Huang et al. Fixed Point Theory and Applications (2015) 2015:43 DOI 10.1186/s13663-015-0289-2

R E S E A R C H Open Access

A note on the equivalence of some metricand H-cone metric fixed point theorems formultivalued contractionsHuaping Huang1, Stojan Radenovic2* and Dragan Ðoric3

*Correspondence:fixedpoint50@gmail.com2Faculty of Mathematics andInformation Technology, Dong ThapUniversity, Dong Thap, VietnamFull list of author information isavailable at the end of the article

AbstractIn this paper, by using Minkowski functional introduced by Kadelburg et al. (Appl.Math. Lett. 24:370-374, 2011) or nonlinear scalarization function introduced by Du(Nonlinear Anal. 72:2259-2261, 2010), we prove some equivalences between vectorialversions of fixed point theorems for H-cone metrics in the sense of Arshad andAhmad and scalar versions of fixed point theorems for (general) Hausdorff-Pompeiumetrics (in usual sense).

MSC: 47H10; 54H25

Keywords: Hausdorff-Pompeiu metric; H-cone metric; cone metric space; fixedpoint; multivalued contraction

1 IntroductionRecently, the investigation of possible equivalence between fixed point results in cone met-ric spaces (or tvs-cone metric spaces) and metric spaces has become a hot topic in manymathematical activities. Namely, by using the properties either of the Minkowski func-tional qe or the nonlinear scalarization function ξe (in particular their monotonicity), somescholars have made a conclusion that many fixed point results in the setting of cone metricspaces or tvs-cone metric spaces can be directly obtained as a consequence of the corre-sponding results in metric spaces (see [–]). However, so far these equivalences havebeen referred to some fixed results only for single valued mappings, whereas, the ones formultivalued mappings have been seldom involved. The aim of this paper is to considersome fixed point theorem equivalences between H-cone metric fixed point theorems formultivalued or generalized multivalued contractions and (usual) metric fixed point theo-rems for (general) multivalued mappings. We mainly establish the equivalences betweenArshad’s and Ahmad’s theorem (see []) and Nadler’s theorem (see []), and betweenÐorić’s theorem (see []) and Achari’s theorem (see []), and Ćirić’s theorem (see []).

Definition . ([]) Let E be a real Banach space and θ be its zero element. Suppose thata nonempty closed subset K of E satisfies the following:

() K �= {θ};() a, b ∈R

+ and x, y ∈ K ⇒ ax + by ∈ K ;() x, –x ∈ K ⇒ x = θ .

© 2015 Huang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly credited.

Huang et al. Fixed Point Theory and Applications (2015) 2015:43 Page 2 of 8

Then K is called a cone. If int K �= ∅, then K is called a solid cone, where int K denotes theinterior of K .

Definition . ([]) Let K be a cone in a real Banach space (E,‖ ·‖). The partial orderings�, ≺, and on E with respect to P are defined as follows, respectively. Let x, y ∈ E. Then

() x � y if y – x ∈ K ;() x ≺ y if x � y and x �= y;() x y if y – x ∈ int K ;() we say that K is normal if there is M > such that θ � x � y ⇒ ‖x‖ ≤ M‖y‖.

Throughout this paper, unless otherwise specified, we always suppose that E is a realBanach space, K is a solid cone in E, �, ≺, are partial orderings with respect to K ,and Y is a locally convex Hausdorff topological vector space (tvs for short) with its zerovector θ .

Definition . ([]) Let X be a nonempty set and (Y , K) be an ordered tvs. Suppose thata vector-valued function d : X × X → Y satisfies:

(i) θ � d(x, y) for all x, y ∈ X and d(x, y) = θ 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(z, y) for all x, y, z ∈ X .Then d is called a tvs-cone metric on X, and (X, d) is called a tvs-cone metric space.

Remark . ([]) If Y = E in Definition ., then d is said to be a cone metric on X, and(X, d) is said to be a cone metric space. In other words, cone metric space is a special caseof tvs-cone metric space.

