 Research
 Open Access
Fixed point and common fixed point results in cone metric space and application to invariant approximation
 Anil Kumar^{1, 2}Email author and
 Savita Rathee^{1}
https://doi.org/10.1186/s1366301502909
© Kumar and Rathee; licensee Springer. 2015
 Received: 14 October 2014
 Accepted: 12 March 2015
 Published: 27 March 2015
Abstract
In this work, the concept of almost contraction for multivalued mappings in the setting of cone metric spaces is defined and then we establish some fixed point and common fixed point results in the setup of cone metric spaces. As an application, some invariant approximation results are obtained. The results of this paper extend and improve the corresponding results of multivalued mapping from metric space theory to cone metric spaces. Further our results improve the recent result of Arshad and Ahmad (Sci. World J. 2013:481601, 2013).
Keywords
 fixed point
 common fixed point
 cone metric space
 multivalued mappings
 best approximation
MSC
 46T99
 47H10
 54H25
1 Introduction
Fixed point theory has many applications in different branches of science. This theory itself is a beautiful mixture of analysis, topology, and geometry. Since the appearance of the Banach contraction mapping principle, there has been a lot of activity in this area and several wellknown fixed point theorems came into existence as a generalization of that principle. Many authors generalized and extended the notion of metric spaces such as bmetric spaces, partial metric spaces, generalized metric spaces, complexvalued metric space etc. For a useful discussion of these generalizations of metric spaces, one may refer to [1].
In 2007, Huang and Zhang [2] introduced the concept of cone metric space as a generalization of metric space, in which they replace the set of real numbers with a real Banach space. Although they proved several fixed point theorems for contractive type mappings on a cone metric space when the underlying cone is normal. Rezapour and Hamlbarani [3] proved such fixed point theorems omitting the assumptions of normality of cone. After that, the study of fixed point theorems in cone metric spaces was followed by many others (e.g., see [4–17] and the references therein).
On the other side, Nadler [18] and Markin [19] initiated the study of fixed point theorems for multivalued mappings and established the multivalued version of the Banach contraction mapping principle. Since the theory of multivalued mappings has many applications, it became a focus of research over the years. Recently, many authors worked out results on multivalued mappings defined on a cone metric space when the underlying cone is normal or regular (see [20–23]). In 2011, Janković et al. [24] showed that most of the fixed point results in the setup of normal cone metric space can be obtained as a consequence of the corresponding results in metric spaces. In the light of this, Arshad and Ahmad [25] improved Wardowski’s results by proving the same without the assumption of the normality of the cones.
Here, the concept of almost contraction for multivalued mappings in the setting of cone metric spaces is defined and then we establish some fixed point and common fixed point results in the setup of cone metric spaces. In this way our results extend the results of Arshad and Ahmad [25] and also improve the corresponding results of both singlevalued and multivalued mappings existing in the literature. Before starting our work we need the following wellknown definitions and results.
Definition 1
 (1)
P is nonempty, closed, and \(P \neq\{\theta\}\), where θ is the zero element of E;
 (2)
for any nonnegative real numbers a, b and for any \(x, y \in P\), one has \(ax+by\in P\);
 (3)
\(x\in P\) and \(x\in P\) implies \(x = \theta\).
Definition 2
[2]
 (d_{1}):

\(\theta\preceq d(x,y) \) for all \(x, y \in X\);
 (d_{2}):

\(d(x,y)=\theta\) if and only if \(x=y\);
 (d_{3}):

\(d(x,y)=d(y,x)\) for all \(x, y \in X\);
 (d_{4}):

\(d(x,y)\preceq d(x,z)+d(z,y)\) for all \(x, y, z \in X\).
Definition 3
[2]
 (1)
\(\{x_{n}\}\) converges to x, if for every \(c\in E\) with \(\theta\ll c\) there exists a positive integer N such that \(d(x_{n}, x)\ll c\), for all \(n\geq N\). We denote this by \(\lim_{n\to\infty} x_{n} = x\).
 (2)
\(\{x_{n}\}\) is said to be Cauchy if for every \(c\in E\) with \(\theta\ll c\) there exists a positive integer N such that \(d(x_{n}, x_{m})\ll c\), for all \(n, m\geq N\).
A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
Lemma 4
[24]
 (1)
If \(c\in \operatorname{int}P\) and \(a_{n}\to\theta\), then there exists a positive integer N such that for all \(n>N\), we have \(a_{n}\ll c\).
 (2)
If \(a\preceq k a\), where \(a\in P\) and \(0\leq k<1\), then \(a=\theta\).
Definition 5
[25]
 (H_{1}):

