Open Access

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

Fixed Point Theory and Applications20122012:79

DOI: 10.1186/1687-1812-2012-79

Received: 25 December 2011

Accepted: 9 May 2012

Published: 9 May 2012

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 Meir-Keeler function (ϕ - φ)-weak contraction mapping (ψ - φ)-weak contraction mapping

1 Introduction and preliminaries

Let (X, d) be a metric space, D a subset of X and f : DX be a map. We say f is contractive if there exists α [0, 1) such that for all x, y D,
d f x , f y α d x , y .
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 : XX on a metric space is called ϕ-contraction if there exists an upper semi-continuous function ϕ : [0, ∞) → [0, ∞) such that
d f x , f y ϕ d x , y for all x , y 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[5] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involing three terms instead of two.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. [611]).

Let (X, d) be a generalized metric space. For γ > 0 and x X, we define
B γ x : = y X | d x , y < γ .

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. (1)

    d(x, x) = 0, for all × X;

     
  2. (2)

    d(x, y) = d(y, x), for all x, y X;

     
  3. (3)

    d(t, 2t) = 3γ;

     
  4. (4)

    d(t, 3t) = d(2t, 3t) = γ;

     
  5. (5)

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

     
  6. (6)

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

     
where γ > 0 is a constant. Then (X, d) be a generalized metric space, but it is not a metric space, because
d t , 2 t = 3 γ > d t , 3 t + d 3 t , 2 t = 2 γ .

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 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 ηt < 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
η > 0 δ > 0 t 0 , η t < δ + η n 0 ϕ t n 0 < η .

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

Example 2 Let ϕ : + + be defined by
ϕ t = 0 , i f t 1 , 3 t , i f 1 < t < 3 , 1 , i f t 3 .

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
η > 0 δ > 0 γ η 0 , 1 t 0 , η t < δ + η ψ t < γ η .

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

Example 3 Let ψ : + 0 , 1 be defined by
ψ d x , y = 2 t 3 t + 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 φ : + + be defined by
φ t = t - 1 , i f t > 1 ; 0 , i f t 1 .

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, ∞), { ϕ n ( t ) } n is decreasing;

(ϕ4) for t n [0, ∞), we have that
  1. (a)

    if limn→∞t n = γ > 0, then limn→∞ϕ(t n ) < γ, and

     
  2. (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 : XX be a function satisfying
φ ( d ( f x , f y ) ) ϕ ( φ ( d ( x , y ) )
(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 μ = f p μ for some p .

Proof. Given x0 and define a sequence {x n } in X by
x n + 1 = f x n for n 0 .
Step 1. We shall prove that
lim n φ d x n , x n + 1 = 0 ,
(2)
lim n φ d x n , x n + 2 = 0 .
(3)
Using the inequality (1), we have that for each n
φ ( d ( x n , x n + 1 ) ) = φ ( d ( f x n 1 , f x n ) ) ϕ ( φ ( d ( x n 1 , x n ) ) ,
and so
φ ( d ( x n , x n + 1 ) ) ϕ ( φ ( d ( x n 1 , x n ) ) ) ϕ ( ϕ ( φ ( d ( x n 2 , x n 1 ) ) ) = ϕ 2 ( φ ( d ( x n 2 , x n 1 ) ) ) ϕ n ( φ ( d ( x 0 , x 1 ) ) ) .
Since { ϕ n ( φ ( d ( x 0 , x 1 ) ) ) } 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 ηφ(d(x0, x1)) < δ + η, there exists n 0 such that ϕ n 0 ( φ ( d ( x 0 , x 1 ) ) ) < η . Since limn→∞ϕ n (φ(d(x0, x1))) = η, there exists p 0 such that ηϕ p (φ(d(x0, x1))) < δ + η, for all pp0. Thus, we conclude that ϕ p 0 + n 0 ( φ ( d ( x 0 , x 1 ) ) ) < η . So we get a contradiction. Therefore limn→∞ϕ n (φ(d(x0, x1))) = 0, that is,
lim n φ d x n , x n + 1 = 0 .
Using the inequality (1), we also have that for each n
φ ( d ( x n , x n + 2 ) ) = φ ( d ( f x n 1 , f x n + 1 ) ) ϕ ( φ ( d ( x n 1 , x n + 1 ) ) ,
and so
φ ( d ( x n , x n + 2 ) ) ϕ ( φ ( d ( x n 1 , x n + 1 ) ) ) ϕ ( ϕ ( φ ( d ( x n 2 , x n ) ) ) = ϕ 2 ( φ ( d ( x n 2 , x n ) ) ) ϕ n ( φ ( d ( x 0 , x 1 ) ) ) .
Since { φ n ( d ( x 0 , x 2 ) ) } n is decreasing, by the same proof process, we also conclude
lim n φ d x n , x n + 2 = 0 .

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

Step 2. Claim that lim n φ d x p n , x q n = 0 , that is, for every ε > 0, there exists n such that if p, qn then φ(d(x p , x q )) < ε.

