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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 407071, 10 pages
http://dx.doi.org/10.1155/2012/407071
Research Article

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

1Department of Mathematics, National Kaohsiung Normal University, Taiwan
2Department of Applied Mathematics, National Hsinchu University of Education, Taiwan
3Department of Applied Mathematics, Chung Yuan Christian University, Taiwan

Received 26 April 2012; Accepted 2 July 2012

Academic Editor: YanshengΒ Liu

Copyright Β© 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.

Abstract

For a tvs-G-cone metric space (𝑋,𝐺) and for the family π’œ of subsets of X, we introduce a new notion of the tvs-β„‹-cone metric β„‹ with respect to G, and we get a fixed result for the π’žβ„¬π’²-tvs-G-cone-type function in a complete tvs-G-cone metric space (π’œ,β„‹). 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 𝑑𝑣𝑠-cone metric and 𝑑𝑣𝑠-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 𝑑𝑣𝑠 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 𝑑𝑣𝑠-valued cone metric spaces can be obtained only in the case of nonnormal cones (for details, see [19]). Further, RadenoviΔ‡ et al. [22] introduced the concept of set-valued contraction of Nadler type in the setting of 𝑑𝑣𝑠-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 𝑑𝑣𝑠-cone metric space with a solid cone 𝑃, and let π’œ be a collection of nonempty subsets of 𝑋. A map β„‹βˆΆπ’œΓ—π’œβ†’πΈ is called a 𝑑𝑣𝑠-β„‹-cone metric with respect to 𝑑 if for any 𝐴1,𝐴2βˆˆπ’œ the following conditions hold:(H1)β„‹(𝐴1,𝐴2)=πœƒβ‡’π΄1=𝐴2,(H2)β„‹(𝐴1,𝐴2)=β„‹(𝐴2,𝐴1),(H3)βˆ€πœ€βˆˆπΈ,πœƒβ‰ͺπœ€βˆ€π‘₯∈𝐴1βˆƒπ‘¦βˆˆπ΄2𝑑(π‘₯,𝑦)β‰Όβ„‹(𝐴1,𝐴2)+πœ€,(H4) one of the following is satisfied:(i)βˆ€πœ€βˆˆπΈ,πœƒβ‰ͺπœ€βˆƒπ‘₯∈𝐴1βˆ€π‘¦βˆˆπ΄2β„‹(𝐴1,𝐴2)≼𝑑(π‘₯,𝑦)+πœ€,(ii)βˆ€πœ€βˆˆπΈ,πœƒβ‰ͺπœ€βˆƒπ‘¦βˆˆπ΄2βˆ€π‘₯∈𝐴1β„‹(𝐴1,𝐴2)≼𝑑(π‘₯,𝑦)+πœ€.

Theorem 1.2 (see [22]). Let (𝑋,𝑑) be a 𝑑𝑣𝑠-cone complete metric space with a solid cone 𝑃 and let π’œ be a collection of nonempty closed subsets of 𝑋, π’œβ‰ πœ™, and let β„‹βˆΆπ’œΓ—π’œβ†’πΈ be a 𝑑𝑣𝑠-β„‹-cone metric with respect to 𝑑. If the mapping π‘‡βˆΆπ‘‹β†’π’œ the condition that exists a πœ†βˆˆ(0,1) such that for all π‘₯,π‘¦βˆˆπ‘‹ holds β„‹(𝑇π‘₯,𝑇𝑦)β‰Όπœ†πΊ(π‘₯,𝑦)(1.1) then 𝑇 has a fixed point in 𝑋.

