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 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

$$d\left(fx,f^{2}x\right)\le k\cdot d\left(x,fx\right),\mathsf{\text{for}}\mathsf{\text{all}}x\in X\mathsf{\text{and}}k\in \left(0,1\right),$$

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 ∪ B → A ∪ 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 ∪ 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.

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 κ _x ∈ (0,1) such that

$$d\left(f^{2n}x,fy\right)\le k_x\cdot d\left(f^{2n-1}x,y\right),n\in \mathbb{N},y\in A.$$

(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 ∪ B → A ∪ 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 : ℝ⁺ → [0,1) that satisfy the simple condition:

$$\alpha \left(t_n\right)\to 1\Rightarrow t_n\to 0.$$

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

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

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 : X → X on a metric space is called Φ-contraction if there exists an upper semi-continuous function ψ : [0,∞)→ [0,∞) such that

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

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 : ℝ⁺ → ℝ⁺ is said to be a Meir-Keeler type mapping (see [6]), if for each η ∈ ℝ⁺, there exists δ > 0 such that for t ∈ ℝ⁺ with η ≤ t < η + δ, we have ψ(t) < η. Subsequently, some authors worked on this notion (for example, [7–10]). 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 : ℝ⁺ → [0,1) a stronger Meir-Keeler type mapping in X if the mapping ψ satisfies the following condition:

$$\forall \eta >0\exists \delta >0\exists {\gamma }_{\eta }\in \left[0,1\right)\forall x,y\in X\left(\eta \le d\left(x,y\right)<\delta +\eta \Rightarrow \psi \left(d\left(x,y\right)\right)<{\gamma }_{\eta }\right).$$

Example 1 Let X = ℝ² and we define d : X × X → ℝ⁺ by

$$d\left(x,y\right)=\left|x_1-y_1\right|+\left|x_2-y_2\right|forallx=\left(x_1,x_2\right),y=\left(y_1,y_2\right)\in X.$$

If $\psi :{ℝ}^{+}\to \left[0,1\right),\psi \left(d\left(x,y\right)\right)=\frac{d\left(x,y\right)}{d\left(x,y\right)+1}$, 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 = ℝ² and we define d : X × X → ℝ⁺ by

$$d\left(x,y\right)=\left|x_1-y_1\right|+\left|x_2-y_2\right|forallx=\left(x_1,x_2\right),y=\left(y_1,y_2\right)\in X.$$

If φ : ℝ⁺ → ℝ⁺,

$$\phi \left(d\left(x,y\right)\right)=\left\{\begin{array}{cc}d\left(x,y\right)-1,\hfill & ifd\left(x,y\right)>1;\hfill \\ 0,\hfill & ifd\left(x,y\right)\le 1,\hfill \end{array}\right.$$

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 : ℝ⁺ → ℝ⁺. Then φ is called a weaker Meir-Keeler type mapping in X, if the mapping φ satisfies the following condition:

$$\forall \eta >0\exists \delta >0\forall x,y\in X\left(\eta \le d\left(x,y\right)<\delta +\eta \Rightarrow \exists n_0\in \mathbb{N}{\phi }^{n_0}\left(d\left(x,y\right)\right)<\eta \right).$$

Example 3 Let X = ℝ² and we define d : X × X → ℝ⁺ by

$$d\left(x,y\right)=\left|x_1-y_1\right|+\left|x_2-y_2\right|forallx=\left(x_1,x_2\right),y=\left(y_1,y_2\right)\in X.$$

If φ : ℝ⁺ → ℝ⁺, $\phi \left(d\left(x,y\right)\right)=\frac{1}{2}d\left(x,y\right)$, 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 = ℝ² and we define d : X × X → ℝ⁺ by

$$d\left(x,y\right)=\left|x_1-y_1\right|+\left|x_2-y_2\right|forallx=\left(x_1,x_2\right),y=\left(y_1,y_2\right)\in X.$$

If φ : ℝ⁺ → ℝ⁺,

$$\phi \left(d\left(x,y\right)\right)=\left\{\begin{array}{cc}0,\hfill & ifd\left(x,y\right)\le 1,\hfill \\ 2\cdot d\left(x,y\right),\hfill & if1<d\left(x,y\right)<2;\hfill \\ 1,\hfill & ifd\left(x,y\right)\ge 2,\hfill \end{array}\right.$$

