Abstract

For a tvs-G-cone metric space (š‘‹,šŗ) and for the family š’œ of subsets of X, we introduce a new notion of the tvs-ā„‹-cone metric ā„‹ with respect to G, and we get a fixed result for the š’žā„¬š’²-tvs-G-cone-type function in a complete tvs-G-cone metric space (š’œ,ā„‹). 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., [2ā€“15]). Recently, Du [16] introduced the concept of š‘”š‘£š‘ -cone metric and š‘”š‘£š‘ -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 [16ā€“21], the authors tried to generalize this approach by using cones in topological vector spaces š‘”š‘£š‘  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 š‘”š‘£š‘ -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 š‘”š‘£š‘ -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 (š‘‹,š‘‘) be a š‘”š‘£š‘ -cone metric space with a solid cone š‘ƒ, and let š’œ be a collection of nonempty subsets of š‘‹. A map ā„‹āˆ¶š’œĆ—š’œā†’šø is called a š‘”š‘£š‘ -ā„‹-cone metric with respect to š‘‘ if for any š“1,š“2āˆˆš’œ the following conditions hold:(H1)ā„‹(š“1,š“2)=šœƒā‡’š“1=š“2,(H2)ā„‹(š“1,š“2)=ā„‹(š“2,š“1),(H3)āˆ€šœ€āˆˆšø,šœƒā‰Ŗšœ€āˆ€š‘„āˆˆš“1āˆƒš‘¦āˆˆš“2š‘‘(š‘„,š‘¦)ā‰¼ā„‹(š“1,š“2)+šœ€,(H4) one of the following is satisfied:(i)āˆ€šœ€āˆˆšø,šœƒā‰Ŗšœ€āˆƒš‘„āˆˆš“1āˆ€š‘¦āˆˆš“2ā„‹(š“1,š“2)ā‰¼š‘‘(š‘„,š‘¦)+šœ€,(ii)āˆ€šœ€āˆˆšø,šœƒā‰Ŗšœ€āˆƒš‘¦āˆˆš“2āˆ€š‘„āˆˆš“1ā„‹(š“1,š“2)ā‰¼š‘‘(š‘„,š‘¦)+šœ€.

Theorem 1.2 (see [22]). Let (š‘‹,š‘‘) be a š‘”š‘£š‘ -cone complete metric space with a solid cone š‘ƒ and let š’œ be a collection of nonempty closed subsets of š‘‹, š’œā‰ šœ™, and let ā„‹āˆ¶š’œĆ—š’œā†’šø be a š‘”š‘£š‘ -ā„‹-cone metric with respect to š‘‘. If the mapping š‘‡āˆ¶š‘‹ā†’š’œ the condition that exists a šœ†āˆˆ(0,1) such that for all š‘„,š‘¦āˆˆš‘‹ holds ā„‹(š‘‡š‘„,š‘‡š‘¦)ā‰¼šœ†šŗ(š‘„,š‘¦)(1.1) then š‘‡ has a fixed point in š‘‹.

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

Let šø be be a real Hausdorff topological vector space (š‘”š‘£š‘  for short) with the zero vector šœƒ. A nonempty subset š‘ƒ of šø is called a convex cone if š‘ƒ+š‘ƒāŠ†š‘ƒ and šœ†š‘ƒāŠ†š‘ƒ for šœ†ā‰„0. A convex cone š‘ƒ is said to be pointed (or proper) if š‘ƒāˆ©(āˆ’š‘ƒ)={šœƒ}; š‘ƒ is normal (or saturated) if šø has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone š‘ƒāŠ†šø, we can define a partial ordering ā‰¼ with respect to š‘ƒ by š‘„ā‰¼š‘¦ if and only if š‘¦āˆ’š‘„āˆˆš‘ƒ; š‘„ā‰ŗš‘¦ will stand for š‘„ā‰¼š‘¦ and š‘„ā‰ š‘¦, while š‘„ā‰Ŗš‘¦ will stand for š‘¦āˆ’š‘„āˆˆintš‘ƒ, where intš‘ƒ denotes the interior of š‘ƒ. The cone š‘ƒ is said to be solid if it has a nonempty interior.

In the sequel, šø will be a locally convex Hausdorff š‘”š‘£š‘  with its zero vector šœƒ, š‘ƒ a proper, closed, and convex pointed cone in šø with intš‘ƒā‰ šœ™ and ā‰¼ a partial ordering with respect to š‘ƒ.