Definition . ([]) Let (X, d) be a cone metric space and let A be a collection ofnonempty subsets of X. A map H : A × A → E is called an H-cone metric in the senseof Wardowski if for any A, B ∈A, the following conditions hold:

(H) H(A, B) = θ ⇒ A = B;(H) H(A, B) = H(B, A);(H) for any ε � θ and each x ∈ A, there exists y ∈ B such that d(x, y) � H(A, B) + ε;(H) one of the following is satisfied:

(i) for any ε � θ , there exists x ∈ A such that for each y ∈ B, H(A, B) � d(x, y) + ε;(ii) for any ε � θ , there exists x ∈ B such that for each y ∈ A, H(A, B) � d(x, y) + ε.

Remark . If we substitute Y for E, then H is called a tvs-H-cone metric (see []).

Definition . ([]) Let (X, d) be a cone metric space and A a collection of nonemptysubsets of X. A map H : A×A → E is called an H-cone metric in the sense of Arshad andAhmad if the following conditions hold:

(H) θ � H(A, B) for all A, B ∈A and H(A, B) = θ if and only if A = B;(H) H(A, B) = H(B, A) for all A, B ∈A;(H) H(A, B) � H(A, C) + H(C, B) for all A, B, C ∈A;(H) if A, B ∈ A, θ ≺ ε ∈ E with H(A, B) ≺ ε, then for each a ∈ A there exists b ∈ B such

that d(a, b) ≺ ε.

Huang et al. Fixed Point Theory and Applications (2015) 2015:43 Page 3 of 8

Example . ([, ]) Let (X, d) be a metric space and let A be a family of all nonemptyclosed bounded subsets of X. Then H : A×A→ R

+ given by the formula

H(A, B) = max{

supx∈A

d(x, B), supy∈B

d(y, A)}

, A, B ∈A, (.)

is an H-cone metric (called a Hausdorff-Pompeiu metric), which satisfies either Defini-tion . or Definition ..

Remark . Compared with Definition ., Definition . minutely modifies Defini-tion . to make it more comparable with a standard metric. The following example indi-cates that Definition . is different from Definition ..

Example . Let X = {a, b, c} and d : X × X → [, +∞) be defined by

d(a, b) = d(b, a) =

, d(a, c) = d(c, a) = d(b, c) = d(c, b) = ,

d(a, a) = d(b, b) = d(c, c) = .

Let A = {{a}, {b}, {c}}, H : A × A → [, +∞) as H({a}, {b}) = H({b}, {a}) = , H({a}, {c}) =H({c}, {a}) = H({b}, {c}) = H({c}, {b}) = , H({a}, {a}) = H({b}, {b}) = H({c}, {c}) = . Then His an H-cone metric which satisfies Definition . but not Definition .. In fact, (H) ofDefinition . does not hold.

Recall (see [] or []) that if V is an absolutely convex and absorbing subset of a tvs Y ,its Minkowski functional is defined by

E � x �→ qV (x) = inf{λ > : x ∈ λV }.

It is a semi-norm on Y and V ⊂ W implies that qW (x) ≤ qV (x) for x ∈ Y . If V is an abso-lutely convex neighborhood of θ in Y , then qV is continuous and

{x ∈ E : qV (x) <

}= int V ⊂ V ⊂ V =

{x ∈ Y : qV (x) ≤

}.

Let (Y , K) be an ordered tvs and e ∈ int K . Then [–e, e] = (K – e) ∩ (e – K) = {z ∈ Y :–e � z � e} is an absolutely convex neighborhood of θ ; its Minkowski functional q[–e,e]

will be denoted by qe. Clearly, int[–e, e] = (int K – e) ∩ (e – int K), qe(x) = inf{λ > : x ≺λe}. Moreover, qe(x) is an increasing function on K . Indeed, if θ � x � y, then {λ : x ∈λ[–e, e]} ⊃ {λ : y ∈ λ[–e, e]} and it follows that qe(x) ≤ qe(y).

