Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 407071 | 10 pages | https://doi.org/10.1155/2012/407071

A Note on tvs-G-Cone Metric Fixed Point Theory

Academic Editor: Yansheng Liu
Received26 Apr 2012
Accepted02 Jul 2012
Published05 Aug 2012

Abstract

For a tvs-G-cone metric space (𝑋,𝐺) and for the family 𝒜 of subsets of X, we introduce a new notion of the tvs-ℋ-cone metric ℋ with respect to G, and we get a fixed result for the ğ’žâ„¬ğ’²-tvs-G-cone-type function in a complete tvs-G-cone metric space (𝒜,ℋ). Our results generalize some recent results in the literature.

1. Introduction and Preliminaries

In 2007, Huang and Zhang [1] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space, and they showed some fixed point theorems of contractive-type mappings on cone metric spaces. The category of cone metric spaces is larger than metric spaces. Subsequently, many authors studied this subject and many results on fixed point theory are proved (see, e.g., [2–15]). Recently, Du [16] introduced the concept of 𝑡𝑣𝑠-cone metric and 𝑡𝑣𝑠-cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [1]. Later, in the papers [16–21], the authors tried to generalize this approach by using cones in topological vector spaces 𝑡𝑣𝑠 instead of Banach spaces. However, it should be noted that an old result shows that if the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space. Thus, proper generalizations when passing from norm-valued cone metric spaces to 𝑡𝑣𝑠-valued cone metric spaces can be obtained only in the case of nonnormal cones (for details, see [19]). Further, Radenović et al. [22] introduced the concept of set-valued contraction of Nadler type in the setting of 𝑡𝑣𝑠-cone spaces and proved a fixed point theorem in the setting of tvs-cone spaces with respect to a solid cone.

Definition 1.1 (see [22]). Let (𝑋,𝑑) be a 𝑡𝑣𝑠-cone metric space with a solid cone 𝑃, and let 𝒜 be a collection of nonempty subsets of 𝑋. A map ℋ∶𝒜×𝒜→𝐸 is called a 𝑡𝑣𝑠-ℋ-cone metric with respect to 𝑑 if for any 𝐴1,𝐴2∈𝒜 the following conditions hold:(H1)ℋ(𝐴1,𝐴2)=𝜃⇒𝐴1=𝐴2,(H2)ℋ(𝐴1,𝐴2)=ℋ(𝐴2,𝐴1),(H3)∀𝜀∈𝐸,𝜃≪𝜀∀𝑥∈𝐴1∃𝑦∈𝐴2𝑑(𝑥,𝑦)≼ℋ(𝐴1,𝐴2)+𝜀,(H4) one of the following is satisfied:(i)∀𝜀∈𝐸,𝜃≪𝜀∃𝑥∈𝐴1∀𝑦∈𝐴2ℋ(𝐴1,𝐴2)≼𝑑(𝑥,𝑦)+𝜀,(ii)∀𝜀∈𝐸,𝜃≪𝜀∃𝑦∈𝐴2∀𝑥∈𝐴1ℋ(𝐴1,𝐴2)≼𝑑(𝑥,𝑦)+𝜀.

Theorem 1.2 (see [22]). Let (𝑋,𝑑) be a 𝑡𝑣𝑠-cone complete metric space with a solid cone 𝑃 and let 𝒜 be a collection of nonempty closed subsets of 𝑋, 𝒜≠𝜙, and let ℋ∶𝒜×𝒜→𝐸 be a 𝑡𝑣𝑠-ℋ-cone metric with respect to 𝑑. If the mapping 𝑇∶𝑋→𝒜 the condition that exists a 𝜆∈(0,1) such that for all 𝑥,𝑦∈𝑋 holds ℋ(𝑇𝑥,𝑇𝑦)≼𝜆𝐺(𝑥,𝑦)(1.1) then 𝑇 has a fixed point in 𝑋.

We recall some definitions and results of the 𝑡𝑣𝑠-cone metric spaces that introduced in [19, 23], which will be needed in the sequel.

Let 𝐸 be be a real Hausdorff topological vector space (𝑡𝑣𝑠 for short) with the zero vector 𝜃. A nonempty subset 𝑃 of 𝐸 is called a convex cone if 𝑃+𝑃⊆𝑃 and 𝜆𝑃⊆𝑃 for 𝜆≥0. A convex cone 𝑃 is said to be pointed (or proper) if 𝑃∩(−𝑃)={𝜃}; 𝑃 is normal (or saturated) if 𝐸 has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone 𝑃⊆𝐸, we can define a partial ordering ≼ with respect to 𝑃 by 𝑥≼𝑦 if and only if 𝑦−𝑥∈𝑃; 𝑥≺𝑦 will stand for 𝑥≼𝑦 and 𝑥≠𝑦, while 𝑥≪𝑦 will stand for 𝑦−𝑥∈int𝑃, where int𝑃 denotes the interior of 𝑃. The cone 𝑃 is said to be solid if it has a nonempty interior.