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

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

(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 : ℝ⁺ → [0,1) be a stronger Meir-Keeler type mapping in X. Suppose f : A ∪ B → A ∪ B is a cyclic orbital stronger Meir-Keeler ψ-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 stronger Meir-Keeler ψ-contraction, there exists x ∈ A satisfying (2), and we also have that for each n ∈ ℕ,

$$\begin{array}{l}d(f^{2n}x,f^{2n+1}x)\le \psi d(f^{2n-1}x,f^{2n}x))\cdot d(f^{2n-1}x,f^{2n}x)\\ \le d(f^{2n-1}x,f^{2n}x),\end{array}$$

and

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

Generally, we have

$$d\left(f^{n}x,f^{n+1}x\right)\le d\left(f^{n-1}x,f^{n}x\right),n\in \mathbb{N}.$$

Thus the sequence {d(fⁿx, fⁿ⁺¹x)} is non-increasing and hence it is convergent. Let lim_n→∞d(fⁿx, fⁿ⁺¹x) = η. Then there exists κ₀ ∈ ℕ 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

$$\psi \left(d\left(f^{k_0+n}x,f^{k_0+n+1}x\right)\right)<{\gamma }_{\eta }\mathsf{\text{for}}\mathsf{\text{all}}n\in \mathbb{N}\cup \left\{0\right\}.$$

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

$$\begin{array}{ll}\hfill d\left(f^{k_0+n}x,f^{k_0+n+1}x\right)& \le \psi \left(d\left(f^{k_0+n-1}x,f^{k_0+n}x\right)\right)\cdot d\left(f^{k_0+n-1}x,f^{k_0+n}x\right)\\ <{\gamma }_{\eta }\cdot d\left(f^{k_0+n-1}x,f^{k_0+n}x\right),\end{array}$$

and it follows that for each n∈ℕ,

$$\begin{array}{ll}\hfill d\left(f^{k_0+n}x,f^{k_0+n+1}x\right)& <{\gamma }_{\eta }\cdot d\left(f^{k_0+n-1}x,f^{k_0+n}x\right)\\ <\cdots \cdots \\ <{\gamma }_{\eta }^n\cdot d\left(f^{k_0}x,f^{k_0+1}x\right).\end{array}$$

So

$$\underset{n\to \infty }{\text{lim}}d\left(f^{k_0+n}x,f^{k_0+n+1}x\right)=0,\mathsf{\text{since}}{\gamma }_{\eta }\in \left[0,1\right).$$

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 ∈ 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, f^mx) → 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 lim_n→∞fⁿx = ν. Now {f²ⁿx} is a sequence in A and {f^2n-1x} 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. Since

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

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 ∈ A ∩ B. Since f is a cyclic orbital stronger Meir-Keeler ψ-contraction, we have

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

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

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

$$d\left(x,y\right)=\left|x-y\right|,forx,y\in X.$$

Define f : X → X by

$$f\left(x\right)=\left\{\begin{array}{cc}0,\hfill & if0\le x<1;\hfill \\ \frac{1}{4},\hfill & ifx\ge 1.\hfill \end{array}\right.$$

and define ψ : ℝ⁺ → [0,1) by

$$\psi \left(t\right)=\left\{\begin{array}{cc}\frac{1}{3},\hfill & if0\le t\le 1;\hfill \\ \frac{t}{t+1},\hfill & ift>1.\hfill \end{array}\right.$$

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 : ℝ⁺ → ℝ⁺ 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 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;

(φ₃) {φⁿ(t)}_n∈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 : ℝ⁺ → ℝ⁺ in X such that

$$d\left(f^{2n}x,fy\right)\le \phi \left(d\left(f^{2n-1}x,y\right)\right),n\in \mathbb{N},y\in A.$$

(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 : ℝ⁺ → ℝ⁺ be a φ-mapping in X. Suppose f : A ∪ B → A ∪ 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 ∈ 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

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

Generally, we have

$$d\left(f^{n}x,f^{n+1}x\right)\le \phi \left(d\left(f^{n-1}x,f^{n}x\right)\right),n\in \mathbb{N}.$$

So we conclude that for each n ∈ ℕ

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

Since {φⁿ(d(x, fx))}_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 type mapping φ in X, there exists δ > 0 such that for x, y ∈ X with η ≤ d(x, y) < δ + η, there exists n₀ ∈ ℕ such that ${\phi }^{n_0}\left(d\left(x,y\right)\right)<\eta$. Since lim_n→∞φⁿ(d(x, fx)) = η, there exists m₀ ∈ ℕ such that η ≤ φ^m(d(x, fx)) < δ + η, for all m > m₀. Thus, we conclude that ${\phi }^{m_0+n_0}(d(x_0,x_1))<\eta$, and we get a contradiction. So lim _n→∞ φⁿ(d(x, fx)) = 0, that is, lim_n→∞d(fⁿx, fⁿ⁺¹x) = 0.