Lemma . ([]) Let (X, d) be a tvs-cone metric space and let e ∈ int K . Let qe be the cor-responding Minkowski functional of [–e, e]. Then dq := qe ◦ d is a metric on X.

Lemma . ([]) Let (X, d) be a tvs-cone metric space and let e ∈ int K . Let ξe : Y → R

be a nonlinear scalarization function defined by ξe(y) = inf{r ∈ R : y ∈ re – K}. Then dξ :X × X → [, +∞) defined by dξ := ξe ◦ d is a metric on X.

For the convenience of the reader, we present some well-known theorems as follows.

Huang et al. Fixed Point Theory and Applications (2015) 2015:43 Page 4 of 8

Theorem . (Nadler []) Let (X, d) be a complete metric space and A be a collectionof nonempty, closed, and bounded subsets of X. Suppose that a mapping T : X → A is amultivalued contraction, that is, there exists λ ∈ [, ) such that for all x, y ∈ X,

H(Tx, Ty) ≤ λd(x, y),

where H(·, ·) is the Hausdorff-Pompeiu metric (.) induced by d. Then T has a fixedpoint.

Theorem . (Arshad and Ahmad []) Let (X, d) be a complete cone metric space. LetA be a collection of nonempty closed subsets of X, and let H : A × A → E be an H-conemetric in the sense of Arshad and Ahmad. If for a map T : X → A there exists λ ∈ [, )such that for all x, y ∈ X,

H(Tx, Ty) � λd(x, y), (.)

then T has a fixed point.

Theorem . (Achari []) Let (X, d) be a complete metric space and let A be a family ofnonempty, closed, and bounded subsets of X. Suppose that T , S : X → A are two multival-ued mappings and suppose that there exists λ ∈ [, ) such that for all x, y ∈ X,

H(Tx, Sy) ≤ λ · max

{d(x, y), d(x, Tx), d(y, Sy),

d(x, Sy) + d(y, Tx)

},

where H(·, ·) is the Hausdorff-Pompeiu metric (.) induced by d. Then T and S have acommon fixed point.

Theorem . (Ðorić []) Let (X, d) be a complete cone metric space. Let A be a family ofnonempty, closed, and bounded subsets of X and let there exist an H-cone metric H : A×A→ E in the sense of Arshad and Ahmad. Suppose that T , S : X →A are two multivaluedmappings and suppose that there is λ ∈ [, ) such that, for all x, y ∈ X, at least one of thefollowing conditions holds:

(C) H(Tx, Sy) � λ · d(x, y);(C) H(Tx, Sy) � λ · d(x, u) for each fixed u ∈ Tx;(C) H(Tx, Sy) � λ · d(y, v) for each fixed v ∈ Sy;(C) H(Tx, Sy) � λ · d(x,v)+d(y,u)

for each fixed v ∈ Sy and each fixed u ∈ Tx.Then T and S have a common fixed point.

Theorem . (Ćirić []) Let (X, d) be a complete metric space and let A be a familyof nonempty, closed, and bounded subsets of X. Suppose that T : X → A is a generalizedmultivalued contraction, that is, there exists λ ∈ [, ) such that for all x, y ∈ X,

H(Tx, Ty) ≤ λ · max

{d(x, y), d(x, Tx), d(y, Ty),

d(x, Ty) + d(y, Tx)

},

where H(·, ·) is the Hausdorff-Pompeiu metric (.) induced by d. Then T has a fixedpoint.

