A note on the equivalence of some metric and Hcone metric fixed point theorems for multivalued contractions
 Huaping Huang^{1},
 Stojan Radenović^{2}Email author and
 Dragan Ðorić^{3}
https://doi.org/10.1186/s1366301502892
© Huang et al.; licensee Springer. 2015
Received: 17 October 2014
Accepted: 6 March 2015
Published: 26 March 2015
Abstract
In this paper, by using Minkowski functional introduced by Kadelburg et al. (Appl. Math. Lett. 24:370374, 2011) or nonlinear scalarization function introduced by Du (Nonlinear Anal. 72:22592261, 2010), we prove some equivalences between vectorial versions of fixed point theorems for Hcone metrics in the sense of Arshad and Ahmad and scalar versions of fixed point theorems for (general) HausdorffPompeiu metrics (in usual sense).
Keywords
MSC
1 Introduction
Recently, the investigation of possible equivalence between fixed point results in cone metric spaces (or tvscone metric spaces) and metric spaces has become a hot topic in many mathematical activities. Namely, by using the properties either of the Minkowski functional \(q_{e}\) or the nonlinear scalarization function \(\xi_{e}\) (in particular their monotonicity), some scholars have made a conclusion that many fixed point results in the setting of cone metric spaces or tvscone metric spaces can be directly obtained as a consequence of the corresponding results in metric spaces (see [1–12]). However, so far these equivalences have been referred to some fixed results only for single valued mappings, whereas, the ones for multivalued mappings have been seldom involved. The aim of this paper is to consider some fixed point theorem equivalences between Hcone metric fixed point theorems for multivalued or generalized multivalued contractions and (usual) metric fixed point theorems for (general) multivalued mappings. We mainly establish the equivalences between Arshad’s and Ahmad’s theorem (see [13]) and Nadler’s theorem (see [14]), and between Ðorić’s theorem (see [15]) and Achari’s theorem (see [16]), and Ćirić’s theorem (see [17]).
Definition 1.1
([18])
 (1)
\(K\neq\{\theta\}\);
 (2)
\(a,b\in\mathbb{R}^{+}\) and \(x,y\in K \Rightarrow ax+by\in K\);
 (3)
\(x,x\in K\Rightarrow x=\theta\).
Definition 1.2
([19])
 (1)
\(x\preceq y\) if \(yx\in K\);
 (2)
\(x\prec y\) if \(x\preceq y\) and \(x\neq y\);
 (3)
\(x\ll y\) if \(yx\in \operatorname{int} K\);
 (4)
we say that K is normal if there is \(M>0\) such that \(\theta \preceq x\preceq y\Rightarrow\x\\leq M\y\\).
Throughout this paper, unless otherwise specified, we always suppose that E is a real Banach 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 zero vector θ.
Definition 1.3
([20])
 (i)
\(\theta\preceq d(x,y)\) for all \(x,y\in X\) and \(d(x,y)=\theta\) if and only if \(x=y\);
 (ii)
\(d(x,y)=d(y,x)\) for all \(x,y\in X\);
 (iii)
\(d(x,y)\preceq d(x,z)+d(z,y)\) for all \(x,y,z\in X\).
Remark 1.4
([8])
If \(Y=E\) in Definition 1.3, 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 case of tvscone metric space.
Definition 1.5
([21])
 (H1)
\(H(A,B)=\theta\Rightarrow A=B\);
 (H2)
\(H(A,B)=H(B,A)\);
 (H3)
for any \(\varepsilon\gg\theta\) and each \(x\in A\), there exists \(y\in B\) such that \(d(x,y)\preceq H(A,B)+\varepsilon\);
 (H4)one of the following is satisfied:
 (i)
for any \(\varepsilon\gg\theta\), there exists \(x\in A\) such that for each \(y\in B\), \(H(A,B)\preceq d(x,y)+\varepsilon\);
 (ii)
for any \(\varepsilon\gg\theta\), there exists \(x\in B\) such that for each \(y\in A\), \(H(A,B)\preceq d(x,y)+\varepsilon\).
 (i)
Remark 1.6
If we substitute Y for E, then H is called a tvsHcone metric (see [20]).
Definition 1.7
([13])
 (H_{1}):

\(\theta\preceq H(A,B)\) for all \(A,B\in\mathcal{A}\) and \(H(A,B)=\theta\) if and only if \(A=B\);
 (H_{2}):

\(H(A,B)=H(B,A)\) for all \(A,B\in\mathcal{A}\);
 (H_{3}):

