Abstract
This manuscript has two aims: first we extend the definitions of compatibility and weakly reciprocally continuity, for a trivariate mapping F and a self-mapping g akin to a compatible mapping as introduced by Choudhary and Kundu (Nonlinear Anal. 73:2524-2531, 2010) for a bivariate mapping F and a self-mapping g. Further, using these definitions we establish tripled coincidence and fixed point results by applying the new concept of an α-series for sequence of mappings, introduced by Sihag et al. (Quaest. Math. 37:1-6, 2014), in the setting of partially ordered metric spaces.
MSC:54H25, 47H10, 54E50.
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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 α-series to give a common fixed point theorem for a sequence of self-mappings. 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 , 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 self-mapping g and a sequence of trivariate self-mapping 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 is called a coupled fixed point of the mapping if and .
Definition 1.2 (see [17])
An element is called a coupled coincidence point of the mappings and if and . In this case, is called a coupled point of coincidence.
Let be a partially ordered set and d be a metric on X such that is a complete metric space. Consider the product with the following partial order: for ,
Definition 1.3 (see [18])
Let be a partially ordered set and . We say that F has the mixed monotone property if is monotone non-decreasing in x and z and is monotone non-increasing in y, that is, for any
Definition 1.4 (see [18])
We call an element a tripled fixed point of mapping if
Definition 1.5 (see [18])
Let be a complete metric space. It is called metric on , the mapping with
Akin to the concept of g-mixed monotone property [17] for a bivariate mapping, and a self-mapping, , Borcut and Berinde [28] introduced the concept of g-mixed monotone property for a trivariate mapping and a self-mapping, in the following way.
Definition 1.6 (see [28])
Let be a partially ordered set and and . We say that F has the g-mixed monotone property if is monotone non-decreasing in x and z, and if it is monotone non-increasing in y, that is, for any ,
Now, we introduce the concept of compatible mapping for a trivariate mapping F and a self-mapping g akin to compatible mapping as introduced by Choudhary and Kundu [11] for a bivariate mapping F and a self-mapping g.
Definition 1.7 Let mapping F and g where and are said to be compatible if
whenever , , and are sequences in X, such that
and
for all .
Definition 1.8 The mappings and are called:
-
(i)
Reciprocally continuous if
and
whenever , and are sequences in X, such that
and
for some .
-
(ii)
Weakly reciprocally continuous if
and
whenever , and are sequences in X, such that
and
for some .
Definition 1.9 Let be a partially ordered metric space. We say that X is regular if the following conditions hold:
-
(i)
if a non-decreasing sequence is such that , then for all ,
-
(ii)
if a non-increasing sequence is such that , then for all .
Definition 1.10 (see [8])
Let be a sequence of non-negative real numbers. We say that a series is an α-series, if there exist and such that for each .
Remark 1.1 (see [8])
Each convergent series of non-negative real terms is an α-series. However, there are also divergent series that are α-series. For example, is an α-series.
2 Main results
Let be a partially ordered set, g be a self-mapping on X and be a sequence of mappings from into X such that and
for with , and .
In the proof of our main theorem, we consider sequences that are constructed in the following way.
Let be such that , and . Since , we can choose such that , and . Again we can choose such that , and . Continuing like this, we can construct three sequences , , and such that
for all .
Now, by using mathematical induction, we prove that
for all . Since , and , in view of , and , we have , , , that is, (3) holds for . We presume that (3) holds for some . Now, by (2) and (3), one deduces that
and
Thus by mathematical induction, we conclude that (3) holds for all . Therefore, we have
and
In view of the above considerations, we revise Definitions 1.7 and 1.8 as follows.
Definition 2.1 Let be a metric space. and g are compatible if
and
whenever , and are sequences in X, such that
and
for some .
Definition 2.2 and g are called weakly reciprocally continuous if
and
whenever , , and are sequences in X, such that
and
for some .
Now, we establish the main result of this manuscript as follows.
Theorem 2.1 Let be a partially ordered metric space. Let g be a self-mapping on X and be a sequence of mappings from into X such that , is a complete subset of X, and g are compatible, weakly reciprocally continuous, g is monotonic non-decreasing, continuous, satisfying condition (1) and the following condition:
for with , , or , , ; for ; . Suppose also that there exists such that , and . If is an α-series and is regular, then and g have a tripled coincidence point, that is, there exists such that , , and for .
Proof We consider the sequences , , and constructed above and denote . 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 and by repeated use of the triangle inequality, one obtains
Let α and be as in Definition 1.10, then, for , and using the fact that the geometric mean of non-negative numbers is less than or equal to the arithmetic mean, it follows that
Now, taking the limit as , one deduces that
which further implies that
Thus , and are Cauchy sequences in X. Since is complete, then there exists , with , and , such that
and
Now, as and g are weakly reciprocally continuous, we have
and
On the other hand, the compatibility of and g yields
and
Then we have
and
Since and are non-decreasing and is non-increasing, using the regularity of X, we have , and for all . Then by (4), one obtains
□
Taking the limit as , we obtain as . Similarly, it can be proved that and . Thus, is a tripled coincidence point of 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 and g have a unique tripled common fixed point, that is, there exists such that , , and for .
Proof From Theorem 2.1, the set of tripled coincidence points is non-empty. Now, we show that if and are tripled coincidence points, that is, if , , , , , and , then , and . Since the set of coincidence points is comparable, applying condition (4) to these points, we get
and so as , it follows that , that is, . Similarly, it can be proved that and . Hence, 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 and g have a unique tripled common fixed point whenever 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 be a complete partially ordered metric space. Let be a sequence of mappings from into X such that satisfies, for , with , , or , , and , the following conditions:
-
(i)
,
-
(ii)
, with and .
Suppose also that there exists such that , and . If is an α-series and X is regular, then has a tripled fixed point, that is, there exists such that , and , for .
Example 2.3 Take endowed with usual metric for all and ⪯ be defined as ‘greater than/equal to’ the be partial order metric space. Let be mapping defined as ; and g is self-mapping defined as .
Clearly, , is a complete subset of X.
By choosing the sequences , and , one can easily observe that and g are compatible, weakly reciprocally continuous; g is monotonic non-decreasing, continuous, as well as satisfying condition (1).
Again by taking and , it is easy to check inequality (4) holds, thus all the hypotheses of Theorem 2.1 are satisfied and , are the tripled coincident points of g and . Moreover, using the same and g in Theorem 2.2, is the unique fixed point of g and .
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.
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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/EMR-II.
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Vats, R.K., Tas, K., Sihag, V. et al. Triple fixed point theorems via α-series in partially ordered metric spaces. J Inequal Appl 2014, 176 (2014). https://doi.org/10.1186/1029-242X-2014-176
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DOI: https://doi.org/10.1186/1029-242X-2014-176