Huang et al. Fixed Point Theory and Applications (2015) 2015:43 Page 5 of 8

Theorem . (Ðorić []) Let (X, d) be a complete cone metric space. Let A be a family ofnonempty, closed, and bounded subsets of X and let there exist an H-cone metric H : A×A → E in the sense of Arshad and Ahmad. Suppose that T : X → A is a cone generalizedmultivalued contraction, that is, there exists λ ∈ [, ) such that, for all x, y ∈ X, one of thefollowing conditions holds:

(D) H(Tx, Ty) � λ · d(x, y);(D) H(Tx, Ty) � λ · d(x, u) for each fixed u ∈ Tx;(D) H(Tx, Ty) � λ · d(y, v) for each fixed v ∈ Ty;(D) H(Tx, Ty) � λ · d(x,v)+d(y,u)

for each fixed v ∈ Ty and each fixed u ∈ Tx.Then T has a fixed point.

2 Main resultsIn what follows, by utilizing the Minkowski functional qe or the nonlinear scalarizationfunction ξe, we present two inequalities. Based on them, we thereupon obtain some equiv-alences between some well-known theorems for multivalued or generalized multivaluedcontractions.

Theorem . Let (X, d) be a cone metric space and A a collection of nonempty subsetsof X. Let H : A × A → E be an H-cone metric in the sense of Arshad and Ahmad and lete ∈ int K and qe be the corresponding Minkowski functional of [–e, e]. If Hq = qe ◦ H anddq = qe ◦ d, then

Hdq (A, B) ≤ Hq(A, B) (A, B ∈A),

where Hdq (A, B) is the Hausdorff-Pompeiu metric induced by dq.

Proof On account of (H)-(H) in Definition ., we conclude that (A, H) is a cone metricspace. Using Lemma ., one finds that dq is a metric on X and Hq is a metric onA. Denote

M ={λ > : H(A, B) ≺ λe

}(A, B ∈A),

N ={λ > : d(x, y) ≺ λe

}(x ∈ A, y ∈ B).

In view of (H), it is not hard to verify that M ⊆ N . Thus, inf M ≥ inf N . Further, we have

Hq(A, B) = qe(H(A, B)

)= inf

{λ > : H(A, B) ≺ λe

}= inf M.

Accordingly, for all A, B ∈A, it follows that

Hdq (A, B) = max{

supx∈A

dq(x, B), supy∈B

dq(y, A)}

= max{

supx∈A

infy∈B

dq(x, y), supy∈B

infx∈A

dq(y, x)}

= max{

supx∈A

infy∈B

inf{λ > : d(x, y) ≺ λe

},

supy∈B

infx∈A

inf{λ > : d(x, y) ≺ λe

}}

Huang et al. Fixed Point Theory and Applications (2015) 2015:43 Page 6 of 8

= max{

supx∈A

infy∈B

inf N , supy∈B

infx∈A

inf N}

≤ max{

supx∈A

infy∈B

inf M, supy∈B

infx∈A

inf M}

= max{

supx∈A

infy∈B

Hq(A, B), supy∈B

infx∈A

Hq(A, B)}

= Hq(A, B). �

Theorem . Let (X, d) be a cone metric space and A a collection of nonempty subsetsof X. Let H : A × A → E be an H-cone metric in the sense of Arshad and Ahmad and lete ∈ int K and ξe be the corresponding nonlinear scalarization function. If Hξ = ξe ◦ H anddξ = ξe ◦ d, then

Hdξ(A, B) ≤ Hξ (A, B) (A, B ∈A),

where Hdξ(A, B) is the Hausdorff-Pompeiu metric induced by dξ .

Proof Similarly as in the proof of Theorem ., by utilizing Lemma ., we obtain theconclusion. �

Theorem . Theorem . is equivalent to Theorem ..

Proof In Theorem ., take E = R, K = [, +∞), and H to be the Hausdorff-Pompeiu met-ric (.) introduced by d, and let A be a collection of nonempty, closed, and bounded sub-sets of X. Then by Theorem ., we easily get Theorem .. Conversely, let Theorem .hold. Applying the Minkowski functional qe to both sides of the inequality (.), we estab-lish that

qe(H(Tx, Ty)

) ≤ qe(λd(x, y)

)= λqe

(d(x, y)

),

that is,

Hq(Tx, Ty) ≤ λdq(x, y).

