Periodic points for the weak contraction mappings in complete generalized metric spaces

Abstract

In this article, we introduce the notions of (ϕ - φ)-weak contraction mappings and (ψ - φ)-weak contraction mappings in complete generalized metric spaces and prove two theorems which assure the existence of a periodic point for these two types of weak contraction.

Mathematical Subject Classification: 47H10; 54C60; 54H25; 55M20.

1 Introduction and preliminaries

Let (X, d) be a metric space, D 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

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

Generalization of the above Banach contraction principle has been a heavily investigated research branch. (see, e.g., [3, 4]).

In 2000, Branciari [5] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involing three terms instead of two. Later, many authors worked on this interesting space (e.g. [6–11]).

Let (X, d) be a generalized metric space. For γ > 0 and x ∈ X, we define

$$B_{\gamma }\left(x\right):=\left\{y\in X|d\left(x,y\right)<\gamma \right\}.$$

Branciari [5] also claimed that {B _γ (x): γ > 0, x ∈ X} is a basis for a topology on X, d is continuous in each of the coordinates and a generalized metric space is a Hausdorff space. We recall some definitions of a generalized metric space, as follows:

Definition 1 [5] Let X be a nonempty set and d : X × X → [0, ∞) be a mapping such that for all x, y ∈ X and for all distinct point u, v ∈ X each of them different from × and y, one has

(i) d(x, y) = 0 if and only if × = y;

(ii) d(x, y) = d(y, x);

(iii) d(x, y) ≤ d(x, u) + d(u, v) + d(v, y) (rectangular inequality).

Then (X, d) is called a generalized metric space (or shortly g.m.s).

We present an example to show that not every generalized metric on a set X is a metric on X.

Example 1 Let X = {t, 2t, 3t, 4t, 5t} with t > 0 is a constant, and we define d : X × X → [0, ∞) by

• (1) d(x, x) = 0, for all × ∈ X;

• (2) d(x, y) = d(y, x), for all x, y ∈ X;

• (3) d(t, 2t) = 3γ;

• (4) d(t, 3t) = d(2t, 3t) = γ;

• (5) d(t, 4t) = d(2t, 4t) = d(3t, 4t) = 2γ;

• (6) $d\left(t,5t\right)=d\left(2t,5t\right)=d\left(3t,5t\right)=\left(4t,5t\right)=\frac{3}{2}\gamma$,

where γ > 0 is a constant. Then (X, d) be a generalized metric space, but it is not a metric space, because

$$d\left(t,\mathsf{\text{2}}t\right)=\mathsf{\text{3}}\gamma >d\left(t,\mathsf{\text{3}}t\right)+d\left(\mathsf{\text{3}}t,\mathsf{\text{2}}t\right)=\mathsf{\text{2}}\gamma \mathsf{\text{.}}$$

Definition 2 [5] Let (X, d) be a g.m.s, {x _n } be a sequence in X and x ∈ X. We say that {x _n } is g.m.s convergent to × if and only if d(x _n , x) → 0 as n → ∞. We denote by x _n → x as n → ∞.

Definition 3 [5] Let (X, d) be a g.m.s, {x _n } be a sequence in X and x ∈ X. We say that {x _n } is g.m.s Cauchy sequence if and only if for each ε > 0, there exists $n_0\in \mathbb{N}$ such that d(x _m , x _n ) < ε for all n > m > n₀.

Definition 4 [5] Let (X, d) be a g.m.s. Then X is called complete g.m.s if every g.m.s Cauchy sequence is g.m.s convergent in X.

In this article, we also recall the notion of Meir-Keeler function (see [12]). 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. Generalization of the above function has been a heavily investigated research branch. Praticularly, in [13, 14], the authors proved the existence and uniqueness of fixed points for various Meir-Keeler type contractive functions. In this study, we introduce the below notions of the weaker Meir-Keeler function ϕ : [0, ∞) → [0, ∞) and stronger Meir-Keeler function ψ : [0, ∞) → [0, 1).

Definition 5 We call ϕ : [0, ∞) → [0, ∞) a weaker Meir-Keeler function if the function ϕ satisfies the following condition

$$\forall \eta >0\exists \delta >0\forall t\in \left[0,\infty \right)\left(\eta \le t<\delta +\eta \Rightarrow \exists n_0\in \mathbb{N}\varphi {\left(t\right)}^{n_0}<\eta \right).$$

The following provides an example of a weaker Meir-Keeler function which is not a Meir-Keeler function.

