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.
Keywords:
Periodic point; MeirKeeler function; (ϕ  φ)weak contraction mapping; (ψ  φ)weak contraction mapping1 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,
The wellknown 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 ϕcontraction. A mapping f : X → X on a metric space is called ϕcontraction if there exists an upper semicontinuous 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[5] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involing three terms instead of two.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. [611]).
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;
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 > n_{0} .
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 MeirKeeler function (see [12]). A function ϕ : [0, ∞) → [0, ∞) is said to be a MeirKeeler 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 MeirKeeler type contractive functions. In this study, we introduce the below notions of the weaker MeirKeeler function ϕ : [0, ∞) → [0, ∞) and stronger MeirKeeler function ψ : [0, ∞) → [0, 1).
Definition 5 We call ϕ : [0, ∞) → [0, ∞) a weaker MeirKeeler function if the function ϕ satisfies the following condition
The following provides an example of a weaker MeirKeeler function which is not a MeirKeeler function.
Then ϕ is a weaker MeirKeeler function which is not a MeirKeeler function.
Definition 6 We call ψ : [0, ∞) → [0, 1) a stronger MeirKeeler function if the function ψ satisfies the following condition
The following provides an example of a stronger MeirKeeler function.
Then ψ is a stronger MeirKeeler function.
The following provides an example of a MeirKeeler function which is not a stronger MeirKeeler function.
Then φ is a MeirKeeler function which is not a stronger MeirKeeler function.
2 Main results
In the sequel, we let the function ϕ : [0, ∞) → [0, ∞) satisfies the following conditions:
(ϕ_{1}) ϕ : [0, ∞) → [0, ∞) is a weaker MeirKeeler function;
(ϕ_{2}) ϕ(t) > 0 for t > 0 and ϕ(0) = 0;
(ϕ_{3}) for all t ∈ (0, ∞), is decreasing;
(ϕ_{4}) 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:
(ψ_{1}) ψ : [0, ∞) → [0, 1) is a stronger MeirKeeler function;
(ψ_{2}) ψ(t) > 0 for t > 0 and ϕ(0) = 0.
And, we let the function φ : [0, ∞) → [0, ∞) satisfies the following conditions:
(φ_{1}) for all t ∈ (0, ∞), lim_{n→∞ }t_{n }= 0 if and only if lim_{n→∞ }φ(t_{n}) = 0;
(φ_{2}) φ(t) > 0 for t > 0 and φ(0) = 0;
(φ_{3}) φ is subadditive, that is, for every μ_{1}, μ_{2 }∈ [0, ∞), φ(μ_{1 }+ μ_{2}) ≤ φ(μ_{1}) + φ(μ_{2}).
Using the functions ϕ and φ, we first introduce the notion of the (ϕφ)weak contraction mapping and prove a theorem which assures the existence of a periodic point for the (ϕφ)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 x_{0 }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 MeirKeeler function ϕ, corresponding to η use, there exists δ > 0 such that for x_{0}, x_{1 }∈ X with η ≤ φ(d(x_{0}, x_{1})) < δ + η, there exists such that . Since lim_{n→∞ }ϕ^{n}(φ(d(x_{0}, x_{1}))) = η, there exists such that η ≤ ϕ^{p}(φ(d(x_{0}, x_{1}))) < δ + η, for all p ≥ p_{0}. Thus, we conclude that . So we get a contradiction. Therefore lim_{n→∞ }ϕ^{n}(φ(d(x_{0}, x_{1}))) = 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 (φ_{1}), 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 (φ_{1}), 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 (ψφ)weak contraction mapping and prove a theorem which assures the existence of a periodic point for the (ψφ)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 x_{0 }and define a sequence {x_{n}} in X by
Step 1. We shall prove that
Taking into account (4) and the definition of stronger MeirKeeler 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 convergent. Let lim_{n → ∞ }φ(d(x_{n}, x_{n+1})) = η ≥ 0. Then there exists and δ > 0 such that for all with n ≥ n_{0}
Taking into account (7) and the definition of stronger MeirKeeler function ψ, corresponding to η use, there exists γ_{η }∈ [0, 1) such that
Thus, we can deduce that for each with n ≥ n_{0 }+ 1
and so
Since γ_{η }∈ [0, 1), we get
Taking into account (4) and the definition of stronger MeirKeeler 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 n_{0 }use, there exists with n ≥ n_{0 }+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 ≥ n_{0 }+ 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 (φ_{1}), 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 (φ_{1}), 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 (ϕφ)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.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.
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