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

  • Chi-Ming Chen1Email author and

    Affiliated with

    • Chao-Hung Chen2

      Affiliated with

      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, yD,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equa_HTML.gif
      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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equb_HTML.gif

      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[5] introduced the following notion of a generalized metric space where the triangle inequality of a metric space had been replaced by an inequality involing three terms instead of two.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 xX, we define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equc_HTML.gif

      Branciari [5] also claimed that {B γ (x): γ > 0, xX} 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, yX and for all distinct point u, vX 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, yX;

      (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) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq1_HTML.gif ,

      where γ > 0 is a constant. Then (X, d) be a generalized metric space, but it is not a metric space, because
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equd_HTML.gif

      Definition 2 [5]Let (X, d) be a g.m.s, {x n } be a sequence in X and xX. 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 xX. We say that {x n } is g.m.s Cauchy sequence if and only if for each ε > 0, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq2_HTML.gif 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 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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Eque_HTML.gif

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

      Example 2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq3_HTML.gif be defined by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equf_HTML.gif

      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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equg_HTML.gif

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

      Example 3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq4_HTML.gif be defined by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equh_HTML.gif

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq5_HTML.gif be defined by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equi_HTML.gif

      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:

      (ϕ 1) ϕ : [0, ∞) → [0, ∞) is a weaker Meir-Keeler function;

      (ϕ 2) ϕ(t) > 0 for t > 0 and ϕ(0) = 0;

      (ϕ 3) for all t ∈ (0, ∞), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq6_HTML.gif 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 Meir-Keeler 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 (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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equ1_HTML.gif
      (1)

      for all x, yX. 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 μX such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq7_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq8_HTML.gif .

      Proof. Given x 0 and define a sequence {x n } in X by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equj_HTML.gif
      Step 1. We shall prove that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equ2_HTML.gif
      (2)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equ3_HTML.gif
      (3)
      Using the inequality (1), we have that for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equk_HTML.gif
      and so
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equl_HTML.gif
      Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq10_HTML.gif 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 x 0, x 1X with ηφ(d(x 0, x 1)) < δ + η, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq11_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq12_HTML.gif . Since lim n→∞ ϕ n (φ(d(x 0, x 1))) = η, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq13_HTML.gif such that ηϕ p (φ(d(x 0, x 1))) < δ + η, for all pp 0. Thus, we conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq14_HTML.gif . So we get a contradiction. Therefore lim n→∞ ϕ n (φ(d(x 0, x 1))) = 0, that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equm_HTML.gif
      Using the inequality (1), we also have that for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equn_HTML.gif
      and so
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equo_HTML.gif
      Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq15_HTML.gif is decreasing, by the same proof process, we also conclude
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equp_HTML.gif

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

      Step 2. Claim that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq16_HTML.gif , that is, for every ε > 0, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif , there are http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq17_HTML.gif with p n > q n n satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equq_HTML.gif
      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq18_HTML.gif . Therefore http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq19_HTML.gif . By the rectangular inequality and (2), (3), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equr_HTML.gif
      Letting n → ∞. Then we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equs_HTML.gif
      On the other hand, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equt_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equu_HTML.gif
      Letting n → ∞. Then we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equv_HTML.gif
      Using the inequality (1), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equw_HTML.gif
      Letting n → ∞, by the definitions of the functions ϕ and φ, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equx_HTML.gif

      So we get a contradiction. Therefore http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq20_HTML.gif , by the condition (φ 1), we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq21_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq22_HTML.gif with pq. By step 2, since X is complete g.m.s, there exists νX such that x n ν. Using the inequality (1), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equy_HTML.gif
      Letting n → ∞, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equz_HTML.gif
      by the condition (φ 1), we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equaa_HTML.gif
      that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equab_HTML.gif