Example 2 Let $\varphi :{ℝ}^{+}\to {ℝ}^{+}$ be defined by

$$\varphi \left(t\right)=\left\{\begin{array}{cc}\hfill 0,\hfill & \hfill ift\le 1,\hfill \\ \hfill 3t,\hfill & \hfill if1<t<3,\hfill \\ \hfill 1,\hfill & \hfill ift\ge 3.\hfill \end{array}\right.$$

Then ϕ is a weaker Meir-Keeler function which is not a Meir-Keeler function.

Definition 6 We call ψ : [0, ∞) → [0, 1) a stronger Meir-Keeler function if the function ψ satisfies the following condition

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

The following provides an example of a stronger Meir-Keeler function.

Example 3 Let $\psi :{ℝ}^{+}\to \left[0,1\right)$ be defined by

$$\psi \left(d\left(x,y\right)\right)=\frac{2t}{3t+1}.$$

Then ψ is a stronger Meir-Keeler function.

The following provides an example of a Meir-Keeler function which is not a stronger Meir-Keeler function.

Example 4 Let $\phi :{ℝ}^{+}\to {ℝ}^{+}$ be defined by

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

Then φ is a Meir-Keeler function which is not a stronger Meir-Keeler function.

2 Main results

In the sequel, we let the function ϕ : [0, ∞) → [0, ∞) satisfies the following conditions:

(ϕ₁) ϕ : [0, ∞) → [0, ∞) is a weaker Meir-Keeler function;

(ϕ₂) ϕ(t) > 0 for t > 0 and ϕ(0) = 0;

(ϕ₃) for all t ∈ (0, ∞), ${\left\{{\varphi }^n\left(t\right)\right\}}_{n\in \mathbb{N}}$ is decreasing;

(ϕ₄) 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.

Let the function ψ : [0, ∞) → [0, 1) satisfies the following conditions:

(ψ₁) ψ : [0, ∞) → [0, 1) is a stronger Meir-Keeler function;

(ψ₂) ψ(t) > 0 for t > 0 and ϕ(0) = 0.

And, we let the function φ : [0, ∞) → [0, ∞) satisfies the following conditions:

(φ₁) for all t ∈ (0, ∞), lim_n→∞t _n = 0 if and only if lim_n→∞φ(t _n ) = 0;

(φ₂) φ(t) > 0 for t > 0 and φ(0) = 0;

(φ₃) φ is subadditive, that is, for every μ₁, μ₂ ∈ [0, ∞), φ(μ₁ + μ₂) ≤ φ(μ₁) + φ(μ₂).

Using the functions ϕ and φ, we first introduce the notion of the (ϕ-φ)-weak contraction mapping and prove a theorem which assures the existence of a periodic point for the (ϕ-φ)-weak contraction mapping.

Definition 7 Let (X, d) be a g.m.s, and let f : X → X be a function satisfying

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

(1)

for all x, y ∈ X. Then f is said to be a (ϕ - φ)-weak contraction mapping.

Theorem 1 Let (X, d) be a Hausdorff and complete g.m.s, and let f be a (ϕ - φ)-weak contraction mapping. Then f has a periodic point μ in X, that is, there exists μ ∈ X such that $\mu =f^p\mu$ for some $p\in \mathbb{N}$.

Proof. Given x₀ and define a sequence {x _n } in X by

$${x_n}_{+\mathsf{\text{1}}}=f{x}_n\mathsf{\text{for}}n\in \mathbb{N}\cup \left\{0\right\}.$$

Step 1. We shall prove that

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

(2)

$$\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_n,{x_n}_{+2}\right)\right)=0.$$

(3)

Using the inequality (1), we have that for each $n\in \mathbb{N}$

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

and so

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

Since ${\left\{{\varphi }^n\left(\phi \left(d\left(x_0,x_{\mathsf{\text{1}}}\right)\right)\right)\right\}}_{n\in \mathbb{N}}$ is 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, corresponding to η use, there exists δ > 0 such that for x₀, x₁ ∈ X with η ≤ φ(d(x₀, x₁)) < δ + η, there exists $n_0\in \mathbb{N}$ such that ${\varphi }^{n_0}\left(\phi \left(d\left(x_0,x_{\mathsf{\text{1}}}\right)\right)\right)<\eta$. Since lim_n→∞ϕⁿ(φ(d(x₀, x₁))) = η, there exists $p_0\in \mathbb{N}$ such that η ≤ ϕ^p(φ(d(x₀, x₁))) < δ + η, for all p ≥ p₀. Thus, we conclude that ${\varphi }^{p_0+n_0}\left(\phi \left(d\left(x_0,x_{\mathsf{\text{1}}}\right)\right)\right)<\eta$. So we get a contradiction. Therefore lim_n→∞ϕⁿ(φ(d(x₀, x₁))) = 0, that is,

