Abstract
In this article, by using the Meir-Keeler type mappings, we obtain some new fixed point theorems for the cyclic orbital stronger (weaker) Meir-Keeler contractions and generalized cyclic stronger (weaker) Meir-Keeler contractions. Our results generalize or improve many recent fixed point theorems in the literature. Mathematical Subject Classification: 54H25; 47H10
Introduction and preliminaries
Throughout this article, by ℝ + , we denote the set of all non-negative numbers, while N is the set of all natural numbers. It is well known and easy to prove that if (X, d) is a complete metric space, and if f: X X is continuous and f satisfies
Introduction and preliminaries
, forall x ∈ X and k ∈ (0, 1), then f has a fixed point in X. Using the above conclusion, Kirk et al. [1] proved the following fixed point theorem. Then A ∩ B is nonempty and f has a unique fixed point in A ∩ B.
Introduction and preliminaries
The following definitions and results will be needed in the sequel. Let A and B be two nonempty subsets of a metric space (X, d). A mapping f : A ∪ B A ∪ B is called a cyclic map if f(A) ⊆ B and f(B) ⊆ A. In the recent, Karpagam and Agrawal [2] introduced the notion of cyclic orbital contraction, and obtained a unique fixed point theorem for such a map.
Introduction and preliminaries
Definition 1 [2] Let A and B be nonempty subsets of a metric space (X, d), f : A ∪ B A ∪ B be a cyclic map such that for some x A, there exists a x (0,1) such that
Introduction and preliminaries
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 (X, d), and let f : A ∪ B A ∪ B be a cyclic orbital contraction. Then f has a fixed point in A ∩ B.
Theorem 2 [2]
Furthermore, Kirk et al. [1] introduced the notion of the generalized cyclic mapping and obtained some fixed point results. Let
Theorem 2 [2]
be nonempty subsets of a metric space (X, d), and let f :
Theorem 2 [2]
Then f is called a generalized cyclic map if f (A i ) ⊆ A i+1 for i = 1, 2,..., k and A k+1 = A 1 . Kirk et al. [1] first extended the question of wherther Edelstein's [3] classical result for contractive mappings, and they obtained the following theorem.
Theorem 2 [2]
be nonempty closed subsets of a complete metric space (X, d), at least one of which is compact, and suppose f :
Theorem 2 [2]
A i satisfies the following conditions (where A k+1 = A 1 ):
Theorem 2 [2]
Then f has a unique fixed point. On the other hand, Kirk et al. [1] took up the question of whether condition (ii) of Theorem 3 can be replaced by contractive conditions which typically arise in extensions of Banachs theorem. The authors began with a condition introduced by Geraghty [4] . Let S denote the class of those functions α : ℝ + [0,1) that satisfy the simple condition:
Theorem 2 [2]
Theorem 4 [4] Let (X, d) be a complete metric space, let f : X X, and suppose that there exists a Î S such that
Theorem 2 [2]
Then f has a unique fixed point z in X and {f n x} converges to z for each x X.
Theorem 2 [2]
Applying Theorem 4, Kirk et al. [1] proved the below theorem.
Theorem 2 [2]
be nonempty closed subsets of a complete metric space (X, d), let α ∈ S, and suppose f :
Theorem 2 [2]
A i satisfies the following conditions (where A k+1 = A 1 ):
Theorem 2 [2]
Then f has a unique fixed point. In 1969, Boyd and Wong [5] introduced the notion of Φ-contraction. A mapping f : X X on a metric space is called Φ-contraction if there exists an upper semi-continuous function ψ : [0,∞) [0,∞) such that
Theorem 2 [2]
be nonempty closed subsets of a complete metric space (X, d).
Theorem 2 [2]
A i satisfies the following conditions (where A k+1 = A 1 ):
Theorem 2 [2]
[0, ∞) is upper semi-contionuous from the right and satisfies 0 ≤ ψ(t) <t for t > 0. Then f has a unique fixed point.
Theorem 2 [2]
In this article, we also recall the notion of the Meir-Keeler type mapping. A function
Theorem 2 [2]
+ is said to be a Meir-Keeler type mapping (see [6] ), if for each h ℝ + , there exists δ > 0 such that for t ℝ + with h ≤ t <h + δ, we have ψ(t) < h. Subsequently, some authors worked on this notion (for example, [7] [8] [9] [10] 
Theorem 2 [2]
Example 1 Let X = ℝ 2 and we define d :
Theorem 2 [2]
, then ψ is a stronger Meir-Keeler type mapping in X.
Theorem 2 [2]
The following provides an example of a Meir-Keeler type mapping which is not a stronger Meir-Keeler type mapping in a metric space (X, d).
Theorem 2 [2]
Example 2 Let X = ℝ 2 and we define d :
Theorem 2 [2]
then is a Meir-Keeler type mapping which is not a stronger Meir-Keeler type mapping in X.
