Abstract
This manuscript has two aims: first we extend the definitions of compatibility and weakly reciprocally continuity, for a trivariate mapping F and a selfmapping g akin to a compatible mapping as introduced by Choudhary and Kundu (Nonlinear Anal. 73:25242531, 2010) for a bivariate mapping F and a selfmapping g. Further, using these definitions we establish tripled coincidence and fixed point results by applying the new concept of an αseries for sequence of mappings, introduced by Sihag et al. (Quaest. Math. 37:16, 2014), in the setting of partially ordered metric spaces.
MSC:54H25, 47H10, 54E50.
Similar content being viewed by others
1 Introduction and preliminaries
The notion of metric space is fundamental in mathematical analysis and the Banach contraction principle is the root of fruitful tree of fixed point theory [1]. In fact, many studies have been done on contractive mappings, e.g., Rhoades [2] presented a comparison of various definitions (more than 100 types varied from 25 basic types) of contractive mappings on complete metric spaces in 1977. See also [3–7]. Up to now, such a study is still going on; proceeding in the same tradition, very recently Sihag et al. [8] introduced the new concept of an αseries to give a common fixed point theorem for a sequence of selfmappings. On the other hand, the concept of a coupled fixed point was introduced in 1991 by Chang and Ma [9]. This concept has been of interest to many researchers in metrical fixed point theory (see for example [3, 10–15]). Recently, Bhaskar and Lakshmikantham [16] established coupled fixed point theorems for a mixed monotone operator in partially ordered metric spaces. Afterward, Lakshmikantham and Ćirić [17] extended the results of [16] by furnishing coupled coincidence and coupled fixed point theorems for two commuting mappings.
Starting from the background of coupled fixed points, recently Berinde and Borcut [18] introduced the notion of tripled fixed points in partially ordered metric spaces, which refer to the operator as F:X\times X\times X\to X, motivated by the fact that through the coupled fixed point technique we cannot solve a system with the following form:
In a subsequent series, Berinde and Borcut [18], introduced the concept of tripled coincidence point and obtained the tripled coincidence point theorems; for more on the tripled fixed point (see [19–27]). Further, Borcut and Berinde [28, 29] established the tripled fixed point theorems by introducing the concept of commuting mappings and also discussed the existence and uniqueness of solution of periodic boundary value problem.
Thus, the purpose of this paper is to prove tripled coincidence and fixed point results in partially ordered metric spaces for a selfmapping g and a sequence {\{{T}_{i}\}}_{n\in \mathbb{N}} of trivariate selfmapping that have some useful properties.
The tripled fixed point theorems we deduce are motivated by the possibilities of solving simultaneous nonlinear equations of the above type.
Now, we collect basic definitions and results regarding coupled and tripled point theory.
Definition 1.1 (see [16])
An element (x,y)\in X\times X is called a coupled fixed point of the mapping F:X\times X\to X if F(x,y)=x and F(y,x)=y.
Definition 1.2 (see [17])
An element (x,y)\in X\times X is called a coupled coincidence point of the mappings F:X\times X\to X and g:X\to X if F(x,y)=g(x) and F(y,x)=g(y). In this case, (g(x),g(y)) is called a coupled point of coincidence.
Let (X,\u2aaf) be a partially ordered set and d be a metric on X such that (X,d) is a complete metric space. Consider the product X\times X\times X with the following partial order: for (x,y,z),(u,v,w)\in X\times X\times X,
Definition 1.3 (see [18])
Let (X,\u2aaf) be a partially ordered set and F:X\times X\times X\to X. We say that F has the mixed monotone property if F(x,y,z) is monotone nondecreasing in x and z and is monotone nonincreasing in y, that is, for any x,y,z\in X
Definition 1.4 (see [18])
We call an element (x,y,z)\in X\times X\times X a tripled fixed point of mapping F:X\times X\times X\to X if
Definition 1.5 (see [18])
Let (X,d) be a complete metric space. It is called metric on X\times X\times X, the mapping d:X\times X\times X\to X with
Akin to the concept of gmixed monotone property [17] for a bivariate mapping, F:X\times X\to X and a selfmapping, g:X\to X, Borcut and Berinde [28] introduced the concept of gmixed monotone property for a trivariate mapping F:X\times X\times X\to X and a selfmapping, g:X\to X in the following way.