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

Using the inequality (1), we also have that for each $n\in \mathbb{N}$

$$\begin{array}{c}\phi \left(d\left(x_n,x_n+2\right)\right)=\phi \left(d\left(f{x}_{n-1},f{x}_{n+1}\right)\right)\\ \le \varphi \left(\phi \left(d(x_{n-1},x_{n+1}\right)\right),\end{array}$$

and so

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

Since ${\left\{{\phi }^n\left(d\left(x_0,x_2\right)\right)\right\}}_{n\in \mathbb{N}}$ is decreasing, by the same proof process, we also conclude

$$\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_n,{x_n}_{+2}\right)\right)=0.$$

Next, we claim that {x _n } is g.m.s Cauchy. We claim that the following result holds:

Step 2. Claim that $\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_{p_n},x_{q_n}\right)\right)=0$, that is, for every ε > 0, there exists $n\in \mathbb{N}$ such that if p, q ≥ n then φ(d(x _p , x _q )) < ε.

Suppose the above statement is false. Then there exists ε > 0 such that for any $n\in \mathbb{N}$, there are $p_n,q_n\in \mathbb{N}$ with p _n > q _n ≥ n satisfying

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

Further, corresponding to q _n ≥ n, we can choose p _n in such a way that it the smallest integer with p _n > q _n ≥ n and $\phi \left(d\left(x_{q_n},x_{p_n}\right)\right)\ge \epsilon$. Therefore $\phi \left(d\left(x_{q_n},x_{p_n-1}\right)\right)<\epsilon$. By the rectangular inequality and (2), (3), we have

$$\begin{array}{c}\epsilon \le \phi \left(d\left(x_{p_n},x_{q_n}\right)\right)\\ \le \phi \left(d(x_{p_n},x_{p_n-2}\right)+d\left(x_{p_n-2},x_{p_n-1}\right)+d\left(x_{p_n-1},x_{q_n})\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n-2}\right)\right)+\phi \left(d\left(x_{p_n-2},x_{p_n-1}\right)\right)+\epsilon .\end{array}$$

Letting n → ∞. Then we get

$$\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_{p_n},x_{q_n}\right)\right)=\epsilon .$$

On the other hand, we have

$$\begin{array}{ll}\hfill \phi \left(d\left(x_{p_n},x_{q_n}\right)\right)& \le \phi \left(d\left(x_{p_n},{x_{p_n-}}_{\mathsf{\text{1}}}\right)+d\left({x_{p_n-}}_{\mathsf{\text{1}}},{x_{q_n-}}_{\mathsf{\text{1}}}\right)+d\left({x_{q_n-}}_1,x_{q_n}\right)\right)\\ \le \phi \left(d\left(x_{p_n},x_{p_n-1}\right)\right)+\phi \left(d\left(x_{p_n-1},x_{q_n-1}\right)\right)+\phi \left(d\left(x_{q_n-1},x_{q_n}\right)\right)\end{array}$$

and

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

Letting n → ∞. Then we get

$$\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_{p_n-1},x_{q_n-1}\right)\right)=\epsilon .$$

Using the inequality (1), we have

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

Letting n → ∞, by the definitions of the functions ϕ and φ, we have

$$\epsilon \le \underset{n\to \infty }{\text{lim}}\varphi \left(\phi \left(d\left(x_{p_n-1},x_{q_n-1}\right)\right)\right)<\epsilon .$$

So we get 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 (φ₁), we have $\underset{n\to \infty }{\text{lim}}d\left(x_{p_n},x_{q_n}\right)=0$. Therefore {x _n } is g.m.s Cauchy.

Step 3. We claim that f has a periodic point in X.