Definition 1.3 (see [16, 18, 19]). Let š‘‹ be a nonempty set and (šø,š‘ƒ) an ordered š‘”š‘£š‘ . A vector-valued function š‘‘āˆ¶š‘‹Ć—š‘‹ā†’šø is said to be a š‘”š‘£š‘ -cone metric, if the following conditions hold:(C1)āˆ€š‘„,š‘¦āˆˆš‘‹,š‘„ā‰ š‘¦šœƒā‰¼š‘‘(š‘„,š‘¦),(C2)āˆ€š‘„,š‘¦āˆˆš‘‹š‘‘(š‘„,š‘¦)=šœƒā‡”š‘„=š‘¦,(C3)āˆ€š‘„,š‘¦āˆˆš‘‹š‘‘(š‘„,š‘¦)=š‘‘(š‘¦,š‘„),(C4)āˆ€š‘„,š‘¦,š‘§āˆˆš‘‹š‘‘(š‘„,š‘§)ā‰¼š‘‘(š‘„,š‘¦)+š‘‘(š‘¦,š‘§). Then the pair (š‘‹,š‘‘) is called a š‘”š‘£š‘ -cone metric space.

Definition 1.4 (see [16, 18, 19]). Let (š‘‹,š‘‘) be a š‘”š‘£š‘ -cone metric space, š‘„āˆˆš‘‹, and {š‘„š‘›} a sequence in š‘‹.(1){š‘„š‘›}š‘”š‘£š‘ -cone converges to š‘„ whenever for every š‘āˆˆšø with šœƒā‰Ŗš‘, there exists š‘›0āˆˆā„• such that š‘‘(š‘„š‘›,š‘„)ā‰Ŗš‘ for all š‘›ā‰„š‘›0. We denote this by cone-limš‘›ā†’āˆžš‘„š‘›=š‘„;(2){š‘„š‘›} is a š‘”š‘£š‘ -cone Cauchy sequence whenever for every š‘āˆˆšø with šœƒā‰Ŗš‘, there exists š‘›0āˆˆā„• such that š‘‘(š‘„š‘›,š‘„š‘š)ā‰Ŗš‘ for all š‘›,š‘šā‰„š‘›0;(3)(š‘‹,š‘‘) is š‘”š‘£š‘ -cone complete if every š‘”š‘£š‘ -cone Cauchy sequence in š‘‹ is š‘”š‘£š‘ -cone convergent in š‘‹.

Remark 1.5. Clearly, a cone metric space in the sense of Huang and Zhang [1] is a special case of š‘”š‘£š‘ -cone metric spaces when (š‘‹,š‘‘) is a š‘”š‘£š‘ -cone metric space with respect to a normal cone š‘ƒ.

Remark 1.6 (see [19, 22, 23]). Let (š‘‹,š‘‘) be a š‘”š‘£š‘ -cone metric space with a solid cone š‘ƒ. The following properties are often used, particularly in the case when the underlying cone is nonnormal.(p1) If š‘¢ā‰¼š‘£ and š‘£ā‰Ŗš‘¤, then š‘¢ā‰Ŗš‘¤,(p2) If š‘¢ā‰Ŗš‘£ and š‘£ā‰¼š‘¤, then š‘¢ā‰Ŗš‘¤,(p3) If š‘¢ā‰Ŗš‘£ and š‘£ā‰Ŗš‘¤, then š‘¢ā‰Ŗš‘¤,(p4) If šœƒā‰¼š‘¢ā‰Ŗš‘ for each š‘āˆˆintš‘ƒ, then š‘¢=šœƒ,(p5) If š‘Žā‰¼š‘+š‘ for each š‘āˆˆintš‘ƒ, then š‘Žā‰¼š‘,(p6) If šø is š‘”š‘£š‘  with a cone š‘ƒ, and if š‘Žā‰¼šœ†š‘Ž where š‘Žāˆˆš‘ƒ and šœ†āˆˆ[0,1), then š‘Ž=šœƒ,(p7) If š‘āˆˆintš‘ƒ, š‘Žš‘›āˆˆšø, and š‘Žš‘›ā†’šœƒ in locally convex š‘”š‘£š‘ šø, then there exists š‘›0āˆˆā„• such that š‘Žš‘›ā‰Ŗš‘ for all š‘›>š‘›0.