In the sequel, 𝐸 will be a locally convex Hausdorff 𝑡𝑣𝑠 with its zero vector 𝜃, 𝑃 a proper, closed, and convex pointed cone in 𝐸 with int𝑃≠𝜙 and ≼ a partial ordering with respect to 𝑃.

Definition 1.3 (see [16, 18, 19]). Let 𝑋 be a nonempty set and (𝐸,𝑃) an ordered 𝑡𝑣𝑠. A vector-valued function 𝑑∶𝑋×𝑋→𝐸 is said to be a 𝑡𝑣𝑠-cone metric, if the following conditions hold:(C1)∀𝑥,𝑦∈𝑋,𝑥≠𝑦𝜃≼𝑑(𝑥,𝑦),(C2)∀𝑥,𝑦∈𝑋𝑑(𝑥,𝑦)=𝜃⇔𝑥=𝑦,(C3)∀𝑥,𝑦∈𝑋𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥),(C4)∀𝑥,𝑦,𝑧∈𝑋𝑑(𝑥,𝑧)≼𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧). Then the pair (𝑋,𝑑) is called a 𝑡𝑣𝑠-cone metric space.

Definition 1.4 (see [16, 18, 19]). Let (𝑋,𝑑) be a 𝑡𝑣𝑠-cone metric space, 𝑥∈𝑋, and {𝑥𝑛} a sequence in 𝑋.(1){𝑥𝑛}𝑡𝑣𝑠-cone converges to 𝑥 whenever for every 𝑐∈𝐸 with 𝜃≪𝑐, there exists 𝑛0∈ℕ such that 𝑑(𝑥𝑛,𝑥)≪𝑐 for all 𝑛≥𝑛0. We denote this by cone-limğ‘›â†’âˆžğ‘¥ğ‘›=𝑥;(2){𝑥𝑛} is a 𝑡𝑣𝑠-cone Cauchy sequence whenever for every 𝑐∈𝐸 with 𝜃≪𝑐, there exists 𝑛0∈ℕ such that 𝑑(𝑥𝑛,𝑥𝑚)≪𝑐 for all 𝑛,𝑚≥𝑛0;(3)(𝑋,𝑑) is 𝑡𝑣𝑠-cone complete if every 𝑡𝑣𝑠-cone Cauchy sequence in 𝑋 is 𝑡𝑣𝑠-cone convergent in 𝑋.

Remark 1.5. Clearly, a cone metric space in the sense of Huang and Zhang [1] is a special case of 𝑡𝑣𝑠-cone metric spaces when (𝑋,𝑑) is a 𝑡𝑣𝑠-cone metric space with respect to a normal cone 𝑃.

Remark 1.6 (see [19, 22, 23]). Let (𝑋,𝑑) be a 𝑡𝑣𝑠-cone metric space with a solid cone 𝑃. The following properties are often used, particularly in the case when the underlying cone is nonnormal.(p1) If 𝑢≼𝑣 and 𝑣≪𝑤, then 𝑢≪𝑤,(p2) If 𝑢≪𝑣 and 𝑣≼𝑤, then 𝑢≪𝑤,(p3) If 𝑢≪𝑣 and 𝑣≪𝑤, then 𝑢≪𝑤,(p4) If 𝜃≼𝑢≪𝑐 for each 𝑐∈int𝑃, then 𝑢=𝜃,(p5) If ğ‘Žâ‰¼ğ‘+𝑐 for each 𝑐∈int𝑃, then ğ‘Žâ‰¼ğ‘,(p6) If 𝐸 is 𝑡𝑣𝑠 with a cone 𝑃, and if ğ‘Žâ‰¼ğœ†ğ‘Ž where ğ‘Žâˆˆğ‘ƒ and 𝜆∈[0,1), then ğ‘Ž=𝜃,(p7) If 𝑐∈int𝑃, ğ‘Žğ‘›âˆˆğ¸, and ğ‘Žğ‘›â†’ğœƒ in locally convex 𝑡𝑣𝑠𝐸, then there exists 𝑛0∈ℕ such that ğ‘Žğ‘›â‰ªğ‘ for all 𝑛>𝑛0.

Metric spaces are playing an important role in mathematics and the applied sciences. To overcome fundamental laws in Dhage’s theory of generalized metric spaces [24]. In 2003, Mustafa and Sims [25] introduced a more appropriate and robust notion of a generalized metric space as follows.