Theorem 2 [2]
We next introduce the below notion of weaker Meir-Keeler type mapping in a metric space.
Theorem 2 [2]
Definition 3 Let (X, d) be a metric space, and :
Theorem 2 [2]
Then is called a weaker Meir-Keeler type mapping in X, if the mapping satisfies the following condition:
Theorem 2 [2]
Example 3 Let X = ℝ 2 and we define d :
Theorem 2 [2]
, then is a weaker Meir-Keeler type mapping in X.
Theorem 2 [2]
The following provides an example of a weaker Meir-Keeler type mapping which is not a Meir-Keeler type mapping in a metric space (X, d).
Theorem 2 [2]
Example 4 Let X = ℝ 2 and we define d :
Theorem 2 [2]
then is a weaker Meir-Keeler type mapping which is not a Meir-Keeler type mapping in X.
The fixed point theorems for cyclic orbital Meir-Keeler contractions
Using the notions of the cyclic orbital contraction (see, Definition 1) and stronger Meir-Keeler type mapping (see, Definition 2), we introduce the below notion of cyclic orbital stronger Meir-Keeler contraction.
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
A ∪ B is a cyclic map such that for some x A, there exists a stronger MeirKeeler type mapping ψ :
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Then f is called a cyclic orbital stronger Meir-Keeler ψ-contraction. Now, we are in a position to state the following theorem. Proof. Since f : A ∪ B A ∪ B is a cyclic orbital stronger Meir-Keeler ψ-contraction, there exists x A satisfying (2), and we also have that for each n N,
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Generally, we have
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Thus the sequence {d(f n x, f n+1 x)} is non-increasing and hence it is convergent. Let
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
x) = h. Then there exists 0 N and δ > 0 such that for all n ≥ 0 , 
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Taking into account the above inequality and the definition of stronger Meir-Keeler type mapping ψ in X, corresponding to h use, there exists γ h [0,1) such that
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Therefore, by (2), we also deduce that for each n N,
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
and it follows that for each n N,
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
We now claim that
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
and hence d(f n x, f m x) 0, since 0 < γ h < 1. So {f n x} is a Cauchy sequence. Since (X, d) is a complete metric space, there exists ν A ∪ B such that lim n ∞ f n x = ν. Now {f 2n x} is a sequence in A and {f 2n-1 x} is a sequence in B, and also both converge to ν.
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Since A and B are closed, ν A ∩ B, and so A ∩ B is nonempty. Since
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
hence ν is a fixed point of f. Finally, to prove the uniqueness of the fixed point, let μ be another fixed point of f. By the cyclic character of f, we have ν,μ A ∩ B. Since f is a cyclic orbital stronger Meir-Keeler ψ-contraction, we have
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
a contradiction. Therefore μ = ν, and so ν is a unique fixed point of f.
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Example 5 Let A = B = X = ℝ + and we define d:
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Define f : X X by
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
and define ψ :
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Then f is a cyclic orbital stronger Meir-Keeler ψ-contraction and 0 is the unique fixed point.
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Using the notions of the cyclic orbital contraction (see, Definition 1) and weaker Meir-Keeler type mapping (see, Definition 3), we next introduce the notion of cyclic orbital weaker Meir-Keeler contraction. We first define the below notion of -mapping.
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Definition 5 Let (X, d) be a metric space. We call :
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
the function satisfies the following conditions:
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
( 1 ) >is a weaker Meir-Keeler type mapping in X with (0) = 0; ( 2 ) (a) if lim n ∞ t n = γ > 0, then lim n ∞ (t n ) < g, and
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Definition 6 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B A ∪ B is a cyclic map such that for some x A, there exists a -mapping :
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Then f is called a cyclic orbital weaker Meir-Keeler -contraction. Now, we are in a position to state the following theorem. Proof. Since f : A ∪ B A ∪ B is a cyclic orbital weaker Meir-Keeler -contraction, there exists x A satisfying (3), and we also have that for each n N,
Definition 4 Let A and B be nonempty subsets of a metric space (X, d). Suppose f : A ∪ B
Generally, we have , fx) ).
So we conclude that for each
Since { n (d(x, fx))} n N is decreasing, it must converge to some h ≥ 0. We claim that 
So we conclude that for each
Next, we let c m = d(f m x, f m+1 x), and we claim that the following result holds:
So we conclude that for each
for each ε > 0, there is n 0 (ε) N such that for all m, n ≥ n 0 (ε),
So we conclude that for each
We shall prove (*) by contradiction. Suppose that (*) is false. Then there exists some ε > 0 such that for all p N, there are m p , n p N with m p > n p ≥ p satisfying: 
So we conclude that for each
Letting p ∞. Then by the condition ( 2 )-(a) of -mapping, we have
So we conclude that for each
a contradiction. So {f n x} is a Cauchy sequence. Since (X, d) is a complete metric space, there exists ν A ∪ B such that lim n ∞ f n x = ν. Now {f 2n x} is a sequence in A and {f 2n-1 x} is a sequence in B, and also both converge to ν. Since A and B are closed, ν A ∩ B, and so A ∩ B is nonempty. By the condition ( 2 )-(b) of -mapping, we have
So we conclude that for each
hence ν is a fixed point of f. Let μ be another fixed point of f. Since f is a cyclic orbital weaker Meir-Keeler -contraction, we have
So we conclude that for each
a contradiction. Therefore, μ = ν. Thus ν is a unique fixed point of f. Example 6 Let A = B = X = ℝ + and we define d :
So we conclude that for each
Define f : X X by
So we conclude that for each
, if x ≥ 1.