We recall some definitions and results of the 𝑑𝑣𝑠-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 (𝑑𝑣𝑠 for short) with the zero vector πœƒ. A nonempty subset 𝑃 of 𝐸 is called a convex cone if 𝑃+π‘ƒβŠ†π‘ƒ and πœ†π‘ƒβŠ†π‘ƒ for πœ†β‰₯0. A convex cone 𝑃 is said to be pointed (or proper) if π‘ƒβˆ©(βˆ’π‘ƒ)={πœƒ}; 𝑃 is normal (or saturated) if 𝐸 has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone π‘ƒβŠ†πΈ, we can define a partial ordering β‰Ό with respect to 𝑃 by π‘₯≼𝑦 if and only if π‘¦βˆ’π‘₯βˆˆπ‘ƒ; π‘₯≺𝑦 will stand for π‘₯≼𝑦 and π‘₯≠𝑦, while π‘₯β‰ͺ𝑦 will stand for π‘¦βˆ’π‘₯∈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 𝑑𝑣𝑠 with its zero vector πœƒ, 𝑃 a proper, closed, and convex pointed cone in 𝐸 with intπ‘ƒβ‰ πœ™ and β‰Ό 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 𝑑𝑣𝑠. A vector-valued function π‘‘βˆΆπ‘‹Γ—π‘‹β†’πΈ is said to be a 𝑑𝑣𝑠-cone metric, if the following conditions hold:(C1)βˆ€π‘₯,π‘¦βˆˆπ‘‹,π‘₯β‰ π‘¦πœƒβ‰Όπ‘‘(π‘₯,𝑦),(C2)βˆ€π‘₯,π‘¦βˆˆπ‘‹π‘‘(π‘₯,𝑦)=πœƒβ‡”π‘₯=𝑦,(C3)βˆ€π‘₯,π‘¦βˆˆπ‘‹π‘‘(π‘₯,𝑦)=𝑑(𝑦,π‘₯),(C4)βˆ€π‘₯,𝑦,π‘§βˆˆπ‘‹π‘‘(π‘₯,𝑧)≼𝑑(π‘₯,𝑦)+𝑑(𝑦,𝑧). Then the pair (𝑋,𝑑) is called a 𝑑𝑣𝑠-cone metric space.

Definition 1.4 (see [16, 18, 19]). Let (𝑋,𝑑) be a 𝑑𝑣𝑠-cone metric space, π‘₯βˆˆπ‘‹, and {π‘₯𝑛} a sequence in 𝑋.(1){π‘₯𝑛}𝑑𝑣𝑠-cone converges to π‘₯ whenever for every π‘βˆˆπΈ with πœƒβ‰ͺ𝑐, there exists 𝑛0βˆˆβ„• such that 𝑑(π‘₯𝑛,π‘₯)β‰ͺ𝑐 for all 𝑛β‰₯𝑛0. We denote this by cone-limπ‘›β†’βˆžπ‘₯𝑛=π‘₯;(2){π‘₯𝑛} is a 𝑑𝑣𝑠-cone Cauchy sequence whenever for every π‘βˆˆπΈ with πœƒβ‰ͺ𝑐, there exists 𝑛0βˆˆβ„• such that 𝑑(π‘₯𝑛,π‘₯π‘š)β‰ͺ𝑐 for all 𝑛,π‘šβ‰₯𝑛0;(3)(𝑋,𝑑) is 𝑑𝑣𝑠-cone complete if every 𝑑𝑣𝑠-cone Cauchy sequence in 𝑋 is 𝑑𝑣𝑠-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 𝑑𝑣𝑠-cone metric spaces when (𝑋,𝑑) is a 𝑑𝑣𝑠-cone metric space with respect to a normal cone 𝑃.

Remark 1.6 (see [19, 22, 23]). Let (𝑋,𝑑) be a 𝑑𝑣𝑠-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 𝑒≼𝑣 and 𝑣β‰ͺ𝑀, then 𝑒β‰ͺ𝑀,(p2) If 𝑒β‰ͺ𝑣 and 𝑣≼𝑀, then 𝑒β‰ͺ𝑀,(p3) If 𝑒β‰ͺ𝑣 and 𝑣β‰ͺ𝑀, then 𝑒β‰ͺ𝑀,(p4) If πœƒβ‰Όπ‘’β‰ͺ𝑐 for each π‘βˆˆint𝑃, then 𝑒=πœƒ,(p5) If π‘Žβ‰Όπ‘+𝑐 for each π‘βˆˆint𝑃, then π‘Žβ‰Όπ‘,(p6) If 𝐸 is 𝑑𝑣𝑠 with a cone 𝑃, and if π‘Žβ‰Όπœ†π‘Ž where π‘Žβˆˆπ‘ƒ and πœ†βˆˆ[0,1), then π‘Ž=πœƒ,(p7) If π‘βˆˆint𝑃, π‘Žπ‘›βˆˆπΈ, and π‘Žπ‘›β†’πœƒ in locally convex 𝑑𝑣𝑠𝐸, then there exists 𝑛0βˆˆβ„• such that π‘Žπ‘›β‰ͺ𝑐 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 πΊβˆΆπ‘‹Γ—π‘‹Γ—π‘‹β†’[0,∞) be a function satisfying the following axioms:(G1)βˆ€π‘₯,𝑦,π‘§βˆˆπ‘‹πΊ(π‘₯,𝑦,𝑧)=0⇔π‘₯=𝑦=𝑧,(G2)βˆ€π‘₯,π‘¦βˆˆπ‘‹,π‘₯≠𝑦𝐺(π‘₯,π‘₯,𝑦)>0,(G3)βˆ€π‘₯,𝑦,π‘§βˆˆπ‘‹πΊ(π‘₯,𝑦,𝑧)β‰₯𝐺(π‘₯,π‘₯,𝑦),(G4)βˆ€π‘₯,𝑦,π‘§βˆˆπ‘‹(π‘₯,𝑦,𝑧)=𝐺(π‘₯,𝑧,𝑦)=𝐺(𝑧,𝑦,π‘₯)=β‹― (symmetric in all three variables),(G5)βˆ€π‘₯,𝑦,𝑧,π‘€βˆˆπ‘‹πΊ(π‘₯,𝑦,𝑧)≀𝐺(π‘₯,𝑀,𝑀)+𝐺(𝑀,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 𝑑𝑣𝑠-cone metrics, we introduce the below notion of 𝑑𝑣𝑠-generalized-cone metrics.