Definition 1.7 (see [25]). Let 𝑋 be a nonempty set, and let 𝐺∶𝑋×𝑋×𝑋→[0,∞) be a function satisfying the following axioms:(G1)∀𝑥,𝑦,𝑧∈𝑋𝐺(𝑥,𝑦,𝑧)=0⇔𝑥=𝑦=𝑧,(G2)∀𝑥,𝑦∈𝑋,𝑥≠𝑦𝐺(𝑥,𝑥,𝑦)>0,(G3)∀𝑥,𝑦,𝑧∈𝑋𝐺(𝑥,𝑦,𝑧)≥𝐺(𝑥,𝑥,𝑦),(G4)∀𝑥,𝑦,𝑧∈𝑋(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑧,𝑦,𝑥)=⋯ (symmetric in all three variables),(G5)∀𝑥,𝑦,𝑧,𝑤∈𝑋𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑤,𝑤)+𝐺(𝑤,𝑦,𝑧). Then the function 𝐺 is called a generalized metric, or, more specifically a 𝐺-metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝐺-metric space.

By using the notions of generalized metrics and 𝑡𝑣𝑠-cone metrics, we introduce the below notion of 𝑡𝑣𝑠-generalized-cone metrics.

Definition 1.8. Let 𝑋 be a nonempty set and (𝐸,𝑃) an ordered 𝑡𝑣𝑠, and let 𝐺∶𝑋×𝑋×𝑋→𝐸 be a function satisfying the following axioms:(G1)∀𝑥,𝑦,𝑧∈𝑋𝐺(𝑥,𝑦,𝑧)=𝜃 if and only if 𝑥=𝑦=𝑧,(G2)∀𝑥,𝑦∈𝑋,𝑥≠𝑦𝜃≪𝐺(𝑥,𝑥,𝑦),(G3)∀𝑥,𝑦,𝑧∈𝑋𝐺(𝑥,𝑥,𝑦)≼𝐺(𝑥,𝑦,𝑧),(G4)∀𝑥,𝑦,𝑧∈𝑋𝐺(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑧,𝑦,𝑥)=⋯ (symmetric in all three variables),(G5)∀𝑥,𝑦,𝑧,𝑤∈𝑋𝐺(𝑥,𝑦,𝑧)≼𝐺(𝑥,𝑤,𝑤)+𝐺(𝑤,𝑦,𝑧). Then the function 𝐺 is called a 𝑡𝑣𝑠-generalized-cone metric, or, more specifically, a 𝑡𝑣𝑠-𝐺-cone metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝑡𝑣𝑠-𝐺-cone metric space.

Definition 1.9. Let (𝑋,𝐺) be a 𝑡𝑣𝑠-𝐺-cone metric space, 𝑥∈𝑋, and {𝑥𝑛} a sequence in 𝑋.(1){𝑥𝑛}𝑡𝑣𝑠-𝐺-cone converges to 𝑥 whenever, for every 𝑐∈𝐸 with 𝜃≪𝑐, there exists 𝑛0∈ℕ such that 𝐺(𝑥𝑛,𝑥𝑚,𝑥)≪𝑐 for all 𝑚,𝑛≥𝑛0. Here 𝑥 is called the limit of the sequence {𝑥𝑛} and is denoted by 𝐺-cone-limğ‘›â†’âˆžğ‘¥ğ‘›=𝑥;(2){𝑥𝑛} is a 𝑡𝑣𝑠-𝐺-cone Cauchy sequence whenever, for every 𝑐∈𝐸 with 𝜃≪𝑐, there exists 𝑛0∈ℕ such that 𝐺(𝑥𝑛,𝑥𝑚,𝑥𝑙)≪𝑐 for all 𝑛,𝑚,𝑙≥𝑛0;(3)(𝑋,𝐺) is 𝑡𝑣𝑠-𝐺-cone complete if every 𝑡𝑣𝑠-𝐺-cone Cauchy sequence in 𝑋 is 𝑡𝑣𝑠-𝐺-cone convergent in 𝑋.

Proposition 1.10. Let (𝑋,𝐺) be a 𝑡𝑣𝑠-𝐺-cone metric space, 𝑥∈𝑋, and {𝑥𝑛} a sequence in 𝑋. The following are equivalent:(i){𝑥𝑛}𝑡𝑣𝑠-𝐺-cone converges to 𝑥,(ii)𝐺(𝑥𝑛,𝑥𝑛,𝑥)→𝜃 as ğ‘›â†’âˆž,(iii)𝐺(𝑥𝑛,𝑥,𝑥)→ as n→∞,(iv)𝐺(𝑥𝑛,𝑥𝑚,𝑥)→𝜃 as 𝑛,ğ‘šâ†’âˆž.