Definition 1.6 (see [28])
Let (X,\u2aaf) be a partially ordered set and F:X\times X\times X\to X and g:X\to X. We say that F has the gmixed monotone property if F(x,y,z) is monotone nondecreasing in x and z, and if it is monotone nonincreasing in y, that is, for any x,y,z\in X,
Now, we introduce the concept of compatible mapping for a trivariate mapping F and a selfmapping g akin to compatible mapping as introduced by Choudhary and Kundu [11] for a bivariate mapping F and a selfmapping g.
Definition 1.7 Let mapping F and g where F:X\times X\times X\to X and g:X\to X are said to be compatible if
whenever \{{x}_{n}\}, \{{y}_{n}\}, and \{{z}_{n}\} are sequences in X, such that
and
for all x,y,z\in X.
Definition 1.8 The mappings F:X\times X\times X\to X and g:X\to X are called:

(i)
Reciprocally continuous if
\begin{array}{c}\underset{n\to +\mathrm{\infty}}{lim}g(F({x}_{n},{y}_{n},{z}_{n}))=g(x)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{n\to +\mathrm{\infty}}{lim}F(g({x}_{n}),g({y}_{n}),g({z}_{n}))=F(x,y,z),\hfill \\ \underset{n\to +\mathrm{\infty}}{lim}g(F({y}_{n},{x}_{n},{y}_{n}))=g(y)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{n\to +\mathrm{\infty}}{lim}F(g({y}_{n}),g({x}_{n}),g({y}_{n}))=F(y,x,y)\hfill \end{array}
and
whenever \{{x}_{n}\}, \{{y}_{n}\} and \{{z}_{n}\} are sequences in X, such that
and
for some x,y,z\in X.

(ii)
Weakly reciprocally continuous if
\begin{array}{c}\underset{n\to +\mathrm{\infty}}{lim}g(F({x}_{n},{y}_{n},{z}_{n}))=g(x)\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\underset{n\to +\mathrm{\infty}}{lim}F(g({x}_{n}),g({y}_{n}),g({z}_{n}))=F(x,y,z),\hfill \\ \underset{n\to +\mathrm{\infty}}{lim}g(F({y}_{n},{x}_{n},{y}_{n}))=g(y)\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\underset{n\to +\mathrm{\infty}}{lim}F(g({y}_{n}),g({x}_{n}),g({y}_{n}))=F(y,x,y)\hfill \end{array}
and
whenever \{{x}_{n}\}, \{{y}_{n}\} and \{{z}_{n}\} are sequences in X, such that
and
for some x,y,z\in X.
Definition 1.9 Let (X,d,\u2aaf) be a partially ordered metric space. We say that X is regular if the following conditions hold:

(i)
if a nondecreasing sequence \{{x}_{n}\} is such that {x}_{n}\to x, then {x}_{n}\u2aafx for all n\ge 0,

(ii)
if a nonincreasing sequence \{{y}_{n}\} is such that {y}_{n}\to y, then y\u2aaf{y}_{n} for all n\ge 0.
Definition 1.10 (see [8])
Let \{{a}_{n}\} be a sequence of nonnegative real numbers. We say that a series {\sum}_{n=1}^{+\mathrm{\infty}}{a}_{n} is an αseries, if there exist 0<\alpha <1 and {n}_{\alpha}\in \mathbb{N} such that {\sum}_{i=1}^{k}{a}_{i}\le \alpha k for each k\ge {n}_{\alpha}.
Remark 1.1 (see [8])
Each convergent series of nonnegative real terms is an αseries. However, there are also divergent series that are αseries. For example, {\sum}_{n=1}^{+\mathrm{\infty}}\frac{1}{n} is an αseries.
2 Main results
Let (X,\u2aaf) be a partially ordered set, g be a selfmapping on X and {\{{T}_{i}\}}_{i\in \mathbb{N}} be a sequence of mappings from X\times X\times X into X such that {T}_{i}(X\times X\times X)\subseteq g(X) and
for x,y,z,u,v,w\in X with g(x)\u2aafg(u), g(v)\u2aafg(y) and g(z)\u2aafg(w).
