# Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces

## Abstract

In this article, we introduce the notions of cyclic weaker ϕ φ-contractions and cyclic weaker (ϕ, φ)-contractions in complete metric spaces and prove two theorems which assure the existence and uniqueness of a fixed point for these two types of contractions. Our results generalize or improve many recent fixed point theorems in the literature.

MSC: 47H10; 54C60; 54H25; 55M20.

## 1 Introduction and preliminaries

Throughout this article, by +, we denote the sets of all nonnegative real numbers and all real numbers, respectively, while is the set of all natural numbers. Let (X, d) be a metric space, D be a subset of X and f: DX be a map. We say f is contractive if there exists α [0,1) such that for all x, y D,

$d\left(fx,fy\right)\le \alpha \cdot d\left(x,y\right).$

The well-known Banach's fixed point theorem asserts that if D = X, f is contractive and (X, d) is complete, then f has a unique fixed point in X. It is well known that the Banach contraction principle  is a very useful and classical tool in nonlinear analysis. In 1969, Boyd and Wong  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 Φ: [0, ∞) → [0, ∞) such that

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

Generalization of the above Banach contraction principle has been a heavily investigated branch research. (see, e.g., [3, 4]). In 2003, Kirk et al.  introduced the following notion of cyclic representation.

Definition 1  Let X be a nonempty set, m and f: XX an operator. Then $X={\cup }_{i=1}^{m}{A}_{i}$ is called a cyclic representation of X with respect to f if

1. (1)

A i , i = 1, 2,..., m are nonempty subsets of X;

2. (2)

f (A1) A2, f (A2) A3,..., f (Am-1) A m , f (A m ) A1.

Kirk et al.  also proved the below theorem.

Theorem 1  Let (X, d) be a complete metric space, m , A1, A2,..., A m , closed nonempty subsets of X and $X={\cup }_{i=1}^{m}{A}_{i}$. Suppose that f satisfies the following condition.

$d\left(fx,fy\right)\le \psi \left(d\left(x,y\right)\right),\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{1em}{0ex}}x\in {A}_{i},\phantom{\rule{1em}{0ex}}y\in {A}_{i+1},\phantom{\rule{1em}{0ex}}i\in \left\{1,2,...,m\right\},$

where ψ: [0, ∞) → [0, ∞) is upper semi-continuous from the right and 0 ≤ ψ(t) < t for t > 0. Then, f has a fixed point $z\in {\cap }_{i=1}^{n}{A}_{i}$.

Recently, the fixed theorems for an operator f: XX that defined on a metric space X with a cyclic representation of X with respect to f had appeared in the literature. (see, e.g., ). In 2010, Pǎcurar and Rus  introduced the following notion of cyclic weaker φ-contraction.

Definition 2  Let (X, d) be a metric space, m , A1, A2,...,A m closed nonempty subsets of X and $X={\cup }_{i=1}^{m}{A}_{i}$. An operator f: XX is called a cyclic weaker φ-contraction if

1. (1)

$X={\cup }_{i=1}^{m}{A}_{i}$ is a cyclic representation of X with respect to f;

2. (2)

there exists a continuous, non-decreasing function φ: [0, ∞) → [0, ∞) with φ(t) > 0 for t (0, ∞) and φ(0) = 0 such that

$d\left(fx,fy\right)\le d\left(x,y\right)-\phi \left(d\left(x,y\right)\right),$

for any x A i , y Ai+1, i = 1,2,...,m where Am+1= A1.

And, Pǎcurar and Rus  proved the below theorem.

Theorem 2  Let (X, d) be a complete metric space, m , A1, A2,..., A m closed nonempty subsets of X and $X={\cup }_{i=1}^{m}{A}_{i}$. Suppose that f is a cyclic weaker φ-contraction. Then, f has a fixed point $z\in {\cap }_{i=1}^{n}{A}_{i}$.