In this paper, we also introduce the below concept of the ğ’žâ„¬ğ’²-𝑡𝑣𝑠-𝐺-cone-type function.

Definition 1.11. One calls𝜑∶int𝑃∪{𝜃}→int𝑃∪{𝜃} a ğ’žâ„¬ğ’²-𝑡𝑣𝑠-𝐺-cone-type function if the function 𝜑 satisfies the following condition(𝜑1)𝜑(𝑡)≪𝑡 for all 𝑡≫𝜃 and 𝜑(𝜃)=𝜃;(𝜑2)limğ‘›â†’âˆžğœ‘ğ‘›(𝑡)=𝜃 for all 𝑡∈int𝑃∪{𝜃}.
In this paeper, for a 𝑡𝑣𝑠-𝐺-cone metric space (𝑋,𝐺) and for the family 𝒜 of subsets of 𝑋, we introduce a new notion of the 𝑡𝑣𝑠-ℋ-cone metric ℋ with respect to 𝐺, and we get a fixed result for the ğ’žâ„¬ğ’²-𝑡𝑣𝑠-𝐺-cone-type function in a complete 𝑡𝑣𝑠-𝐺-cone metric space (𝒜,ℋ). Our results generalize some recent results in the literature.

2. Main Results

Let 𝐸 be a locally convex Hausdorff 𝑡𝑣𝑠 with its zero vector 𝜃, 𝑃 a proper, closed and convex pointed cone in 𝐸 with int𝑃≠𝜙, and ≼ a partial ordering with respect to 𝑃. We introduce the below notion of the 𝑡𝑣𝑠-ℋ-cone metric ℋ with respect to 𝑡𝑣𝑠-𝐺-cone metric 𝐺.

Definition 2.1. Let (𝑋,𝐺) be a 𝑡𝑣𝑠-𝐺-cone metric space with a solid cone 𝑃, and let 𝒜 be a collection of nonempty subsets of 𝑋. A map ℋ∶𝒜×𝒜×𝒜→𝐸 is called a 𝑡𝑣𝑠-ℋ-cone metric with respect to 𝐺 if for any 𝐴1,𝐴2,𝐴3∈𝒜 the following conditions hold:(H1)ℋ(𝐴1,𝐴2,𝐴3)=𝜃⇒𝐴1=𝐴2=𝐴3,(H2)ℋ(𝐴1,𝐴2,𝐴3)=ℋ(𝐴2,𝐴1,𝐴3)=ℋ(𝐴1,𝐴3,𝐴2)=⋯(symmetry in all variables),(H3)ℋ(𝐴1,𝐴1,𝐴2)≼ℋ(𝐴1,𝐴2,𝐴3),(H4)∀𝜀∈𝐸,𝜃≪𝜀∀𝑥∈𝐴1,𝑦∈𝐴2∃𝑧∈𝐴3𝐺(𝑥,𝑦,𝑧)≼ℋ(𝐴1,𝐴2,𝐴3)+𝜀,(H5) one of the following is satisfied:(i)∀𝜀∈𝐸,𝜃≪𝜀∃𝑥∈𝐴1∀𝑦∈𝐴2,𝑧∈𝐴3ℋ(𝐴1,𝐴2,𝐴3)≼𝐺(𝑥,𝑦,𝑧)+𝜀,(ii)∀𝜀∈𝐸,𝜃≪𝜀∃𝑦∈𝐴2∀𝑥∈𝐴1,𝑧∈𝐴3ℋ(𝐴1,𝐴2,𝐴3)≼𝐺(𝑥,𝑦,𝑧)+𝜀,(iii)∀𝜀∈𝐸,𝜃≪𝜀∃𝑧∈𝐴3∀𝑦∈𝐴2,𝑥∈𝐴1ℋ(𝐴1,𝐴2,𝐴3)≼𝐺(𝑥,𝑦,𝑧)+𝜀.

We will prove that a 𝑡𝑣𝑠-ℋ-cone metric satisfies the conditions of (𝐺1)-(𝐺5).

Lemma 2.2. Let (𝑋,𝐺) be a 𝑡𝑣𝑠-𝐺-cone metric space with a solid cone 𝑃, and let 𝒜 be a collection of nonempty subsets of 𝑋, 𝒜≠𝜙. If ℋ∶𝒜×𝒜×𝒜→𝐸 is a 𝑡𝑣𝑠-ℋ-cone metric with respect to 𝐺, then pair (𝒜,ℋ) is a 𝑡𝑣𝑠-𝐺-cone metric space.