      As (X, d) is Hausdorff, we have ν = , a contradiction with our assumption that f has no periodic point. Therefore, there exists νX such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq23_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq24_HTML.gif . 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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equ4_HTML.gif
      (4)

      for all x, yX. 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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equac_HTML.gif
      Step 1. We shall prove that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equ5_HTML.gif
      (5)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equ6_HTML.gif
      (6)
      Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equad_HTML.gif
      Thus the sequence {φ(d(x n , x n+1))} is descreasing and bounded below and hence it is con-vergent. Let lim n → ∞ φ(d(x n , x n+1)) = η ≥ 0. Then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq25_HTML.gif and δ > 0 such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif with nn 0
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equ7_HTML.gif
      (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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equae_HTML.gif
      Thus, we can deduce that for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif with nn 0 + 1
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equaf_HTML.gif
      and so
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equag_HTML.gif
      Since γ η ∈ [0, 1), we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equah_HTML.gif
      Taking into account (4) and the definition of stronger Meir-Keeler function ψ, we have that for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equai_HTML.gif
      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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equaj_HTML.gif

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

      Step 2. Claim that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq26_HTML.gif , that is, for every ε > 0, corresponding to above n 0 use, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif with nn 0 +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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq9_HTML.gif , there are http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq27_HTML.gif with p n > q n nn 0 + 1 satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equak_HTML.gif
      Following from Theorem 1, we have that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equal_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equam_HTML.gif
      Using the inequality (4), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equan_HTML.gif
      Letting n → ∞, by the definitions of the functions ψ and φ, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equao_HTML.gif

      So we get a contradiction. Therefore http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq28_HTML.gif , by the condition (φ 1), we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq29_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq30_HTML.gif with pq. By step 2, since X is complete g.m.s, there exists νX such that x n ν. Using the inequality (4), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equap_HTML.gif
      Letting n → ∞, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equaq_HTML.gif
      by the condition (φ 1), we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equar_HTML.gif
      that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_Equas_HTML.gif

      As (X, d) is Hausdorff, we have ν = , a contradiction with our assumption that f has no periodic point. Therefore, there exists νX such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq31_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-79/MediaObjects/13663_2011_170_IEq32_HTML.gif . 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

      References

      1. Banach S: Sur les operations dans les ensembles abstraits et leur application aux equations integerales. Fund Math 1922, 3:133–181.MATH
      2. Boyd DW, Wong SW: On nonlinear contractions. Proc Am Math Soc 1969, 20:45864.MathSciNetView Article
      3. Aydi H, Karapinar E, Shatnawi W: Coupled fixed point results for (( ψ - φ )-weakly contractive condition in ordered partial metric spaces. Comput Math Appl 2011,62(12):4449–4460.MathSciNetMATHView Article
      4. Karapinar E: Weak φ -contraction on partial metric spaces and existence of fixed points in partially ordered sets. Math Aeterna 2011,1(4):237–244.MathSciNet
      5. Branciari A: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ Math Debrecen 2000, 57:31–37.MathSciNetMATH
      6. Azam A, Arshad M: Kannan fixed point theorem on generalized metric spaces. J Nonlinear Sci Appl 2008,1(1):45–48.MathSciNetMATH
      7. Das P: A fixed point theorem on a class of generalized metric spaces. Korea J Math Sci 2002, 9:29–33.
      8. Mihet D: On Kannan fixed point principle in generalized metric spaces. J Nonlinear Sci Appl 2009,2(2):92–96.MathSciNetMATH
      9. Samet B: A fixed point theorem in a generalized metric space for mappings satisfying a contractive condition of integral type. Int J Math Anal 2009,26(3):1265–1271.
      10. Samet B: Disscussion on: a fixed point theorem of Banach-Caccioppli type on a class of generalized metric spaces. Publ Math Debrecen 2010,76(4):493–494.MathSciNetMATH
      11. Lakzian H, Samet B: Fixed points for ( ψ , φ )-weakly contractive mappings in general-ized metric spaces. Appl Math Lett 25(5):902–906.
      12. Meir A, Keeler E: A theorem on contraction mappings. J Math Anal Appl 1969, 28:326–329.MathSciNetMATHView Article
      13. Anthony Eldred A, Veeramani P: Existence and convergence of best proximity points. J Math Anal Appl 2006, 323:1001–1006.MathSciNetMATHView Article
      14. De la Sen M: Linking contractive self-mappings and cyclic Meir-Keeler contractions with Kannan self-mappings. Fixed Point Theory Appl 2010, 2010:23. Article ID 572057 doi:10.1155/2010/572057

      Copyright

      © Chen and Chen; licensee Springer. 2012