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MizoguchiTakahashitype theorems in tvscone metric spaces
 Wasfi Shatanawi^{1},
 Vesna Ćojbašić Rajić^{2},
 Stojan Radenović^{3} and
 Ahmed AlRawashdeh^{4}Email author
https://doi.org/10.1186/168718122012106
© Shatanawi et al; licensee Springer. 2012
 Received: 24 February 2012
 Accepted: 25 June 2012
 Published: 25 June 2012
Abstract
In this paper, the concepts of a setvalued contraction of MizoguchiTakahashi type in the context of topological vector space (tvs)cone metric spaces are introduced and a fixed point theorem in the context of tvscone metric spaces with respect to a solid cone is proved. We obtained results which extend and generalize the main results of S. H. Cho with J. S. Bae, Mizoguchi with Takahashi and S. B. Nadler Jr. Two examples are given to illustrate the usability of our results.
2010 MSC: 47H10, 54H25.
Keywords
 tvscone metric space
 fixed point
 Nadler's fixed point theorem
 MizoguchiTakahashi fixed point theorem
 locally convex space
Introduction and preliminaries
Huang and Zhang introduced in [1] the concept of cone metric spaces as a generalization of metric spaces. They have replaced the real numbers (as the codomain of a metric) by an ordered Banach space. They described the convergence in cone metric spaces, introduced their completeness and proved some fixed point theorems for contractive mappings on cone metric spaces. The concept of cone metric space in the sense of HuangZhang is characterized by AlRawashdeh, Shatanawi and Khandaqji in [2]. Indeed (X, d) is a cone metric space if and only if (X, d^{ E } ) is an Emetric space, where E is a normed ordered space, with Int(E^{+}) ≠ ∅ ([2], Theorem 3.8). Recently in [3–28] many authors proved fixed point theorems in cone metric spaces.
Du in [13] introduced the concept of topological vector space (tvs)cone metric and tvscone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [1]. In [7, 9, 13, 14] the authors tried to generalize this approach using cones in tvs 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 normvalued cone metric spaces to tvsvalued cone metric spaces can be obtained only in the case of nonnormal cones (for more details see [14]).
We recall some definitions and results from [14, 15], which will be needed in the sequel.
Let E be a tvs with its zero vector θ. A nonempty subset P of E is called a convex cone if P + P ⊆ P and λP ⊆ P for λ ≥ 0. A convex cone P is said to be pointed (or proper) if P ∩ (−P) = {θ}; and P is a normal (or saturated) if E has a base of neighborhoods of zero consists of orderconvex subsets. For a given cone P ⊆ E, we define a partial ordering ≼ with respect to P by x ≼ y if and only if y −x ∈ P; x ≺ y will stand for x ≼ y and x ≠ y, while x ≪ y stand for y − x ∈ 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 tvs with its zero vector θ, P is a proper, closed and convex pointed cone in E with intP ≠ ∅ and ≼ denotes the induced partial ordering with respect to P .
Definition 1.1. [7, 13, 14] Let X be a nonempty set and (E, P ) be an ordered tvs. A vectorvalued function d : X × X → E is said to be a tvscone metric, if the following conditions hold:
(C_{1}) θ ≼ d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y;
(C_{2}) d(x, y) = d(y, x) for all x, y ∈ X;
(C_{3}) d(x, z) ≼ d(x, y) + d(y, z) for all x, y, z ∈ X.
The pair (X, d) is then called a tvscone metric space.
Remark 1.2. The concept of a cone metric space [1] (E is a real Banach space and d : X ×X → E satisfies (C_{1}), (C_{2}) and (C_{3})) is more general than that of a metric space, because each metric space is a cone metric space, where E = and P = [0, +∞) (see [1, Example 1]). Clearly, a cone metric space in the sense of Huang and Zhang is a special case of tvscone metric spaces when (X, d) is tvscone metric space with respect to a normal cone P.
 (i)
{x_{ n } } tvscone converges to x whenever for every c ∈ E with θ ≪ c there is a natural number n _{0} such that d(x_{ n } , x) ≪ c, for all n ≥ n _{0}. We denote this by cone− lim_{n→∞}x_{n}= x;
 (ii)
{x_{ n } } is a tvscone Cauchy sequence whenever for every c ∈ E with θ ≪ c there is a natural number n _{0} such that d(x_{ n } , x_{ m } ) ≪ c, for all n, m ≥ n _{0};
 (iii)
(X, d) is tvscone complete if every tvscone Cauchy sequence in X is tvscone convergent.
Let (X, d) be a tvscone metric space. The following properties are often used, particularly in the case when the underlying cone is nonnormal. The only assumption is that the cone P has a nonempty interior (i.e. P is a solid). For more details about these properties see [14] and [15].