Metric spaces are playing an important role in mathematics and the applied sciences. To overcome fundamental laws in Dhageā€™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 š‘‹ be a nonempty set, and let šŗāˆ¶š‘‹Ć—š‘‹Ć—š‘‹ā†’[0,āˆž) be a function satisfying the following axioms:(G1)āˆ€š‘„,š‘¦,š‘§āˆˆš‘‹šŗ(š‘„,š‘¦,š‘§)=0ā‡”š‘„=š‘¦=š‘§,(G2)āˆ€š‘„,š‘¦āˆˆš‘‹,š‘„ā‰ š‘¦šŗ(š‘„,š‘„,š‘¦)>0,(G3)āˆ€š‘„,š‘¦,š‘§āˆˆš‘‹šŗ(š‘„,š‘¦,š‘§)ā‰„šŗ(š‘„,š‘„,š‘¦),(G4)āˆ€š‘„,š‘¦,š‘§āˆˆš‘‹(š‘„,š‘¦,š‘§)=šŗ(š‘„,š‘§,š‘¦)=šŗ(š‘§,š‘¦,š‘„)=ā‹Æ (symmetric in all three variables),(G5)āˆ€š‘„,š‘¦,š‘§,š‘¤āˆˆš‘‹šŗ(š‘„,š‘¦,š‘§)ā‰¤šŗ(š‘„,š‘¤,š‘¤)+šŗ(š‘¤,š‘¦,š‘§). Then the function šŗ is called a generalized metric, or, more specifically a šŗ-metric on š‘‹, and the pair (š‘‹,šŗ) is called a šŗ-metric space.

By using the notions of generalized metrics and š‘”š‘£š‘ -cone metrics, we introduce the below notion of š‘”š‘£š‘ -generalized-cone metrics.

Definition 1.8. Let š‘‹ be a nonempty set and (šø,š‘ƒ) an ordered š‘”š‘£š‘ , and let šŗāˆ¶š‘‹Ć—š‘‹Ć—š‘‹ā†’šø be a function satisfying the following axioms:(G1)āˆ€š‘„,š‘¦,š‘§āˆˆš‘‹šŗ(š‘„,š‘¦,š‘§)=šœƒ if and only if š‘„=š‘¦=š‘§,(G2)āˆ€š‘„,š‘¦āˆˆš‘‹,š‘„ā‰ š‘¦šœƒā‰Ŗšŗ(š‘„,š‘„,š‘¦),(G3)āˆ€š‘„,š‘¦,š‘§āˆˆš‘‹šŗ(š‘„,š‘„,š‘¦)ā‰¼šŗ(š‘„,š‘¦,š‘§),(G4)āˆ€š‘„,š‘¦,š‘§āˆˆš‘‹šŗ(š‘„,š‘¦,š‘§)=šŗ(š‘„,š‘§,š‘¦)=šŗ(š‘§,š‘¦,š‘„)=ā‹Æ (symmetric in all three variables),(G5)āˆ€š‘„,š‘¦,š‘§,š‘¤āˆˆš‘‹šŗ(š‘„,š‘¦,š‘§)ā‰¼šŗ(š‘„,š‘¤,š‘¤)+šŗ(š‘¤,š‘¦,š‘§). Then the function šŗ is called a š‘”š‘£š‘ -generalized-cone metric, or, more specifically, a š‘”š‘£š‘ -šŗ-cone metric on š‘‹, and the pair (š‘‹,šŗ) is called a š‘”š‘£š‘ -šŗ-cone metric space.

Definition 1.9. Let (š‘‹,šŗ) be a š‘”š‘£š‘ -šŗ-cone metric space, š‘„āˆˆš‘‹, and {š‘„š‘›} a sequence in š‘‹.(1){š‘„š‘›}š‘”š‘£š‘ -šŗ-cone converges to š‘„ whenever, for every š‘āˆˆšø with šœƒā‰Ŗš‘, there exists š‘›0āˆˆā„• such that šŗ(š‘„š‘›,š‘„š‘š,š‘„)ā‰Ŗš‘ for all š‘š,š‘›ā‰„š‘›0. Here š‘„ is called the limit of the sequence {š‘„š‘›} and is denoted by šŗ-cone-limš‘›ā†’āˆžš‘„š‘›=š‘„;(2){š‘„š‘›} is a š‘”š‘£š‘ -šŗ-cone Cauchy sequence whenever, for every š‘āˆˆšø with šœƒā‰Ŗš‘, there exists š‘›0āˆˆā„• such that šŗ(š‘„š‘›,š‘„š‘š,š‘„š‘™)ā‰Ŗš‘ for all š‘›,š‘š,š‘™ā‰„š‘›0;(3)(š‘‹,šŗ) is š‘”š‘£š‘ -šŗ-cone complete if every š‘”š‘£š‘ -šŗ-cone Cauchy sequence in š‘‹ is š‘”š‘£š‘ -šŗ-cone convergent in š‘‹.