Suppose, on contrary, f has no periodic point. Then {x _n } is a sequence of distinct points, that is, x _p ≠ x _q for all $p,q\in \mathbb{N}$ with p ≠ q. By step 2, since X is complete g.m.s, there exists ν ∈ X such that x _n → ν. Using the inequality (1), we have

$$\phi \left(d\left(f{x}_n,f\nu \right)\right)\le \varphi \left(\phi \left(d\left(x_n\mathsf{\text{,}}\nu \right)\right)\right)$$

Letting n → ∞, we have

$$\phi \left(d\left(f{x}_n,f\nu \right)\right)\to 0,\mathsf{\text{as}}n\to \infty ,$$

by the condition (φ₁), we get

$$d\left(f{x}_n,f\nu \right)\to 0,\mathsf{\text{as}}n\to \infty ,$$

that is,

$${x_n}_{+\mathsf{\text{1}}}=f{x}_n\to f\nu ,\mathsf{\text{as}}n\to \infty .$$

As (X, d) is Hausdorff, we have ν = fν, a contradiction with our assumption that f has no periodic point. Therefore, there exists ν ∈ X such that$v=f^p\left(v\right)$ for some $p\in \mathbb{N}$. So f has a periodic point in X.   □

Using the functions ψ and φ, we next introduce the notion of the (ψ-φ)-weak contraction mapping and prove a theorem which assures the existence of a periodic point for the (ψ-φ)-weak contraction mapping.

Definition 8 Let (X, d) be a g.m.s, and let f : X → X be a function satisfying

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

(4)

for all x, y ∈ X. Then f is said to be a (ψ - φ)-weak contraction mapping.

Theorem 2 Let (X, d) be a Hausdorff and complete g.m.s, and let f be a (ψ - φ)-weak contraction mapping. Then f has a periodic point μ in X.

Proof. Given x₀ and define a sequence {x _n } in X by

$${x_n}_{+\mathsf{\text{1}}}=f{x}_n\mathsf{\text{for}}n\in \mathbb{N}\cup \left\{0\right\}.$$

Step 1. We shall prove that

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

(5)

$$\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_n,x_{n+2}\right)\right)=0.$$

(6)

Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each $n\in \mathbb{N}$

$$\begin{array}{c}\phi \left(d\left(x_n,x_n+1\right)\right)=\phi \left(d\left(f{x}_{n-1},f{x}_n\right)\right)\\ \le \psi \left(\phi \left(d(x_{n-1},x_n\right)\right)\cdot \phi \left(d(x_{n-1},x_n\right)\\ <\phi \left(d(x_{n-1},x_n\right).\end{array}$$

Thus the sequence {φ(d(x _n , x_n+1))} is descreasing and bounded below and hence it is con-vergent. Let lim_{n → ∞}φ(d(x _n , x_n+1)) = η ≥ 0. Then there exists $n_0\in \mathbb{N}$ and δ > 0 such that for all $n\in \mathbb{N}$ with n ≥ n₀

$$\eta \le \phi \left(d\left(x_n,x_{n+1}\right)\right)<\eta +\delta .$$

(7)

Taking into account (7) and the definition of stronger Meir-Keeler function ψ, corresponding to η use, there exists γ _η ∈ [0, 1) such that

$$\psi \left(\phi \left(d\left(x_n,{x_n}_{+\mathsf{\text{1}}}\right)\right)\right)<{\gamma }_n\mathsf{\text{for all}}n\ge n_0.$$

Thus, we can deduce that for each $n\in \mathbb{N}$ with n ≥ n₀ + 1

$$\begin{array}{c}\phi \left(d\left(x_n,x_n+1\right)\right)=\phi \left(d\left(f{x}_{n-1},f{x}_n\right)\right)\\ \le \psi \left(\phi \left(d(x_{n-1},x_n\right)\right)\cdot \phi \left(d(x_{n-1},x_n\right)\\ <{\gamma }_{\eta }\cdot \phi \left(d\left(x_{n-1},x_n\right)\right),\end{array}$$

and so

$$\begin{array}{ll}\hfill \phi \left(d\left(x_n,x_{n+1}\right)\right)& \le {\gamma }_{\eta }\cdot \phi \left(d\left(x_{n-1},x_n\right)\right)\\ \le {\gamma }_{\eta }^2\cdot \phi \left(d\left(x_{n-2},x_{n_0-1}\right)\right)\\ \le \cdots \\ \le {\gamma }_{\eta }^{n-n_0}\cdot \phi \left(d\left(x_{n_0},x_{n_0+1}\right)\right).\end{array}$$

