# Fixed point theorems for cyclic Meir-Keeler type mappings in complete metric spaces

## 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

## 1 Introduction and preliminaries

Throughout this article, by +, we denote the set of all non-negative numbers, while 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: XX is continuous and f satisfies

$d\left(fx,{f}^{2}x\right)\le k\cdot d\left(x,fx\right),\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{all}}\phantom{\rule{2.77695pt}{0ex}}x\in X\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}k\in \left(0,1\right),$

then f has a fixed point in X. Using the above conclusion, Kirk et al. [1] proved the following fixed point theorem.

Theorem 1 [1] Let A and B be two nonempty closed subsets of a complete metric space (X, d), and suppose f: A BA B satisfies

(i) f(A) B and f(B) A,

(ii) d(fx, fy) ≤ k d(x, y) for all x A, y B and k (0,1).

Then A ∩ B is nonempty and f has a unique fixed point in A ∩ B.

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 BA 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.

Definition 1 [2] Let A and B be nonempty subsets of a metric space (X, d), f : A BA B be a cyclic map such that for some x A, there exists a κ x (0,1) such that

$d\left({f}^{2n}x,fy\right)\le {k}_{x}\cdot d\left({f}^{2n-1}x,y\right),\phantom{\rule{1em}{0ex}}n\in ℕ,\phantom{\rule{1em}{0ex}}y\in A.$
(1)

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 BA B be a cyclic orbital contraction. Then f has a fixed point in A ∩ B.

Furthermore, Kirk et al. [1] introduced the notion of the generalized cyclic mapping and obtained some fixed point results. Let ${\left\{{A}_{i}\right\}}_{i=1}^{k}$ be nonempty subsets of a metric space (X, d), and let $f:{\cup }_{i=1}^{k}{A}_{i}\to {\cup }_{i=1}^{k}{A}_{i}$ Then f is called a generalized cyclic map if f(A i ) A i+ 1 for i = 1, 2,..., k and A k+ 1 = A1. 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 3 [1] Let ${\left\{{A}_{i}\right\}}_{i=1}^{k}$ be nonempty closed subsets of a complete metric space (X, d), at least one of which is compact, and suppose $f:{\cup }_{i=1}^{k}{A}_{i}\to {\cup }_{i=1}^{k}{A}_{i}$ satisfies the following conditions (where A k+ 1 = A1):

(i) f(A i ) A i+ 1 for i = 1,2,...,k,

(ii) d(fx, fy) < d(x, y) whenever x A i , y A i+ 1 and x ≠ y, (i = 1, 2,..., k).

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 : + → [0,1) that satisfy the simple condition:

$\alpha \left({t}_{n}\right)\to 1\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}{t}_{n}\to 0.$

Theorem 4 [4] Let (X, d) be a complete metric space, let f : XX, and suppose that there exists α S such that

$d\left(fx,fy\right)\le \alpha \left(d\left(x,y\right)\right)\cdot d\left(x,y\right),\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}x,y\in X.$

Then f has a unique fixed point z in X and {fnx} converges to z for each x X.

Applying Theorem 4, Kirk et al. [1] proved the below theorem.

Theorem 5 [1] Let ${\left\{{A}_{i}\right\}}_{i=1}^{k}$ be nonempty closed subsets of a complete metric space (X, d), let $\alpha \in \mathcal{S}$, and suppose $f:{\cup }_{i=1}^{k}{A}_{i}\to {\cup }_{i=1}^{k}{A}_{i}$ satisfies the following conditions (where A k+ 1 = A1):

(i) f(A i ) A i +1 for i = 1,2,...,k,

(ii) d(fx, fy) ≤ α(d(x, y)) d(x, y) for all x A i , y A i+ 1, i= 1,2,...,k.

Then f has a unique fixed point.

In 1969, Boyd and Wong [5] introduced the notion of Φ-contraction. A mapping f : XX on a metric space is called Φ-contraction if there exists an upper semi-continuous function ψ : [0,∞)→ [0,∞) such that

$d\left(fx,fy\right)\le \Phi \left(d\left(x,y\right)\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{all}}\phantom{\rule{0.3em}{0ex}}x,y\in X.$

Kirk et al. [1] also proved the below theorem.

Theorem 6 [1] Let ${\left\{{A}_{i}\right\}}_{i=1}^{k}$ be nonempty closed subsets of a complete metric space (X, d). Suppose $f:{\cup }_{i=1}^{k}{A}_{i}\to {\cup }_{i=1}^{k}{A}_{i}$ satisfies the following conditions (where A k+ 1 = A1):

(i) f(A i ) A i +1 for i = 1,2,...,k,

(ii) d(fx, fy) ≤ Φ(d(x, y)) for all x A i , y A i+ 1, i = 1,2,...,k,

where Φ : [0, ∞) → [0, ∞) is upper semi-contionuous from the right and satisfies 0 ≤ ψ(t) < t for t > 0. Then f has a unique fixed point.

In this article, we also recall the notion of the Meir-Keeler type mapping. A function ψ : ++ is said to be a Meir-Keeler type mapping (see [6]), if for each η +, there exists δ > 0 such that for t + with ηt < η + δ, we have ψ(t) < η. Subsequently, some authors worked on this notion (for example, [710]). This article will deal with two new mappings of the stronger Meir-Keeler type and weaker Meir-Keeler type in a metric space (X,d). We first introduce the below notion of stronger Meir-Keeler type mapping in a metric space.