Proposition 1.10. Let (š‘‹,šŗ) be a š‘”š‘£š‘ -šŗ-cone metric space, š‘„āˆˆš‘‹, and {š‘„š‘›} a sequence in š‘‹. The following are equivalent:(i){š‘„š‘›}š‘”š‘£š‘ -šŗ-cone converges to š‘„,(ii)šŗ(š‘„š‘›,š‘„š‘›,š‘„)ā†’šœƒ as š‘›ā†’āˆž,(iii)šŗ(š‘„š‘›,š‘„,š‘„)ā†’ as nā†’āˆž,(iv)šŗ(š‘„š‘›,š‘„š‘š,š‘„)ā†’šœƒ as š‘›,š‘šā†’āˆž.

In this paper, we also introduce the below concept of the š’žā„¬š’²-š‘”š‘£š‘ -šŗ-cone-type function.

Definition 1.11. One callsšœ‘āˆ¶intš‘ƒāˆŖ{šœƒ}ā†’intš‘ƒāˆŖ{šœƒ} a š’žā„¬š’²-š‘”š‘£š‘ -šŗ-cone-type function if the function šœ‘ satisfies the following condition(šœ‘1)šœ‘(š‘”)ā‰Ŗš‘” for all š‘”ā‰«šœƒ and šœ‘(šœƒ)=šœƒ;(šœ‘2)limš‘›ā†’āˆžšœ‘š‘›(š‘”)=šœƒ for all š‘”āˆˆintš‘ƒāˆŖ{šœƒ}.
In this paeper, for a š‘”š‘£š‘ -šŗ-cone metric space (š‘‹,šŗ) and for the family š’œ of subsets of š‘‹, we introduce a new notion of the š‘”š‘£š‘ -ā„‹-cone metric ā„‹ with respect to šŗ, and we get a fixed result for the š’žā„¬š’²-š‘”š‘£š‘ -šŗ-cone-type function in a complete š‘”š‘£š‘ -šŗ-cone metric space (š’œ,ā„‹). Our results generalize some recent results in the literature.

2. Main Results

Let šø be a locally convex Hausdorff š‘”š‘£š‘  with its zero vector šœƒ, š‘ƒ a proper, closed and convex pointed cone in šø with intš‘ƒā‰ šœ™, and ā‰¼ a partial ordering with respect to š‘ƒ. We introduce the below notion of the š‘”š‘£š‘ -ā„‹-cone metric ā„‹ with respect to š‘”š‘£š‘ -šŗ-cone metric šŗ.

Definition 2.1. Let (š‘‹,šŗ) be a š‘”š‘£š‘ -šŗ-cone metric space with a solid cone š‘ƒ, and let š’œ be a collection of nonempty subsets of š‘‹. A map ā„‹āˆ¶š’œĆ—š’œĆ—š’œā†’šø is called a š‘”š‘£š‘ -ā„‹-cone metric with respect to šŗ if for any š“1,š“2,š“3āˆˆš’œ the following conditions hold:(H1)ā„‹(š“1,š“2,š“3)=šœƒā‡’š“1=š“2=š“3,(H2)ā„‹(š“1,š“2,š“3)=ā„‹(š“2,š“1,š“3)=ā„‹(š“1,š“3,š“2)=ā‹Æ(symmetry in all variables),(H3)ā„‹(š“1,š“1,š“2)ā‰¼ā„‹(š“1,š“2,š“3),(H4)āˆ€šœ€āˆˆšø,šœƒā‰Ŗšœ€āˆ€š‘„āˆˆš“1,š‘¦āˆˆš“2āˆƒš‘§āˆˆš“3šŗ(š‘„,š‘¦,š‘§)ā‰¼ā„‹(š“1,š“2,š“3)+šœ€,(H5) one of the following is satisfied:(i)āˆ€šœ€āˆˆšø,šœƒā‰Ŗšœ€āˆƒš‘„āˆˆš“1āˆ€š‘¦āˆˆš“2,š‘§āˆˆš“3ā„‹(š“1,š“2,š“3)ā‰¼šŗ(š‘„,š‘¦,š‘§)+šœ€,(ii)āˆ€šœ€āˆˆšø,šœƒā‰Ŗšœ€āˆƒš‘¦āˆˆš“2āˆ€š‘„āˆˆš“1,š‘§āˆˆš“3ā„‹(š“1,š“2,š“3)ā‰¼šŗ(š‘„,š‘¦,š‘§)+šœ€,(iii)āˆ€šœ€āˆˆšø,šœƒā‰Ŗšœ€āˆƒš‘§āˆˆš“3āˆ€š‘¦āˆˆš“2,š‘„āˆˆš“1ā„‹(š“1,š“2,š“3)ā‰¼šŗ(š‘„,š‘¦,š‘§)+šœ€.