\(H(A,B)\preceq H(A,C)+H(C,B)\) for all \(A,B,C\in\mathcal{A}\);
 (H_{4}):

if \(A,B\in\mathcal{A}\), \(\theta\prec\varepsilon\in E\) with \(H(A,B)\prec\varepsilon\), then for each \(a\in A\) there exists \(b\in B\) such that \(d(a,b)\prec\varepsilon\).
Example 1.8
Remark 1.9
Compared with Definition 1.5, Definition 1.7 minutely modifies Definition 1.5 to make it more comparable with a standard metric. The following example indicates that Definition 1.7 is different from Definition 1.5.
Example 1.10
Let \((Y,K)\) be an ordered tvs and \(e\in \operatorname{int} K\). Then \([e,e]=(Ke)\cap(eK)=\{z\in Y: e\preceq z\preceq e\}\) is an absolutely convex neighborhood of θ; its Minkowski functional \(q_{[e,e]}\) will be denoted by \(q_{e}\). Clearly, \(\operatorname{int} [e,e]=(\operatorname{int} Ke)\cap(e\operatorname{int} K)\), \(q_{e}(x)=\inf\{\lambda>0:x\prec\lambda e\}\). Moreover, \(q_{e}(x)\) is an increasing function on K. Indeed, if \(\theta\preceq x\preceq y\), then \(\{\lambda:x\in\lambda[e,e]\}\supset\{\lambda:y\in\lambda [e,e]\}\) and it follows that \(q_{e}(x)\leq q_{e}(y)\).
Lemma 1.11
([9])
Let \((X,d)\) be a tvscone metric space and let \(e\in \operatorname{int} K\). Let \(q_{e}\) be the corresponding Minkowski functional of \([e,e]\). Then \(d_{q}:=q_{e}\circ d\) is a metric on X.
Lemma 1.12
([8])
Let \((X,d)\) be a tvscone metric space and let \(e\in \operatorname{int} K\). Let \(\xi_{e}:Y\rightarrow\mathbb{R}\) be a nonlinear scalarization function defined by \(\xi_{e}(y)=\inf\{r\in\mathbb{R}:y\in reK\}\). Then \(d_{\xi}:X\times X\rightarrow[0,+\infty)\) defined by \(d_{\xi}:=\xi_{e} \circ d\) is a metric on X.
For the convenience of the reader, we present some wellknown theorems as follows.
Theorem 1.13
(Nadler [14])
Theorem 1.14
(Arshad and Ahmad [13])
Theorem 1.15
(Achari [16])
Theorem 1.16
(Ðorić [15])
 (C1)
\(H(Tx,Sy)\preceq\lambda\cdot d(x,y)\);
 (C2)
\(H(Tx,Sy)\preceq\lambda\cdot d(x,u)\) for each fixed \(u\in Tx\);
 (C3)
\(H(Tx,Sy)\preceq\lambda\cdot d(y,v)\) for each fixed \(v\in Sy\);
 (C4)
\(H(Tx,Sy)\preceq\lambda\cdot\frac{d(x,v)+d(y,u)}{2}\) for each fixed \(v\in Sy\) and each fixed \(u\in Tx\).
Theorem 1.17
(Ćirić [17])
Theorem 1.18
(Ðorić [15])
 (D1)
\(H(Tx,Ty)\preceq\lambda\cdot d(x,y)\);
 (D2)
\(H(Tx,Ty)\preceq\lambda\cdot d(x,u)\) for each fixed \(u\in Tx\);
 (D3)
\(H(Tx,Ty)\preceq\lambda\cdot d(y,v)\) for each fixed \(v\in Ty\);
 (D4)
\(H(Tx,Ty)\preceq\lambda\cdot\frac{d(x,v)+d(y,u)}{2}\) for each fixed \(v\in Ty\) and each fixed \(u\in Tx\).
2 Main results
In what follows, by utilizing the Minkowski functional \(q_{e}\) or the nonlinear scalarization function \(\xi_{e}\), we present two inequalities. Based on them, we thereupon obtain some equivalences between some wellknown theorems for multivalued or generalized multivalued contractions.
Theorem 2.1
Proof
Theorem 2.2
Proof
Similarly as in the proof of Theorem 2.1, by utilizing Lemma 1.12, we obtain the conclusion. □
Proof
Proof
Proof
If one takes \(S=T\) in Theorem 1.16 and Theorem 1.15, then by Theorem 2.4, the proof is completed. □
Remark 2.6
According to Theorem 2.3, Theorem 2.4 and Corollary 2.5, we can easily see that the vectorial versions of Nadler’s theorem, Achari’s theorem and Ćirić’s theorem are just equivalent to their scalar versions, respectively. It is worth mentioning that it is possible to obtain the same conclusion using the nonlinear scalarization function \(\xi_{e}\).
We finally pose the following problems:
Declarations
Acknowledgements
The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and kind comments. The third author is thankful to the Ministry of Education, Science and Technological Development of Serbia.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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