Next, we let c _m = d(f^mx, f^m+1x), and we claim that the following result holds:

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

$$d(f^{m}x,f^{m+1}x)<\epsilon .(*)$$

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:

• (i) m _p is even and n _p is odd,

• (ii) $d\left(f^{m_p}x,f^{n_p}x\right)\ge \epsilon$, and

• (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

$$\begin{array}{ll}\hfill \epsilon & \le d\left(f^{m_p}x,f^{n_p}x\right)\\ \le d\left(f^{m_p}x,f^{m_p+1}x\right)+d\left(f^{m_p+1}x,f^{n_p+1}x\right)+d\left(f^{n_p+1}x,f^{n_p}x\right)\\ \le d\left(f^{m_p}x,f^{m_p+1}x\right)+\phi \left(d\left(f^{m_p}x,f^{n_p}x\right)\right)+d\left(f^{n_p+1}x,f^{n_p}x\right).\end{array}$$

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 lim_n→∞f _n x = ν. Now {f²ⁿx} is a sequence in A and {f^2n-1x} 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 (φ₂)-(b) of φ-mapping, we have

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

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

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

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 : X × X → ℝ⁺ by

$$d\left(x,y\right)=\left|x-y\right|,forx,y\in X.$$

Define f : X → X by

$$f\left(x\right)=\left\{\begin{array}{cc}0,\hfill & if0\le x<1;\hfill \\ \frac{1}{4},\hfill & ifx\ge 1.\hfill \end{array}\right.$$

and define φ : ℝ⁺ → ℝ⁺ by

$$\phi \left(t\right)=\frac{1}{3}tfort\in {ℝ}^{+}.$$

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 : ℝ⁺ → [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 : ℝ⁺ → [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 ∈ ℕ. Since f is a generalized cyclic stronger Meir-Keeler ψ-contraction, we have that for each n ∈ ℕ

$$\begin{array}{ll}\hfill d\left(x_n,x_{n+1}\right)& =d\left(f^n{x}_0,f^{n+1}{x}_0\right)\\ \le \psi \left(d\left(f^{n-1}{x}_0,f^n{x}_0\right)\right)\cdot d\left(f^{n-1}{x}_0,f^n{x}_0\right)\\ \le d\left(f^{n-1}{x}_0,f^n{x}_0\right)=d\left(x_{n-1},x_n\right).\end{array}$$

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}) = η ≥ 0. Then there exists κ₀ ∈ ℕ 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 ∈ ℕ ∪ {0}. Thus, we can deduce that for each n ∈ ℕ

$$\begin{array}{ll}\hfill d\left(x_{k_0+n},x_{k_0+n+1}\right)& =d\left(f^{k_0+n}{x}_0,f^{k_0+n+1}{x}_0\right)\\ \le \psi \left(d\left(f^{k_0+n-1}{x}_0,f^{k_0+n}{x}_0\right)\right)\cdot d\left(f^{k_0+n-1}{x}_0,f^{k_0+n}{x}_0\right)\\ <{\gamma }_{\eta }d\left(f^{k_0+n-1}{x}_0,f^{k_0+n}{x}_0\right),\end{array}$$

and it follows that for each n ∈ ℕ

$$\begin{array}{ll}\hfill d\left(x_{k_0+n},x_{k_0+n+1}\right)& <{\gamma }_{\eta }d\left(f^{k_0+n-1}{x}_0,f^{k_0+n}{x}_0\right)\\ <\cdots \\ <{\gamma }_{\eta }^{n}d\left(f^{k_0+1}{x}_0,f^{k_0+2}{x}_0\right).\end{array}$$

So

$$\underset{n\to \infty }{\text{lim}}d\left(x_{k_0+n},x_{k_0+n+1}\right)=0,\mathsf{\text{since}}{\gamma }_{\eta }<1.$$

