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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 : 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[5] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involing three terms instead of two.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→∞ϕn(φ(d(x0, x1))) = η, there exists p 0 such that ηϕp(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→∞ϕn(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 thatv= 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.

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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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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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