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A fixed point of generalized T_{ F }contraction mappings in cone metric spaces
 Farshid Khojasteh^{1}Email author,
 Abdolrahman Razani^{1} and
 Sirous Moradi^{2}
https://doi.org/10.1186/16871812201114
© Khojasteh et al; licensee Springer. 2011
 Received: 1 March 2011
 Accepted: 8 July 2011
 Published: 8 July 2011
Abstract
In this paper, the existence of a fixed point for T_{ F } contractive mappings on complete metric spaces and cone metric spaces is proved, where T : X → X is a one to one and closed graph function and F : P → P is nondecreasing and right continuous, with F^{1}(0) = {0} and F(t_{ n } ) → 0 implies t_{ n } → 0. Our results, extend previous results given by Meir and Keeler (J. Math. Anal. Appl. 28, 326329, 1969), Branciari (Int. J. Math. sci. 29, 531536, 2002), Suzuki (J. Math. Math. Sci. 2007), Rezapour et al. (J. Math. Anal. Appl. 345, 719724, 2010), Moradi et al. (Iran. J. Math. Sci. Inf. 5, 2532, 2010) and Khojasteh et al. (Fixed Point Theory Appl. 2010).
MSC(2000): 47H10; 54H25; 28B05.
Keywords
 integral type contraction; regular cone
 MeirKeeler contraction
 T_{ F } contraction
 function
1 Introduction
In 2007, Huang et al. [1], introduced the cone metric spaces and proved some fixed point theorems. Recently, Many results closely related to cone metric spaces are given (see [2–6]). In addition, some topological properties of these spaces are surveyed.
Example 1.2. Suppose is endowed with the Euclidean metric. Consider two mappings T, f : X → X defined by and fx = 2x, respectively. Obviously, f is not a contraction but it is a T_{ F }contraction, where F(x) ≡ x.
Definition 1.3. Let (X, d) be a metric space. A mapping T : X → X is said to be closed graph, if for every sequence {x_{ n }} such that , there exists b ∈ X such that Tb = a. For example, the identity function on X is closed graph.
In 2010, Moradi et al. [8] proved the following fixed point theorem.
Theorem 1.4. Let (X, d) be a complete metric space, α ∈ [0, 1) and T, f : X → X be two mappings such that T is onetoone and closed graph, and f is T_{ F }  contraction, respectively, where F ∈ Ψ. Then, f has a unique fixed point a ∈ X. Also, for every x ∈ X, the sequence of iterates {Tf^{ n }x} converges to Ta.
2 Cone metric space
Let E be a real Banach space. A subset P of E is called a cone, if and only if, the following hold:

P is closed, nonempty, and P ≠ {0},

a, b ∈ ℝ, a, b ≥ 0, and x, y ∈ P imply that ax + by ∈ P,

x ∈ P and x ∈ P imply that x = 0.
Given a cone P ⊂ E, we define a partial ordering ≤ with respect to P by x ≤ y, if and only if, y x ∈ P. We write x < y to indicate that x ≤ y but x ≠ y, while x ≪ y stand for y  x ∈ intP, where intP denotes the interior of P. The cone P is called normal, if there exist a number K > 0 such that, 0 ≤ x ≤ y implies x ≤ K y, for all x, y ∈ E. The least positive number satisfying this, called the normal constant [1].
The cone P is called regular, if every increasing sequence which is bounded from above is convergent. That is, if {x_{ n }}_{n≥1}is a sequence such that x_{1} ≤ x_{2} ≤ ··· ≤ y for some y ∈ E, then there exist x ∈ E such that lim_{n→∞}x_{ n } x = 0. Equivalently, the cone P is regular, if and only if, every decreasing sequence which is bounded from below is convergent [1]. Also, every regular cone is normal [5]. Following example shows that the converse is not true.
Example 2.1. [5]Suppose with the norm  f  =  f _{∞} +  f' _{∞}, and consider the cone P = { f ∈ E : f ≥ 0}. For each K ≥ 1, put f(x) = x and g(x) = x^{2K}. Then, 0 ≤ g ≤ f,  f  = 2, and g = 2K + 1. Since K f  < g, K is not normal constant of P.
In this paper, E denotes a real Banach space, P denotes a cone in E with intP ≠ ∅ and ≤ denotes partial ordering with respect to P. Let X be a nonempty set. A function d : X × X → E is called a cone metric on X, if it satisfies the following conditions:
(I) d(x, y) ≥ 0 for all x, y ∈ X and d(x, y) = 0, if and only if, x = y,
(II) d(x, y) = d(y, x), for all x, y ∈ X,
(III) d(x, y) ≤ d(x, z) + d(y, z), for all x, y, z ∈ X.
Then, (X, d) is called a cone metric space (see [1]).
Example 2.2. [5]Suppose E = ℓ^{1}, P = {{x_{ n }}_{n∈ℕ}∈ E : x_{ n }≥ 0, for all n} and (X, ρ) be a metric space. Suppose d : X × X → E is defined by . Then, (X, d) is a cone metric space and the normal constant of P is equal to 1.