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 ∈ ℕ with m > n, we have

$$\begin{array}{l}d(x_{k_0+n},x_{k_0+m})=d(f^{k_0+n}{x}_0,f^{k_0+m}{x}_0)\\ \le {\displaystyle \sum _{i=n}^{m-1}d(f^{k_0+i}{x}_0,f^{k_0+i+1}{x}_0)}\\ <\frac{{\gamma }_{\eta }^{m-1}}{1-{\gamma }_{\eta }}d(f^{k_0}{x}_0,f^{k_0+1}{x}_0)),\end{array}$$

and hence d(fⁿx₀, f^mx₀) → 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 lim_n→∞fⁿx₀ = ν. Now for all i = 0,1, 2,..., k - 1, {f^kn-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

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

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

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

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

Example 7 Let X = ℝ³ and we define d : X × X → ℝ⁺by

$$d\left(x,y\right)=\left|x_1-y_1\right|+\left|x_2-y_2\right|+\left|x_3-y_3\right|,forx=\left(x_1,x_2,x_3\right),y=\left(y_1,y_2,y_3\right)\in X,$$

and let A = {(x, 0,0): x ∈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

$$\begin{array}{c}f\left(\left(x,0,0\right)\right)=\left(0,x,0\right);forallx\in ℝ;\\ f\left(\left(0,y,0\right)\right)=\left(0,0,y\right);forally\in ℝ;\\ f\left(\left(0,0,z\right)\right)=\left(z,0,0\right);forallz\in ℝ.\end{array}$$

and define ψ : ℝ⁺ → [0,1) by

$$\psi \left(t\right)=\frac{t}{t+1};fort\in {ℝ}^{+}.$$

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 : ℝ⁺ → ℝ⁺ 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 : ℝ⁺ → ℝ⁺ 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 ∈ ℕ. Since f is a generalized cyclic weaker Meir-Keeler φ-contraction, we have that for each n ∈ ℕ

$$\begin{array}{ll}\hfill d\left(x_n,x_{n+1}\right)& =d\left(f^n{x}_0,f^{n+1}{x}_0\right)\\ \le \phi \left(d\left(f^{n-1}{x}_0,f^n{x}_0\right)\right)=\phi \left(d\left(x_{n-1},x_n\right)\right)\\ \le \cdots \cdots \\ \le {\phi }^n\left(d\left(x_0,x_1\right)\right).\end{array}$$

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 type mapping φ in X, there exists δ > 0 such that for x, y ∈ X with η ≤ d(x, y) < δ + η, there exists n₀ ∈ ℕ such that ${\phi }^{n_0}\left(d\left(x,y\right)\right)<\eta$ < η. Since lim_n→∞φⁿ(d(x₀,x₁)) = η, there exists m₀ ∈ ℕ such that η < φ^m(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 lim_n→∞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) ∈ ℕ such that for all m, n ≥ n₀(ε),

$$d\left(x_m,x_n\right)<\epsilon ,\left(**\right)$$

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:

• (i) $d\left(x_{m_p},x_{n_p}\right)\ge \epsilon$, and

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

Since

$$\begin{array}{ll}\hfill \epsilon & \le d\left(x_{m_p},x_{n_p}\right)\\ \le d\left(x_{m_p},x_{m_{p-1}}\right)+d\left(x_{m_{p-1}},x_{n_p}\right)\\ \le d\left(x_{m_p},x_{m_{p-1}}\right)+\epsilon ,\end{array}$$

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 ≤ i ≤ k - 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,

$$\begin{array}{ll}\hfill \epsilon & \le d\left(x_{m_p},x_{n_p}\right)\\ \le d\left(x_{m_p},x_{m_p+1}\right)+d\left(x_{m_p+1},x_{n_p+1}\right)+d\left(x_{n_p+1},x_{n_p}\right)\\ \le d\left(x_{m_p},x_{m_p+1}\right)+\phi \left(d\left(x_{m_p},x_{n_p}\right)\right)+d\left(x_{n_p+1},x_{n_p}\right).\end{array}$$

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 lim_n→∞x _n = ν. Now for all i = 0, 1, 2, ..., k - 1, {f^kn-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

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

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

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

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

Example 8 Let X = ℝ³ and we define d : X × X → ℝ⁺ by

$$d\left(x,y\right)=\left|x_1-y_1\right|+\left|x_2-y_2\right|+\left|x_3-y_3\right|,forx=\left(x_1,x_2,x_3\right),y=\left(y_1,y_2,y_3\right)\in X,$$

and let A = {(x,0,0): x ∈ 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

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

and define φ : ℝ⁺ → ℝ⁺ by

$$\phi \left(t\right)=\frac{1}{3}t;fort\in {ℝ}^{+}.$$

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