- Open Access
Coupled coincidence point theorems for α-ψ-contractive type mappings in partially ordered metric spaces
© Kaushik et al.; licensee Springer. 2013
Received: 19 March 2013
Accepted: 22 October 2013
Published: 28 November 2013
In this paper, we introduce the notion of α-ψ-contractions and -admissibility for a pair of mappings. In fact, our theorem is a generalization of the result of Mursaleen et al. (Fixed Point Theory Appl. 2012, doi:10.1186/1687-1812-2012-228). At the end, we shall provide an example in support of our main result.
MSC:47H10, 54H25, 34B15.
Fixed point problems of contractive mappings in metric spaces endowed with a partial order have been studied in a number of works. Bhaskar and Lakshmikantham  established coupled fixed point results for mixed monotone operators in partially ordered metric spaces. Afterwards, Lakshmikantham and Ćirić  proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces. Choudhury and Kundu , Ćirić et al. , Luong and Thuan , Nieto and López [5, 6], Ran and Reurings  and Samet  presented some new results for contractions in partially ordered metric spaces. Ilić and Rakočević  determined some common fixed point theorems by considering the maps on cone metric spaces. For more details on fixed point theory and related concepts, we refer to [4, 8–18] and the references therein.
The aim of this paper is to determine some coupled coincidence point theorems for generalized contractive mappings in the framework of partially ordered metric spaces.
Before proceeding to our main result, we give some preliminaries.
Definition 2.1 
Definition 2.2 
Definition 2.3 
Definition 2.4 
Definition 2.5 
Let X be a non-empty set and and . We say F and g are commutative if .
Definition 2.6 
for all , and
for all .
Lemma 2.7 
If is non-decreasing and right continuous, then as for all if and only if for all .
Definition 2.8 
Definition 2.9 
Now, we will introduce our notions.
for with and .
Definition 2.11 Let , and be mappings.
3 Main results
Recently, Mursaleen et al.  proved the coupled fixed point theorem with α-ψ-contractive conditions in a partially ordered metric space as follows.
Theorem 3.1 
F is -admissible.
- (ii)There exist such that
F is continuous.
for all with and .
F and g are -admissible.
- (ii)There exist such that
, g is continuous and commutes with F.
F is continuous.
Let be such that and .
Thus (3.2) holds for . Now suppose that (3.2) holds for some fixed .
Thus, by mathematical induction, we conclude that (3.2) holds for all .
then, obviously, and , i.e., F has a coupled coincidence point.
Now, we assume that for all .
hence and are Cauchy sequences in .
Since is complete, therefore, and are convergent in .
There exist such that and .
Hence, we have proved that F and g have a coupled coincidence point. □
Now, we will replace the continuity of F in Theorem 3.2 by a condition on sequences.
conditions (i), (ii) and (iii) of Theorem 3.2 and (3.1),
- (ii)if and are sequences in X such that
Proof Proceeding along the same lines as in the proof of Theorem 3.2, we know that and are Cauchy sequences in the complete metric space . Then there exist such that and .
Thus, and . □
Example 3.4 Let and be a standard metric.
Define a mapping by and by for all .
Thus (3.1) holds for for all , and we also see that F satisfies the g-mixed monotone property as well as F and g commute. Therefore, all the hypotheses of Theorem 3.2 are fulfilled. Then there exists a coupled coincidence point of F and g. In this case, is a coupled coincidence point of F and g.
One of the authors (Sanjay Kumar) is thankful to UGC for providing Major Research Project F.No. 39-41/2010(SR). Also, the authors thank the editors and referees for their insightful comments. Moreover, the third author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support under grant No. NRU56000508.
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