Definition 1.8. Let 𝑋 be a nonempty set and (𝐸,𝑃) an ordered 𝑑𝑣𝑠, and let πΊβˆΆπ‘‹Γ—π‘‹Γ—π‘‹β†’πΈ be a function satisfying the following axioms:(G1)βˆ€π‘₯,𝑦,π‘§βˆˆπ‘‹πΊ(π‘₯,𝑦,𝑧)=πœƒ if and only if π‘₯=𝑦=𝑧,(G2)βˆ€π‘₯,π‘¦βˆˆπ‘‹,π‘₯β‰ π‘¦πœƒβ‰ͺ𝐺(π‘₯,π‘₯,𝑦),(G3)βˆ€π‘₯,𝑦,π‘§βˆˆπ‘‹πΊ(π‘₯,π‘₯,𝑦)≼𝐺(π‘₯,𝑦,𝑧),(G4)βˆ€π‘₯,𝑦,π‘§βˆˆπ‘‹πΊ(π‘₯,𝑦,𝑧)=𝐺(π‘₯,𝑧,𝑦)=𝐺(𝑧,𝑦,π‘₯)=β‹― (symmetric in all three variables),(G5)βˆ€π‘₯,𝑦,𝑧,π‘€βˆˆπ‘‹πΊ(π‘₯,𝑦,𝑧)≼𝐺(π‘₯,𝑀,𝑀)+𝐺(𝑀,𝑦,𝑧). Then the function 𝐺 is called a 𝑑𝑣𝑠-generalized-cone metric, or, more specifically, a 𝑑𝑣𝑠-𝐺-cone metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝑑𝑣𝑠-𝐺-cone metric space.

Definition 1.9. Let (𝑋,𝐺) be a 𝑑𝑣𝑠-𝐺-cone metric space, π‘₯βˆˆπ‘‹, and {π‘₯𝑛} a sequence in 𝑋.(1){π‘₯𝑛}𝑑𝑣𝑠-𝐺-cone converges to π‘₯ whenever, for every π‘βˆˆπΈ with πœƒβ‰ͺ𝑐, there exists 𝑛0βˆˆβ„• such that 𝐺(π‘₯𝑛,π‘₯π‘š,π‘₯)β‰ͺ𝑐 for all π‘š,𝑛β‰₯𝑛0. Here π‘₯ is called the limit of the sequence {π‘₯𝑛} and is denoted by 𝐺-cone-limπ‘›β†’βˆžπ‘₯𝑛=π‘₯;(2){π‘₯𝑛} is a 𝑑𝑣𝑠-𝐺-cone Cauchy sequence whenever, for every π‘βˆˆπΈ with πœƒβ‰ͺ𝑐, there exists 𝑛0βˆˆβ„• such that 𝐺(π‘₯𝑛,π‘₯π‘š,π‘₯𝑙)β‰ͺ𝑐 for all 𝑛,π‘š,𝑙β‰₯𝑛0;(3)(𝑋,𝐺) is 𝑑𝑣𝑠-𝐺-cone complete if every 𝑑𝑣𝑠-𝐺-cone Cauchy sequence in 𝑋 is 𝑑𝑣𝑠-𝐺-cone convergent in 𝑋.