Example 2.3. Let E = ℝ^{2}, P = {(x, y) ∈ E  x, y ≥ 0}, x = ℝ. Suppose d : X × X → E is defined by d(x, y) = (x  y, αx  y), where α ≥ 0 is a constant. Then, (X, d) is a cone metric space.
The following definitions and lemmas have been chosen from [1].
Definition 2.4. Let (X, d) be a cone metric space and {x_{ n }}_{n∈ℕ}be a sequence in x and x ∈ X. If for all c ∈ E with 0 ≪ c, there is n_{0} ∈ ℕ such that for all n > n_{0}, d(x_{ n }, x_{0}) ≪ c, then {x_{ n }}_{n∈ℕ}is said to be convergent and {x_{ n }}_{n∈ℕ}converges to x and x is the limit of {x_{ n }}_{n∈ℕ}.
Definition 2.5. Let (X, d) be a cone metric space and {x_{ n } }_{n∈ℕ}be a sequence in X. If for all c ∈ E with 0 ≪ c, there is n_{0} ∈ ℕ such that for all m, n > n_{0}, d(x_{ n }, x_{ m } ) ≪ c, then {x_{ n } }_{n∈ℕ}is called a Cauchy sequence in X.
Definition 2.6. Let (X, d) be a cone metric space. If every Cauchy sequence is convergent in X, then X is called a complete cone metric space.
Definition 2.7. Let (X, d) be a cone metric space. A selfmap T on X is said to be continuous, if lim_{n→∞}x_{ n }= x implies lim_{n→∞}T (x_{ n }) = T (x) for all sequence {x_{ n }}_{n∈ℕ}in X.
We use the following lemmas in the proof of the main result and refer to [1] for their proofs.
Lemma 2.9. Let (X, d) be a cone metric space and {x_{ n } }_{n∈ℕ}be a sequence in X. If {x_{ n } }_{n∈ℕ}is convergent, then it is a Cauchy sequence.
In 1969, Meir and Keeler [4] introduced a new type of fixed point theorem by defining MeirKeeler contraction (KMC) as a new contractive condition in complete metric spaces. It is as follows:
for all x, y ∈ X. Then, f has a unique fixed point.
In 2006, Suzuki [10] proved the integral type contraction (which has been introduced by Branciari [11]) is a special case of KMC (see also[12]). In 2010, Rezapour et al. [13] extended MeirKeeler's theorem to cone metric spaces as follows:
for all x, y ∈ X. Then, f has a unique fixed point.
3 Cone integration
We recall the following definitions and lemmas of cone integration and refer to [7] for their proofs.
Definition 3.2. The set {a = x_{0}, x_{1},···, x_{ n } = b} is called a partition for [a, b], if and only if, the intervals are pairwise disjoint and . Denote as the collection of all partitions of [a, b].
respectively. Also, we denote Δ(Q) = sup{x_{ i } x_{i1}, x_{ i }∈ Q}.
which S^{ Con } must be unique.
We denote the set of all cone integrable function ϕ : [a, b] → E by .
Lemma 3.5. Let M be a subset of P. The following conditions hold:
(1)
(2)
In 2010, Khojasteh et al. [7] introduced the following fixed point theorem in cone metric spaces.
for all x, y ∈ X, and for some α ∈ (0, 1). Then, f has a unique fixed point in X.
Also, they proved the following lemma:
The rest of the paper is organized as follows: In Section 4, we extend Theorems 1.4 and 3.7 in cone metric spaces. Many authors avoid of using the normality condition of P (see [13–15]). Here, we avoid of using such condition and the subadditivity assumption (Theorem 4.7). In addition, a new generalization of Theorems 1.4 and 3.7 which has a closer relative with KMC (see [4, 10]), is given. In Section 5, an example is given to illustrate our result is a generalization of the results given by Moradi et al. [8] and Khojasteh et al. [7].
4 Some extensions of recent results
The following definitions play a crucial role to state the main results.
where ε_{ n } → 0 and (x_{ n }  y_{ n } ) → 0, then F (x_{ n } )  F(y_{ n } ) → 0.
Definition 4.2. A mapping F : P → P is bounded, if for each bounded subset Q ⊂ P with respect to norm of E, F(Q) is a bounded subset.
Definition 4.3. Let P be a cone in E. Let Ω be the set of all mappings F : P → P such that
(I) F^{1}(0) = {0}.
(II) For each sequence {t_{ n } } ⊂ P, F (t_{ n } ) → 0 implies that t_{ n } → 0.
(III) F is bounded and nondecreasing in a sense that F(a) ≤ F(b) if a ≤ b, for every a, b ∈ P.
(IV) F be right continuous as declared in Definition 4.1.
Definition 4.4. ψ: P → P is called a function, if for each ε ≫ 0, there exists δ ≫ 0 such that ψ(t) ≤ ε for each ε ≤ t ≤ ε + δ. Suppose denote the set of all functions on P into itself.
