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

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.

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,

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

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[5] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involing three terms instead of two.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

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) ,

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

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 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

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

Example 2 Let be defined by

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

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

Example 3 Let be defined by

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 be defined by

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

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, ∞), 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

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 for some .

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

Step 1. We shall prove that

Using the inequality (1), we have that for each

and so

Since 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 such that . Since lim_n→∞ ϕⁿ(φ(d(x₀, x₁))) = η, there exists such that η ≤ ϕ^p(φ(d(x₀, x₁))) < δ + η, for all p ≥ p₀. Thus, we conclude that . So we get a contradiction. Therefore lim_n→∞ ϕⁿ(φ(d(x₀, x₁))) = 0, that is,

Using the inequality (1), we also have that for each

and so

Since is decreasing, by the same proof process, we also conclude

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

Step 2. Claim that , that is, for every ε > 0, there exists 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 , there are with p_n > q_n ≥ n satisfying

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 . Therefore . By the rectangular inequality and (2), (3), we have

Letting n → ∞. Then we get

On the other hand, we have

and

Letting n → ∞. Then we get

Using the inequality (1), we have

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

So we get a contradiction. Therefore , by the condition (φ₁), we have . 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 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

Letting n → ∞, we have

by the condition (φ₁), we get

that is,

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 for some . 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

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

Step 1. We shall prove that

Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each

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 and δ > 0 such that for all with n ≥ n₀

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

Thus, we can deduce that for each with n ≥ n₀ + 1

and so

Since γ_η ∈ [0, 1), we get

Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each

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

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

Step 2. Claim that , that is, for every ε > 0, corresponding to above n₀ use, there exists 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 , there are with p_n > q_n ≥ n ≥ n₀ + 1 satisfying

Following from Theorem 1, we have that

and

Using the inequality (4), we have

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

So we get a contradiction. Therefore , by the condition (φ₁), we have . 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 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

Letting n → ∞, we have

by the condition (φ₁), we get

that is,

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 for some . 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.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

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

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