Proposition 1.10. Let (𝑋,𝐺) be a 𝑑𝑣𝑠-𝐺-cone metric space, π‘₯βˆˆπ‘‹, and {π‘₯𝑛} a sequence in 𝑋. The following are equivalent:(i){π‘₯𝑛}𝑑𝑣𝑠-𝐺-cone converges to π‘₯,(ii)𝐺(π‘₯𝑛,π‘₯𝑛,π‘₯)β†’πœƒ as π‘›β†’βˆž,(iii)𝐺(π‘₯𝑛,π‘₯,π‘₯)β†’ as nβ†’βˆž,(iv)𝐺(π‘₯𝑛,π‘₯π‘š,π‘₯)β†’πœƒ as 𝑛,π‘šβ†’βˆž.

In this paper, we also introduce the below concept of the π’žβ„¬π’²-𝑑𝑣𝑠-𝐺-cone-type function.

Definition 1.11. One callsπœ‘βˆΆint𝑃βˆͺ{πœƒ}β†’int𝑃βˆͺ{πœƒ} a π’žβ„¬π’²-𝑑𝑣𝑠-𝐺-cone-type function if the function πœ‘ satisfies the following condition(πœ‘1)πœ‘(𝑑)β‰ͺ𝑑 for all π‘‘β‰«πœƒ and πœ‘(πœƒ)=πœƒ;(πœ‘2)limπ‘›β†’βˆžπœ‘π‘›(𝑑)=πœƒ for all π‘‘βˆˆint𝑃βˆͺ{πœƒ}.
In this paeper, for a 𝑑𝑣𝑠-𝐺-cone metric space (𝑋,𝐺) and for the family π’œ of subsets of 𝑋, we introduce a new notion of the 𝑑𝑣𝑠-β„‹-cone metric β„‹ with respect to 𝐺, and we get a fixed result for the π’žβ„¬π’²-𝑑𝑣𝑠-𝐺-cone-type function in a complete 𝑑𝑣𝑠-𝐺-cone metric space (π’œ,β„‹). Our results generalize some recent results in the literature.

2. Main Results

Let 𝐸 be a locally convex Hausdorff 𝑑𝑣𝑠 with its zero vector πœƒ, 𝑃 a proper, closed and convex pointed cone in 𝐸 with intπ‘ƒβ‰ πœ™, and β‰Ό a partial ordering with respect to 𝑃. We introduce the below notion of the 𝑑𝑣𝑠-β„‹-cone metric β„‹ with respect to 𝑑𝑣𝑠-𝐺-cone metric 𝐺.

Definition 2.1. Let (𝑋,𝐺) be a 𝑑𝑣𝑠-𝐺-cone metric space with a solid cone 𝑃, and let π’œ be a collection of nonempty subsets of 𝑋. A map β„‹βˆΆπ’œΓ—π’œΓ—π’œβ†’πΈ is called a 𝑑𝑣𝑠-β„‹-cone metric with respect to 𝐺 if for any 𝐴1,𝐴2,𝐴3βˆˆπ’œ the following conditions hold:(H1)β„‹(𝐴1,𝐴2,𝐴3)=πœƒβ‡’π΄1=𝐴2=𝐴3,(H2)β„‹(𝐴1,𝐴2,𝐴3)=β„‹(𝐴2,𝐴1,𝐴3)=β„‹(𝐴1,𝐴3,𝐴2)=β‹―(symmetry in all variables),(H3)β„‹(𝐴1,𝐴1,𝐴2)β‰Όβ„‹(𝐴1,𝐴2,𝐴3),(H4)βˆ€πœ€βˆˆπΈ,πœƒβ‰ͺπœ€βˆ€π‘₯∈𝐴1,π‘¦βˆˆπ΄2βˆƒπ‘§βˆˆπ΄3𝐺(π‘₯,𝑦,𝑧)β‰Όβ„‹(𝐴1,𝐴2,𝐴3)+πœ€,(H5) one of the following is satisfied:(i)βˆ€πœ€βˆˆπΈ,πœƒβ‰ͺπœ€βˆƒπ‘₯∈𝐴1βˆ€π‘¦βˆˆπ΄2,π‘§βˆˆπ΄3β„‹(𝐴1,𝐴2,𝐴3)≼𝐺(π‘₯,𝑦,𝑧)+πœ€,(ii)βˆ€πœ€βˆˆπΈ,πœƒβ‰ͺπœ€βˆƒπ‘¦βˆˆπ΄2βˆ€π‘₯∈𝐴1,π‘§βˆˆπ΄3β„‹(𝐴1,𝐴2,𝐴3)≼𝐺(π‘₯,𝑦,𝑧)+πœ€,(iii)βˆ€πœ€βˆˆπΈ,πœƒβ‰ͺπœ€βˆƒπ‘§βˆˆπ΄3βˆ€π‘¦βˆˆπ΄2,π‘₯∈𝐴1β„‹(𝐴1,𝐴2,𝐴3)≼𝐺(π‘₯,𝑦,𝑧)+πœ€.