In the proof of our main theorem, we consider sequences that are constructed in the following way.
Let {x}_{0},{y}_{0},{z}_{0}\in X be such that g({x}_{0})\u2aaf{T}_{0}({x}_{0},{y}_{0},{z}_{0}), g({y}_{0})\u2ab0{T}_{0}({y}_{0},{x}_{0},{y}_{0}) and g({z}_{0})\u2aaf{T}_{0}({z}_{0},{y}_{0},{x}_{0}). Since {T}_{0}(X\times X\times X)\subseteq g(X), we can choose {x}_{1},{y}_{1},{z}_{1}\in X such that g({x}_{1})={T}_{0}({x}_{0},{y}_{0},{z}_{0}), g({y}_{2})={T}_{0}({y}_{0},{x}_{0},{y}_{0}) and g({z}_{2})={T}_{0}({z}_{0},{y}_{0},{x}_{0}). Again we can choose {x}_{2},{y}_{2},{z}_{2}\in X such that g({x}_{2})={T}_{1}({x}_{1},{y}_{1},{z}_{1}), g({y}_{2})={T}_{1}({y}_{1},{x}_{1},{y}_{1}) and g({z}_{2})={T}_{1}({z}_{1},{y}_{1},{x}_{1}). Continuing like this, we can construct three sequences \{{x}_{n}\}, \{{y}_{n}\}, and \{{x}_{n}\} such that
for all n\ge 0.
Now, by using mathematical induction, we prove that
for all n\ge 0. Since g({x}_{0})\u2aaf{T}_{0}({x}_{0},{y}_{0},{z}_{0}), g({y}_{0})\u2ab0{T}_{0}({y}_{0},{x}_{0},{y}_{0}) and g({z}_{0})\u2aaf{T}_{0}({z}_{0},{y}_{0},{x}_{0}), in view of g({x}_{1})={T}_{0}({x}_{0},{y}_{0},{z}_{0}), g({y}_{1})={T}_{0}({y}_{0},{x}_{0},{z}_{0}) and g({z}_{1})={T}_{0}({z}_{0},{y}_{0},{x}_{0}), we have g({x}_{0})\u2aafg({x}_{1}), g({y}_{0})\u2ab0g({y}_{1}), g({z}_{0})\u2aafg({z}_{1}), that is, (3) holds for n=0. We presume that (3) holds for some n>0. Now, by (2) and (3), one deduces that
and
Thus by mathematical induction, we conclude that (3) holds for all n\ge 0. Therefore, we have
and
In view of the above considerations, we revise Definitions 1.7 and 1.8 as follows.
Definition 2.1 Let (X,d) be a metric space. {\{{T}_{i}\}}_{i\in \mathbb{N}} and g are compatible if
and
whenever \{{x}_{n}\}, \{{y}_{n}\} and \{{z}_{n}\} are sequences in X, such that
and
for some x,y,z\in X.
Definition 2.2 {\{{T}_{i}\}}_{i\in \mathbb{N}} and g are called weakly reciprocally continuous if
and
whenever \{{x}_{n}\}, \{{y}_{n}\}, and \{{z}_{n}\} are sequences in X, such that
and
for some x,y,z\in X.
Now, we establish the main result of this manuscript as follows.
