- Open Access
Triple fixed point theorems via α-series in partially ordered metric spaces
© Vats et al.; licensee Springer. 2014
Received: 29 November 2013
Accepted: 23 April 2014
Published: 12 May 2014
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.
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 . In fact, many studies have been done on contractive mappings, e.g., Rhoades  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.  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 . This concept has been of interest to many researchers in metrical fixed point theory (see for example [3, 10–15]). Recently, Bhaskar and Lakshmikantham  established coupled fixed point theorems for a mixed monotone operator in partially ordered metric spaces. Afterward, Lakshmikantham and Ćirić  extended the results of  by furnishing coupled coincidence and coupled fixed point theorems for two commuting mappings.
In a subsequent series, Berinde and Borcut , 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 )
An element is called a coupled fixed point of the mapping if and .
Definition 1.2 (see )
An element is called a coupled coincidence point of the mappings and if and . In this case, is called a coupled point of coincidence.
Definition 1.3 (see )
Definition 1.4 (see )
Definition 1.5 (see )
Akin to the concept of g-mixed monotone property  for a bivariate mapping, and a self-mapping, , Borcut and Berinde  introduced the concept of g-mixed monotone property for a trivariate mapping and a self-mapping, in the following way.
Definition 1.6 (see )
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  for a bivariate mapping F and a self-mapping g.
for all .
- (i)Reciprocally continuous if
- (ii)Weakly reciprocally continuous if
for some .
if a non-decreasing sequence is such that , then for all ,
if a non-increasing sequence is such that , then for all .
Definition 1.10 (see )
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 )
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
for with , and .
In the proof of our main theorem, we consider sequences that are constructed in the following way.
for all .
In view of the above considerations, we revise Definitions 1.7 and 1.8 as follows.
for some .
for some .
Now, we establish the main result of this manuscript as follows.
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 .
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 .
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.
, 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.
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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