In this article, we also recall the notion of Meir-Keeler function (see ). A function ϕ: [0, ∞) → [0, ∞) is said to be a Meir-Keeler function if for each η > 0, there exists δ > 0 such that for t [0, ∞) with ηt < η + δ, we have ϕ (t) < η. We now introduce the notion of weaker Meir-Keeler function ϕ: [0, ∞) → [0,∞), as follows:

Definition 3 We call ϕ: [0, ∞) → [0, ∞) a weaker Meir-Keeler function if for each η > 0, there exists δ > 0 such that for t [0, ∞) with ηt < η + δ, there exists n0 such that ${\varphi }^{{n}_{0}}\left(t\right)<\eta$.

## 2 Fixed point theory for the cyclic weaker ϕ○φ-contractions

The main purpose of this section is to present a generalization of Theorem 1. In the section, we let ϕ: [0, ∞) → [0, ∞) be a weaker Meir-Keeler function satisfying the following conditions:

• (ϕ1) ϕ(t) > 0 for t > 0 and ϕ (0) = 0;

• (ϕ2) for all t (0, ∞), {ϕn(t)}nis decreasing;

• (ϕ3) for t n [0, ∞), we have that

1. (a)

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

2. (b)

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

And, let φ: [0, ∞) → [0, ∞) be a non-decreasing and continuous function satisfying

• (φ1) φ(t) > 0 for t > 0 and φ(0) = 0;

• (φ2) φ is subadditive, that is, for every μ1, μ2 [0, ∞), φ( μ1 + μ2) ≤ φ(μ1) + φ(μ2);

• (φ3) for all t (0, ∞), limn→∞t n = 0 if and only if limn→∞φ(t n ) = 0.

We state the notion of cyclic weaker ϕ φ-contraction, as follows:

Definition 4 Let (X, d) be a metric space, m , A1, A2,..., A m nonempty subsets of X and $X={\cup }_{i=1}^{m}{A}_{i}$. An operator f: XX is called a cyclic weaker ϕ φ-contraction if

1. (i)

$X={\cup }_{i=1}^{m}{A}_{i}$ is a cyclic representation of X with respect to f;

2. (ii)

for any x A i , y Ai+1, i = 1, 2,..., m,

$\phi \left(d\left(fx,fy\right)\right)\le \varphi \left(\phi \left(d\left(x,y\right)\right)\right),$

where Am+1= A1.

Theorem 3 Let (X, d) be a complete metric space, m , A1, A2, ..., A m nonempty subsets of X and $X={\cup }_{i=1}^{m}{A}_{i}$. Let f: XX be a cyclic weaker ϕ φ-contraction. Then, f has a unique fixed point $z\in {\cap }_{i=1}^{m}{A}_{i}$.

Proof. Given x0 and let xn+1= fx n = fn+1x0, for n {0}. If there exists n0 {0} such that ${x}_{{n}_{0}+1}={x}_{{n}_{0}}$, then we finished the proof. Suppose that xn+1x n for any n {0}. Notice that, for any n > 0, there exists i n {1,2,...,m} such that ${x}_{n-1}\in {A}_{{i}_{n}}$ and ${x}_{n}\in {A}_{{i}_{n}+1}$. Since f: XX is a cyclic weaker ϕ φ-contraction, we have that for all n

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

and so

$\begin{array}{c}\phi \left(d\left({x}_{n},{x}_{n+1}\right)\right)\le \varphi \left(\phi \left(d\left({x}_{n-1},{x}_{n}\right)\right)\right)\\ \le \varphi \left(\varphi \left(\phi \left(d\left({x}_{n-2},{x}_{n-1}\right)\right)\right)={\varphi }^{2}\left(\phi \left(\left(d\left({x}_{n-2},{x}_{n-1}\right)\right)\right)\\ \le \dots \dots \\ \le {\varphi }^{n}\left(\phi \left(d\left({x}_{0},{x}_{1}\right)\right)\right).\end{array}$