Proof. Let {𝜀𝑛}⊂𝐸 be a sequence such that 𝜃≪𝜀𝑛 for all 𝑛∈ℕ and 𝐺-𝑐𝑜𝑛𝑒-limğ‘›â†’âˆžğœ€ğ‘›=𝜃. Take any 𝐴1,𝐴2,𝐴3∈𝒜 and 𝑥∈𝐴1, 𝑦∈𝐴2. From (H4), for each 𝑛∈ℕ, there exists 𝑧𝑛∈𝐴3 such that 𝐺𝑥,𝑦,𝑧𝑛𝐴≼ℋ1,𝐴2,𝐴3+𝜀𝑛.(2.1) Therefore, ℋ(𝐴1,𝐴2,𝐴3)+𝜀𝑛∈𝑃 for each 𝑛∈ℕ. By the closedness of 𝑃, we conclude that 𝜃≼ℋ(𝐴1,𝐴2,𝐴3).
Assume that 𝐴1=𝐴2=𝐴3. From H5, we obtain ℋ(𝐴1,𝐴2,𝐴3)≼𝜀𝑛 for any 𝑛∈ℕ. So ℋ(𝐴1,𝐴2,𝐴3)=𝜃.
Let 𝐴1,𝐴2,𝐴3,𝐴4∈𝒜. Assume that 𝐴1,𝐴2,𝐴3 satisfy condition (H5)(i). Then, for each 𝑛∈ℕ, there exists 𝑥𝑛∈𝐴1 such that ℋ(𝐴1,𝐴2,𝐴3)≼𝐺(𝑥𝑛,𝑦,𝑧)+𝜀𝑛 for all 𝑦∈𝐴2 and 𝑧∈𝐴3. From (H4), there exists a sequence {𝑤𝑛}⊂𝐴4 satisfying 𝐺(𝑥𝑛,𝑤𝑛,𝑤𝑛)≼ℋ(𝐴1,𝐴4,𝐴4)+𝜀𝑛 for every 𝑛∈ℕ. Obviously, for any 𝑦∈𝐴2 and any 𝑧∈𝐴3 and 𝑛∈ℕ, we have ℋ𝐴1,𝐴2,𝐴3𝑥≼𝐺𝑛,𝑦,𝑧+𝜀𝑛𝑥≼𝐺𝑛,𝑤n,𝑤𝑛𝑤+𝐺𝑛,𝑦,𝑧+𝜀𝑛.(2.2) Now for each 𝑛∈ℕ, there exists 𝑦𝑛∈𝐴2, 𝑧𝑛∈𝐴3 such that 𝐺(𝑤𝑛,𝑦𝑛,𝑧𝑛)≼ℋ(𝐴4,𝐴2,𝐴3)+𝜀𝑛. Consequently, we obtain that for each 𝑛∈ℕℋ𝐴1,𝐴2,𝐴3𝐴≼ℋ1,𝐴4,𝐴4𝐴+ℋ4,𝐴2,𝐴3+3𝜀𝑛.(2.3) Therefore, ℋ𝐴1,𝐴2,𝐴3𝐴≼ℋ1,𝐴4,𝐴4𝐴+ℋ4,𝐴2,𝐴3.(2.4) In the case when (H5)(ii) or (H5)(iii) holds, we use the analogous method.

In the sequel, we denote by Θ the class of functions 𝜑∶int𝑃∪{𝜃}→int𝑃∪{𝜃} satisfying the following conditions:(C1)𝜑 is a ğ’žâ„¬ğ’²-𝑡𝑣𝑠-𝐺-cone-type-function;(C2)𝜑 is subadditive, that is, 𝜑(𝑢1+𝑢2)≼𝜑(𝑢1)+𝜑(𝑢2) for all 𝑢1,𝑢2∈int𝑃.

Our main result is the following.

Theorem 2.3. Let (𝑋,𝐺) be a 𝑡v𝑠-𝐺-cone complete metric space with a solid cone 𝑃, let 𝒜 be a collection of nonempty closed subsets of 𝑋, 𝒜≠𝜙, and let ℋ∶𝒜×𝒜×𝒜→𝐸 be a 𝑡𝑣𝑠-ℋ-cone metric with respect to 𝐺. If the mapping 𝑇∶𝑋→𝒜 satisfies the condition that exists a 𝜑∈Θ such that for all 𝑥,𝑦,𝑧∈𝑋 holds ℋ(𝑇𝑥,𝑇𝑦,𝑇𝑧)≼𝜑(𝐺(𝑥,𝑦,𝑧)),(2.5) then 𝑇 has a fixed point in 𝑋.

