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

Chi-Ming Chen1* and Chao-Hung Chen2

Author Affiliations

1 Department of Applied Mathematics, National Hsinchu University of Education, Hsin-Chu, Taiwan

2 Department of Applied Mathematics, Chung Yuan Christian University, Chungli City, Taiwan

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Fixed Point Theory and Applications 2012, 2012:79  doi:10.1186/1687-1812-2012-79

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/79


Received:25 December 2011
Accepted:9 May 2012
Published:9 May 2012

© 2012 Chen and Chen; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 : D X be a map. We say f is contractive if there exists α ∈ [0, 1) such that for all x, y D,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M1">View MathML</a>

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M2">View MathML</a>

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M3">View MathML</a>

Branciari [5] also claimed that {Bγ(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) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M4">View MathML</a>,

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M5">View MathML</a>

Definition 2 [5]Let (X, d) be a g.m.s, {xn} be a sequence in X and x X. We say that {xn} is g.m.s convergent to × if and only if d(xn, x) → 0 as n → ∞. We denote by xn x as n → ∞.

Definition 3 [5]Let (X, d) be a g.m.s, {xn} be a sequence in X and x X. We say that {xn} is g.m.s Cauchy sequence if and only if for each ε > 0, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M6">View MathML</a>such that d(xm, xn) < 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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M7">View MathML</a>

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

Example 2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M8">View MathML</a>be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M9">View MathML</a>

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M10">View MathML</a>

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

Example 3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M11">View MathML</a>be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M12">View MathML</a>

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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M13">View MathML</a>be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M14">View MathML</a>

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, ∞), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M15">View MathML</a> is decreasing;

(ϕ4) for tn ∈ [0, ∞), we have that

(a) if limn→∞ tn = γ > 0, then limn→∞ ϕ(tn) < γ, and

(b) if limn→∞ tn = 0, then limn→∞ ϕ(tn) = 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→∞ tn = 0 if and only if limn→∞ φ(tn) = 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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M16">View MathML</a>

(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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M17">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M18">View MathML</a> .

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M19">View MathML</a>

Step 1. We shall prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M20">View MathML</a>

(2)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M21">View MathML</a>

(3)

Using the inequality (1), we have that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M23">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M24">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M25">View MathML</a> 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M26">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M27">View MathML</a>. Since limn→∞ ϕn(φ(d(x0, x1))) = η, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M28">View MathML</a> such that η ϕp(φ(d(x0, x1))) < δ + η, for all p p0. Thus, we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M29">View MathML</a>. So we get a contradiction. Therefore limn→∞ ϕn(φ(d(x0, x1))) = 0, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M30">View MathML</a>

Using the inequality (1), we also have that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M31">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M32">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M33">View MathML</a> is decreasing, by the same proof process, we also conclude

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M34">View MathML</a>

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

Step 2. Claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M35">View MathML</a>, that is, for every ε > 0, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a> such that if p, q n then φ(d(xp, xq)) < ε.

Suppose the above statement is false. Then there exists ε > 0 such that for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a>, there are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M36">View MathML</a> with pn > qn n satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M37">View MathML</a>

Further, corresponding to qn n, we can choose pn in such a way that it the smallest integer with pn > qn n and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M38">View MathML</a>. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M39">View MathML</a>. By the rectangular inequality and (2), (3), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M40">View MathML</a>

Letting n → ∞. Then we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M41">View MathML</a>

On the other hand, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M42">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M43">View MathML</a>

Letting n → ∞. Then we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M44">View MathML</a>

Using the inequality (1), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M45">View MathML</a>

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M46">View MathML</a>

So we get a contradiction. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M47">View MathML</a>, by the condition (φ1), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M48">View MathML</a>. Therefore {xn} 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 {xn} is a sequence of distinct points, that is, xp xq for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M49">View MathML</a> with p q. By step 2, since X is complete g.m.s, there exists ν X such that xn ν. Using the inequality (1), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M50">View MathML</a>

Letting n → ∞, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M51">View MathML</a>

by the condition (φ1), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M52">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M53">View MathML</a>

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<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M54">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M55">View MathML</a>. 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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M56">View MathML</a>

(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 {xn} in X by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M57">View MathML</a>

Step 1. We shall prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M58">View MathML</a>

(5)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M59">View MathML</a>

(6)

Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M60">View MathML</a>

Thus the sequence {φ(d(xn, xn+1))} is descreasing and bounded below and hence it is con-vergent. Let limn → ∞ φ(d(xn, xn+1)) = η ≥ 0. Then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M61">View MathML</a> and δ > 0 such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a> with n n0

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M62">View MathML</a>

(7)

Taking into account (7) and the definition of stronger Meir-Keeler function ψ, corresponding to η use, there exists γη ∈ [0, 1) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M63">View MathML</a>

Thus, we can deduce that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a> with n n0 + 1

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M64">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M65">View MathML</a>

Since γη ∈ [0, 1), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M66">View MathML</a>

Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M67">View MathML</a>

Thus the sequence {φ(d(xn, xn+2))} is descreasing and bounded below and hence it is convergent. By the same proof process, we also conclude

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M68">View MathML</a>

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

Step 2. Claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M69">View MathML</a>, that is, for every ε > 0, corresponding to above n0 use, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a> with n n0 +1 such that if p, q n then φ(d(xp, xq)) < ε.

Suppose the above statement is false. Then there exists ε > 0 such that for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M22">View MathML</a>, there are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M70">View MathML</a> with pn > qn n n0 + 1 satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M71">View MathML</a>

Following from Theorem 1, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M72">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M73">View MathML</a>

Using the inequality (4), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M74">View MathML</a>

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M75">View MathML</a>

So we get a contradiction. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M76">View MathML</a>, by the condition (φ1), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M77">View MathML</a>. Therefore {xn} 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 {xn} is a sequence of distinct points, that is, xp xq for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M78">View MathML</a> with p q. By step 2, since X is complete g.m.s, there exists ν X such that xn ν. Using the inequality (4), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M79">View MathML</a>

Letting n → ∞, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M80">View MathML</a>

by the condition (φ1), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M81">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M82">View MathML</a>

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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M83">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/79/mathml/M84">View MathML</a>. 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.

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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