Theorem 2.1 Let (X,d,\u2aaf) be a partially ordered metric space. Let g be a selfmapping on X and {\{{T}_{i}\}}_{i\in \mathbb{N}} be a sequence of mappings from X\times X\times X into X such that {T}_{i}(X\times X\times X)\subseteq g(X), g(X) is a complete subset of X, {\{{T}_{i}\}}_{i\in \mathbb{N}} and g are compatible, weakly reciprocally continuous, g is monotonic nondecreasing, continuous, satisfying condition (1) and the following condition:
for x,y,z,u,v,w\in X with g(x)\u2aafg(u), g(v)\u2aafg(y), g(z)\u2aafg(w) or g(x)\u2ab0g(u), g(v)\u2ab0g(y), g(z)\u2ab0g(w); 0\le {\beta}_{i,j},{\gamma}_{i,j}<1 for i,j\in N; {lim}_{n\to +\mathrm{\infty}}sup{\beta}_{i,n}<1. Suppose also that there exists ({x}_{0},{y}_{0},{z}_{0})\in X\times X\times X such that g({x}_{0})\u2aaf{T}_{0}({x}_{0},{y}_{0},{z}_{0}), g({y}_{0})\u2ab0{T}_{0}({y}_{0},{x}_{0},{y}_{0}) and g({z}_{0})\u2aaf{T}_{0}({z}_{0},{y}_{0},{x}_{0}). If {\sum}_{i=1}^{+\mathrm{\infty}}(\frac{{\beta}_{i,i+1}+{\gamma}_{i,i+1}}{1{\beta}_{i,i+1}}) is an αseries and g(X) is regular, then {\{{T}_{i}\}}_{i\in \mathbb{N}} and g have a tripled coincidence point, that is, there exists (x,y,z)\in X\times X\times X such that g(x)={T}_{i}(x,y,z), g(y)={T}_{i}(y,x,y), and g(z)={T}_{i}(z,y,x) for i\in \mathbb{N}.
Proof We consider the sequences \{{x}_{n}\}, \{{y}_{n}\}, and \{{z}_{n}\} constructed above and denote {\delta}_{n}=d(g({x}_{n}),g({x}_{n+1}))+d(g({y}_{n}),g({y}_{n+1}))+d(g({z}_{n}),g({z}_{n+1})). Then, by (4), we get
It follows that
or, equivalently,
Also, one obtains
Repeating the above procedure, we have
Using similar arguments as above, one can also show that
and
Adding (5), (6), and (7), we have
Moreover, for p>0 and by repeated use of the triangle inequality, one obtains
Let α and {n}_{\alpha} be as in Definition 1.10, then, for n\ge {n}_{\alpha}, and using the fact that the geometric mean of nonnegative numbers is less than or equal to the arithmetic mean, it follows that
Now, taking the limit as n\to +\mathrm{\infty}, one deduces that
which further implies that
Thus \{g({x}_{n})\}, \{g({y}_{n})\} and \{g({z}_{n})\} are Cauchy sequences in X. Since g(X) is complete, then there exists (r,s,t)\in X\times X\times X, with g(r)=x, g(s)=y and g(t)=z, such that
and
Now, as {\{{T}_{i}\}}_{i\in \mathbb{N}} and g are weakly reciprocally continuous, we have
and
On the other hand, the compatibility of {\{{T}_{i}\}}_{i\in \mathbb{N}} and g yields
and
Then we have
and
Since \{g({x}_{n})\} and \{g({z}_{n})\} are nondecreasing and \{g({y}_{n})\} is nonincreasing, using the regularity of X, we have g({x}_{n})\u2aafx, y\u2aafg({y}_{n}) and g({z}_{n})\u2aafz for all n\ge 0. Then by (4), one obtains
□
Taking the limit as n\to +\mathrm{\infty}, we obtain {T}_{i}(x,y,z)=g(x) as {\beta}_{i,n}<1. Similarly, it can be proved that g(y)={T}_{i}(y,x,y) and g(z)={T}_{i}(z,y,x). Thus, (x,y,z) is a tripled coincidence point of {\{{T}_{i}\}}_{i\in \mathbb{N}} and g.
Now, we give useful conditions for the existence and uniqueness of a tripled common fixed point.
Theorem 2.2 In addition to the hypotheses of Theorem 2.1, suppose that the set of coincidence points is comparable with respect to g, then {\{{T}_{i}\}}_{i\in \mathbb{N}} and g have a unique tripled common fixed point, that is, there exists (x,y,z)\in X\times X\times X such that x=g(x)={T}_{i}(x,y,z), y=g(y)={T}_{i}(y,x,y), and z=g(z)={T}_{i}(z,y,x) for i\in \mathbb{N}.