We will prove that a š‘”š‘£š‘ -ā„‹-cone metric satisfies the conditions of (šŗ1)-(šŗ5).

Lemma 2.2. Let (š‘‹,šŗ) be a š‘”š‘£š‘ -šŗ-cone metric space with a solid cone š‘ƒ, and let š’œ be a collection of nonempty subsets of š‘‹, š’œā‰ šœ™. If ā„‹āˆ¶š’œĆ—š’œĆ—š’œā†’šø is a š‘”š‘£š‘ -ā„‹-cone metric with respect to šŗ, then pair (š’œ,ā„‹) is a š‘”š‘£š‘ -šŗ-cone metric space.

Proof. Let {šœ€š‘›}āŠ‚šø be a sequence such that šœƒā‰Ŗšœ€š‘› for all š‘›āˆˆā„• and šŗ-š‘š‘œš‘›š‘’-limš‘›ā†’āˆžšœ€š‘›=šœƒ. Take any š“1,š“2,š“3āˆˆš’œ and š‘„āˆˆš“1, š‘¦āˆˆš“2. From (H4), for each š‘›āˆˆā„•, there exists š‘§š‘›āˆˆš“3 such that šŗī€·š‘„,š‘¦,š‘§š‘›ī€øī€·š“ā‰¼ā„‹1,š“2,š“3ī€ø+šœ€š‘›.(2.1) Therefore, ā„‹(š“1,š“2,š“3)+šœ€š‘›āˆˆš‘ƒ for each š‘›āˆˆā„•. By the closedness of š‘ƒ, we conclude that šœƒā‰¼ā„‹(š“1,š“2,š“3).
Assume that š“1=š“2=š“3. From H5, we obtain ā„‹(š“1,š“2,š“3)ā‰¼šœ€š‘› for any š‘›āˆˆā„•. So ā„‹(š“1,š“2,š“3)=šœƒ.
Let š“1,š“2,š“3,š“4āˆˆš’œ. Assume that š“1,š“2,š“3 satisfy condition (H5)(i). Then, for each š‘›āˆˆā„•, there exists š‘„š‘›āˆˆš“1 such that ā„‹(š“1,š“2,š“3)ā‰¼šŗ(š‘„š‘›,š‘¦,š‘§)+šœ€š‘› for all š‘¦āˆˆš“2 and š‘§āˆˆš“3. From (H4), there exists a sequence {š‘¤š‘›}āŠ‚š“4 satisfying šŗ(š‘„š‘›,š‘¤š‘›,š‘¤š‘›)ā‰¼ā„‹(š“1,š“4,š“4)+šœ€š‘› for every š‘›āˆˆā„•. Obviously, for any š‘¦āˆˆš“2 and any š‘§āˆˆš“3 and š‘›āˆˆā„•, we have ā„‹ī€·š“1,š“2,š“3ī€øī€·š‘„ā‰¼šŗš‘›ī€ø,š‘¦,š‘§+šœ€š‘›ī€·š‘„ā‰¼šŗš‘›,š‘¤n,š‘¤š‘›ī€øī€·š‘¤+šŗš‘›ī€ø,š‘¦,š‘§+šœ€š‘›.(2.2) Now for each š‘›āˆˆā„•, there exists š‘¦š‘›āˆˆš“2, š‘§š‘›āˆˆš“3 such that šŗ(š‘¤š‘›,š‘¦š‘›,š‘§š‘›)ā‰¼ā„‹(š“4,š“2,š“3)+šœ€š‘›. Consequently, we obtain that for each š‘›āˆˆā„•ā„‹ī€·š“1,š“2,š“3ī€øī€·š“ā‰¼ā„‹1,š“4,š“4ī€øī€·š“+ā„‹4,š“2,š“3ī€ø+3šœ€š‘›.(2.3) Therefore, ā„‹ī€·š“1,š“2,š“3ī€øī€·š“ā‰¼ā„‹1,š“4,š“4ī€øī€·š“+ā„‹4,š“2,š“3ī€ø.(2.4) In the case when (H5)(ii) or (H5)(iii) holds, we use the analogous method.