Example 4.5. For each x ∈ P define ψ(x) = αx, which α ∈ [0, 1). Suppose ε ≫ 0 is given. Taking implies that ψ(x) ≤ ε for each ε ≤ x ≤ ε + δ. Thus, ψ is a function.
The following theorem extends the previous result given by Moradi et al. [8] and Khojasteh et al. [7] without assuming to be subadditive.
Theorem 4.7. Let (X, d) be a complete cone metric space, α ∈ [0, 1) and T, f : X → X be two mappings such that T is onetoone and closed graph, and f is T_{ F }  contraction, respectively, where F ∈ Ω. Then, f has a unique fixed point a ∈ X. Also, for every x_{0} ∈ X, the sequence of iterates {Tf^{ n }x_{0}} converges to Ta.
Proof. Uniqueness of the fixed point follows from (4.1). Let x_{0} ∈ X, x_{n+1}= fx_{ n }and y_{ n }= Tx_{ n }for all n ∈ ℕ. We break the argument into four steps.
Since F ∈ Ω,
Step 2. {y_{ n } }is a bounded sequence.
Hence, the sequence {d(y_{n(k)}, y_{n(k+1)})} is bounded.
Since 1  α > 0 and (4.10) holds, then F(d(y_{n(k)1}, y_{n(k+1)1})) → 0. So from (4.8), e_{ k } → 0 and this is a contradiction because e_{ k }  = 1.
Step 3. {y_{ n } } is Cauchy sequence.
Since {y_{ n } } is bounded and (4.13) holds, . This means that, {y_{ n } } is a Cauchy sequence.
Step 4. f has a fixed point.
This shows F(d(y_{n+1}, Tf (a))) → 0. So d(y_{n+1}, Tf (a)) → 0. Therefore, y_{ n }→ Tf(a), i.e., Tf(a) = Ta. Since T is one to one, thus fa = a.□
Lemma 4.8. Define , where ϕ : P → P is a nonvanishing mapping and subadditive cone integrable on each [a, b] ⊂ P such that for each ε ≫ 0, and the mapping F(x) by (x ≥ 0), has a continuous inverse. Then, F satisfies all conditions of Definition 4.3.
Therefore, (4.20) contradicts (4.19).
for some α ∈ (0, 1), then f has a unique fixed point in X.
Proof. Set in Theorem 4.7 and by using Lemma 4.8, the desired result is obtained.
Remark 4.11. Theorem 4.7 is an extension of Theorem 1.4 and 3.7 in cone metric spaces.
Corollary 4.12. Let (X, d) be a complete metric space, α ∈ [0, 1) and T, f : X → X be two mappings such that T is onetoone and closed graph, and f is T_{ F }contraction, respectively, where F ∈ Ω. Then, f has a unique fixed point a ∈ X. Also, for every x_{0} ∈ X, the sequence of iterates {Tf^{ n }x_{0}} converges to Ta.
Proof. By the same proof asserted in Theorem 4.7, the result is obtained.□
The following theorem is a diverse generalization of the results given by Moradi et al. [8], Khojasteh et al. [7], Suzuki [10], MeirKeeler [4] and Rezapour et al. [13].
Theorem 4.13. Let (X, d) be a complete regular cone metric space and f be a mapping on X. Let T : X → X be a mapping such that T is one to one and closed graph. Assume that there exists a function θ from P into itself satisfying the following:
(I) θ(0) = 0 and θ(t) ≫ 0 for all t ≫ 0.
(II) θ is nondecreasing and continuous function. Moreover, its inverse is continuous.
Then, f has a unique fixed point.
for all x ≠ y.
Remark 4.14. The following notations are considerable:
The following theorem is a direct result of Theorem 4.13.
for all x, y ∈ X, respectively, where θ : P → P satisfies in (I), (II) and (IV) of Theorem 4.13 and (see Definition 4.4). Then, f has a unique fixed point a ∈ X.
This means that, the condition (III) of Theorem 4.13 holds and so f has a unique fixed point.□
for all x, y ∈ X, respectively, where ψ is a function and is a nonvanishing integrable mapping on each such that for each ε > 0, . Then, f has a unique fixed point a∈ X.
Proof. By taking and in Theorem 4.15, the desired result is obtained.
5 An example
In this section, we give an example to illustrate our results.
has the continuous inverse at zero. We show, f does not satisfy in Theorem 3.7 with ϕ defined as above.
This means that, q > 1 and this is a contradiction. Therefore, we can't apply Theorem 3.7 for f.
To prove our claim we need to consider the following cases:
Thus, the desired result is obtained.
Case (2). If and , where m, n are odd.
Case (3). If and , where m is odd and n is even.
Proof of the Case (2) and (3) are similar to the argument as in the Case (1).
From and , the desired result is obtained.
From and , the desired result is obtained. Therefore, one can apply Theorem 4.10 for the mapping f.
Remark 5.2. Example 5.1 shows Corollary 4.10 is an extension of Theorem 3.7.
Declarations
Acknowledgements
The authors would like to thank and the anonymous referees for their respective helpful discussions and suggestions in preparation of this article.
Authors’ Affiliations
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