We will prove that a 𝑑𝑣𝑠-β„‹-cone metric satisfies the conditions of (𝐺1)-(𝐺5).

Lemma 2.2. Let (𝑋,𝐺) be a 𝑑𝑣𝑠-𝐺-cone metric space with a solid cone 𝑃, and let π’œ be a collection of nonempty subsets of 𝑋, π’œβ‰ πœ™. If β„‹βˆΆπ’œΓ—π’œΓ—π’œβ†’πΈ is a 𝑑𝑣𝑠-β„‹-cone metric with respect to 𝐺, then pair (π’œ,β„‹) is a 𝑑𝑣𝑠-𝐺-cone metric space.

Proof. Let {πœ€π‘›}βŠ‚πΈ be a sequence such that πœƒβ‰ͺπœ€π‘› for all π‘›βˆˆβ„• and 𝐺-π‘π‘œπ‘›π‘’-limπ‘›β†’βˆžπœ€π‘›=πœƒ. Take any 𝐴1,𝐴2,𝐴3βˆˆπ’œ and π‘₯∈𝐴1, π‘¦βˆˆπ΄2. From (H4), for each π‘›βˆˆβ„•, there exists π‘§π‘›βˆˆπ΄3 such that 𝐺π‘₯,𝑦,𝑧𝑛𝐴≼ℋ1,𝐴2,𝐴3ξ€Έ+πœ€π‘›.(2.1) Therefore, β„‹(𝐴1,𝐴2,𝐴3)+πœ€π‘›βˆˆπ‘ƒ for each π‘›βˆˆβ„•. By the closedness of 𝑃, we conclude that πœƒβ‰Όβ„‹(𝐴1,𝐴2,𝐴3).
Assume that 𝐴1=𝐴2=𝐴3. From H5, we obtain β„‹(𝐴1,𝐴2,𝐴3)β‰Όπœ€π‘› for any π‘›βˆˆβ„•. So β„‹(𝐴1,𝐴2,𝐴3)=πœƒ.
Let 𝐴1,𝐴2,𝐴3,𝐴4βˆˆπ’œ. Assume that 𝐴1,𝐴2,𝐴3 satisfy condition (H5)(i). Then, for each π‘›βˆˆβ„•, there exists π‘₯π‘›βˆˆπ΄1 such that β„‹(𝐴1,𝐴2,𝐴3)≼𝐺(π‘₯𝑛,𝑦,𝑧)+πœ€π‘› for all π‘¦βˆˆπ΄2 and π‘§βˆˆπ΄3. From (H4), there exists a sequence {𝑀𝑛}βŠ‚π΄4 satisfying 𝐺(π‘₯𝑛,𝑀𝑛,𝑀𝑛)β‰Όβ„‹(𝐴1,𝐴4,𝐴4)+πœ€π‘› for every π‘›βˆˆβ„•. Obviously, for any π‘¦βˆˆπ΄2 and any π‘§βˆˆπ΄3 and π‘›βˆˆβ„•, we have ℋ𝐴1,𝐴2,𝐴3ξ€Έξ€·π‘₯≼𝐺𝑛,𝑦,𝑧+πœ€π‘›ξ€·π‘₯≼𝐺𝑛,𝑀n,𝑀𝑛𝑀+𝐺𝑛,𝑦,𝑧+πœ€π‘›.(2.2) Now for each π‘›βˆˆβ„•, there exists π‘¦π‘›βˆˆπ΄2, π‘§π‘›βˆˆπ΄3 such that 𝐺(𝑀𝑛,𝑦𝑛,𝑧𝑛)β‰Όβ„‹(𝐴4,𝐴2,𝐴3)+πœ€π‘›. Consequently, we obtain that for each π‘›βˆˆβ„•β„‹ξ€·π΄1,𝐴2,𝐴3𝐴≼ℋ1,𝐴4,𝐴4𝐴+β„‹4,𝐴2,𝐴3ξ€Έ+3πœ€π‘›.(2.3) Therefore, ℋ𝐴1,𝐴2,𝐴3𝐴≼ℋ1,𝐴4,𝐴4𝐴+β„‹4,𝐴2,𝐴3ξ€Έ.(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 πœ‘βˆΆint𝑃βˆͺ{πœƒ}β†’int𝑃βˆͺ{πœƒ} satisfying the following conditions:(C1)πœ‘ is a π’žβ„¬π’²-𝑑𝑣𝑠-𝐺-cone-type-function;(C2)πœ‘ is subadditive, that is, πœ‘(𝑒1+𝑒2)β‰Όπœ‘(𝑒1)+πœ‘(𝑒2) for all 𝑒1,𝑒2∈int𝑃.

