1 by taking g to be an identity mapping. � Now, we give useful conditions for existence and uniqueness of a n-tupled common fixed point. Theorem 3. In addition to the hypotheses of Theorem 1, suppose that the set of coinci- dence points is comparable with respect to g, then {Ti}i∈N and g have a unique r-tupled common fixed point. Proof. It is clear from Theorem 1 that the set of r-tupled coincidence points is nonempty. Let (x1, x2, . . . , xr) and (y1, y2, . . . , yr) ∈ ∏r λ=1X λ be two r-tupled coinci- dence points of {Ti}i∈N and g. Then g(x1) = Ti(x 1, x2, . . . , xr) g(x2) = Ti(x 2, x3, . . . , xr, x1) M. Grewal, R. Kumar, A. Kumar / Eur. J. Pure Appl. Math, 10 (2) (2017), 295-311 308 ... g(xr) = Ti(x r, x1, . . . , xr−1). and g(y1) = Ti(y 1, y2, . . . , yr) g(y2) = Ti(y 2, y3, . . . , yr, x1) ... g(yr) = Ti(y r, y1, . . . , yr−1). Now our aim is to show that g(x1) = g(y1) g(x2) = g(y2) ... g(xr) = g(yr). Since the set of coincidence points is comparable, using property (A) to these points we obtain, d(g(x1), g(y1)) = d(Ti(x 1, x2, . . . xr)), Tj(y 1, y2, . . . yr) ≤ βi,j [d(g(x1), Ti(x 1, x2, . . . xr)) +d(g(y1), Tj(y 1, y2, . . . yr))] +γi,j [d(g(y1), g(x1))] i.e., (1− γi,j)d(g(x1), g(y1)) ≤ βi,j [d(g(x1), Ti(x 1, x2, . . . xr)) +d(g(y1), Tj(y 1, y2, . . . yr))] ⇒ d(g(x1), g(y1)) ≤ βi,j (1− γi,j) [d(g(x1), Ti(x 1, x2, . . . xr)) +d(g(y1), Tj(y 1, y2, . . . yr))]. Also γi,j < 1 and the elements of coincidence point are comparable, thus d(g(x1), g(y1)) = 0 ⇒ g(x1) = g(y1). Using the argument analogous to those used above, one can show that g(x2) = g(y2) g(x3) = g(y3) ... g(xr) = g(yr). Hence {Ti}i∈N and g have a unique r-tupled coincidence point. It is well-known that two compatible mappings are also weakly compatible, (i.e., they commute at their coincidence REFERENCES 309 points). Thus, {Ti}i∈N and g have a unique r-tupled common fixed point. Hence the result. � Example 4. Take X = [0, 1] endowed with usual metric d = |x−y| for all x, y ∈ X and � be defined as “greater equal” the (X, d,�) be ordered metric space. Let Ti : ∏r λ=1X λ → X be mapping defined as Ti(x 1, x2, . . . , xr) = 1 r [ x1 + x2 + . . .+ xr i ]; i ∈ N and g is a self mapping defined as g(x) = x2. By choosing the sequences {x1m} = 1 m {x2m} = 1 m+ 1 ... {x2m} = 1 m+ r − 1 . One can easily observe that (1) {Ti}i∈N have g-mixed monotone property; (2) {Ti}i∈N and g are compatible, weakly reciprocally continuous (3) g is continuous. By taking 0 < βi,j < 1 and 0 ≤ γi,j < 1 it is easy to verify property (A) and (B). Thus all the hypotheses of Theorem 1 are satisfied and (0, 0, 0, . . . , 0) is the only r-tupled coincident point of g and Ti for all i, i ∈ N. The following remarks depicts that if we take Ti to be a single mapping rather than the sequence of mappings {Ti}i∈N, then the pair Ti and g may have more than one r-tupled coincident points which depends upon the value of i. Remark 3. 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