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Fixed point theorems of generalized cyclic orbital Meir-Keeler contractions
Fixed Point Theory and Applications volume 2013, Article number: 91 (2013)
Abstract
In this paper, we introduce two class of generalized cyclic orbital Meir-Keeler contractions and we study the existence and uniqueness of fixed points for these mappings. Our results in this paper extend and generalize several existing fixed-point theorems in the literature.
MSC:47H10, 54C60, 54H25, 55M20.
1 Introduction and preliminaries
Throughout this paper, by , we denote the set of all non-negative numbers, while â„• is the set of all natural numbers. It is well known and easy to prove that if is a complete metric space, and if is continuous and f satisfies
then f has a fixed point in X. Using the above conclusion, Kirk, Srinivasan and Veeramani [1] proved the following fixed-point theorem.
Theorem 1 [1]
Let A and B be two nonempty closed subsets of a complete metric space , and suppose satisfies
-
(i)
and ,
-
(ii)
for all , and .
Then is nonempty and f has a unique fixed point in .
The following definitions and results will be needed in the sequel. Let A and B be two nonempty subsets of a metric space . A mapping is called a cyclic map if and . In 2010, Karpagam and Agrawal [2] introduced the notion of cyclic orbital contraction, and obtained a unique fixed point theorem for such a map.
Definition 1 [2]
Let A and B be nonempty subsets of a metric space , be a cyclic map such that for some , there exists a such that
Then f is called a cyclic orbital contraction.
Theorem 2 [2]
Let A and B be two nonempty closed subsets of a complete metric space , and let be a cyclic orbital contraction. Then f has a fixed point in .
Further, many results dealing with cyclic contractions have appeared in the literature (see, e.g., [3–16]).
In 2012, Chen [17] introduced the below notion of cyclic orbital stronger Meir-Keeler contraction, and obtained a unique fixed-point theorem for such class of mappings.
Definition 2 [17]
Let be a metric space. We call a stronger Meir-Keeler type mapping in X if the mapping ψ satisfies the following condition:
Definition 3 [17]
Let A and B be nonempty subsets of a metric space . Suppose is a cyclic map such that for some , there exists a stronger Meir-Keeler type mapping in X such that
for all and . Then f is called a cyclic orbital stronger Meir-Keeler -contraction.
Clearly, if is a cyclic orbital contraction, then f is a cyclic orbital stronger Meir-Keeler -contraction, where for all .
Theorem 3 [17]
Let A and B be two nonempty closed subsets of a complete metric space , and let be a stronger Meir-Keeler type mapping in X. Suppose is a cyclic orbital stronger Meir-Keeler -contraction. Then is nonempty and f has a unique fixed point in .
Chen [17] also introduced the below notion of cyclic orbital weaker Meir-Keeler contraction, and obtained a unique fixed-point theorem for such class of mappings.
Definition 4 [17]
Let be a metric space, and . Then ψ is called a weaker Meir-Keeler type mapping in X, if the mapping ψ satisfies the following condition:
Definition 5 [17]
Let be a metric space. We call a ψ-mapping in X if the function f satisfies the following conditions:
() f is a weaker Meir-Keeler type mapping in X with ;
()
-
(a)
if , then , and
-
(b)
if , then ;
() is decreasing, for each .
Definition 6 [17]
Let A and B be nonempty subsets of a metric space . Suppose is a cyclic map such that for some , there exists a ψ-mapping in X such that
for all and . Then f is called a cyclic orbital weaker Meir-Keeler -contraction.
Theorem 4 [17]
Let A and B be two nonempty closed subsets of a complete metric space , and let be a ψ-mapping in X. Suppose is a cyclic orbital weaker Meir-Keeler -contraction. Then is nonempty and f has a unique fixed point in .
2 Fixed-point theorems (I)
In this section, we will introduce the class of generalized cyclic orbital stronger Meir-Keeler -contraction and we study the existence and uniqueness of fixed points for such mappings. Our results in this section extend and generalize several existing fixed-point theorems in the literature, including Theorem 2 and Theorem 3.