Our main result is the following.

Theorem 2.3. Let (𝑋,𝐺) be a 𝑑v𝑠-𝐺-cone complete metric space with a solid cone 𝑃, let π’œ be a collection of nonempty closed subsets of 𝑋, π’œβ‰ πœ™, and let β„‹βˆΆπ’œΓ—π’œΓ—π’œβ†’πΈ be a 𝑑𝑣𝑠-β„‹-cone metric with respect to 𝐺. If the mapping π‘‡βˆΆπ‘‹β†’π’œ satisfies the condition that exists a πœ‘βˆˆΞ˜ such that for all π‘₯,𝑦,π‘§βˆˆπ‘‹ holds β„‹(𝑇π‘₯,𝑇𝑦,𝑇𝑧)β‰Όπœ‘(𝐺(π‘₯,𝑦,𝑧)),(2.5) then 𝑇 has a fixed point in 𝑋.

Proof. Let us choose πœ€βˆˆint𝑃 arbitrarily, and let πœ€π‘›βˆˆπΈ be a sequence such that πœƒβ‰ͺπœ€π‘› and πœ€π‘›β‰Όπœ€/3𝑛. Let us choose π‘₯0βˆˆπ‘‹ arbitrarily and π‘₯1βˆˆπ‘‡π‘₯0. If 𝐺(π‘₯0,π‘₯0,π‘₯1)=πœƒ, then π‘₯0=π‘₯1βˆˆπ‘‡(π‘₯0), and we are done. Assume that 𝐺(π‘₯0,π‘₯0,π‘₯1)β‰«πœƒ. Taking into account (2.5) and (H4), there exists π‘₯2βˆˆπ‘‡π‘₯1 such that 𝐺π‘₯1,π‘₯1,π‘₯2≼ℋ𝑇π‘₯0,𝑇π‘₯0,𝑇π‘₯1ξ€Έ+πœ€1𝐺π‘₯β‰Όπœ‘0,π‘₯0,π‘₯1ξ€Έξ€Έ+πœ€1.(2.6) Taking into account (2.5), (2.6), and (H4) and since πœ‘βˆˆΞ˜, there exists π‘₯3βˆˆπ‘‡π‘₯2 such that 𝐺π‘₯2,π‘₯2,π‘₯3≼ℋ𝑇π‘₯1,𝑇π‘₯1,𝑇π‘₯2ξ€Έ+πœ€2𝐺π‘₯β‰Όπœ‘1,π‘₯1,π‘₯2ξ€Έξ€Έ+πœ€2ξ€·πœ‘ξ€·πΊξ€·π‘₯β‰Όπœ‘0,π‘₯0,π‘₯1ξ€Έξ€Έ+πœ€1ξ€Έ+πœ€2ξ€·πœ‘ξ€·πΊξ€·π‘₯β‰Όπœ‘0,π‘₯0,π‘₯1ξ€·πœ€ξ€Έξ€Έξ€Έ+πœ‘1ξ€Έ+πœ€2β‰ͺπœ‘2𝐺π‘₯0,π‘₯0,π‘₯1ξ€Έξ€Έ+πœ€1+πœ€2β‰Όπœ‘2𝐺π‘₯0,π‘₯0,π‘₯1+πœ€ξ€Έξ€Έ3+πœ€32.(2.7) We continue in this manner. In general, for π‘₯𝑛, π‘›βˆˆβ„•, π‘₯𝑛+1 is chosen such that π‘₯𝑛+1βˆˆπ‘‡π‘₯𝑛 and 𝐺π‘₯𝑛,π‘₯𝑛,π‘₯𝑛+1≼ℋ𝑇π‘₯π‘›βˆ’1,𝑇π‘₯π‘›βˆ’1,𝑇π‘₯𝑛+πœ€π‘›ξ€·πΊξ€·π‘₯β‰ͺπœ‘π‘›βˆ’1,π‘₯π‘›βˆ’1,π‘₯𝑛+πœ€π‘›β‰Όπœ‘2𝐺π‘₯π‘›βˆ’2,π‘₯π‘›βˆ’2,π‘₯π‘›βˆ’1ξ€Έξ€Έ+πœ€π‘›βˆ’1+πœ€π‘›β‰Όβ‹―β‹―β‰Όπœ‘π‘›ξ€·πΊξ€·π‘₯0,π‘₯0,π‘₯1+𝑛𝑖=1πœ€π‘–β‰Όπœ‘π‘›ξ€·πΊξ€·π‘₯0,π‘₯0,π‘₯1+𝑛𝑖=1πœ€3π‘–β‰Όπœ‘π‘›ξ€·πΊξ€·π‘₯0,π‘₯0,π‘₯1+1ξ€Έξ€Έ2πœ€.