In the sequel, we denote by Ī˜ the class of functions šœ‘āˆ¶intš‘ƒāˆŖ{šœƒ}ā†’intš‘ƒāˆŖ{šœƒ} satisfying the following conditions:(C1)šœ‘ is a š’žā„¬š’²-š‘”š‘£š‘ -šŗ-cone-type-function;(C2)šœ‘ is subadditive, that is, šœ‘(š‘¢1+š‘¢2)ā‰¼šœ‘(š‘¢1)+šœ‘(š‘¢2) for all š‘¢1,š‘¢2āˆˆintš‘ƒ.

Our main result is the following.

Theorem 2.3. Let (š‘‹,šŗ) be a š‘”vš‘ -šŗ-cone complete metric space with a solid cone š‘ƒ, let š’œ be a collection of nonempty closed subsets of š‘‹, š’œā‰ šœ™, and let ā„‹āˆ¶š’œĆ—š’œĆ—š’œā†’šø be a š‘”š‘£š‘ -ā„‹-cone metric with respect to šŗ. If the mapping š‘‡āˆ¶š‘‹ā†’š’œ satisfies the condition that exists a šœ‘āˆˆĪ˜ such that for all š‘„,š‘¦,š‘§āˆˆš‘‹ holds ā„‹(š‘‡š‘„,š‘‡š‘¦,š‘‡š‘§)ā‰¼šœ‘(šŗ(š‘„,š‘¦,š‘§)),(2.5) then š‘‡ has a fixed point in š‘‹.