\(\theta\preceq H(A, B)\) for all \(A, B\in C(X)\).
 (H_{2}):

\(H(A, B) = \theta\) if and only if \(A= B\).
 (H_{3}):

\(H(A, B)= H(B,A)\) for all \(A, B\in C(X)\).
 (H_{4}):

\(H(A, B)\preceq H(A, C)+ H(C, B)\) for all \(A, B, C\in C(X)\).
 (H_{5}):

If \(A, B\in C(X)\), \(\theta\prec\epsilon\in E\) with \(H(A, B)\prec\epsilon\), then for each \(a\in A\) there exists \(b\in B\) such that \(d(a,b)\prec\epsilon\).
Example 6
It is to be noted that \((C(X), H)\) is a complete metric space whenever \((X,d)\) is a complete metric space.
Definition 7
 (i)
a fixed point of T, if \(x\in Tx\);
 (ii)
a common fixed point of T and f, if \(x=fx\in Tx\);
 (iii)
a coincidence point of T and f, if \(w=fx\in Tx\), and w is called the point of coincidence of T and f.
We denote \(C(f,T)=\{x\in X : fx\in Tx\}\), the set of coincidence point of f and T. The set of fixed point of T and the set of common fixed point of f and T is denoted by \(F(T)\) and \(F(f,T)\), respectively.
Definition 8
[26]
Let X be a nonempty set, \(T: X\to C(X)\) be a multivalued mapping, and \(f:X\to X\). Then f is called Tweakly commuting at \(x\in X\) if \(ffx\in Tfx\).
2 Main result
We start this section with the following definition.
Definition 9
Theorem 10
Let \((X,d)\) be a complete cone metric space and let there exist an Hcone metric on \(C(X)\) induced by d. Suppose \(T:X\to C(X)\) is a multivalued almost contraction. Then T has a fixed point in X.
Proof
Now we present an example in support of the proved result.
Example 11
Now, from all the cases, it is concluded that the multivalued mapping T satisfies the inequality (2.1) for \(\lambda=\frac{1}{2}\) and \(L=5\). Hence, T is an almost multivalued contraction that satisfies all the hypotheses of Theorem 10. Thus, the mapping T has a fixed point. Here \(x=0\) is such a fixed point.
Remark 12
 (i)
Theorem 3.1 of Arshad and Ahmad [25], Theorem 2.4 of Dorić [27], and Theorem 3.1 of Wardowski [20] are direct consequences of Theorem 10.
 (ii)
In Example 11, for \(x=\frac{1}{2}\) and \(y=\frac {2}{3}\), we get \(Tx=[0,\frac{1}{4}]\), \(Ty=[\frac{2}{3}, \frac {13}{18}]\), therefore \(H(Tx,Ty)=\frac{17}{36}e^{t}\). Then it can easily be checked that there does not exist any \(\lambda\in(0,1)\) such that the mapping T satisfies the conditions (D1), (D2), (D3), (D4) given in Definition 2.1 of Dorić [27]. Hence, Theorem 2.4 of Dorić [27] cannot be applied to Example 11. It is also to be noted that Theorem 3.1 of Arshad and Ahmad [25] and Theorem 3.1 of Wardowski [20] are not applicable to Example 11.
In [28] Haghi et al. proved the following lemma.
Lemma 13
Let X be a nonempty set and \(f: X\to X\) be a function. Then there exists a subset \(E\subseteq X\) such that \(f(E)=f(X)\) and \(f: E\to X\) is one to one.
Theorem 14
Proof
Theorem 15
 (i)
\(\delta K\subseteq fK\);
 (ii)
\((\bigcup_{x\in K}Tx)\cap K\subseteq fK\);
 (iii)
\(fx\in\delta K \Rightarrow Tx\subseteq K\);
 (iv)
fK is closed in X.
Proof
Let \(x\in\delta K\). We construct two sequences \(\{x_{n}\}\) in K and \(\{ y_{n}\}\) in fK in the following way. Since \(\delta K\subseteq fK\), there exists \(x_{0}\in K\) such that \(fx_{0}=x\in\delta K\). So, by (iii) we get \(Tx_{0}\subseteq K\). Since \((\bigcup_{x\in K}Tx)\cap K\subseteq fK\), we have \(Tx_{0}\subseteq fK\). Let \(y_{1}\in Tx_{0}\), then there exists \(x_{1}\in K\) such that \(y_{1}=fx_{1}\). Consider the element \(H(Tx_{0}, Tx_{1})\in E\). If the righthand side of (2.13) is θ at \(x=x_{0}\) and \(y=x_{1}\), then, as \(fx_{1}\in Tx_{0}\), we have \(d(fx_{0}, fx_{1})=\theta\) and hence \(fx_{1}=fx_{0}\). This and \(fx_{1}\in Tx_{0}\) imply \(fx_{0}\in Tx_{0}\). Thus, \(x_{0}\) is coincidence point of f and T.