(2.8) Since πœ€ is arbitrary, letting πœ€β†’πœƒ and by the definition of the π’žβ„¬π’²-𝑑𝑣𝑠-𝐺-cone-type function, we obtain that limπ‘›β†’βˆžπΊξ€·π‘₯𝑛,π‘₯𝑛,π‘₯𝑛+1ξ€Έ=πœƒ.(2.9)
Next, we let π‘π‘š=𝐺(π‘₯π‘š,π‘₯π‘š+1,π‘₯π‘š+1), and we claim that the following result holds: for each π›Ύβ‰«πœƒ,  there is 𝑛0(πœ€)βˆˆπ‘ such that for all m, 𝑛β‰₯𝑛0(𝛾), 𝐺π‘₯π‘š,π‘₯π‘š+1,π‘₯π‘š+1ξ€Έβ‰ͺ𝛾,(2.10) We will prove (2.10) by contradiction. Suppose that (2.10) is false. Then there exists some π›Ύβ‰«πœƒ such that for all π‘βˆˆβ„•, there are π‘šπ‘,π‘›π‘βˆˆβ„• with π‘šπ‘>𝑛𝑝β‰₯𝑝 satisfying(i)π‘šπ‘ is even and 𝑛𝑝 is odd,(ii)𝐺(π‘₯π‘šπ‘,π‘₯𝑛𝑝,π‘₯𝑛𝑝)≽𝛾, and(iii)π‘šπ‘ is the smallest even number such that conditions (i), (ii) hold.Since π‘π‘šβ†“πœƒ, by (ii), we have that limπ‘β†’βˆžπΊ(π‘₯π‘šπ‘,π‘₯𝑛𝑝,π‘₯𝑛𝑝)=𝛾 and ξ‚€π‘₯π›Ύβ‰ΌπΊπ‘šπ‘,π‘₯𝑛𝑝,π‘₯𝑛𝑝π‘₯β‰ΌπΊπ‘šπ‘,π‘₯π‘šπ‘+1,π‘₯π‘šπ‘+1π‘₯+πΊπ‘šπ‘+1,π‘₯𝑛𝑝+1,π‘₯𝑛𝑝+1π‘₯+𝐺𝑛𝑝+1,π‘₯𝑛𝑝,π‘₯𝑛𝑝.(2.11) It follows from (H4); let us choose πœ€βˆˆπΈ arbitrarily such that 𝐺π‘₯𝑛𝑝+1,π‘₯𝑛𝑝+1,π‘₯π‘šπ‘+1≼ℋ𝑇π‘₯𝑛𝑝+1,𝑇π‘₯𝑛𝑝+1,𝑇π‘₯π‘šπ‘+1+πœ€.(2.12) Taking into account (2.5), (2.11), and (2.12), we have that ξ‚€π‘₯π›Ύβ‰ΌπΊπ‘šπ‘,π‘₯𝑛𝑝,π‘₯𝑛𝑝π‘₯β‰ΌπΊπ‘šπ‘,π‘₯π‘šπ‘+1,π‘₯π‘šπ‘+1+ℋ𝑇π‘₯𝑛𝑝+1,𝑇π‘₯𝑛𝑝+1,𝑇π‘₯π‘šπ‘+1π‘₯+πœ€+𝐺𝑛𝑝+1,π‘₯𝑛𝑝,π‘₯𝑛𝑝π‘₯β‰ΌπΊπ‘šπ‘,π‘₯π‘šπ‘+1,π‘₯π‘šπ‘+1𝐺π‘₯+πœ‘π‘›π‘,𝐺π‘₯𝑛𝑝,𝐺π‘₯π‘šπ‘ξ‚€π‘₯+πœ€+𝐺𝑛𝑝+1,π‘₯𝑛𝑝,π‘₯𝑛𝑝π‘₯β‰ͺπΊπ‘šπ‘,π‘₯π‘šπ‘+1,π‘₯π‘šπ‘+1π‘₯+𝐺𝑛𝑝,𝐺π‘₯𝑛𝑝,𝐺π‘₯π‘šπ‘ξ‚ξ‚€π‘₯+πœ€+𝐺𝑛𝑝+1,π‘₯𝑛𝑝,π‘₯𝑛𝑝.