Since {ϕn(φ(d(x0, x 1)))}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 function ϕ, there exists δ > 0 such that for x0, x1 X with ηφ(d(x0, x1)) < δ + η, there exists n0 such that ${\varphi }^{{n}_{0}}\left(\phi \left(d\left({x}_{0},{x}_{1}\right)\right)\right)<\eta$. Since limn→∞ϕn(φ(d(x0, x1))) = η, there exists p0 such that ηϕp(φ(d(x0, x1)) < δ + η, for all pp0. Thus, we conclude that ${\varphi }^{{p}_{0}+{n}_{0}}\left(\phi \left(d\left({x}_{0},{x}_{1}\right)\right)\right)<\eta$. So we get a contradiction. Therefore limn→∞ϕn(φ(d(x0, x1))) = 0, that is,

$\underset{n\to \infty }{\text{lim}}\phi \left(d\left({x}_{n},{x}_{n+1}\right)\right)=0.$

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

Claim: for each ε > 0, there is n0(ε) such that for all p, qn0(ε),

$\phi \left(d\left({x}_{p},{x}_{q}\right)\right)<\epsilon ,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}\left(*\right)$

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

1. (i)

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

2. (ii)

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

Since

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

hence we conclude $\underset{p\to \infty }{\text{lim}}\phi \left(d\left({x}_{{p}_{n}},{x}_{{q}_{n}}\right)\right)=\epsilon$. Since φ is subadditive and nondecreasing, we conclude

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

and so

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

Letting n → ∞, we also have

$\underset{n\to \infty }{\text{lim}}\phi \left(d\left({x}_{{p}_{n}+1},{x}_{{q}_{n}}\right)\right)=\epsilon .$

Thus, there exists i, 0 ≤ im - 1 such that p n - q n + i = 1 mod m for infinitely many n. If i = 0, then we have that for such n,

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

Letting n → ∞. Then by, we have

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

a contradiction. Therefore $\underset{n\to \infty }{\text{lim}}\phi \left(d\left({x}_{{p}_{n}},{x}_{{q}_{n}}\right)\right)=0$, by the condition (φ3), we also have $\underset{n\to \infty }{\text{lim}}d\left({x}_{{p}_{n}},{x}_{{q}_{n}}\right)=0$. The case i ≠ 0 is similar. Thus, {x n } is a Cauchy sequence. Since X is complete, there exists $\nu \in {\cup }_{i=1}^{m}{A}_{i}$ such that limn→∞x n = ν. Now for all i = 0, 1, 2,..., m - 1, {fx mn-i } is a sequence in A i and also all converge to ν. Since A i is clsoed for all i = 1, 2,..., m, we conclude $\nu \in {\cup }_{i=1}^{m}{A}_{i}$, and also we conclude that ${\cap }_{i=1}^{m}{A}_{i}\ne \varphi$. Since

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

hence φ(d(ν, fν)) = 0, that is, d(ν, fν) = 0, 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 $\mu ,\nu \in {\cap }_{i=1}^{n}{A}_{i}$. Since f is a cyclic weaker ϕ φ-contraction, we have

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

and this is a contradiction unless φ(d(ν, μ)) = 0, that is, μ = ν. Thus ν is a unique fixed point of f.

Example 1 Let X = 3 and we define d: X × X → [0,∞) by d(x,y) = |x1-y1 |+| x2-y2 |+| x3-y3|, for x = (x1, x2, x3), y = (y1, y2, y3) 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

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

We define φ: [0, ∞) → [0, ∞) by

$\varphi \left(t\right)=\frac{1}{3}t\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}t\in \left[0,\infty \right),$

and φ: [0, ∞) → [0, ∞) by

$\phi \left(t\right)=\frac{1}{2}t\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}t\in \left[0,\infty \right).$

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

## 3 Fixed point theory for the cyclic weaker (ϕ, φ-contractions

The main purpose of this section is to present a generalization of Theorem 2. In the section, we let ϕ: [0, ∞) → [0, ∞) be a weaker Meir-Keeler function satisfying the following conditions:

• (ϕ1) ϕ (t) > 0 for t > 0 and ϕ(0) = 0;

• (ϕ2) for all t (0, ∞), {ϕn(t)}nis decreasing;

• (ϕ3) for t n [0, ∞), if limn→∞t n = γ, then limn→∞ϕ(t n ) ≤ γ.