Proof. Let us choose 𝜀∈int𝑃 arbitrarily, and let 𝜀𝑛∈𝐸 be a sequence such that 𝜃≪𝜀𝑛 and 𝜀𝑛≼𝜀/3𝑛. Let us choose 𝑥0∈𝑋 arbitrarily and 𝑥1∈𝑇𝑥0. If 𝐺(𝑥0,𝑥0,𝑥1)=𝜃, then 𝑥0=𝑥1∈𝑇(𝑥0), and we are done. Assume that 𝐺(𝑥0,𝑥0,𝑥1)≫𝜃. Taking into account (2.5) and (H4), there exists 𝑥2∈𝑇𝑥1 such that 𝐺𝑥1,𝑥1,𝑥2≼ℋ𝑇𝑥0,𝑇𝑥0,𝑇𝑥1+𝜀1𝐺𝑥≼𝜑0,𝑥0,𝑥1+𝜀1.(2.6) Taking into account (2.5), (2.6), and (H4) and since 𝜑∈Θ, there exists 𝑥3∈𝑇𝑥2 such that 𝐺𝑥2,𝑥2,𝑥3≼ℋ𝑇𝑥1,𝑇𝑥1,𝑇𝑥2+𝜀2𝐺𝑥≼𝜑1,𝑥1,𝑥2+𝜀2𝜑𝐺𝑥≼𝜑0,𝑥0,𝑥1+𝜀1+𝜀2𝜑𝐺𝑥≼𝜑0,𝑥0,𝑥1𝜀+𝜑1+𝜀2≪𝜑2𝐺𝑥0,𝑥0,𝑥1+𝜀1+𝜀2≼𝜑2𝐺𝑥0,𝑥0,𝑥1+𝜀3+𝜀32.(2.7) We continue in this manner. In general, for 𝑥𝑛, 𝑛∈ℕ, 𝑥𝑛+1 is chosen such that 𝑥𝑛+1∈𝑇𝑥𝑛 and 𝐺𝑥𝑛,𝑥𝑛,𝑥𝑛+1≼ℋ𝑇𝑥𝑛−1,𝑇𝑥𝑛−1,𝑇𝑥𝑛+𝜀𝑛𝐺𝑥≪𝜑𝑛−1,𝑥𝑛−1,𝑥𝑛+𝜀𝑛≼𝜑2𝐺𝑥𝑛−2,𝑥𝑛−2,𝑥𝑛−1+𝜀𝑛−1+𝜀𝑛≼⋯⋯≼𝜑𝑛𝐺𝑥0,𝑥0,𝑥1+𝑛𝑖=1𝜀𝑖≼𝜑𝑛𝐺𝑥0,𝑥0,𝑥1+𝑛𝑖=1𝜀3𝑖≼𝜑𝑛𝐺𝑥0,𝑥0,𝑥1+12𝜀.(2.8) Since 𝜀 is arbitrary, letting 𝜀→𝜃 and by the definition of the ğ’žâ„¬ğ’²-𝑡𝑣𝑠-𝐺-cone-type function, we obtain that limğ‘›â†’âˆžğºî€·ğ‘¥ğ‘›,𝑥𝑛,𝑥𝑛+1=𝜃.(2.9)
Next, we let 𝑐𝑚=𝐺(𝑥𝑚,𝑥𝑚+1,𝑥𝑚+1), and we claim that the following result holds: for each 𝛾≫𝜃,  there is 𝑛0(𝜀)∈𝑁 such that for all m, 𝑛≥𝑛0(𝛾), 𝐺𝑥𝑚,𝑥𝑚+1,𝑥𝑚+1≪𝛾,(2.10) We will prove (2.10) by contradiction. Suppose that (2.10) is false. Then there exists some 𝛾≫𝜃 such that for all 𝑝∈ℕ, there are 𝑚𝑝,𝑛𝑝∈ℕ with 𝑚𝑝>𝑛𝑝≥𝑝 satisfying(i)𝑚𝑝 is even and 𝑛𝑝 is odd,(ii)𝐺(𝑥𝑚𝑝,𝑥𝑛𝑝,𝑥𝑛𝑝)≽𝛾, and(iii)𝑚𝑝 is the smallest even number such that conditions (i), (ii) hold.Since 𝑐𝑚↓𝜃, by (ii), we have that limğ‘â†’âˆžğº(𝑥𝑚𝑝,𝑥𝑛𝑝,𝑥𝑛𝑝)=𝛾 and 𝑥𝛾≼𝐺𝑚𝑝,𝑥𝑛𝑝,𝑥𝑛𝑝𝑥≼𝐺𝑚𝑝,𝑥𝑚𝑝+1,𝑥𝑚𝑝+1𝑥+𝐺𝑚𝑝+1,𝑥𝑛𝑝+1,𝑥𝑛𝑝+1𝑥+𝐺𝑛𝑝+1,𝑥𝑛𝑝,𝑥𝑛𝑝.