Proof. Let us choose šœ€āˆˆintš‘ƒ arbitrarily, and let šœ€š‘›āˆˆšø be a sequence such that šœƒā‰Ŗšœ€š‘› and šœ€š‘›ā‰¼šœ€/3š‘›. Let us choose š‘„0āˆˆš‘‹ arbitrarily and š‘„1āˆˆš‘‡š‘„0. If šŗ(š‘„0,š‘„0,š‘„1)=šœƒ, then š‘„0=š‘„1āˆˆš‘‡(š‘„0), and we are done. Assume that šŗ(š‘„0,š‘„0,š‘„1)ā‰«šœƒ. Taking into account (2.5) and (H4), there exists š‘„2āˆˆš‘‡š‘„1 such that šŗī€·š‘„1,š‘„1,š‘„2ī€øī€·ā‰¼ā„‹š‘‡š‘„0,š‘‡š‘„0,š‘‡š‘„1ī€ø+šœ€1ī€·šŗī€·š‘„ā‰¼šœ‘0,š‘„0,š‘„1ī€øī€ø+šœ€1.(2.6) Taking into account (2.5), (2.6), and (H4) and since šœ‘āˆˆĪ˜, there exists š‘„3āˆˆš‘‡š‘„2 such that šŗī€·š‘„2,š‘„2,š‘„3ī€øī€·ā‰¼ā„‹š‘‡š‘„1,š‘‡š‘„1,š‘‡š‘„2ī€ø+šœ€2ī€·šŗī€·š‘„ā‰¼šœ‘1,š‘„1,š‘„2ī€øī€ø+šœ€2ī€·šœ‘ī€·šŗī€·š‘„ā‰¼šœ‘0,š‘„0,š‘„1ī€øī€ø+šœ€1ī€ø+šœ€2ī€·šœ‘ī€·šŗī€·š‘„ā‰¼šœ‘0,š‘„0,š‘„1ī€·šœ€ī€øī€øī€ø+šœ‘1ī€ø+šœ€2ā‰Ŗšœ‘2ī€·šŗī€·š‘„0,š‘„0,š‘„1ī€øī€ø+šœ€1+šœ€2ā‰¼šœ‘2ī€·šŗī€·š‘„0,š‘„0,š‘„1+šœ€ī€øī€ø3+šœ€32.(2.7) We continue in this manner. In general, for š‘„š‘›, š‘›āˆˆā„•, š‘„š‘›+1 is chosen such that š‘„š‘›+1āˆˆš‘‡š‘„š‘› and šŗī€·š‘„š‘›,š‘„š‘›,š‘„š‘›+1ī€øī€·ā‰¼ā„‹š‘‡š‘„š‘›āˆ’1,š‘‡š‘„š‘›āˆ’1,š‘‡š‘„š‘›ī€ø+šœ€š‘›ī€·šŗī€·š‘„ā‰Ŗšœ‘š‘›āˆ’1,š‘„š‘›āˆ’1,š‘„š‘›ī€øī€ø+šœ€š‘›ā‰¼šœ‘2ī€·šŗī€·š‘„š‘›āˆ’2,š‘„š‘›āˆ’2,š‘„š‘›āˆ’1ī€øī€ø+šœ€š‘›āˆ’1+šœ€š‘›ā‰¼ā‹Æā‹Æā‰¼šœ‘š‘›ī€·šŗī€·š‘„0,š‘„0,š‘„1+ī€øī€øš‘›ī“š‘–=1šœ€š‘–ā‰¼šœ‘š‘›ī€·šŗī€·š‘„0,š‘„0,š‘„1+ī€øī€øš‘›ī“š‘–=1šœ€3š‘–ā‰¼šœ‘š‘›ī€·šŗī€·š‘„0,š‘„0,š‘„1+1ī€øī€ø2šœ€.(2.8) Since šœ€ is arbitrary, letting šœ€ā†’šœƒ and by the definition of the š’žā„¬š’²-š‘”š‘£š‘ -šŗ-cone-type function, we obtain that limš‘›ā†’āˆžšŗī€·š‘„š‘›,š‘„š‘›,š‘„š‘›+1ī€ø=šœƒ.(2.9)
Next, we let š‘š‘š=šŗ(š‘„š‘š,š‘„š‘š+1,š‘„š‘š+1), and we claim that the following result holds: for each š›¾ā‰«šœƒ,ā€‰ā€‰there is š‘›0(šœ€)āˆˆš‘ such that for all m, š‘›ā‰„š‘›0(š›¾), šŗī€·š‘„š‘š,š‘„š‘š+1,š‘„š‘š+1ī€øā‰Ŗš›¾,(2.10) We will prove (2.10) by contradiction. Suppose that (2.10) is false. Then there exists some š›¾ā‰«šœƒ such that for all š‘āˆˆā„•, there are š‘šš‘,š‘›š‘āˆˆā„• with š‘šš‘>š‘›š‘ā‰„š‘ satisfying(i)š‘šš‘ is even and š‘›š‘ is odd,(ii)šŗ(š‘„š‘šš‘,š‘„š‘›š‘,š‘„š‘›š‘)ā‰½š›¾, and(iii)š‘šš‘ is the smallest even number such that conditions (i), (ii) hold.Since š‘š‘šā†“šœƒ, by (ii), we have that limš‘ā†’āˆžšŗ(š‘„š‘šš‘,š‘„š‘›š‘,š‘„š‘›š‘)=š›¾ and ī‚€š‘„š›¾ā‰¼šŗš‘šš‘,š‘„š‘›š‘,š‘„š‘›š‘ī‚ī‚€š‘„ā‰¼šŗš‘šš‘,š‘„š‘šš‘+1,š‘„š‘šš‘+1ī‚ī‚€š‘„+šŗš‘šš‘+1,š‘„š‘›š‘+1,š‘„š‘›š‘+1ī‚ī‚€š‘„+šŗš‘›š‘+1,š‘„š‘›š‘,š‘„š‘›š‘ī‚.