And, let φ: [0, ∞) → [0, ∞) be a non-decreasing and continuous function satisfying φ(t) > 0 for t > 0 and φ(0) = 0.

We now state the notion of cyclic weaker (ϕ, φ)-contraction, as follows:

Definition 5 Let (X, d) be a metric space, m , A1, A2,..., A m nonempty subsets of X and $X={\cup }_{i=1}^{m}{A}_{i}$. An operator f: XX is called a cyclic weaker (ϕ,φ)-contraction if

1. (i)

$X={\cup }_{i=1}^{m}{A}_{i}$ is a cyclic representation of X with respect to f;

2. (ii)

for any x A i , y Ai+1, i = 1, 2,..., m,

$d\left(fx,fy\right)\le \varphi \left(d\left(x,y\right)\right)-\phi \left(d\left(x,y\right)\right),$

where Am+ 1= A1.

Theorem 4 Let (X, d) be a complete metric space, m , A1, A2,..., A m nonempty subsets of X and $X={\cup }_{i=1}^{m}{A}_{i}$. Let f: XX be a cyclic weaker (ϕ, φ)-contraction. Then f has a unique fixed point $z\in {\cap }_{i=1}^{m}{A}_{i}$.

Proof. Given x0 and let xn+1= fx n = fn+1x0, for n {0}. If there exists n {0} such that ${x}_{{n}_{0}+1}={x}_{{n}_{0}}$, then we finished the proof. Suppose that xn+ 1x n for any n {0}. Notice that, for any n > 0, there exists i n {1,2,...,m} such that ${x}_{n-1}\in {A}_{{i}_{n}}$ and ${x}_{n}\in {A}_{{i}_{n}+1}$. Since f: XX is a cyclic weaker (ϕ, φ)-contraction, we have that n

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

and so

$\begin{array}{c}d\left({x}_{n},{x}_{n+1}\right)\le \varphi \left(d\left({x}_{n-1},{x}_{n}\right)\right)\\ \le \varphi \left(\varphi \left(d\left({x}_{n-2},{x}_{n-1}\right)\right)={\varphi }^{2}\left(d\left({x}_{n-2},{x}_{n-1}\right)\right)\\ \le \dots \dots \\ \le {\varphi }^{n}\left(d\left({x}_{0},{x}_{1}\right)\right).\end{array}$

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 function ϕ, there exists δ > 0 such that for x0, x1 X with ηd(x0, x1) < δ + η, there exists n0 such that ${\varphi }^{{n}_{0}}\left(d\left({x}_{0},{x}_{1}\right)\right)<\eta$. Since limn→∞, ϕn(d(x0, x1)) = η, there exists p0 such that ηϕp(d(x0, x1)) < δ + η, for all pp0. Thus, we conclude that ${\varphi }^{{p}_{0}+{n}_{0}}\left(d\left({x}_{0},{x}_{1}\right)\right)<\eta$. So we get a contradiction. Therefore limn→∞ϕn(d(x0, x1)) = 0, that is,

$\underset{n\to \infty }{\text{lim}}d\left({x}_{n},{x}_{n+1}\right)=0.$

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

Claim: For every ε > 0, there exists n such that if p, qn with p-q = 1 mod m, then d(x p , x q ) < ε.

Suppose the above statement is false. Then there exists ϵ > 0 such that for any n , there are p n , q n with p n > q n n with p n - q n = 1 mod m satisfying

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

Now, we let n > 2m. Then corresponding to q n n use, we can choose p n in such a way, that it is the smallest integer with p n > q n n satisfying p n - q n = 1 mod m and $d\left({x}_{{q}_{n}},{x}_{{p}_{n}}\right)\ge \epsilon$. Therefore $d\left({x}_{{q}_{n}},{x}_{{p}_{n}-m}\right)\le \epsilon$ and

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

Letting n → ∞ , we obtain that

$\underset{n\to \infty }{\text{lim}}d\left({x}_{{q}_{n}},{x}_{{p}_{n}}\right)=\epsilon .$