Definition 2 Let (X, d) be a metric space. We call ψ : + → [0,1) a stronger Meir-Keeler type mapping in X if the mapping ψ satisfies the following condition:

$\forall \eta >0\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\exists \delta >0\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\exists {\gamma }_{\eta }\in \left[0,1\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall x,y\in X\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(\eta \le d\left(x,y\right)<\delta +\eta \phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\psi \left(d\left(x,y\right)\right)<{\gamma }_{\eta }\right).$

Example 1 Let X = 2 and we define d : X × X+ by

$d\left(x,y\right)=\left|{x}_{1}-{y}_{1}\right|+\left|{x}_{2}-{y}_{2}\right|\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}x=\left({x}_{1},{x}_{2}\right),\phantom{\rule{1em}{0ex}}y=\left({y}_{1},{y}_{2}\right)\in X.$

If $\psi :{ℝ}^{+}\to \left[0,1\right),\phantom{\rule{2.77695pt}{0ex}}\psi \left(d\left(x,y\right)\right)=\frac{d\left(x,y\right)}{d\left(x,y\right)+1}$, then ψ is a stronger Meir-Keeler type mapping in X.

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).

Example 2 Let X = 2 and we define d : X × X+ by

$d\left(x,y\right)=\left|{x}_{1}-{y}_{1}\right|+\left|{x}_{2}-{y}_{2}\right|\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}x=\left({x}_{1},{x}_{2}\right),\phantom{\rule{1em}{0ex}}y=\left({y}_{1},{y}_{2}\right)\in X.$

If φ : ++,

$\phi \left(d\left(x,y\right)\right)=\left\{\begin{array}{cc}d\left(x,y\right)-1,\hfill & if\phantom{\rule{0.3em}{0ex}}d\left(x,y\right)>1;\hfill \\ 0,\hfill & if\phantom{\rule{0.3em}{0ex}}d\left(x,y\right)\le 1,\hfill \end{array}\right\$

then φ is a Meir-Keeler type mapping which is not a stronger Meir-Keeler type mapping in X.

We next introduce the below notion of weaker Meir-Keeler type mapping in a metric space.

Definition 3 Let (X, d) be a metric space, and φ : ++. Then φ is called a weaker Meir-Keeler type mapping in X, if the mapping φ satisfies the following condition:

$\forall \eta >0\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\exists \delta >0\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall x,y\in X\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(\eta \le d\left(x,y\right)<\delta +\eta \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}⇒\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\exists {n}_{0}\in ℕ\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\phi }^{{n}_{0}}\left(d\left(x,y\right)\right)<\eta \right).$

Example 3 Let X = 2 and we define d : X × X+ by

$d\left(x,y\right)=\left|{x}_{1}-{y}_{1}\right|+\left|{x}_{2}-{y}_{2}\right|\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}x=\left({x}_{1},{x}_{2}\right),\phantom{\rule{1em}{0ex}}y=\left({y}_{1},{y}_{2}\right)\in X.$

If φ : ++, $\phi \left(d\left(x,y\right)\right)=\frac{1}{2}d\left(x,y\right)$, then φ is a weaker Meir-Keeler type mapping in X.

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).

Example 4 Let X = 2 and we define d : X × X+ by

$d\left(x,y\right)=\left|{x}_{1}-{y}_{1}\right|+\left|{x}_{2}-{y}_{2}\right|\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}x=\left({x}_{1},{x}_{2}\right),\phantom{\rule{1em}{0ex}}y=\left({y}_{1},{y}_{2}\right)\in X.$

If φ : ++,

$\phi \left(d\left(x,y\right)\right)=\left\{\begin{array}{cc}0,\hfill & if\phantom{\rule{0.3em}{0ex}}d\left(x,y\right)\le 1,\hfill \\ 2\cdot d\left(x,y\right),\hfill & if\phantom{\rule{0.3em}{0ex}}1

then φ is a weaker Meir-Keeler type mapping which is not a Meir-Keeler type mapping in X.

## 2 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 BA B is a cyclic map such that for some x A, there exists a stronger Meir-Keeler type mapping ψ : + → [0,1) in X such that

$d\left({f}^{2n}x,fy\right)\le \psi \left(d\left({f}^{2n-1}x,y\right)\right)\cdot d\left({f}^{2n-1}x,y\right),\phantom{\rule{1em}{0ex}}n\in ℕ,\phantom{\rule{1em}{0ex}}y\in A.$
(2)

Then f is called a cyclic orbital stronger Meir-Keeler ψ-contraction.

Now, we are in a position to state the following theorem.

Theorem 7 Let A and B be two nonempty closed subsets of a complete metric space (X, d), and let ψ : + → [0,1) be a stronger Meir-Keeler type mapping in X. Suppose f : A BA B is a cyclic orbital stronger Meir-Keeler ψ-contraction. Then AB is nonempty and f has a unique fixed point in AB.

