A New Technique to Compute Coupled Coincidence Points

We compute coupled coincidence points without assuming the condition of compatibility of the pair of maps and relaxing the continuity condition of both the maps. In fact, our technique improves the technique introduced by Sintunavarat et al. (2011) which was then used by Hussain et al. (2012) to obtain coupled coincidence points.


Introduction and Preliminaries
In recent times, the study of common fixed points of mappings under contractive conditions has developed rapidly.The Banach contraction principle [1] is an important tool in nonlinear analysis for solving problems concerning fixed points.Different authors extended and generalized this principle in various spaces by using more general contractive conditions in different ways.References [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] are some examples of these works.Nowadays, fixed point theory has been receiving much attention in partially ordered metric spaces, that is, metric spaces endowed with a partial ordering.Ran and Reurings [22] were the first to establish the results in this direction.The results were then extended by Nieto and Rodríguez-López [23] for nondecreasing mappings.The work in [23] was illustrated by showing the existence of unique solution for a first-order ordinary differential equation with periodic boundary conditions.Works noted in [24][25][26][27][28][29][30] are some examples in this direction.
Bhaskar and Lakshmikantham [31] developed some coupled fixed point theorems for a mapping satisfying mixed monotone property in partially ordered metric spaces.As an application, they discussed the existence and uniqueness of solution for a periodic boundary value problem.Lakshmikantham and Ćiri ć [32] extended the notion of mixed monotone property to mixed -monotone property and generalized the results of Bhaskar and Lakshmikantham [31] by establishing the existence of coupled coincidence point results using a pair of commutative maps.Choudhury and Kundu [33] further generalized these results to a pair of compatible maps.Alotaibi and Alsulami [34] extended the results of Luong and Thuan [35] for a compatible pair.Recently, Haghi et al. [36] introduced a new technique that generalized several results present in the literature.This technique was further extended by Sintunavarat et al. [37] to obtain coupled coincidence points of mappings satisfying contractive conditions without the need for commutative condition in intuitionistic fuzzy normed spaces, which was then used by Hussain et al. [38] to generalize the results noted in [32][33][34] by replacing the assumption of compatibility (and hence of commutativity) and completeness of the space  by assuming the completeness of the range subspace of the map .
The purpose of this paper is to provide a technique that generalizes and improves the technique introduced by Sintunavarat et al. [37] and then used by Hussain et al. [38], in the sense that we are not first proving any result for a single mapping and then extending the obtained result for a pair of maps; rather, we give direct proof to obtain coupled coincidence points.In order to produce and compare our technique with the technique used by Hussain et al. [38], we use the same contractive conditions used by Hussain et al. [38] in their Theorems 2.9 and 2.11.Our technique is based on the iteration argument and provides a tool to generalize the coupled coincidence point results under different contractive conditions present in the literature of fixed point theory.

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Chinese Journal of Mathematics We also provide an example that illustrates that the assumption of continuity of the mapping  :  →  in coupled coincidence point results is not necessary.Now we give the following definitions which are useful in our study.
Definition 6 (see [33]).The mappings  :  ×  →  and  :  →  are said to be compatible if whenever {  } and Now we state a lemma which will be an important and powerful tool for us in computing the coupled coincidence points without assuming the condition of commutativity or minimal commutativity of the pair of maps and assuming subspace to be complete instead of completeness of the space.
Lemma 7 (see [36]).Let X be a nonempty set and  :  →  a mapping.Then there exists a subset  ⊆  such that () = () and the mapping  :  →  is one-to-one.

Main Results
We now give our technique and improve Theorem 2.9 proved in [38].
Thus we proved that  and  have a coupled coincidence point in .This completes the proof.
In order to improve Theorem 2.11 proved in [38], we need the following.
Then, as in the proof of Theorem 3.1 [34], it is easy to show that the sequences {  } and {  } are Cauchy sequences.Since () is complete, there exist ,  ∈  such that lim We now show that  = (, ) and  = (, ).Suppose that assumption (a) holds.Now, using Lemma 7, there exists a subset  ⊆  such that () = () and the mapping  :  →  is one-toone.Let us define a mapping  : () × () →  by for all ,  ∈ () = ().

Remark 10. (i)
The technique of Sintunavarat et al. [37] used by Hussain et al. [38] first requires to prove the result for a single mapping and, then, extends the obtained result to a pair of mappings but our technique yields a direct method to compute coupled coincidence points for a pair of mappings.
(ii) Case (b) of Theorems 8 and 9 proved in this paper not only relaxes the continuity assumption of the mapping  but also relaxes the continuity of the mapping  which has not been relaxed in Case (b) of Theorems 2.9 and 2.11 of Hussain et al. [38].
In view of this discussion, we can conclude that our technique improves the technique used in [37,38].
Remark 13.Theorems 2.9 and 2.11 in [38] cannot be applied to Example 12 since  is not continuous but using Theorems 8 and 9 we obtained coupled coincidence points under the same contractive conditions as used in [38,Theorems 2.9 and 2.11].This shows that the results presented in this paper are true generalizations of the results in [38].