In the sequel, we denote by Θ the class of functions satisfying the following conditions:
() φ is a strictly increasing, continuous function in each coordinate;
() for all , , , , , and .
Example 1 Let denote
Then φ satisfies the above conditions and .
We now denote the below notion of generalized cyclic orbital stronger Meir-Keeler -contraction.
Definition 7 Let A and B be nonempty subsets of a metric space . Suppose is a cyclic map such that for some , there exist a stronger Meir-Keeler type mapping in X and such that
where
for all and . Then f is called a generalized cyclic orbital stronger Meir-Keeler -contraction.
Our main result is the following.
Theorem 5 Let A and B be two nonempty closed subsets of a complete metric space , and let be a stronger Meir-Keeler type mapping in X and . Suppose is a generalized cyclic orbital stronger Meir-Keeler -contraction. Then is nonempty and f has a unique fixed point in .
Proof Since is a generalized cyclic orbital stronger Meir-Keeler -contraction and for , we have . Put , for . Then we have that for each

and by the conditions of the function φ, we get
and
Similarly, we put and for each

and by the conditions of the function φ, we get
and
Using inequalities (2.1) and (2.2), we deduce that is a decreasing sequence and hence it is convergent. Let . Then there exists and such that for all ,
Taking into account the above inequality and the definition of stronger Meir-Keeler type mapping in X, corresponding to η use, there exists such that
Put , where is the integer part of . It follows from (2.1), (2.2) and (2.3) that we deduce that for all ,
and
It follows from (2.4) and (2.5) that for each
Since , we get
For with , we have
and hence , since . So, is a Cauchy sequence. Since is a complete metric space, A and B are closed, , there exists such that . Now is a sequence in A and is a sequence in B, and also both converge to ν. Since A and B are closed, , and so is nonempty. Next, we want to show that ν is a fixed point of f. Suppose that ν is not a fixed point of f. Then . Since and
where
we obtain that
This leads to a contradiction. So, , that is, ν is a fixed point of f.
Finally, we want to show the uniqueness of the fixed point. Let μ be another fixed point of f. By the cyclic character of f, we have . Since f is a generalized cyclic orbital stronger Meir-Keeler -contraction, we have
and
where
It follows from (2.7), (2.8) and the condition of the mapping φ that
This leads to a contradiction. Therefore, , and so ν is the unique fixed point of f. □
We give the following example to illustrate Theorem 5.
Example 2 Let and we define by
and let denote
We next define by
and let denote
Then f is a generalized cyclic orbital stronger Meir-Keeler -contraction and 0 is the unique fixed point.
3 Fixed-point theorems (II)
In this section, we will introduce the class of generalized cyclic orbital weaker Meir-Keeler -contraction and we study the existence and uniqueness of fixed points for such mappings.
In the sequel, we denote by Φ the class of functions satisfying the following conditions:
() Ï• is lower semi-continuous, and
() if and only if .
Definition 8 Let A and B be nonempty subsets of a metric space . Suppose is a cyclic map such that for some , there exist a ψ-mapping in X and such that
Then f is called a generalized cyclic orbital weaker Meir-Keeler -contraction.
Our second main result is the following.
Theorem 6 Let A and B be two nonempty closed subsets of a complete metric space , and let be a ψ-mapping in X and . Suppose is a generalized cyclic orbital weaker Meir-Keeler -contraction. Then is nonempty and f has a unique fixed point in .
Proof Since is a generalized cyclic orbital weaker Meir-Keeler -contraction and for , there exist a ψ-mapping in X and such that (3.1) is satisfied. Put for all . Then we have that for each
and
Generally, we have that for each
and so we conclude that for each
Since is decreasing, it must converge to some . We claim that . On the contrary, assume that . Then by the definition of weaker Meir-Keeler type mapping in X, there exists such that for with , there exists such that . Since , there exists such that , for all . Thus, we conclude that , and we get a contradiction. So, , that is,
We now claim that is a Cauchy sequence. It is sufficient to show that is a Cauchy sequence. Suppose is not Cauchy. Then there exists such that for all , there are with satisfying:
-
(i)
, and
-
(ii)
is the smallest number greater than such that the condition (i) holds.