Suppose the above statement is false. Then there exists ε > 0 such that for any n , there are p n , q n with p n > q n n satisfying
φ d x q n , x p n ε .
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 φ d x q n , x p n ε . Therefore φ d x q n , x p n - 1 < ε . By the rectangular inequality and (2), (3), we have
ε φ ( d ( x p n , x q n ) ) φ ( d ( x p n , x p n 2 ) + d ( x p n 2 , x p n 1 ) + d ( x p n 1 , x q n ) ) φ ( d ( x p n , x p n 2 ) ) + φ ( d ( x p n 2 , x p n 1 ) ) + ε .
Letting n → ∞. Then we get
lim n φ d x p n , x q n = ε .
On the other hand, we have
φ d x p n , x q n φ d x p n , x p n - 1 + d x p n - 1 , x q n - 1 + d x q n - 1 , x q n φ d x p n , x p n - 1 + φ d x p n - 1 , x q n - 1 + φ d x q n - 1 , x q n
and
φ d x p n - 1 , x q n - 1 φ d x p n - 1 , x p n + d x p n , x q n + d x q n , x q n - 1 φ d x p n - 1 , x p n + φ d x p n , x q n + φ d x q n , x q n - 1 .
Letting n → ∞. Then we get
lim n φ d x p n - 1 , x q n - 1 = ε .
Using the inequality (1), we have
φ d x p n , x q n = φ d f x p n - 1 f x q n - 1 ϕ φ d x p n - 1 , x q n - 1 ,
Letting n → ∞, by the definitions of the functions ϕ and φ, we have
ε lim n ϕ φ d x p n - 1 , x q n - 1 < ε .

So we get a contradiction. Therefore lim n φ d x p n , x q n = 0 , by the condition (φ1), we have lim n d x p n , x q n = 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 with pq. 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
φ d f x n , f ν ϕ φ d x n , ν
Letting n → ∞, we have
φ d f x n , f ν 0 , as n ,
by the condition (φ1), we get
d f x n , f ν 0 , as n ,
that is,
x n + 1 = f x n f ν , as n .

As (X, d) is Hausdorff, we have ν = , a contradiction with our assumption that f has no periodic point. Therefore, there exists ν X such that v = f p ( v ) for some p . 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 : XX be a function satisfying
φ ( d ( f x , f y ) ) ψ ( φ ( d ( x , y ) ) φ ( 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 + 1 = f x n for n { 0 } .
Step 1. We shall prove that
lim n φ d x n , x n + 1 = 0 ,
(5)
lim n φ d x n , x n + 2 = 0 .
(6)
Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each n
φ ( d ( x n , x n + 1 ) ) = φ ( d ( f x n 1 , f x n ) ) ψ ( φ ( d ( x n 1 , x n ) ) φ ( d ( x n 1 , x n ) < φ ( d ( x n 1 , x n ) .
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 n 0 and δ > 0 such that for all n with nn0
η φ d x n , x n + 1 < η + δ .
(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
ψ φ d x n , x n + 1 < γ n for all n n 0 .
Thus, we can deduce that for each n with nn0 + 1
φ ( d ( x n , x n + 1 ) ) = φ ( d ( f x n 1 , f x n ) ) ψ ( φ ( d ( x n 1 , x n ) ) φ ( d ( x n 1 , x n ) < γ η φ ( d ( x n 1 , x n ) ) ,
and so
φ d x n , x n + 1 γ η φ d x n - 1 , x n γ η 2 φ d x n - 2 , x n 0 - 1 γ η n - n 0 φ d x n 0 , x n 0 + 1 .
Since γ η [0, 1), we get
lim n φ d x n , x n + 1 = 0 .
Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each n
φ ( d ( x n , x n + 2 ) ) = φ ( d ( f x n 1 , f x n + 1 ) ) ψ ( φ ( d ( x n 1 , x n + 1 ) ) φ ( d ( x n 1 , x n + 1 ) < φ ( d ( x n 1 , x n + 1 ) .
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
lim n φ d x n , x n + 2 = 0 .

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

Step 2. Claim that lim n φ d x p n , x q n = 0 , that is, for every ε > 0, corresponding to above n0 use, there exists n with nn0 +1 such that if p, qn then φ(d(x p , x q )) < ε.

Suppose the above statement is false. Then there exists ε > 0 such that for any n , there are p n , q n with p n > q n nn0 + 1 satisfying
φ d x q n , x p n ε .
Following from Theorem 1, we have that
lim n φ d x p n , x q n = ε .
and
lim n φ d x p n - 1 , x q n - 1 = ε .
Using the inequality (4), we have
φ d x p n , x q n = φ d f x p n - 1 , f x q n - 1 ψ φ d x p n - 1 , x q n - 1 φ d x p n - 1 , x q n - 1 < γ η φ d x p n - 1 , x q n - 1 ,
Letting n → ∞, by the definitions of the functions ψ and φ, we have
ε < lim n γ η φ d x p n - 1 , x q n - 1 < γ η ε < ε .

So we get a contradiction. Therefore lim n φ d x p n , x q n = 0 , by the condition (φ1), we have lim n d x p n , x q n = 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 with pq. 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
φ d f x n , f ν ψ φ d x n , ν φ d x n , ν
Letting n → ∞, we have
φ d f x n , f ν 0 , as n ,
by the condition (φ1), we get
d f x n , f ν 0 , as n ,
that is,
x n + 1 = f x n f ν , as n .

As (X, d) is Hausdorff, we have ν = , a contradiction with our assumption that f has no periodic point. Therefore, there exists ν X such that v = f p ( v ) for some p . 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.

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Applied Mathematics, National Hsinchu University of Education
(2)
Department of Applied Mathematics, Chung Yuan Christian University

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© Chen and Chen; licensee Springer. 2012

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