On Strong Coupled Coincidence Points of g-Couplings and an Application

The concept of a coupled fixed point was first introduced by Guo et al. [1]. Later, many coupled fixed and coupled coincidence point results were given. For more details, see [2–22]. Kirk et al. [23] gave the concept of cyclic mappings. Recently, Choudhury et al. [24] introduced the concept of couplings, which are actually coupled cyclic mappings with respect to two given subsets of a metric space. In [24], they also proved the existence of strong coupled fixed points for Banach and Chatterjea couplings. Very recently, Aydi et al. [25, 26] proved some existence and uniqueness results of a strong coupled fixed point for nonlinear couplings in (partial) metric spaces. In this paper, we extend the concept of the Banach (resp., Chatterjea) type coupling to the Banach (resp., Chatterjea) type g-couplings. We generalize the results of Choudhury et al. [24]. First, we recall some known definitions.


Introduction and Preliminaries
The concept of a coupled fixed point was first introduced by Guo et al. [1].Later, many coupled fixed and coupled coincidence point results were given.For more details, see .Kirk et al. [23] gave the concept of cyclic mappings.Recently, Choudhury et al. [24] introduced the concept of couplings, which are actually coupled cyclic mappings with respect to two given subsets of a metric space.In [24], they also proved the existence of strong coupled fixed points for Banach and Chatterjea couplings.Very recently, Aydi et al. [25,26] proved some existence and uniqueness results of a strong coupled fixed point for nonlinear couplings in (partial) metric spaces.In this paper, we extend the concept of the Banach (resp., Chatterjea) type coupling to the Banach (resp., Chatterjea) type -couplings.We generalize the results of Choudhury et al. [24].
First, we recall some known definitions.
Definition 2 (strong coupled fixed point, [28]).Let  be a nonempty set.An element (, ) ∈  ×  is called a strong coupled fixed point of the mapping  :  ×  →  if (, ) is a coupled fixed point and  = , that is, if (, ) = .

Main Results
First of all, we introduce some definitions.
Now, we give an example illustrating the concept of couplings.
This shows that  is a -coupling with respect to  and .
Remark 12. Every -coupling is a coupling.For convenience, let  : × →  be a -coupling.Then by Definition 14, we have (, ) ∈ (() ∩ ) ⊂   (, ) ∈ (() ∩ ) ⊂ .Then  is a coupling (with respect to  and ).But, the converse is not true in general.Now, we give an example which shows that every coupling need not be a -coupling.
We present the following examples.

Now, we
give some examples where Theorem 15 works, but the result of Choudhury et al. [24] is not applicable.
Definition 22.Let  and  be any two nonempty subsets of a metric space (, ) and  :  →  be a self-mapping.Let  :  ×  →  be a -coupling with respect to  and .Then  is said to be a Chatterjea type -coupling with respect to  and , if it satisfies  ( (, ) ,  (, V)) ≤  [ ( () ,  (, )) +  ( (V) ,  (, V))] (60) where , V ∈  and ,  ∈ , where  ∈ (0, 1/2 Proceeding similarly to the proof of Theorem 15, we get that (, ) = () and (, ) = ().Hence (, ) is a coupled coincidence point of  and  in  × .Again, if  is one to one, it is obvious that (, ) is the unique strong coupled coincidence point of  and .Now, we report some examples where again the result of Choudhury et al. [24] can not be applied.