(p_{1}) If u ≼ v and v ≪ w, then u ≪ w.
(p_{2}) If u ≪ v and v ≼ w, then u ≪ w.
(p_{3}) If u ≪ v and v ≪ w, then u ≪ w.
(p_{4}) If θ ≼ u ≪ c for each c ∈ intP, then u = θ.
(p_{5}) If a ≼ b + c, for each c ∈ intP, then a ≼ b.
(p_{6}) If E is a tvs cone metric space with a cone P, and if a ≼ λa, where a ∈ P and 0 ≤ λ < 1, then a = θ.
(p_{7}) If c ∈ intP, a_{ n } ∈ E and a_{ n } → θ in locally convex Hausdorff tvs E, then there exists an n_{0} such that, for all n > n_{0}, we have a_{ n } ≪ c.
In [11], the concept of a setvalued contraction of MizoguchiTakahashi type was introduced and a fixed point theorem in setting of a normal cone was proved. In this article, we prove the same theorem in the setting of a tvscone metric space. We generalize results of [11], by omitting the assumption of normality in the results, that is the normality of P is not a necessary. We use only the definition of convergence in terms of the relation "≪". The only assumption is that the interior of the cone P in locally convex Hausdorff tvs E is nonempty, so we neither use continuity of the vector metric d, nor Sandwich Theorem. In such a way, we generalize results of [11, 29, 30].
Main results
The following lemma will be used to prove Theorem 2.3.
 (1)
For all p, q ∈ E. If p ≼ q, then s(q) ⊂ s(p).
 (2)
For all x ∈ X and $A\in \mathcal{A}$. If θ ∈ s(x, A), then x ∈ A.
 (3)
For all q ∈ P and $A,B\in \mathcal{A}$ and a∈ A. If q ∈ s(A, B), then q ∈ s(a, B).
 (4)
For all q ∈ P and $A,B\in \mathcal{A}$. Then q ∈ s(A, B) if and only if there exist a ∈ A and b ∈ B such that d(a, b) ≼ q.
Remark 2.2. Let (X, d) be a tvscone metric space. If $E=\mathbb{R}$ and P = [0, +∞), then (X, d) is a metric space. Moreover, for A, B ∈ CB(X), H(A, B) = inf s(A, B) is the Hausdorff distance induced by d. Also, s({x}, {y}) = s(d(x, y)), for all x, y ∈ X.
Now let us prove the following main results of this article.
for all x, y ∈ X (x ≠ y), then T has a fixed point in X.
Now by Lemma 2.1(3), we have φ(d(x_{0}, x_{1}))d(x_{0}, x_{1}) ∈ s(x_{1}, Tx_{1}). By definition, we can take x_{2} ∈ Tx_{1} such that φ(d(x_{0}, x_{1}))d(x_{0}, x_{1}) ∈ s(d(x_{1}, x_{2})). So, d(x_{1}, x_{2}) ≼ φ(d(x_{0}, x_{1}))d(x_{0}, x_{1}).
If x_{ n } = x_{ n }_{+1} for some $n\in \mathbb{N}$, then T has a fixed point.
Since λ^{ n } → 0 as n → ∞, we obtain that ${\lambda}^{n}\left[\frac{1}{{{\lambda}^{n}}^{{0}_{}}\left(1\lambda \right)}\right]d\left({x}_{{n}_{0}},{x}_{{n}_{0}+1}\right)\to \theta $ in the locally convex space E, as n → ∞. Now, according to (p_{7}) and (p_{1}), we can conclude that for every c ∈ E with θ ≪ c there is a natural number n_{1} such that d (x_{ n }, x_{ m }) ≪ c for all m, n ≥ max {n_{0}, n_{1}} , so {x_{ n }} is a tvscone Cauchy sequence. Since (X, d) is tvscone complete, then {x_{ n }} is tvscone convergent in X and conelim_{ n→∞ } x_{ n } = x, that is, for every c ∈ E with θ ≪ c, there is a natural number k such that d(x_{ n }, x) ≪ c for all n ≥ k.
Hence, according to Definition 1.3(i), we have that conelim_{ n→∞ } y_{ n } = x. As Tx is closed, then x ∈ Tx, hence x is a fixed point of T and this ends the proof. □
The next example shows that Theorem 2.3 is a proper generalization of the main result from [11]. Indeed, as in Example 2.4, the cone P is nonnormal, so Theorem 2.1 of [11] is not applicable.
Now, taking $a=\frac{1}{3}x$, and $b=\frac{1}{3}y$, we obtain that the hypothesis (2) is satisfied. Hence using Theorem 2.3, it follows that T has a fixed point.
