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.
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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,
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[5] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involing three terms instead of two.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.
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, ∞), is decreasing;
(ϕ4) for t n ∈ [0, ∞), we have that
-
(a)
if limn→∞t n = γ > 0, then limn→∞ϕ(t n ) < γ, and
-
(b)
if limn→∞t n = 0, then limn→∞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, ∞), limn→∞t n = 0 if and only if limn→∞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 limn→∞ϕn(φ(d(x0, x1))) = η, there exists such that η ≤ ϕp(φ(d(x0, x1))) < δ + η, for all p ≥ p0. Thus, we conclude that . So we get a contradiction. Therefore limn→∞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 , xn+1))} is descreasing and bounded below and hence it is con-vergent. Let limn → ∞φ(d(x n , xn+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 , xn+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.
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The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.
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Chen, CM., Chen, CH. Periodic points for the weak contraction mappings in complete generalized metric spaces. Fixed Point Theory Appl 2012, 79 (2012). https://doi.org/10.1186/1687-1812-2012-79
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DOI: https://doi.org/10.1186/1687-1812-2012-79