 (a)
\(y_{n+1}\in Tx_{n}\), for each \(n\in\mathbb{N}\cup\{0\}\);
 (b)
\(d(y_{n}, y_{n+1})\preceq H(Tx_{n1}, Tx_{n})+\lambda^{n}e\);
 (c)
if \(y_{n}\in K\), then \(y_{n}=fx_{n}\);
 (d)if \(y_{n}\notin K\), then \(fx_{n}\in\delta K\) with$$ d(fx_{n1}, fx_{n})+ d(fx_{n}, y_{n})=d(fx_{n1}, y_{n}). $$(2.18)
If we let \(f=I\) (identity map) in Theorem 15, we obtain the following result as an extension of Theorem 9 of [29] to a cone metric space.
Corollary 16
Now, we present a nontrivial example which shows the generality of Corollary 16 over the corresponding existing theorems.
Example 17
Remark 18
 (i)
 (ii)
3 Application to invariant approximation
Since the appearance of Meinardus’ result in best approximation theory, several authors have obtained best approximation results for singlevalued maps as an application of fixed point and common fixed point results. The best approximation results for multivalued mappings was obtained by Kamran [26], AlThagafi and Shahzad [31], Beg et al. [32], O’Regan and Shahzad [33], and Markin and Shahzad [34]. Further, best approximation results in the setting of cone metric space were for the first time considered by Rezapour [35] (see also [36]).
In this section the best approximation results for a multivalued mapping in the setting of cone metric spaces are obtained.
Definition 19
Let M be a nonempty subset of a cone metric space X. A point \(y\in M\) is said to be a best approximation to \(p\in X\), if \(d(y,p)\preceq d(z,p)\) for all \(z\in M\). The set of best approximations to p in M is denoted by \(B_{M}(p)\).
As an application of Theorem 14, we obtain the following theorem, which ensures the existence of a best approximation.
Theorem 20
 (i)
\(f(B_{M}(p))=B_{M}(p)\).
 (ii)
\(ffv=fv\) for \(v\in C(f,T)\cap B_{M}(p)\).
 (iii)
\(d(y,p)\preceq d(fx,p)\) for all \(x\in B_{M}(p)\) and \(y\in Tx\).
 (iv)
\(f(B_{M}(p))\) is complete.
Proof
First we show that \(T_{B_{M}(p)}: B_{M}(p)\to C(B_{M}(p))\) is a multivalued mapping. For this let \(x\in B_{M}(p)\) and \(u\in Tx\). Then, as \(f(B_{M}(p))= B_{M}(p)\), we get \(fx\in B_{M}(p)\) and hence \(d(fx,p)\preceq d(z,p)\) for all \(z\in M\).
Corollary 21
 (i)
\(d(y,p)\preceq d(x,p)\) for all \(x\in B_{M}(p)\) and \(y\in Tx\).
 (ii)
\(B_{M}(p)\) is complete.
Theorem 22
Proof
The following theorem ensures the existence of a fixed point from the set of best approximations.
Theorem 23
Let M be a subset of a cone metric space \((X,d)\) and let there exist an Hcone metric on \(C(X)\) induced by d. Suppose \(T: X\to C(X)\), \(p\in X\), and \(B_{M}(p)\) is nonempty, compact, and it has a joint contractive family \(F=\{h_{A} : A\in C(B_{M}(p))\}\). If T is continuous on \(B_{M}(p)\), (3.3) holds for all \(x,y\in B_{M}(p)\) and also \(d(y,p)\preceq d(x,p)\) for all \(x\in B_{M}(p)\) and \(y\in Tx\). Then \(B_{M}(p)\cap F(T)\neq\phi\).
Proof
Remark 24
All results in this paper hold as well in the frame of tvscone metric spaces (see [16]).
Declarations
Acknowledgements
The authors are grateful to the referees for their valuable comments in modifying the first version of this paper.
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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