Example 2.5. Let $E={C}_{\mathbb{R}}^{1}\left[0,1\right]$ with a norm u = u_{∞} + u'_{∞}, u ∈ E and let P = {u ∈ E : u(t) ≥ 0, t ∈ 0[1]}. It is well known that this cone is solid but it is not normal. Now consider the space $E={C}_{\mathbb{R}}^{1}\left[0,1\right]$ endowed with the strongest locally convex topology t*. Then P is also t* solid (it has nonempty t*interior), but not t* normal. (For more details, see [31], Example 2.2).
Then (X, d) is a complete tvscone metric space over the nonnormal cone P . Now, consider the mapping T : X → X which is given by Ta = {a, b} , Tb = {a, c} and Tc = {a, b, c}. Let $\phi \left(c\right)=\frac{1}{2}$, for all c ∈ P. It is clear that the hypothesis (1) is satisfied. So let us prove that (2) is also satisfied, that is $\frac{1}{2}d\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)\in s\left(Tx,\phantom{\rule{2.77695pt}{0ex}}Ty\right)$, for allx, y ∈ X (x ≠ y). Now, we have the following:
${1}^{0}\frac{1}{2}d\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right)\in s\left(Ta,Tb\right)=s(\left\{a,\phantom{\rule{2.77695pt}{0ex}}b\right\}$, $\phantom{\rule{2.77695pt}{0ex}}\left\{a,\phantom{\rule{2.77695pt}{0ex}}c\right\})\iff \exists {a}_{1}\in Ta,\exists {b}_{1}\in Tb$ such that $d\left({a}_{1},\phantom{\rule{2.77695pt}{0ex}}{b}_{1}\right)\preccurlyeq \frac{1}{2}d\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right)$ Take a_{1} = b_{1} = a;
${2}^{0}\frac{1}{2}d\left(a,\phantom{\rule{2.77695pt}{0ex}}c\right)\in s\left(Ta,Tc\right)=s(\left\{a,\phantom{\rule{2.77695pt}{0ex}}b\right\}$, $\left\{a,\phantom{\rule{2.77695pt}{0ex}}b,\phantom{\rule{2.77695pt}{0ex}}c\right\})\iff \exists {a}_{2}\in Ta,\exists {b}_{2}\in Tc$ such that $d\left({a}_{2},\phantom{\rule{2.77695pt}{0ex}}{b}_{2}\right)\preccurlyeq \frac{1}{2}d\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right)$. Take a_{2} = b_{2} = a;
${3}^{0}\frac{1}{2}d\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right)\in s\left(Tb,\phantom{\rule{2.77695pt}{0ex}}Tc\right)=s(\left\{a,\phantom{\rule{2.77695pt}{0ex}}c\right\}$, $\left\{a,\phantom{\rule{2.77695pt}{0ex}}b,\phantom{\rule{2.77695pt}{0ex}}c\right\})\iff \exists {a}_{3}\in Tb,\exists {b}_{3}\in Tc$ such that $d\left({a}_{3},{b}_{3}\right)\preccurlyeq \frac{1}{2}d\left(a,b\right)$. Take a_{3} = b_{3} = a.
Therefore, all conditions of Theorem 2.3 are satisfied and hence T has a fixed point. Precisely, x = a and x = c are the fixed points of T.
Finally, we finish our paper by introducing the following consequence corollaries of our main theorem. let (X, d) be a given metric space, and let us define the following:

CB (X) = {A : A is a nonempty closed and bounded subset of X},

D (a, B) = inf {d (a, b): b ∈ B X}, for a ∈ X,

H (A, B) = max {sup {D (a, B): a ∈ A} , sup {D (b, A): b ∈ B}}.
for all x, y ∈ X. A point x ∈ X is called a fixed point of T , if x ∈ Tx. Then as a consequence of Theorem 2.3 and in particular by taking $E=\mathbb{R}$, P = [0, +∞), $\mathcal{A}=CB\left(X\right)$,φ(c) = λ, for all c ∈ P , we obtain the following corollary.
Corollary 2.6. (Nadler [30]) Let (X, d) be a complete metric space and let T : X → CB(X) be a multivalued contraction mapping. Then T has a fixed point.
Also, according to Remark 2.2, we obtain the following corollary.
for all x, y ∈ X(x ≠ y), then T has a fixed point in X.
Declarations
Acknowledgments
The authors would like to thank the referee and the editor for their valuable comments and suggestions. Vesna Ćojbašić Rajić and Stojan Radenović are thankful to the Ministry of Science and Technological Development of Serbia
Authors’ Affiliations
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