(2.11) It follows from (H4); let us choose 𝜀∈𝐸 arbitrarily such that 𝐺𝑥𝑛𝑝+1,𝑥𝑛𝑝+1,𝑥𝑚𝑝+1≼ℋ𝑇𝑥𝑛𝑝+1,𝑇𝑥𝑛𝑝+1,𝑇𝑥𝑚𝑝+1+𝜀.(2.12) Taking into account (2.5), (2.11), and (2.12), we have that 𝑥𝛾≼𝐺𝑚𝑝,𝑥𝑛𝑝,𝑥𝑛𝑝𝑥≼𝐺𝑚𝑝,𝑥𝑚𝑝+1,𝑥𝑚𝑝+1+ℋ𝑇𝑥𝑛𝑝+1,𝑇𝑥𝑛𝑝+1,𝑇𝑥𝑚𝑝+1𝑥+𝜀+𝐺𝑛𝑝+1,𝑥𝑛𝑝,𝑥𝑛𝑝𝑥≼𝐺𝑚𝑝,𝑥𝑚𝑝+1,𝑥𝑚𝑝+1𝐺𝑥+𝜑𝑛𝑝,𝐺𝑥𝑛𝑝,𝐺𝑥𝑚𝑝𝑥+𝜀+𝐺𝑛𝑝+1,𝑥𝑛𝑝,𝑥𝑛𝑝𝑥≪𝐺𝑚𝑝,𝑥𝑚𝑝+1,𝑥𝑚𝑝+1𝑥+𝐺𝑛𝑝,𝐺𝑥𝑛𝑝,𝐺𝑥𝑚𝑝𝑥+𝜀+𝐺𝑛𝑝+1,𝑥𝑛𝑝,𝑥𝑛𝑝.(2.13) Since 𝜀 is arbitrarily, letting 𝜀→𝜃 and by letting ğ‘â†’âˆž, we have 𝛾≪𝜃+limğ‘â†’âˆžğºî‚€ğ‘¥ğ‘šğ‘,𝑥𝑛𝑝,𝑥𝑛𝑝+𝜃+𝜃=𝛾,(2.14) a contradiction. So {𝑥𝑛} is a 𝑡𝑣𝑠-𝐺-cone Cauchy sequence. Since (𝑋,𝐺)is a 𝑡𝑣𝑠-𝐺-cone complete metric space, {𝑥𝑛} is 𝑡𝑣𝑠-𝐺-cone convergent in 𝑋 and 𝐺-cone-limğ‘›â†’âˆžğ‘¥ğ‘›=𝑥. Thus, for every 𝜏∈int𝑃 and sufficiently large 𝑛, we have that ℋ𝑇𝑥𝑛,𝑇𝑥𝑛𝐺𝑥,𝑇𝑥≼𝜑𝑛,𝑥𝑛𝑥,𝑥≪𝐺𝑛,𝑥𝑛≪𝜏,𝑥3.(2.15) Since for 𝑛∈ℕ∪{0}, 𝑥𝑛+1∈𝑇𝑥𝑛, by (H4) we obtain that for all 𝑛∈ℕ there exists𝑦𝑛∈𝑇𝑥 such that 𝐺𝑥𝑛+1,𝑥𝑛+1,𝑦𝑛+1≼ℋ𝑇𝑥𝑛,𝑇𝑥𝑛,𝑇𝑥+𝜀𝑛+1𝐺𝑥≼𝜑𝑛,𝑥𝑛+𝜀,𝑥3𝑛+1𝑥≪𝐺𝑛,𝑥𝑛+𝜀,𝑥3𝑛+1.(2.16) Since 𝜀/3𝑛+1→𝜃, then for sufficiently large 𝑛, we obtain that 𝐺𝑦𝑛+1𝑦,𝑥,𝑥≼𝐺𝑛+1,𝑥𝑛+1,𝑥𝑛+1𝑥+𝐺𝑛+1≪,𝑥,𝑥2𝜏3+𝜏3=𝜏,(2.17) which implies 𝐺-cone-limğ‘›â†’âˆžğ‘¦ğ‘›=𝑥. Since 𝑇𝑥 is closed, we obtain that 𝑥∈𝑇𝑥.