Using (3.2), we have
Let , we get
On the other hand, applying (3.1) with for all , we get
Since for each
and
taking and using the inequalities (3.3), (3.5) and (3.6), we get
and
Taking into account the inequalities (3.4), (3.7) and (3.8), and by the definitions of the functions Ï• and , we get
which implies that . Thus, is a Cauchy sequence.
Since is a complete metric space, A and B are closed, , there exists such that . Now is a sequence in A and is a sequence in B, and also both converge to ν. Since A and B are closed, , and so is nonempty. On the other hand, since and
taking ,we obtain that
and hence , that is, ν is a fixed point of f.
Finally, we want to show the uniqueness of the fixed point. Let μ be another fixed point of f. By the cyclic character of f, we have . Since f is a generalized cyclic orbital weaker Meir-Keeler -contraction, we have
Letting , and by the definitions of the functions Ï• and , we obtain that
which implies that . Therefore, , and so ν is the unique fixed point of f. □
We give the following example to illustrate Theorem 6.
Example 3 Let and we define by
Define by
and define by
Then f is a generalized cyclic orbital weaker Meir-Keeler -contraction and 0 is the unique fixed point.
References
Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4(1):79–89.
Karpagam S, Agrawal S: Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps. Nonlinear Anal. 2010, 74: 1040–1046.
Chen CM, Chen CH: Best proximity point theorems for some new cyclic mappings. J. Appl. Math. 2012., 2012: Article ID 643729. doi:10.1155/2012/643729
Chen CM, Lin CJ: Best periodic proximity points for cyclic weaker Meir-Keeler contractions. J. Appl. Math. 2012., 2012: Article ID 782389. doi:10.1155/2012/782389
Chen CM: Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 17
Karpagam S, Agrawal S: Existence of best proximity points of p -cyclic contractions. Fixed Point Theory 2012, 13(1):99–105.
Karapinar E: Fixed point theory for cyclic weak ϕ -contraction. Appl. Math. Lett. 2011, 24: 822–825. 10.1016/j.aml.2010.12.016
Karapinar E: Best proximity points of cyclic mappings. Appl. Math. Lett. 2012, 25: 1761–1766. 10.1016/j.aml.2012.02.008
Karapinar E, Erhan IM, Ulus AY: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci. 2012, 6(1):239–244.
Karapinar E, Sadarangani K:Fixed point theory for cyclic -contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69
Karapinar E, Erhan IM: Cyclic contractions and fixed point theorems. Filomat 2012, 26(4):777–782. 10.2298/FIL1204777K
Karapinar E, Petrusel G, Tas K: Best proximity point theorems for KT -types cyclic orbital contraction mappings. Fixed Point Theory 2012, 13(2):537–546.
Karapinar E, Jleli M, Samet B:Fixed point results for almost generalized cyclic -weak contractive type mappings with applications. Abstr. Appl. Anal. 2012., 2012: Article ID 917831
Karapinar E, Romaguera S, Tas K: Fixed points for cyclic orbital generalized contractions on complete metric spaces. Cent. Eur. J. Math. 2013, 11(3):552–560. 10.2478/s11533-012-0145-0
Pǎcurar M, Rus IA: Fixed point theory for cyclic φ -contractions. Nonlinear Anal. 2010, 72(3–4):1181–1187. 10.1016/j.na.2009.08.002
Rezapour S, Derafshpour M, Shahzad N: Best proximity point of cyclic φ -contractions in ordered metric spaces. Topol. Methods Nonlinear Anal. 2011, 37(1):193–202.
Chen CM: Fixed point theorems for cyclic Meir-Keeler type mappings in complete metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 41
Acknowledgements
This research was supported by the National Science Council of the Republic of China. The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.
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Chen, CM. Fixed point theorems of generalized cyclic orbital Meir-Keeler contractions. Fixed Point Theory Appl 2013, 91 (2013). https://doi.org/10.1186/1687-1812-2013-91
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DOI: https://doi.org/10.1186/1687-1812-2013-91