Proof From Theorem 2.1, the set of tripled coincidence points is nonempty. Now, we show that if (x,y,z) and (r,s,t) are tripled coincidence points, that is, if g(x)={T}_{i}(x,y,z), g(y)={T}_{i}(y,x,y), g(z)={T}_{i}(z,y,x), g(r)={T}_{i}(r,s,t), g(s)={T}_{i}(s,r,s), and g(t)={T}_{i}(t,s,r), then g(x)=g(r), g(y)=g(s) and g(z)=g(t). Since the set of coincidence points is comparable, applying condition (4) to these points, we get
and so as {\gamma}_{i,j}<1, it follows that d(g(x),g(r))=0, that is, g(x)=g(r). Similarly, it can be proved that g(y)=g(s) and g(z)=g(t). Hence, {\{{T}_{i}\}}_{i\in \mathbb{N}} and g have a unique tripled point of coincidence. It is well known that two compatible mappings are also weakly compatible, that is, they commute at their coincidence points. Thus, it is clear that {\{{T}_{i}\}}_{i\in \mathbb{N}} and g have a unique tripled common fixed point whenever {\{{T}_{i}\}}_{i\in \mathbb{N}} and g are weakly compatible. This finishes the proof. □
If g is the identity mapping, as a consequence of Theorem 2.1, we state the following corollary.
Corollary 2.3 Let (X,d,\u2aaf) be a complete partially ordered metric space. Let {\{{T}_{i}\}}_{i\in \mathbb{N}} be a sequence of mappings from X\times X\times X into X such that {\{{T}_{i}\}}_{i\in \mathbb{N}} satisfies, for x,y,z,u,v,w\in X, with x\u2aafu, v\u2aafy, z\u2aafw or u\u2aafx, y\u2aafv, and w\u2aafz, the following conditions:

(i)
{T}_{n}(x,y,z)\u2aaf{T}_{n+1}(u,v,w),

(ii)
d({T}_{i}(x,y,z),{T}_{j}(u,v,w))\le {\beta}_{i,j}[d(x,{T}_{i}(x,y,z))+d(u,{T}_{j}(u,v,w))]+{\gamma}_{i,j}d(u,x), with 0\le {\beta}_{i,j},{\gamma}_{i,j}<1 and i,j\in \mathbb{N}.
Suppose also that there exists ({x}_{0},{y}_{0},{z}_{0})\in X\times X\times X such that {x}_{0}\u2aaf{T}_{0}({x}_{0},{y}_{0},{z}_{0}), {y}_{0}\u2ab0{T}_{0}({y}_{0},{x}_{0},{y}_{0}) and {z}_{0}\u2aaf{T}_{0}({z}_{0},{y}_{0},{z}_{0}). If {\sum}_{i=1}^{+\mathrm{\infty}}(\frac{{\beta}_{i,i+1}+{\gamma}_{i,i+1}}{1{\beta}_{i,i+1}}) is an αseries and X is regular, then {\{{T}_{i}\}}_{i\in \mathbb{N}} has a tripled fixed point, that is, there exists (x,y,z)\in X\times X\times X such that x={T}_{i}(x,y,z), y={T}_{i}(y,x,y) and z={T}_{i}(z,y,x), for i\in N.
Example 2.3 Take X=[0,1] endowed with usual metric d=xy for all x,y\in X and ⪯ be defined as ‘greater than/equal to’ the (X,d,\u2aaf) be partial order metric space. Let {T}_{i}:{X}^{3}\to X be mapping defined as {T}_{i}(x,y,z)=\frac{x+y+z}{3i}; i\in \mathbb{N} and g is selfmapping defined as g(x)=x.
Clearly, {T}_{i}(x,y,z)\subseteq g(X), g(X) is a complete subset of X.
By choosing the sequences \{{x}_{n}\}=\frac{1}{n}, \{{y}_{n}\}=\frac{1}{n+1} and \{{z}_{n}\}=\frac{1}{n+2}, one can easily observe that {\{{T}_{i}\}}_{i\in \mathbb{N}} and g are compatible, weakly reciprocally continuous; g is monotonic nondecreasing, continuous, as well as satisfying condition (1).
Again by taking 0<{\beta}_{i,j}<1 and 0\le {\gamma}_{i,j}<1, it is easy to check inequality (4) holds, thus all the hypotheses of Theorem 2.1 are satisfied and (0,0,0), (1,1,1) are the tripled coincident points of g and {T}_{i}. Moreover, using the same {T}_{i} and g in Theorem 2.2, (0,0,0) is the unique fixed point of g and {T}_{i}.