(2.11) It follows from (H4); let us choose šœ€āˆˆšø arbitrarily such that šŗī‚€š‘„š‘›š‘+1,š‘„š‘›š‘+1,š‘„š‘šš‘+1ī‚ī‚€ā‰¼ā„‹š‘‡š‘„š‘›š‘+1,š‘‡š‘„š‘›š‘+1,š‘‡š‘„š‘šš‘+1ī‚+šœ€.(2.12) Taking into account (2.5), (2.11), and (2.12), we have that ī‚€š‘„š›¾ā‰¼šŗš‘šš‘,š‘„š‘›š‘,š‘„š‘›š‘ī‚ī‚€š‘„ā‰¼šŗš‘šš‘,š‘„š‘šš‘+1,š‘„š‘šš‘+1ī‚ī‚€+ā„‹š‘‡š‘„š‘›š‘+1,š‘‡š‘„š‘›š‘+1,š‘‡š‘„š‘šš‘+1ī‚ī‚€š‘„+šœ€+šŗš‘›š‘+1,š‘„š‘›š‘,š‘„š‘›š‘ī‚ī‚€š‘„ā‰¼šŗš‘šš‘,š‘„š‘šš‘+1,š‘„š‘šš‘+1ī‚ī‚€šŗī‚€š‘„+šœ‘š‘›š‘,šŗš‘„š‘›š‘,šŗš‘„š‘šš‘ī‚€š‘„ī‚ī‚+šœ€+šŗš‘›š‘+1,š‘„š‘›š‘,š‘„š‘›š‘ī‚ī‚€š‘„ā‰Ŗšŗš‘šš‘,š‘„š‘šš‘+1,š‘„š‘šš‘+1ī‚ī‚€š‘„+šŗš‘›š‘,šŗš‘„š‘›š‘,šŗš‘„š‘šš‘ī‚ī‚€š‘„+šœ€+šŗš‘›š‘+1,š‘„š‘›š‘,š‘„š‘›š‘ī‚.(2.13) Since šœ€ is arbitrarily, letting šœ€ā†’šœƒ and by letting š‘ā†’āˆž, we have š›¾ā‰Ŗšœƒ+limš‘ā†’āˆžšŗī‚€š‘„š‘šš‘,š‘„š‘›š‘,š‘„š‘›š‘ī‚+šœƒ+šœƒ=š›¾,(2.14) a contradiction. So {š‘„š‘›} is a š‘”š‘£š‘ -šŗ-cone Cauchy sequence. Since (š‘‹,šŗ)is a š‘”š‘£š‘ -šŗ-cone complete metric space, {š‘„š‘›} is š‘”š‘£š‘ -šŗ-cone convergent in š‘‹ and šŗ-cone-limš‘›ā†’āˆžš‘„š‘›=š‘„. Thus, for every šœāˆˆintš‘ƒ and sufficiently large š‘›, we have that ā„‹ī€·š‘‡š‘„š‘›,š‘‡š‘„š‘›ī€øī€·šŗī€·š‘„,š‘‡š‘„ā‰¼šœ‘š‘›,š‘„š‘›ī€·š‘„,š‘„ī€øī€øā‰Ŗšŗš‘›,š‘„š‘›ī€øā‰Ŗšœ,š‘„3.(2.15) Since for š‘›āˆˆā„•āˆŖ{0}, š‘„š‘›+1āˆˆš‘‡š‘„š‘›, by (H4) we obtain that for all š‘›āˆˆā„• there existsš‘¦š‘›āˆˆš‘‡š‘„ such that šŗī€·š‘„š‘›+1,š‘„š‘›+1,š‘¦š‘›+1ī€øī€·ā‰¼ā„‹š‘‡š‘„š‘›,š‘‡š‘„š‘›ī€ø,š‘‡š‘„+šœ€š‘›+1ī€·šŗī€·š‘„ā‰¼šœ‘š‘›,š‘„š‘›+šœ€,š‘„ī€øī€ø3š‘›+1ī€·š‘„ā‰Ŗšŗš‘›,š‘„š‘›ī€ø+šœ€,š‘„3š‘›+1.(2.16) Since šœ€/3š‘›+1ā†’šœƒ, then for sufficiently large š‘›, we obtain that šŗī€·š‘¦š‘›+1ī€øī€·š‘¦,š‘„,š‘„ā‰¼šŗš‘›+1,š‘„š‘›+1,š‘„š‘›+1ī€øī€·š‘„+šŗš‘›+1ī€øā‰Ŗ,š‘„,š‘„2šœ3+šœ3=šœ,(2.17) which implies šŗ-cone-limš‘›ā†’āˆžš‘¦š‘›=š‘„. Since š‘‡š‘„ is closed, we obtain that š‘„āˆˆš‘‡š‘„.

For the case šœ‘(š‘”)=š‘˜š‘”, š‘˜āˆˆ(0,1), then šœ‘āˆˆĪ˜ and it is easy to get the following corollary.

Corollary 2.4. Let (š‘‹,šŗ) be a š‘”š‘£š‘ -šŗ-cone complete metric space with a solid cone š‘ƒ, let š’œ be a collection of nonempty closed subsets of š‘‹, š’œā‰ šœ™, and let ā„‹āˆ¶š’œĆ—š’œĆ—š’œā†’šø be a š‘”š‘£š‘ -ā„‹-cone metric with respect to šŗ. If the mapping š‘‡āˆ¶š‘‹ā†’š’œ satisfies the condition that exists š‘˜āˆˆ(0,1) such that for all š‘„,š‘¦,š‘§āˆˆš‘‹ holds ā„‹(š‘‡š‘„,š‘‡š‘¦,š‘‡š‘§)ā‰¼š‘˜ā‹…šŗ(š‘„,š‘¦,š‘§),(2.18) then š‘‡ has a fixed point in š‘‹.

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.