Since γ _η ∈ [0, 1), we get

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

Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each $n\in \mathbb{N}$

$$\begin{array}{c}\phi \left(d\left(x_n,x_{n+2}\right)\right)=\phi \left(d\left(f{x}_{n-1},f{x}_{n+1}\right)\right)\\ \le \psi \left(\phi \left(d(x_{n-1},x_{n+1}\right)\right)\cdot \phi \left(d(x_{n-1},x_{n+1}\right)\\ <\phi \left(d(x_{n-1},x_{n+1}\right).\end{array}$$

Thus the sequence {φ(d(x _n , x_n+2))} is descreasing and bounded below and hence it is convergent. By the same proof process, we also conclude

$$\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_n,x_{n+2}\right)\right)=0.$$

Next, we claim that {x _n } is g.m.s Cauchy.

Step 2. Claim that $\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_{p_n},x_{q_n}\right)\right)=0$, that is, for every ε > 0, corresponding to above n₀ use, there exists $n\in \mathbb{N}$ with n ≥ n₀ +1 such that if p, q ≥ n then φ(d(x _p , x _q )) < ε.

Suppose the above statement is false. Then there exists ε > 0 such that for any $n\in \mathbb{N}$, there are $p_n,q_n\in \mathbb{N}$ with p _n > q _n ≥ n ≥ n₀ + 1 satisfying

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

Following from Theorem 1, we have that

$$\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_{p_n},x_{q_n}\right)\right)=\epsilon .$$

and

$$\underset{n\to \infty }{\text{lim}}\phi \left(d\left(x_{p_n-1},x_{q_n-1}\right)\right)=\epsilon .$$

Using the inequality (4), we have

$$\begin{array}{ll}\hfill \phi \left(d\left(x_{p_n},x_{q_n}\right)\right)& =\phi \left(d\left(\left.f{x}_{p_n-1},f{x}_{q_n-1}\right)\right)\right)\\ \le \psi \left(\phi \left(d\left(x_{p_n-1},x_{q_n-1}\right)\right)\right)\cdot \phi \left(d\left(x_{p_n-1},x_{q_n-1}\right)\right)\\ <{\gamma }_{\eta }\cdot \phi \left(d\left(x_{p_n-1},x_{q_n-1}\right)\right),\end{array}$$

Letting n → ∞, by the definitions of the functions ψ and φ, we have

$$\epsilon <\underset{n\to \infty }{\text{lim}}{\gamma }_{\eta }\cdot \phi \left(d\left(x_{p_n-1},x_{q_n-1}\right)\right)<{\gamma }_{\eta }\cdot \epsilon <\epsilon .$$

So we get 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 (φ₁), we have $\underset{n\to \infty }{\text{lim}}d\left(x_{p_n},x_{q_n}\right)=0$. Therefore {x _n } is g.m.s Cauchy.

Step 3. We claim that f has a periodic point in X.

Suppose, on contrary, f has no periodic point. Then {x _n } is a sequence of distinct points, that is, x _p ≠ x _q for all $p,q\in \mathbb{N}$ with p ≠ q. By step 2, since X is complete g.m.s, there exists ν ∈ X such that x _n → ν. Using the inequality (4), we have

$$\phi \left(d\left(f{x}_n,f\nu \right)\right)\le \psi \left(\phi \left(d\left(x_n,\nu \right)\right)\right)\cdot \phi \left(d\left(x_n,\nu \right)\right)$$

Letting n → ∞, we have

$$\phi \left(d\left(f{x}_n,f\nu \right)\right)\to 0,\mathsf{\text{as}}n\to \infty ,$$

by the condition (φ₁), we get

$$d\left(f{x}_n,f\nu \right)\to 0,\mathsf{\text{as}}n\to \infty ,$$

that is,

$${x_n}_{+\mathsf{\text{1}}}=f{x}_n\to f\nu ,\mathsf{\text{as}}n\to \infty .$$

As (X, d) is Hausdorff, we have ν = fν, a contradiction with our assumption that f has no periodic point. Therefore, there exists ν ∈ X such that $v=f^p\left(v\right)$ for some $p\in \mathbb{N}$. So f has a periodic point in X.   □

In conclusion, by using the new concepts of (ϕ-φ)-weak contraction mappings and (ψ - φ)-weak contraction mappings, we obtain two theorems (Theorems 1 and 2) which assure the existence of a periodic point for these two types of weak contraction in complete generalized metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.