On the other hand, we can conclude that

$\begin{array}{l}\epsilon \le d\left({x}_{{q}_{n}},{x}_{{p}_{n}}\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({x}_{{q}_{n}},{x}_{{q}_{n+1}}\right)+d\left({x}_{{q}_{n+1}},{x}_{{p}_{n+1}}\right)+d\left({x}_{{x}_{{p}_{n+1}},{p}_{n}}\right)\phantom{\rule{2em}{0ex}}\\ \le d\left({x}_{{q}_{n}},{x}_{{q}_{n+1}}\right)+d\left({x}_{{q}_{n+1}},{x}_{{q}_{n}}\right)+d\left({x}_{{q}_{n}},{x}_{{p}_{n}}\right)+d\left({x}_{{p}_{n}},{x}_{{p}_{n+1}}\right)+d\left({x}_{{x}_{{p}_{n+1}},pn}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Letting n → ∞, we obtain that

$\underset{n\to \infty }{\text{lim}}d\left({x}_{{q}_{n+1}},{x}_{{p}_{n+1}}\right)=\epsilon .$

Since ${x}_{{q}_{n}}$ and ${x}_{{p}_{n}}$ lie in different adjacently labeled sets A i and Ai+1for certain 1 ≤ im, by using the fact that f is a cyclic weaker (ϕ, φ)-contraction, we have

$d\left({x}_{{q}_{n+1}},{x}_{{p}_{n+1}}\right)=d\left(f{x}_{{q}_{n}},f{x}_{{p}_{n}}\right)\le \varphi \left(d\left({x}_{{q}_{n}},{x}_{{p}_{n}}\right)\right)-\phi \left(d\left({x}_{{q}_{n}},{x}_{{p}_{n}}\right)\right).$

Letting n → ∞, by using the condition ϕ3 of the function ϕ, we obtain that

$\epsilon \le \epsilon -\phi \left(\epsilon \right),$

and consequently, φ (ϵ) = 0. By the definition of the function φ, we get ϵ = 0 which is contraction. Therefore, our claim is proved.

In the sequel, we shall show that {x n } is a Cauchy sequence. Let ε > 0 be given. By our claim, there exists n1 such that if p, qn1 with p - q = 1 mod m, then

$d\left({x}_{p},{x}_{q}\right)\le \frac{\epsilon }{2}.$

Since limn→∞d(x n , xn+1) = 0, there exists n2 such that

$d\left({x}_{n},{x}_{n+1}\right)\le \frac{\epsilon }{2m},$

for any nn2.

Let p, q ≥ max{n 1, n2} and p > q. Then there exists k {1, 2,..., m} such that p -q = k mod m. Therefore, p - q + j = 1 mod m for j = m - k + 1, and so we have

$\begin{array}{ll}\hfill d\left({x}_{q},{x}_{p}\right)& \le d\left({x}_{q},{x}_{p+j}\right)+d\left({x}_{p+j},{x}_{p+j-1}\right)+\cdots +d\left({x}_{p+1},{x}_{p}\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{\epsilon }{2}+j×\frac{\epsilon }{2m}\phantom{\rule{2em}{0ex}}\\ \le \frac{\epsilon }{2}+m×\frac{\epsilon }{2m}=\epsilon .\phantom{\rule{2em}{0ex}}\end{array}$

Thus, {x n } is a Cauchy sequence. Since X is complete, there exists $\nu \in {\cup }_{i=1}^{m}{A}_{i}$ such that limn→∞x n = ν. Since $X={\cup }_{i=1}^{m}{A}_{i}$ is a cyclic representation of X with respect to f, the sequence {x n } has infinite terms in each A i for i {1,2,...,m}. Now for all i = 1,2,...,m, we may take a subsequence $\left\{{x}_{{n}_{k}}\right\}$ of {x n } with ${x}_{{n}_{k}}\in {A}_{i-1}$ and also all converge to ν. Since

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

Letting k → ∞ , we have

$d\left(\nu ,f\nu \right)\le 0,$

and so ν = fν.