(2.13) Since πœ€ is arbitrarily, letting πœ€β†’πœƒ and by letting π‘β†’βˆž, we have 𝛾β‰ͺπœƒ+limπ‘β†’βˆžπΊξ‚€π‘₯π‘šπ‘,π‘₯𝑛𝑝,π‘₯𝑛𝑝+πœƒ+πœƒ=𝛾,(2.14) a contradiction. So {π‘₯𝑛} is a 𝑑𝑣𝑠-𝐺-cone Cauchy sequence. Since (𝑋,𝐺)is a 𝑑𝑣𝑠-𝐺-cone complete metric space, {π‘₯𝑛} is 𝑑𝑣𝑠-𝐺-cone convergent in 𝑋 and 𝐺-cone-limπ‘›β†’βˆžπ‘₯𝑛=π‘₯. Thus, for every 𝜏∈int𝑃 and sufficiently large 𝑛, we have that ℋ𝑇π‘₯𝑛,𝑇π‘₯𝑛𝐺π‘₯,𝑇π‘₯β‰Όπœ‘π‘›,π‘₯𝑛π‘₯,π‘₯ξ€Έξ€Έβ‰ͺ𝐺𝑛,π‘₯𝑛β‰ͺ𝜏,π‘₯3.(2.15) Since for π‘›βˆˆβ„•βˆͺ{0}, π‘₯𝑛+1βˆˆπ‘‡π‘₯𝑛, by (H4) we obtain that for all π‘›βˆˆβ„• there existsπ‘¦π‘›βˆˆπ‘‡π‘₯ such that 𝐺π‘₯𝑛+1,π‘₯𝑛+1,𝑦𝑛+1≼ℋ𝑇π‘₯𝑛,𝑇π‘₯𝑛,𝑇π‘₯+πœ€π‘›+1𝐺π‘₯β‰Όπœ‘π‘›,π‘₯𝑛+πœ€,π‘₯ξ€Έξ€Έ3𝑛+1ξ€·π‘₯β‰ͺ𝐺𝑛,π‘₯𝑛+πœ€,π‘₯3𝑛+1.(2.16) Since πœ€/3𝑛+1β†’πœƒ, then for sufficiently large 𝑛, we obtain that 𝐺𝑦𝑛+1𝑦,π‘₯,π‘₯≼𝐺𝑛+1,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝐺𝑛+1ξ€Έβ‰ͺ,π‘₯,π‘₯2𝜏3+𝜏3=𝜏,(2.17) which implies 𝐺-cone-limπ‘›β†’βˆžπ‘¦π‘›=π‘₯. Since 𝑇π‘₯ is closed, we obtain that π‘₯βˆˆπ‘‡π‘₯.

For the case πœ‘(𝑑)=π‘˜π‘‘, π‘˜βˆˆ(0,1), then πœ‘βˆˆΞ˜ and it is easy to get the following corollary.

Corollary 2.4. Let (𝑋,𝐺) be a 𝑑𝑣𝑠-𝐺-cone complete metric space with a solid cone 𝑃, let π’œ be a collection of nonempty closed subsets of 𝑋, π’œβ‰ πœ™, and let β„‹βˆΆπ’œΓ—π’œΓ—π’œβ†’πΈ be a 𝑑𝑣𝑠-β„‹-cone metric with respect to 𝐺. If the mapping π‘‡βˆΆπ‘‹β†’π’œ satisfies the condition that exists π‘˜βˆˆ(0,1) such that for all π‘₯,𝑦,π‘§βˆˆπ‘‹ holds β„‹(𝑇π‘₯,𝑇𝑦,𝑇𝑧)β‰Όπ‘˜β‹…πΊ(π‘₯,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.

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