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 ,

$\begin{array}{l}d\left({f}^{2n}x,{f}^{2n+1}x\right)\le \psi d\left({f}^{2n-1}x,{f}^{2n}x\right)\right)\cdot d\left({f}^{2n-1}x,{f}^{2n}x\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\le d\left({f}^{2n-1}x,{f}^{2n}x\right),\end{array}$

and

$\begin{array}{ll}\hfill d\left({f}^{2n+1}x,{f}^{2n+2}x\right)& =d\left({f}^{2n+2}x,{f}^{2n+1}x\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(d\left({f}^{2n+1}x,{f}^{2n}x\right)\right)\cdot d\left({f}^{2n+1}x,{f}^{2n}x\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({f}^{2n+1}x,{f}^{2n}x\right)=d\left({f}^{2n}x,{f}^{2n+1}x\right).\phantom{\rule{2em}{0ex}}\end{array}$

Generally, we have

$d\left({f}^{n}x,{f}^{n+1}x\right)\le d\left({f}^{n-1}x,{f}^{n}x\right),\phantom{\rule{1em}{0ex}}n\in ℕ.$

Thus the sequence {d(fnx, fn+1x)} is non-increasing and hence it is convergent. Let limn→∞d(fnx, fn+1x) = η. Then there exists κ0 and δ > 0 such that for all n ≥ κ0,

$\eta \le d\left({f}^{n}x,{f}^{n+1}x\right)<\eta +\delta .$

Taking into account the above inequality and the definition of stronger Meir-Keeler type mapping ψ in X, corresponding to η use, there exists γ η [0,1) such that

$\psi \left(d\left({f}^{{k}_{0}+n}x,{f}^{{k}_{0}+n+1}x\right)\right)\phantom{\rule{2.77695pt}{0ex}}<{\gamma }_{\eta }\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{all}}\phantom{\rule{1em}{0ex}}n\in ℕ\cup \left\{0\right\}.$

Therefore, by (2), we also deduce that for each n ,

$\begin{array}{ll}\hfill d\left({f}^{{k}_{0}+n}x,{f}^{{k}_{0}+n+1}x\right)& \le \psi \left(d\left({f}^{{k}_{0}+n-1}x,{f}^{{k}_{0}+n}x\right)\right)\cdot d\left({f}^{{k}_{0}+n-1}x,{f}^{{k}_{0}+n}x\right)\phantom{\rule{2em}{0ex}}\\ <{\gamma }_{\eta }\cdot d\left({f}^{{k}_{0}+n-1}x,{f}^{{k}_{0}+n}x\right),\phantom{\rule{2em}{0ex}}\end{array}$

and it follows that for each n,

$\begin{array}{ll}\hfill d\left({f}^{{k}_{0}+n}x,{f}^{{k}_{0}+n+1}x\right)& <{\gamma }_{\eta }\cdot d\left({f}^{{k}_{0}+n-1}x,{f}^{{k}_{0}+n}x\right)\phantom{\rule{2em}{0ex}}\\ <\cdots \cdots \phantom{\rule{2em}{0ex}}\\ <{\gamma }_{\eta }^{n}\cdot d\left({f}^{{k}_{0}}x,{f}^{{k}_{0}+1}x\right).\phantom{\rule{2em}{0ex}}\end{array}$

So

$\underset{n\to \infty }{\text{lim}}d\left({f}^{{k}_{0}+n}x,{f}^{{k}_{0}+n+1}x\right)=0,\phantom{\rule{1em}{0ex}}\mathsf{\text{since}}\phantom{\rule{1em}{0ex}}{\gamma }_{\eta }\in \left[0,1\right).$

We now claim that $\underset{n\to \infty }{\text{lim}}d\left({f}^{{k}_{0}+n}x,{f}^{{k}_{0}+m}x\right)=0$ for m > n. For m, n with m > n, we have

$d\left({f}^{{k}_{0}+n}x,{f}^{{k}_{0}+m}x\right)\le \sum _{i=n}^{m-1}d\left({f}^{{k}_{0}+i}x,{f}^{{k}_{0}+i+1}x\right)<\frac{{\gamma }_{\eta }^{m-1}}{1-{\gamma }_{\eta }}d\left({f}^{{k}_{0}}x,{f}^{{k}_{0}+1}x\right),$

and hence d(fnx, fmx) → 0, since 0 < γ η < 1. So {fnx} is a Cauchy sequence. Since (X, d) is a complete metric space, there exists ν A B such that limn→∞fnx = ν. Now {f2nx} is a sequence in A and {f2n-1x} is a sequence in B, and also both converge to ν. Since A and B are closed, ν AB, and so AB is nonempty. Since

$\begin{array}{ll}\hfill d\left(\nu ,f\nu \right)& =\underset{n\to \infty }{\text{lim}}d\left({f}^{2n}x,f\nu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\left[\psi \left(d\left({f}^{2n-1}x,\nu \right)\right)\cdot d\left({f}^{2n-1}x,\nu \right)\right]\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\left[{\gamma }_{\eta }\cdot d\left({f}^{2n-1}x,\nu \right)\right]=0,\phantom{\rule{2em}{0ex}}\end{array}$

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 AB. Since f is a cyclic orbital stronger Meir-Keeler ψ-contraction, we have

$\begin{array}{ll}\hfill d\left(\nu ,\mu \right)& =d\left(\nu ,f\mu \right)=\underset{n\to \infty }{\text{lim}}d\left({f}^{2n}x,f\mu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\left[\psi \left(d\left({f}^{2n-1}x,\mu \right)\right)\cdot d\left({f}^{2n-1}x,\mu \right)\right]\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\left[{\gamma }_{\eta }\cdot d\left({f}^{2n-1}x,\mu \right)\right]\phantom{\rule{2em}{0ex}}\\ \le {\gamma }_{\eta }\cdot d\left(\nu ,\mu \right)

a contradiction. Therefore μ = ν, and so ν is a unique fixed point of f.

Example 5 Let A = B = X = + and we define d: X × X+ by

$d\left(x,y\right)=\left|x-y\right|,\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}x,y\in X.$

Define f : XX by

$f\left(x\right)=\left\{\begin{array}{cc}0,\hfill & if\phantom{\rule{0.3em}{0ex}}0\le x<1;\hfill \\ \frac{1}{4},\hfill & if\phantom{\rule{0.3em}{0ex}}x\ge 1.\hfill \end{array}\right\$

and define ψ : + → [0,1) by

$\psi \left(t\right)=\left\{\begin{array}{cc}\frac{1}{3},\hfill & if\phantom{\rule{0.3em}{0ex}}0\le t\le 1;\hfill \\ \frac{t}{t+1},\hfill & if\phantom{\rule{0.3em}{0ex}}t>1.\hfill \end{array}\right\$

Then f is a cyclic orbital stronger Meir-Keeler ψ-contraction and 0 is the unique fixed point.

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 φ-mapping.

Definition 5 Let (X, d) be a metric space. We call φ : ++ a φ-mapping in X if the function φ satisfies the following conditions:

(φ1) φ >is a weaker Meir-Keeler type mapping in X with φ(0) = 0;

(φ2) (a) if limn→∞t n = γ > 0, then limn→∞φ(t n ) < γ, and

(b) if limn→∞t n = 0, then limn→∞φ(t n ) = 0;

(φ3) {φn(t)}nis decreasing.

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 φ-mapping φ : ++ in X such that

$d\left({f}^{2n}x,fy\right)\le \phi \left(d\left({f}^{2n-1}x,y\right)\right),\phantom{\rule{1em}{0ex}}n\in ℕ,\phantom{\rule{1em}{0ex}}y\in A.$
(3)

Then f is called a cyclic orbital weaker Meir-Keeler φ -contraction.

Now, we are in a position to state the following theorem.

Theorem 8 Let A and B be two nonempty closed subsets of a complete metric space (X, d), and let φ : ++ be a φ-mapping in X. Suppose f : A BA B is a cyclic orbital weaker Meir-Keeler φ -contraction. Then A ∩ B is nonempty and f has a unique fixed point in A ∩ B.

Proof. Since f : A B → A B is a cyclic orbital weaker Meir-Keeler φ-contraction, there exists x A satisfying (3), and we also have that for each n ,

$d\left({f}^{2n}x,{f}^{2n+1}x\right)\le \phi \left(d\left({f}^{2n-1}x,{f}^{2n}x\right)\right),$

and

$\begin{array}{ll}\hfill d\left({f}^{2n+1}x,{f}^{2n+2}x\right)& =d\left({f}^{2n+2}x,{f}^{2n+1}x\right)\phantom{\rule{2em}{0ex}}\\ \le \phi \left(d\left({f}^{2n+1}x,{f}^{2n}x\right)\right).\phantom{\rule{2em}{0ex}}\end{array}$

Generally, we have

$d\left({f}^{n}x,{f}^{n+1}x\right)\le \phi \left(d\left({f}^{n-1}x,{f}^{n}x\right)\right),\phantom{\rule{1em}{0ex}}n\in ℕ.$

So we conclude that for each n

$\begin{array}{ll}\hfill d\left({f}^{n}x,{f}^{n+1}x\right)& \le \phi \left(d\left({f}^{n-1}x,{f}^{n}x\right)\right)\phantom{\rule{2em}{0ex}}\\ \le {\phi }^{2}\left(d\left({f}^{n-2}x,{f}^{n-1}x\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \cdots \cdots \phantom{\rule{2em}{0ex}}\\ \le {\phi }^{n}\left(d\left(x,fx\right)\right).\phantom{\rule{2em}{0ex}}\end{array}$

Since {φn(d(x, fx))}nis decreasing, it must converge to some η ≥ 0. We claim that η = 0. On the contrary, assume that η > 0. Then by the definition of weaker Meir-Keeler type mapping φ in X, there exists δ > 0 such that for x, y X with ηd(x, y) < δ + η, there exists n0 such that ${\phi }^{{n}_{0}}\left(d\left(x,y\right)\right)<\eta$. Since limn→∞φn(d(x, fx)) = η, there exists m0 such that ηφm(d(x, fx)) < δ + η, for all m > m0. Thus, we conclude that ${\phi }^{{m}_{0}+{n}_{0}}\left(d\left({x}_{0},{x}_{1}\right)\right)<\eta$, and we get a contradiction. So lim n→∞ φn(d(x, fx)) = 0, that is, limn→∞d(fnx, fn+1x) = 0.