Remark 2.1 Open problem: In this paper, we prove tripled fixed point results. The idea can be extended to multidimensional cases. But the technicalities in the proofs therein will be different. We consider this as an open problem.
References
Agarwal RP, Meehan M, O’Regan D: Fixed Point Theory and Application. Cambridge University Press, Cambridge; 2001.
Rhoades BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1977,226(2):257–290.
Luong NV, Thuan NX: Coupled fixed point theorems in partially ordered metric spaces. Bull. Math. Anal. Appl. 2010, 2: 16–24.
Nieto JJ, RodriguezLopez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation. Order 2005, 22: 223–239. 10.1007/s1108300590185
Nieto JJ, RodriguezLopez R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007,23(12):2205–2212. 10.1007/s1011400507690
Paesano D, Vetro P: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl. 2012,159(3):911–920. 10.1016/j.topol.2011.12.008
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002993903072204
Sihag V, Vetro C, Vats RK: A fixed point theorem in G metric spaces via α series. Quaest. Math. 2014, 37: 1–6. 10.2989/16073606.2013.779961
Chang SS, Ma YH: Coupled fixed point for mixed monotone condensing operators and an existence theorem of the solutions for a class of functional equations arising in dynamic programming. J. Math. Anal. Appl. 1991, 160: 468–479. 10.1016/0022247X(91)90319U
Agarwal RP, Kadelburg Z, Radenovic S: On coupled fixed point results in asymmetric G metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 528
Choudhary BS, Kundu A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 2010, 73: 2524–2531. 10.1016/j.na.2010.06.025
Berinde V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2010, 74: 7347–7355.
Samet B: Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026
Karapinar E, Agarwal RP: Further fixed point results on G metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 154
Agarwal RP, Karapinar E: Remarks on some coupled fixed point theorems in G metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 2
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric space and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011,74(15):4889–4897. 10.1016/j.na.2011.03.032
Aydi H, Karapinar E, Shatanawi W: Tripled common fixed point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 101
Aydi H, Karapinar E, Vetroc C: MeirKeeler type contractions for tripled fixed points. Acta Math. Sci. 2012,32(6):2119–2130. 10.1016/S02529602(12)601647
Aydi H, Karapinar E, Shatanawi W: Tripled fixed point results in generalized metric spaces. J. Appl. Math. 2012., 2012: Article ID 314279
Aydi H, Karapinar E, Postolache M: Tripled coincidence point theorems for weak φ contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 44
Aydi H, Karapinar E, Radenovic S: Tripled coincidence fixed point results for BoydWong and Matkowski type contractions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. a Mat. 2013,107(2):339–353. 10.1007/s1339801200773
Aydi H, Abbas M, Sintunavarat W, Kumam P: Tripled fixed point of W compatible mappings in abstract metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 134
Abbas M, Ali B, Sintunavarat W, Kumam P: Tripled fixed point and tripled coincidence point theorems in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 187
Abbas M, Aydi H, Karapinar E: Tripled fixed points of multivalued nonlinear contraction mappings in partially ordered metric spaces. Abstr. Appl. Anal. 2011., 2011: Article ID 812690
Karapinar E, Aydi H, Mustafa Z: Some tripled coincidence point theorems for almost generalized contractions in ordered metric spaces. Tamkang J. Math. 2013,44(3):233–251.
Borcut M, Berinde V: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput. 2012,218(10):5929–5936. 10.1016/j.amc.2011.11.049
Borcut M: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput. 2012,218(14):7339–7346. 10.1016/j.amc.2012.01.030
Acknowledgements
The authors gratefully acknowledge the learned referees for providing a suggestion to improve the manuscript. The first author also acknowledges the Council of Scientific and Industrial Research, Government of India, for providing financial assistance under research project no. 25(0197)/11/EMRII.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Vats, R.K., Tas, K., Sihag, V. et al. Triple fixed point theorems via αseries in partially ordered metric spaces. J Inequal Appl 2014, 176 (2014). https://doi.org/10.1186/1029242X2014176
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2014176