Finally, to prove the uniqueness of the fixed point, let μ be the another fixed point of f. By the cyclic character of f, we have $\mu ,\nu \in {\cap }_{i=1}^{n}{A}_{i}$. Since f is a cyclic weaker (ϕ, φ)-contraction, we have

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

and we can conclude that

$\phi \left(d\left(\nu ,\mu \right)\right)=0.$

So we have μ = ν. We complete the proof.

Example 2 Let X = [-1,1] with the usual metric. Suppose that A1 = [-1,0] = A3 and A2 = [0,1] = A4. Define f: XX by $f\left(x\right)=\frac{-x}{6}$ for all x X, and let ϕ, φ: [0,∞) → [0, ∞) be $\varphi \left(t\right)=\frac{1}{2},\phi \left(t\right)=\frac{t}{4}$. Then f is a cyclic weaker (ϕ, φ)-contraction and 0 is the unique fixed point.

Example 3 Let X = + with the metric d:X × X+ given by

$d\left(x,y\right)=\text{max}\left\{x,y\right\},\phantom{\rule{1em}{0ex}}for\phantom{\rule{1em}{0ex}}x,y\in X.$

Let A1 = A2 = ... = A m = +. Define f: XX by

$f\left(x\right)=\frac{{x}^{2}}{77}\phantom{\rule{1em}{0ex}}for\phantom{\rule{1em}{0ex}}x\in X,$

and let ϕ, φ: [0, ∞) → [0,∞) be $\phi \left(t\right)=\frac{{t}^{3}}{2\left(t+2\right)}$ and

$\varphi \left(t\right)=\left\{\begin{array}{cc}\frac{2{t}^{3}}{3t+8},\hfill & \hfill if\phantom{\rule{0.3em}{0ex}}t\ge 1;\hfill \\ \frac{{t}^{2}}{2},\hfill & \hfill if\phantom{\rule{2.77695pt}{0ex}}t<1.\hfill \end{array}\right\$

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

Example 4 Let X = 3 and we define d: X × X → [0, ∞) by

$d\left(x,y\right)=\text{max}\left\{\left|{x}_{1}-{y}_{1}\right|,\left|{x}_{2}-{y}_{2}\right|,\left|{x}_{3}-{y}_{3}\right|,\right\}$

for x = (x1,x2,x3), y = (y1, y2, y3) X, and let A = {(x,0,0): x [0,1]}, B = {(0,y,0): y [0,1]}, C = {(0,0, z): z [0,1]} be three subsets of X.

Define f: A B CA B C by

$\begin{array}{ll}\hfill f\left(\left(x,0,0\right)\right)& =\left(0,\frac{1}{8}{x}^{2},0\right);\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}x\in \left[0,1\right];\phantom{\rule{2em}{0ex}}\\ \hfill f\left(\left(0,y,0\right)\right)& =\left(0,0,\frac{1}{8}{y}^{2}\right);\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}y\in \left[0,1\right];\phantom{\rule{2em}{0ex}}\\ \hfill f\left(\left(0,0,z\right)\right)& =\left(\frac{1}{8}{z}^{2},0,0\right);\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}z\in \left[0,1\right].\phantom{\rule{2em}{0ex}}\end{array}$

We define φ: [0, ∞) → [0,∞) by

$\varphi \left(t\right)=\frac{{t}^{2}}{t+1}\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}t\in \left[0,\infty \right),$

and φ: [0, ∞) → [0,∞) by

$\phi \left(t\right)=\frac{{t}^{2}}{t+2}\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{2.77695pt}{0ex}}\in \left[0,\infty \right).$

Then f is a cyclic weaker (ϕ, 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 theory for the cyclic weaker Meir-Keeler function in complete metric spaces. Fixed Point Theory Appl 2012, 17 (2012). https://doi.org/10.1186/1687-1812-2012-17

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### Keywords

• fixed point theory
• weaker Meir-Keeler function
• cyclic weaker ϕ φ-contraction
• cyclic weaker (ϕ, φ)-contraction 