Next, we let c m = d(fmx, fm+1x), and we claim that the following result holds:

for each ε > 0, there is n0(ε) such that for all m, n ≥ n0(ε),

$d\left({f}^{m}x,{f}^{m+1}x\right)<\epsilon .\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\left(*\right)$

We shall prove (*) by contradiction. Suppose that (*) is false. Then there exists some ε > 0 such that for all p , there are m p , n p with m p > n p p satisfying:

1. (i)

m p is even and n p is odd,

2. (ii)

$d\left({f}^{{m}_{p}}x,{f}^{{n}_{p}}x\right)\ge \epsilon$, and

3. (iii)

m p is the smallest even number such that the conditions (i), (ii) hold.

Since c m 0, by (ii), we have $\underset{k\to \infty }{\text{lim}}d\left({f}^{{m}_{p}}x,{f}^{{n}_{p}}x\right)=\epsilon$, and

$\begin{array}{ll}\hfill \epsilon & \le d\left({f}^{{m}_{p}}x,{f}^{{n}_{p}}x\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({f}^{{m}_{p}}x,{f}^{{m}_{p}+1}x\right)+d\left({f}^{{m}_{p}+1}x,{f}^{{n}_{p}+1}x\right)+d\left({f}^{{n}_{p}+1}x,{f}^{{n}_{p}}x\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({f}^{{m}_{p}}x,{f}^{{m}_{p}+1}x\right)+\phi \left(d\left({f}^{{m}_{p}}x,{f}^{{n}_{p}}x\right)\right)+d\left({f}^{{n}_{p}+1}x,{f}^{{n}_{p}}x\right).\phantom{\rule{2em}{0ex}}\end{array}$

Letting p → ∞. Then by the condition (φ2)-(a) of φ-mapping, we have

$\epsilon \le 0+\underset{p\to \infty }{\text{lim}}\phi \left(d\left({f}^{{m}_{p}}x,{f}^{{n}_{p}}x\right)\right)+0<\epsilon ,$

a contradiction. So {fnx} is a Cauchy sequence. Since (X, d) is a complete metric space, there exists ν A B such that limn→∞f n x = ν. Now {f2nx} is a sequence in A and {f2n-1x} is a sequence in B, and also both converge to ν. Since A and B are closed, ν AB, and so AB is nonempty. By the condition (φ2)-(b) of φ-mapping, we have

$\begin{array}{ll}\hfill d\left(\nu ,f\nu \right)& =\underset{n\to \infty }{\text{lim}}d\left({f}^{2n}x,f\nu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\phi \left(d\left({f}^{2n-1}x,\nu \right)\right)=0,\phantom{\rule{2em}{0ex}}\end{array}$

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 φ-contraction, we have

$\begin{array}{ll}\hfill d\left(\nu ,\mu \right)& =d\left(\nu ,f\mu \right)=\underset{n\to \infty }{\text{lim}}d\left({f}^{2n}x,f\mu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\phi \left(d\left({f}^{2n-1}x,\mu \right)\right)\phantom{\rule{2em}{0ex}}\\

a contradiction. Therefore, μ = ν. Thus ν is a unique fixed point of f.

Example 6 Let A = B = X = + and we define d : X × X+ by

$d\left(x,y\right)=\left|x-y\right|,\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}x,y\in X.$

Define f : XX by

$f\left(x\right)=\left\{\begin{array}{cc}0,\hfill & if\phantom{\rule{2.77695pt}{0ex}}0\le x<1;\hfill \\ \frac{1}{4},\hfill & if\phantom{\rule{2.77695pt}{0ex}}x\ge 1.\hfill \end{array}\right\$

and define φ : ++ by

$\phi \left(t\right)=\frac{1}{3}t\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}t\in {ℝ}^{+}.$

Then f is a cyclic orbital weaker Meir-Keeler φ -contraction and 0 is the unique fixed point.

## 3 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.

Definition 7 Let ${\left\{{A}_{i}\right\}}_{i=1}^{k}$ be nonempty subsets of a metric space (X, d), let ψ : + → [0,1) be a stronger Meir-Keeler type mapping in X, and suppose $f:{\cup }_{i=1}^{k}{A}_{i}\to {\cup }_{i=1}^{k}{A}_{i}$ satisfies the following conditions (where A k+ 1 = A1):

(i) f(A i ) A i +1 for i = 1,2,...,k;

(ii) d(fx, fy) ≤ ψ(d(x, y)) d(x, y) for all x A i , y A i+ 1, i= 1,2,...,k.

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:

Theorem 9 Let ${\left\{{A}_{i}\right\}}_{i=1}^{k}$ be nonempty closed subsets of a complete metric space (X, d), let ψ : + → [0,1) be a stronger Meir-Keeler type mapping in X, and let $f:{\cup }_{i=1}^{k}{A}_{i}\to {\cup }_{i=1}^{k}{A}_{i}$ be a generalized cyclic stronger Meir-Keeler ψ-contraction. Then f has a unique fixed point in ${\cap }_{i=1}^{k}{A}_{i}$.