For the case 𝜑(𝑡)=𝑘𝑡, 𝑘∈(0,1), then 𝜑∈Θ and it is easy to get the following corollary.

Corollary 2.4. Let (𝑋,𝐺) be a 𝑡𝑣𝑠-𝐺-cone complete metric space with a solid cone 𝑃, let 𝒜 be a collection of nonempty closed subsets of 𝑋, 𝒜≠𝜙, and let ℋ∶𝒜×𝒜×𝒜→𝐸 be a 𝑡𝑣𝑠-ℋ-cone metric with respect to 𝐺. If the mapping 𝑇∶𝑋→𝒜 satisfies the condition that exists 𝑘∈(0,1) such that for all 𝑥,𝑦,𝑧∈𝑋 holds ℋ(𝑇𝑥,𝑇𝑦,𝑇𝑧)≼𝑘⋅𝐺(𝑥,𝑦,𝑧),(2.18) then 𝑇 has a fixed point in 𝑋.

Remark 2.5. Following Corollary 2.4, it is easy to get Theorem 1.2. So our results generalize some recent results in the literature (e.g., [22]).

Acknowledgment

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

References

  1. L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 416–420, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. S. Janković, Z. Kadelburg, S. Radonevic, and B. E. Rhoades, “Assad-Kirktype fixed point theorems for a pair of nonself mappings on cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 761086, 16 pages, 2009. View at: Publisher Site | Google Scholar
  4. Sh. Rezapour and R. Hamlbarani, “Some notes on the paper: ‘Cone metric spaces and fixed point theorems of contractive mappings’,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 719–724, 2008. View at: Publisher Site | Google Scholar
  5. M. Arshad, A. Azam, and P. Vetro, “Some common fixed point results in cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 493965, 11 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. A. Azam and M. Arshad, “Common fixed points of generalized contractive maps in cone metric spaces,” Iranian Mathematical Society. Bulletin, vol. 35, no. 2, pp. 255–264, 2009. View at: Google Scholar | Zentralblatt MATH
  7. C. Di Bari and P. Vetro, “ϕ-pairs and common fixed points in cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 57, no. 2, pp. 279–285, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. C. Di Bari and P. Vetro, “Weakly ϕ-pairs and common fixed points in cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 58, no. 1, pp. 125–132, 2009. View at: Publisher Site | Google Scholar
  9. J. Harjani and K. Sadarangani, “Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1188–1197, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. R. H. Haghi and Sh. Rezapour, “Fixed points of multifunctions on regular cone metric spaces,” Expositiones Mathematicae, vol. 28, no. 1, pp. 71–77, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. D. Klim and D. Wardowski, “Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5170–5175, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. Z. Kadelburg, S. Radenović, and V. Rakočević, “Remarks on ‘Quasi-contraction on a cone metric space’,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1674–1679, 2009. View at: Publisher Site | Google Scholar
  13. Z. Kadelburg, M. Pavlović, and S. Radenović, “Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3148–3159, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. S. Rezapour, H. Khandani, and S. M. Vaezpour, “Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions,” Rendiconti del Circolo Matematico di Palermo, vol. 59, no. 2, pp. 185–197, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. Sh. Rezapour and R. H. Haghi, “Fixed point of multifunctions on cone metric spaces,” Numerical Functional Analysis and Optimization, vol. 30, no. 7-8, pp. 825–832, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. W.-S. Du, “A note on cone metric fixed point theory and its equivalence,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 5, pp. 2259–2261, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  17. A. Azam, I. Beg, and M. Arshad, “Fixed point in topological vector space-valued cone metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 604084, 9 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  18. I. Beg, A. Azam, and M. Arshad, “Common fixed points for maps on topological vector space valued cone metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 560264, 8 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  19. Z. Kadelburg, S. Radenović, and V. Rakočević, “Topological vector space-valued cone metric spaces and fixed point theorems,” Fixed Point Theory and Applications, vol. 2011, Article ID 170253, 17 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. Z. Kadelburg and S. Radenović, “Coupled fixed point results under tvs–cone metric and w-cone-distance,” Advanced in Fixed Point Theory, vol. 2, no. 1, pp. 29–46, 2012. View at: Google Scholar
  21. L. J. Ćirić, H. Lakzian, and V. Rakoćević, “Fixed point theorems for w-cone distance contraction mappings in tvs-cone metric spaces,” Fixed Point Theory and Applications, vol. 2012, p. 3, 2012. View at: Publisher Site | Google Scholar
  22. S. Radenović, S. Simić, N. Cakić, and Z. Golubović, “A note on tvs-cone metric fixed point theory,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2418–2422, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. Z. Kadelburg, S. Radenović, and V. Rakočević, “A note on the equivalence of some metric and cone metric fixed point results,” Applied Mathematics Letters, vol. 24, no. 3, pp. 370–374, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  24. B. C. Dhage, “Generalized metric space and mapping with fixed point,” Bulletin of the Calcutta Mathematical Society, vol. 84, pp. 329–336, 1992. View at: Google Scholar
  25. Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006. View at: Google Scholar | Zentralblatt MATH

Copyright © 2012 Ing-Jer Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

528 Views | 396 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder