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 ℝ⁺, ℝ we denote the sets of all nonnegative real numbers and all real numbers, respectively, while ℕ is the set of all natural numbers. Let (X, d) be a metric space, D be a subset of X and f: D → X 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 [1] is a very useful and classical tool in nonlinear analysis. In 1969, Boyd and Wong [2] 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 Φ: [0, ∞) → [0, ∞) such that

$$d\left(fx,fy\right)\le \Phi \left(d\left(x,y\right)\right)\mathsf{\text{for}}\mathsf{\text{all}}x,y\in X.$$

Generalization of the above Banach contraction principle has been a heavily investigated branch research. (see, e.g., [3, 4]). In 2003, Kirk et al. [5] introduced the following notion of cyclic representation.

Definition 1 [5] Let X be a nonempty set, m ∈ ℕ and f: X → X an operator. Then $X={\cup }_{i=1}^m{A}_i$ is called a cyclic representation of X with respect to f if

• (1) A _i , i = 1, 2,..., m are nonempty subsets of X;

• (2) f (A₁) ⊂ A₂, f (A₂) ⊂ A₃,..., f (A_m-1) ⊂ A _m , f (A _m ) ⊂ A₁.

Kirk et al. [5] also proved the below theorem.

Theorem 1 [5] Let (X, d) be a complete metric space, m ∈ ℕ, A₁, A₂,..., A _m , closed nonempty subsets of X and $X={\cup }_{i=1}^m{A}_i$. Suppose that f satisfies the following condition.

$$d\left(fx,fy\right)\le \psi \left(d\left(x,y\right)\right),forallx\in A_i,y\in A_{i+1},i\in \left\{1,2,...,m\right\},$$

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: X → X 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., [6–10]). In 2010, Pǎcurar and Rus [7] introduced the following notion of cyclic weaker φ-contraction.

Definition 2 [7] Let (X, d) be a metric space, m ∈ ℕ, A₁, A₂,...,A _m closed nonempty subsets of X and $X={\cup }_{i=1}^m{A}_i$. An operator f: X → X is called a cyclic weaker φ-contraction if

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

• (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 ∈ A_i+1, i = 1,2,...,m where A_m+1= A₁.

And, Pǎcurar and Rus [7] proved the below theorem.

Theorem 2 [7] Let (X, d) be a complete metric space, m ∈ ℕ, 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 [11]). 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 η ≤ t < 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 η ≤ t < η + δ, there exists n₀ ∈ ℕ 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)}_n∈ℕis decreasing;

• (ϕ₃) for t _n ∈ [0, ∞), we have that

  • (a) if lim_n→∞t _n = γ > 0, then lim_n→∞ϕ (t _n ) < γ, and

  • (b) if lim_n→∞t _n = 0, then lim_n→∞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, ∞), lim_n→∞t _n = 0 if and only if lim_n→∞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 ∈ ℕ, A₁, A₂,..., A _m nonempty subsets of X and $X={\cup }_{i=1}^m{A}_i$. An operator f: X → X is called a cyclic weaker ϕ ○ φ-contraction if

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

• (ii) for any x ∈ A _i , y ∈ A_i+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 A_m+1= A₁.

Theorem 3 Let (X, d) be a complete metric space, m ∈ ℕ, A₁, A₂, ..., A _m nonempty subsets of X and $X={\cup }_{i=1}^m{A}_i$. Let f: X → X 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 x_n+1= fx _n = fⁿ⁺¹x₀, for n ∈ ℕ∪{0}. If there exists n₀ ∈ ℕ ∪ {0} such that $x_{n_0+1}=x_{n_0}$, then we finished the proof. Suppose that x_n+1≠ x _n for any n ∈ 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: X → X is a cyclic weaker ϕ ○ φ-contraction, we have that for all n ∈ ℕ

$$\begin{array}{ll}\hfill \phi \left(d\left(x_n,x_{n+1}\right)\right)& =\phi \left(d\left(f{x}_{n-1},f{x}_n\right)\right)\\ \le \varphi \left(\phi \left(d\left(x_{n-1},x_n\right)\right)\right),\end{array}$$

and so

$$\begin{array}{c}\phi (d(x_n,x_{n+1}))\le \varphi (\phi (d(x_{n-1},x_n)))\\ \le \varphi (\varphi (\phi (d(x_{n-2},x_{n-1})))={\varphi }^2(\phi ((d(x_{n-2},x_{n-1})))\\ \le \dots \dots \\ \le {\varphi }^n(\phi (d(x_0,x_1))).\end{array}$$

Since {ϕⁿ(φ(d(x₀, x 1)))}_n∈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 η ≤ φ(d(x₀, x₁)) < δ + η, there exists n₀ ∈ ℕ such that ${\varphi }^{n_0}\left(\phi \left(d\left(x_0,x_1\right)\right)\right)<\eta$. Since lim_n→∞ϕⁿ(φ(d(x₀, x₁))) = η, there exists p₀ ∈ ℕ such that η ≤ ϕ^p(φ(d(x₀, x₁)) < δ + η, for all p ≥ p₀. 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 lim_n→∞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) ∈ ℕ such that for all p, q ≥ n₀(ε),

$$\phi \left(d\left(x_p,x_q\right)\right)<\epsilon ,\left(*\right)$$

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

• (i) $\phi \left(d\left(x_{p_n},x_{q_n}\right)\right)\ge \epsilon$, and

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

Since

$$\begin{array}{ll}\hfill \epsilon & \le \phi \left(d\left(x_{p_n},x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n-1}\right)+d\left(x_{p_n-1},x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n-1}\right)\right)+\phi \left(d\left(x_{p_n-1},x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n-1}\right)\right)+\epsilon ,\end{array}$$

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

$$\begin{array}{ll}\hfill \phi \left(d\left(x_{p_n},x_{q_n}\right)\right)& \le \phi \left(d\left(x_{p_n},x_{q_n+1}\right)+d\left(x_{p_n+1},x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{q_n+1}\right)\right)+\phi \left(d\left(x_{p_n+1},x_{q_n}\right)\right),\end{array}$$

and so

$$\begin{array}{ll}\hfill \phi \left(d\left(x_{p_n},x_{q_n}\right)\right)-\phi \left(d\left(x_{p_n},x_{p_n+1}\right)\right)& \le \phi \left(d\left(x_{p_n+1},x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n+1}\right)+d\left(x_{p_n},x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n+1}\right)\right)+\phi \left(d\left(x_{p_n},x_{q_n}\right)\right).\end{array}$$

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 ≤ i ≤ m - 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,

$$\begin{array}{ll}\hfill \epsilon & \le \phi \left(d\left(x_{p_n},x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n+1}\right)+d\left(x_{p_n+1},{x_q}_{n+1}\right)+d\left(x_{q_n+1},x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n+1}\right)\right)+\phi \left(d\left(x_{p_n+1},x_{q_n+1}\right)\right)+\phi \left(d\left(x_{q_n+1},x_{q_n}\right)\right)\\ =\phi \left(d\left(x_{p_n},x_{p_n+1}\right)\right)+\phi \left(d\left(f{x}_{p_n},f{x}_{q_n}\right)\right)+\phi \left(d\left(x_{q_n+1},x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n+1}\right)\right)+\varphi \left(\phi \left(d\left(x_{p_n},x_{q_n}\right)\right)\right)+\phi \left(d\left(x_{q_n+1},x_{q_n}\right)\right).\end{array}$$

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 lim_n→∞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

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

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

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

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 = ℝ³ 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 ∈ ℝ}, B = {(0,y,0):y ∈ ℝ},C = {(0,0, z): z ∈ ℝ} be three subsets of X. Define f: A ∪ B ∪ C → A ∪ B ∪ C by

$$\begin{array}{ll}\hfill f\left(\left(x,0,0\right)\right)& =\left(0,\frac{1}{4}x,0\right);forallx\in ℝ;\\ \hfill f\left(\left(0,y,0\right)\right)& =\left(0,0,\frac{1}{4}y\right);forally\in ℝ;\\ \hfill f\left(\left(0,0,z\right)\right)& =\left(\frac{1}{4}z,0,0\right);forallz\in ℝ.\end{array}$$

We define φ: [0, ∞) → [0, ∞) by

$$\varphi \left(t\right)=\frac{1}{3}tfort\in \left[0,\infty \right),$$

and φ: [0, ∞) → [0, ∞) by

$$\phi \left(t\right)=\frac{1}{2}tfort\in \left[0,\infty \right).$$

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)}_n∈ℕis decreasing;

• (ϕ₃) for t _n ∈ [0, ∞), if lim_n→∞t _n = γ, then lim_n→∞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 ∈ ℕ, A₁, A₂,..., A _m nonempty subsets of X and $X={\cup }_{i=1}^m{A}_i$. An operator f: X → X is called a cyclic weaker (ϕ,φ)-contraction if

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

• (ii) for any x ∈ A _i , y ∈ A_i+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 A_{m+ 1}= A₁.

Theorem 4 Let (X, d) be a complete metric space, m ∈ ℕ, A₁, A₂,..., A _m nonempty subsets of X and $X={\cup }_{i=1}^m{A}_i$. Let f: X → X 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 x_n+1= fx _n = fⁿ⁺¹x₀, for n ∈ ℕ ∪{0}. If there exists n ∈ ℕ ∪{0} such that $x_{n_0+1}=x_{n_0}$, then we finished the proof. Suppose that x_{n+ 1}≠ x _n for any n ∈ 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: X → X is a cyclic weaker (ϕ, φ)-contraction, we have that n ∈ℕ

$$\begin{array}{ll}\hfill d\left(x_n,x_{n+1}\right)& =d\left(f{x}_{n-1},f{x}_n\right)\\ \le \varphi \left(d\left(x_{n-1},x_n\right)\right)-\phi \left(d\left(x_{n-1},x_n\right)\right)\\ \le \varphi \left(d\left(x_{n-1},x_n\right)\right),\end{array}$$

and so

$$\begin{array}{c}d(x_n,x_{n+1})\le \varphi (d(x_{n-1},x_n))\\ \le \varphi (\varphi (d(x_{n-2},x_{n-1}))={\varphi }^2(d(x_{n-2},x_{n-1}))\\ \le \dots \dots \\ \le {\varphi }^n(d(x_0,x_1)).\end{array}$$

Since {ϕⁿ(d(x₀, x₁))}_n∈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 η ≤ d(x₀, x₁) < δ + η, there exists n₀ ∈ ℕ such that ${\varphi }^{n_0}\left(d\left(x_0,x_1\right)\right)<\eta$. Since lim_n→∞, ϕⁿ(d(x₀, x₁)) = η, there exists p₀ ∈ ℕ such that η ≤ ϕ^p(d(x₀, x₁)) < δ + η, for all p ≥ p₀. 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 lim_n→∞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 ∈ ℕ such that if p, q ≥ n 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 ∈ℕ, there are p _n , q _n ∈ 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

$$\begin{array}{ll}\hfill \epsilon & \le d\left(x_{q_n},x_{p_n}\right)\\ \le d\left(x_{q_n},x_{p_n-m}\right)+\sum _{i=1}^{m}d\left(x_{p_{n-i}},x_{p_{n-i+1}}\right)\\ <\epsilon +\sum _{i=1}^{m}d\left(x_{p_{n-i}},x_{p_{n-i+1}}\right).\end{array}$$

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

$$\begin{array}{l}\epsilon \le d\left(x_{q_n},x_{p_n}\right)\\ \le d\left(x_{q_n},x_{q_{n+1}}\right)+d\left(x_{q_{n+1}},x_{p_{n+1}}\right)+d\left(x_{x_{p_{n+1}},p_n}\right)\\ \le d\left(x_{q_n},x_{q_{n+1}}\right)+d\left(x_{q_{n+1}},x_{q_n}\right)+d\left(x_{q_n},x_{p_n}\right)+d\left(x_{p_n},x_{p_{n+1}}\right)+d\left(x_{x_{p_{n+1}},pn}\right).\end{array}$$

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 A_i+1for certain 1 ≤ i ≤ m, 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 ∈ ℕ such that if p, q ≥ n₁ with p - q = 1 mod m, then

$$d\left(x_p,x_q\right)\le \frac{\epsilon }{2}.$$

Since lim_n→∞d(x _n , x_n+1) = 0, there exists n₂ ∈ ℕsuch that

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

for any n ≥ n₂.

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

$$\begin{array}{ll}\hfill d\left(x_q,x_p\right)& \le d\left(x_q,x_{p+j}\right)+d\left(x_{p+j},x_{p+j-1}\right)+\cdots +d\left(x_{p+1},x_p\right)\\ \le \frac{\epsilon }{2}+j\times \frac{\epsilon }{2m}\\ \le \frac{\epsilon }{2}+m\times \frac{\epsilon }{2m}=\epsilon .\end{array}$$

Thus, {x _n } is a Cauchy sequence. Since X is complete, there exists $\nu \in {\cup }_{i=1}^m{A}_i$ such that lim_n→∞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

$$\begin{array}{ll}\hfill d\left(x_{n_{k+1}},f\nu \right)& =d\left(f{x}_{n_k},f\nu \right)\\ \le \varphi \left(d\left(x_{n_k},\nu \right)\right)-\phi \left(d\left(x_{n_k},\nu \right)\right)\\ \le \varphi \left(d\left(x_{n_k},\nu \right)\right).\end{array}$$

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

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

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: X → X 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 = ℝ⁺ with the metric d:X × X → ℝ⁺ given by

$$d\left(x,y\right)=\text{max}\left\{x,y\right\},forx,y\in X.$$

Let A₁ = A₂ = ... = A _m = ℝ⁺. Define f: X → X by

$$f\left(x\right)=\frac{x^2}{77}forx\in X,$$

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

$$\varphi \left(t\right)=\left\{\begin{array}{cc}\frac{2t^3}{3t+8},\hfill & \hfill ift\ge 1;\hfill \\ \frac{t^2}{2},\hfill & \hfill ift<1.\hfill \end{array}\right.$$

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

Example 4 Let X = ℝ³ 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 ∪ C → A ∪ B ∪ C by

$$\begin{array}{ll}\hfill f\left(\left(x,0,0\right)\right)& =\left(0,\frac{1}{8}{x}^2,0\right);forallx\in \left[0,1\right];\\ \hfill f\left(\left(0,y,0\right)\right)& =\left(0,0,\frac{1}{8}{y}^2\right);forally\in \left[0,1\right];\\ \hfill f\left(\left(0,0,z\right)\right)& =\left(\frac{1}{8}{z}^2,0,0\right);forallz\in \left[0,1\right].\end{array}$$

We define φ: [0, ∞) → [0,∞) by

$$\varphi \left(t\right)=\frac{t^2}{t+1}fort\in \left[0,\infty \right),$$

and φ: [0, ∞) → [0,∞) by

$$\phi \left(t\right)=\frac{t^2}{t+2}fort\in \left[0,\infty \right).$$

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