Proof. Given x0 X and let x n = fnx0, n . Since f is a generalized cyclic stronger Meir-Keeler ψ-contraction, we have that for each n

$\begin{array}{ll}\hfill d\left({x}_{n},{x}_{n+1}\right)& =d\left({f}^{n}{x}_{0},{f}^{n+1}{x}_{0}\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(d\left({f}^{n-1}{x}_{0},{f}^{n}{x}_{0}\right)\right)\cdot d\left({f}^{n-1}{x}_{0},{f}^{n}{x}_{0}\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({f}^{n-1}{x}_{0},{f}^{n}{x}_{0}\right)=d\left({x}_{n-1},{x}_{n}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Thus the sequence {d(x n , x n+ 1)} is non-increasing and hence it is convergent. Let limn→∞d(x n , x n+ 1) = η ≥ 0. Then there exists κ0 and δ > 0 such that for all n ≥ κ0

$\eta \le d\left({x}_{n},{x}_{n+1}\right)<\eta +\delta .$

Taking into account the above inequality and the definition of stronger Meir-Keeler type mapping ψ in X, corresponding to η use, there exists γ n [0,1) such that

$\psi \left(d\left({x}_{{k}_{0}+n},{x}_{{k}_{0}+n+1}\right)\right)<{\gamma }_{\eta },$

for all n {0}. Thus, we can deduce that for each n

$\begin{array}{ll}\hfill d\left({x}_{{k}_{0}+n},{x}_{{k}_{0}+n+1}\right)& =d\left({f}^{{k}_{0}+n}{x}_{0},{f}^{{k}_{0}+n+1}{x}_{0}\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(d\left({f}^{{k}_{0}+n-1}{x}_{0},{f}^{{k}_{0}+n}{x}_{0}\right)\right)\cdot d\left({f}^{{k}_{0}+n-1}{x}_{0},{f}^{{k}_{0}+n}{x}_{0}\right)\phantom{\rule{2em}{0ex}}\\ <{\gamma }_{\eta }d\left({f}^{{k}_{0}+n-1}{x}_{0},{f}^{{k}_{0}+n}{x}_{0}\right),\phantom{\rule{2em}{0ex}}\end{array}$

and it follows that for each n

$\begin{array}{ll}\hfill d\left({x}_{{k}_{0}+n},{x}_{{k}_{0}+n+1}\right)& <{\gamma }_{\eta }d\left({f}^{{k}_{0}+n-1}{x}_{0},{f}^{{k}_{0}+n}{x}_{0}\right)\phantom{\rule{2em}{0ex}}\\ <\cdots \phantom{\rule{2em}{0ex}}\\ <{\gamma }_{\eta }^{n}d\left({f}^{{k}_{0}+1}{x}_{0},{f}^{{k}_{0}+2}{x}_{0}\right).\phantom{\rule{2em}{0ex}}\end{array}$

So

$\underset{n\to \infty }{\text{lim}}d\left({x}_{{k}_{0}+n},{x}_{{k}_{0}+n+1}\right)=0,\phantom{\rule{1em}{0ex}}\mathsf{\text{since}}\phantom{\rule{1em}{0ex}}{\gamma }_{\eta }<1.$

We now claim that $\underset{n\to \infty }{\text{lim}}d\left({x}_{{k}_{0}+n},{x}_{{k}_{0}+m}\right)=0$ for m > n. For m, n with m > n, we have

$\begin{array}{l}d\left({x}_{{k}_{0}+n},{x}_{{k}_{0}+m}\right)=d\left({f}^{{k}_{0}+n}{x}_{0},{f}^{{k}_{0}+m}{x}_{0}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\le \sum _{i=n}^{m-1}d\left({f}^{{k}_{0}+i}{x}_{0},{f}^{{k}_{0}+i+1}{x}_{0}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}<\frac{{\gamma }_{\eta }^{m-1}}{1-{\gamma }_{\eta }}d\left({f}^{{k}_{0}}{x}_{0},{f}^{{k}_{0}+1}{x}_{0}\right)\right),\end{array}$

and hence d(fnx0, fmx0) → 0, since 0 < γ η < 1. So {fnx0} is a Cauchy sequence. Since X is complete, there exists $\nu \in {\cup }_{i=1}^{k}{A}_{i}$ such that limn→∞fnx0 = ν. Now for all i = 0,1, 2,..., k - 1, {fkn-ix} 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 $\nu \in {\cap }_{i=1}^{k}{A}_{i}$, and also we conclude that ${\cap }_{i=1}^{k}{A}_{i}\ne \varphi$ Since

$\begin{array}{ll}\hfill d\left(\nu ,f\nu \right)& =\underset{n\to \infty }{\text{lim}}d\left({f}^{kn}x,f\nu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\left[\psi \left(d\left({f}^{kn-1}x,\nu \right)\right)\cdot d\left({f}^{kn-1}x,\nu \right)\right]\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\left[{\gamma }_{\eta }\cdot d\left({f}^{kn-1}x,\nu \right)\right]=0,\phantom{\rule{2em}{0ex}}\end{array}$

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 $\mu \in {\cap }_{i=1}^{k}{A}_{i}$. Since f is a generalized cyclic stronger Meir-Keeler ψ-contraction, we have

$\begin{array}{ll}\hfill d\left(\nu ,\mu \right)& =d\left(\nu ,f\mu \right)=\underset{n\to \infty }{\text{lim}}d\left({f}^{kn}x,f\mu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\left[\psi \left(d\left({f}^{kn-1}x,\mu \right)\right)\cdot d\left({f}^{kn-1}x,\mu \right)\right]\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\left[{\gamma }_{\eta }\cdot d\left({f}^{kn-1}x,\mu \right)\right]\phantom{\rule{2em}{0ex}}\\ \le {\gamma }_{\eta }\cdot d\left(\nu ,\mu \right)

a contradiction. Therefore, μ = ν. Thus ν is a unique fixed point of f.