Here, Hq = qe ◦ H and dq = qe ◦ d are metrics from Lemma .. By using Theorem ., forall x, y ∈ X, it follows that

Hdq (Tx, Ty) ≤ λdq(x, y).

Hence by Theorem ., T has a fixed point. �

Theorem . Theorem . is equivalent to Theorem ..

Proof In Theorem ., take E = R, K = [, +∞), and let H be the Hausdorff-Pompeiumetric (.) introduced by d. Then Theorem . is valid. Indeed, in this case, from (C)-

Huang et al. Fixed Point Theory and Applications (2015) 2015:43 Page 7 of 8

(C) of Theorem ., we conclude that

H(Tx, Sy) ≤ λ

⎧⎪⎪⎪⎨⎪⎪⎪⎩

d(x, y),d(x, Tx),d(y, Sy), (d(y, Tx) + d(x, Sy)),

where d(a, B) = infb∈B d(a, b). Hence,

H(Tx, Sy) ≤ λ · max

{d(x, y), d(x, Tx), d(y, Sy),

d(x, Sy) + d(y, Tx)

}.

That is to say, we obtain Theorem ..Conversely, let Theorem . hold. Then by (C)-(C), it follows that

H(Tx, Sy) � λ

⎧⎪⎪⎪⎨⎪⎪⎪⎩

d(x, y),d(x, u), u ∈ Tx,d(y, v), v ∈ Sy, (d(y, u) + d(x, v)), u ∈ Tx, v ∈ Sy.

Applying the Minkowski functional qe to both sides of the above inequalities, we establishthat

Hq(Tx, Sy) ≤ λ

⎧⎪⎪⎪⎨⎪⎪⎪⎩

dq(x, y),infu∈Tx dq(x, u),infv∈Sy dq(y, v), (infu∈Tx dq(y, u) + infv∈Sy dq(x, v)).

That is to say,

Hq(Tx, Sy) ≤ λ · max

{dq(x, y), dq(x, Tx), dq(y, Sy),

dq(x, Sy) + dq(y, Tx)

},

where dq(a, B) = infb∈B dq(a, b). Hq = qe ◦ H and dq = qe ◦ d are metrics from Lemma ..Thus by Theorem ., for all x, y ∈ X, we have

Hdq (Tx, Sy) ≤ λ · max

{dq(x, y), dq(x, Tx), dq(y, Sy),

dq(x, Sy) + dq(y, Tx)

}.

Therefore, by Theorem ., T and S have a common fixed point. �

Corollary . Theorem . is equivalent to Theorem ..

Proof If one takes S = T in Theorem . and Theorem ., then by Theorem ., theproof is completed. �

Remark . According to Theorem ., Theorem . and Corollary ., we can easilysee that the vectorial versions of Nadler’s theorem, Achari’s theorem and Ćirić’s theoremare just equivalent to their scalar versions, respectively. It is worth mentioning that it ispossible to obtain the same conclusion using the nonlinear scalarization function ξe.

Huang et al. Fixed Point Theory and Applications (2015) 2015:43 Page 8 of 8

We finally pose the following problems:

Problem Does Definition . imply Definition .?

Problem Is Theorem . of [] equivalent to Theorem .?

Problem Is Theorem . of [] equivalent to Theorem .?

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThe authors contributed equally and significantly in writing this paper. All authors read and approved the finalmanuscript.

Author details1School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China. 2Faculty of Mathematics andInformation Technology, Dong Thap University, Dong Thap, Vietnam. 3Faculty of Organizational Sciences, University ofBelgrade, Jove Ilica 154, Beograd, Serbia.

AcknowledgementsThe authors would like to express their sincere appreciation to the referees for their very helpful suggestions and kindcomments. The third author is thankful to the Ministry of Education, Science and Technological Development of Serbia.

Received: 17 October 2014 Accepted: 6 March 2015

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