So we conclude that for each
Then f is a cyclic orbital weaker Meir-Keeler -contraction and 0 is the unique fixed point.
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Using the notions of the generalized cyclic contraction [1] and stronger Meir-Keeler type mapping, we introduce the below notion of generalized cyclic stronger Meir-Keeler contraction. 
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Then we call f a generalized cyclic stronger Meir-Keeler ψ-contraction. We state the main fixed point theorem for the generalized cyclic stronger Meir-Keeler ψ-contraction, as follows:
The fixed point theorems for generalized cyclic Meir-Keeler contractions
be nonempty closed subsets of a complete metric space (X, d), let ψ : ℝ + [0,1) be a stronger Meir-Keeler type mapping in X, and let
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Proof. Given x 0 X and let x n = f n x 0 , n N. Since f is a generalized cyclic stronger
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Thus the sequence {d(x n , x n+1 )} is non-increasing and hence it is convergent. Let lim n ∞ d(x n , x n+1 ) = h ≥ 0. Then there exists 0 N and δ > 0 such that for all n ≥ 0
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Taking into account the above inequality and the definition of stronger Meir-Keeler type mapping ψ in X, corresponding to h use, there exists γ n [0,1) such that
The fixed point theorems for generalized cyclic Meir-Keeler contractions
for all n N ∪ {0}. Thus, we can deduce that for each n N
The fixed point theorems for generalized cyclic Meir-Keeler contractions
and it follows that for each n N
The fixed point theorems for generalized cyclic Meir-Keeler contractions
We now claim that lim n→∞ d(x k 0 +n , x k 0 +m ) = 0 for m > n. For m, n N with m > n, we have
The fixed point theorems for generalized cyclic Meir-Keeler contractions
and hence d(f n x 0 , f m x 0 ) 0, since 0 < γ h < 1. So {f n x 0 } is a Cauchy sequence. Since X is complete, there exists ν ∈ ∪ k i=1 A i such that lim n ∞ f n x 0 = ν. Now for all i = 0,1, 2,...,
The fixed point theorems for generalized cyclic Meir-Keeler contractions
x} is a sequence in A i and also all converge to ν. Since A i is clsoed for all i = 1, 2,..., k, we conclude ν ∈ ∩ k i=1 A i , and also we conclude that
The fixed point theorems for generalized cyclic Meir-Keeler contractions
hence ν is a fixed point of f. Finally, to prove the uniqueness of the fixed point, let μ be another fixed point of f.
The fixed point theorems for generalized cyclic Meir-Keeler contractions
By the cyclic character of f, we have μ ∈ ∩ k i=1 A i . Since f is a generalized cyclic stronger Meir-Keeler ψ-contraction, we have
The fixed point theorems for generalized cyclic Meir-Keeler contractions
a contradiction. Therefore, μ = ν. Thus ν is a unique fixed point of f. Example 7 Let X = ℝ 3 and we define d : 
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Then f is a generalized cyclic stronger Meir-Keeler ψ-contraction and (0,0,0) is the unique fixed point.
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Using the notions of the generalized cyclic contraction and weaker Meir-Keeler type mapping, we introduce the below notion of generalized cyclic weaker Meir-Keeler contraction. 
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Then we call f a generalized cyclic weaker Meir-Keeler -contraction. Now, we are in a position to state the following theorem.
The fixed point theorems for generalized cyclic Meir-Keeler contractions
be nonempty closed subsets of a complete metric space (X, d),
The fixed point theorems for generalized cyclic Meir-Keeler contractions
+ be a -mapping in X, and let f : cup
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Proof. Given x 0 X and let x n = f n x 0 , n N. Since f is a generalized cyclic weaker
The fixed point theorems for generalized cyclic Meir-Keeler contractions
Meir-Keeler -contraction, we have that for each n N Next, we claim that {x n } is a Cauchy sequence. We claim that the following result holds:
The fixed point theorems for generalized cyclic Meir-Keeler contractions
for each ε > 0, there is n 0 (ε) N such that for all m, n ≥ n 0 (ε),
The fixed point theorems for generalized cyclic Meir-Keeler contractions
We shall prove (**) by contradiction. Suppose that (**) is false. Then there exists some ε > 0 such that for all p N, there are m p , n p N with m p > n p ≥ p satisfying: 

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