Example 7 Let X = 3 and we define d : X × X+by

$d\left(x,y\right)=\left|{x}_{1}-{y}_{1}\right|+\left|{x}_{2}-{y}_{2}\right|+\left|{x}_{3}-{y}_{3}\right|,\phantom{\rule{1em}{0ex}}for\phantom{\rule{0.3em}{0ex}}x=\left({x}_{1},{x}_{2},{x}_{3}\right),y=\left({y}_{1},{y}_{2},{y}_{3}\right)\in X,$

and let A = {(x, 0,0): x }, B = {(0,y,0): y }, C = {(0,0, z): z } be three subsets of X. Define f: A B CA B C by

$\begin{array}{c}f\left(\left(x,0,0\right)\right)=\left(0,x,0\right);\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}x\in ℝ;\\ f\left(\left(0,y,0\right)\right)=\left(0,0,y\right);\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}y\in ℝ;\\ f\left(\left(0,0,z\right)\right)=\left(z,0,0\right);\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}z\in ℝ.\end{array}$

and define ψ : + → [0,1) by

$\psi \left(t\right)=\frac{t}{t+1}\phantom{\rule{1em}{0ex}};for\phantom{\rule{2.77695pt}{0ex}}t\in {ℝ}^{+}.$

Then f is a generalized cyclic stronger Meir-Keeler ψ -contraction and (0,0,0) is the unique fixed point.

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.

Definition 8 Let ${\left\{{A}_{i}\right\}}_{i=1}^{k}$ be nonempty subsets of a metric space (X, d), let φ : ++ be a φ-mapping in X, and suppose $f:{\cup }_{i=1}^{k}{A}_{i}\to {\cup }_{i=1}^{k}{A}_{i}$ satisfies the following conditions (where A k+ 1 = A1):

(i) f(A i ) A i+ 1 for i = 1,2,...,k;

(ii) d(fx, fy) ≤ φ (d(x, y)) for all x A i , y A i+ 1, i = 1,2,...,k.

Then we call f a generalized cyclic weaker Meir-Keeler φ -contraction.

Now, we are in a position to state the following theorem.

Theorem 10 Let ${\left\{{A}_{i}\right\}}_{i=1}^{k}$ be nonempty closed subsets of a complete metric space (X, d), let φ : ++ be a φ-mapping in X, and let $f:cu{p}_{i=1}^{k}{A}_{i}\to {\cup }_{i=1}^{k}{A}_{i}$ be a generalized cyclic weaker Meir-Keeler φ -contraction. Then f has a unique fixed point in ${\cap }_{i=1}^{k}{A}_{i}$.

Proof. Given x0 X and let x n = fnx0, n . Since f is a generalized cyclic weaker Meir-Keeler φ-contraction, we have that for each n

$\begin{array}{ll}\hfill d\left({x}_{n},{x}_{n+1}\right)& =d\left({f}^{n}{x}_{0},{f}^{n+1}{x}_{0}\right)\phantom{\rule{2em}{0ex}}\\ \le \phi \left(d\left({f}^{n-1}{x}_{0},{f}^{n}{x}_{0}\right)\right)=\phi \left(d\left({x}_{n-1},{x}_{n}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \cdots \cdots \phantom{\rule{2em}{0ex}}\\ \le {\phi }^{n}\left(d\left({x}_{0},{x}_{1}\right)\right).\phantom{\rule{2em}{0ex}}\end{array}$

Since {φn(d(x0,x1))}nis decreasing, it must converge to some η ≥ 0. We claim that η = 0. On the contrary, assume that η > 0. Then by the definition of weaker Meir-Keeler type mapping φ in X, there exists δ > 0 such that for x, y X with ηd(x, y) < δ + η, there exists n0 such that ${\phi }^{{n}_{0}}\left(d\left(x,y\right)\right)<\eta$ < η. Since limn→∞φn(d(x0,x1)) = η, there exists m0 such that η < φm(d(x0,x1)) < δ + η, for all m > m0. Thus, we conclude that ${\phi }^{{m}_{0}+{n}_{0}}\left(d\left({x}_{0},{x}_{1}\right)\right)<\eta$, a contradiction. So limn→∞φn(d(x0,x1)) = 0, that is, lim n →∞d(x n , x n+ 1) = 0.

Next, we claim that {x n } is a Cauchy sequence. We claim that the following result holds:

for each ε > 0, there is n0(ε) such that for all m, n ≥ n0(ε),

$d\left({x}_{m},{x}_{n}\right)<\epsilon ,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(**\right)$

We shall prove (**) by contradiction. Suppose that (**) is false. Then there exists some ε > 0 such that for all p , there are m p , n p with m p > n p ≥ p satisfying:

1. (i)

$d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)\ge \epsilon$, and

2. (ii)

m p is the smallest number greater than n p such that the condition (i) holds.

Since

$\begin{array}{ll}\hfill \epsilon & \le d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({x}_{{m}_{p}},{x}_{{m}_{p-1}}\right)+d\left({x}_{{m}_{p-1}},{x}_{{n}_{p}}\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({x}_{{m}_{p}},{x}_{{m}_{p-1}}\right)+\epsilon ,\phantom{\rule{2em}{0ex}}\end{array}$

hence we conclude $\underset{p\to \infty }{\text{lim}}d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)=\epsilon$. Since

$d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)-d\left({x}_{{m}_{p}},{x}_{{m}_{p+1}}\right)\le d\left({x}_{{m}_{p}+1},{x}_{{n}_{p}}\right)\le d\left({x}_{{m}_{p}},{x}_{{m}_{p}+1}\right)+d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right),$

we also conclude $\underset{p\to \infty }{\text{lim}}d\left({x}_{{m}_{p}+1},{x}_{{n}_{p}}\right)=\epsilon$. Thus, there exists i, 0 ≤ ik - 1 such that m p -n p + i = 1 mod k for infinitely many p. If i = 0, then we have that for such p,

$\begin{array}{ll}\hfill \epsilon & \le d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({x}_{{m}_{p}},{x}_{{m}_{p}+1}\right)+d\left({x}_{{m}_{p}+1},{x}_{{n}_{p}+1}\right)+d\left({x}_{{n}_{p}+1},{x}_{{n}_{p}}\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({x}_{{m}_{p}},{x}_{{m}_{p}+1}\right)+\phi \left(d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)\right)+d\left({x}_{{n}_{p}+1},{x}_{{n}_{p}}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Letting p → ∞. Then by the condition (φ2)-(a) of φ-mapping, we have

$\epsilon \le 0+\underset{p\to \infty }{\text{lim}}\phi \left(d\left({x}_{{m}_{p}},{x}_{{n}_{p}}\right)\right)+0<\epsilon ,$

a contradiction. The case i ≠ 0 similar. Thus, {x n } is a Cauchy sequence. Since X is complete, there exists $\nu \in {\cup }_{i=1}^{k}{A}_{i}$ such that limn→∞x n = ν. Now for all i = 0, 1, 2, ..., k - 1, {fkn-ix} is a sequence in A i and also all converge to ν. Since A i is closed for all i = 1, 2,..., k, we conclude $\nu \in {\cup }_{i=1}^{k}{A}_{i}$, and also we conclude that ${\cap }_{i=1}^{k}{A}_{i}\ne \varphi$. By the condition (φ2)-(b) of φ-mapping, we have

$\begin{array}{ll}\hfill d\left(\nu ,f\nu \right)& =\underset{n\to \infty }{\text{lim}}d\left({f}^{kn}x,f\nu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\phi \left(d\left({f}^{kn-1}x,\nu \right)\right)=0,\phantom{\rule{2em}{0ex}}\end{array}$

hence ν is a fixed point of f. Let μ be another fixed point of f. Since f is a generalized cyclic weaker Meir-Keeler φ-contraction, we have

$\begin{array}{ll}\hfill d\left(\nu ,\mu \right)& =d\left(\nu ,f\mu \right)=\underset{n\to \infty }{\text{lim}}d\left({f}^{kn}x,f\mu \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{\text{lim}}\phi \left(d\left({f}^{kn-1}x,\mu \right)\right)\phantom{\rule{2em}{0ex}}\\

a contradiction. Therefore, μ = ν. Thus ν is a unique fixed point of f.

Example 8 Let X = 3 and we define d : X × X+ by

$d\left(x,y\right)=\left|{x}_{1}-{y}_{1}\right|+\left|{x}_{2}-{y}_{2}\right|+\left|{x}_{3}-{y}_{3}\right|,\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}x=\left({x}_{1},{x}_{2},{x}_{3}\right),y=\left({y}_{1},{y}_{2},{y}_{3}\right)\in X,$

and let A = {(x,0,0): x }, B = {(0,y,0): y }, C = {(0,0, z): z } be three subsets of X. Define f : A B C → A B C by

$\begin{array}{ll}\hfill f\left(\left(x,0,0\right)\right)& =\left(0,\frac{1}{4}x,0\right);\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}x\in ℝ;\phantom{\rule{2em}{0ex}}\\ \hfill f\left(\left(0,y,0\right)\right)& =\left(0,0,\frac{1}{4}y\right);\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}y\in ℝ;\phantom{\rule{2em}{0ex}}\\ \hfill f\left(\left(0,0,z\right)\right)& =\left(\frac{1}{4}z,0,0\right);\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}z\in ℝ.\phantom{\rule{2em}{0ex}}\end{array}$

and define φ : ++ by

$\phi \left(t\right)=\frac{1}{3}t\phantom{\rule{1em}{0ex}};for\phantom{\rule{2.77695pt}{0ex}}t\in {ℝ}^{+}.$

Then f is a generalized cyclic weaker Meir-Keeler φ -contraction and (0, 0, 0) is the unique fixed point.

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## Acknowledgements

The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the article.

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Correspondence to Chi-Ming Chen.

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Chen, CM. Fixed point theorems for cyclic Meir-Keeler type mappings in complete metric spaces. Fixed Point Theory Appl 2012, 41 (2012). https://doi.org/10.1186/1687-1812-2012-41