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Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces

Wasfi Shatanawi1, Vesna Cojbasic Rajic2, Stojan Radenovic3 and Ahmed Al-Rawashdeh4*

* Correspondence:

aalrawashdeh@uaeu.ac.ae

4Department of Mathematical

Sciences, UAEU, 17551 Al-Ain,

United Arab Emirates

Full list of author information is

available at the end of the article

Abstract

In this paper, the concepts of a set-valued contraction of Mizoguchi-Takahashi type in the context of topological vector space (tvs)-cone metric spaces are introduced and a fixed point theorem in the context of tvs-cone 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: tvs-cone metric space, fixed point, Nadler's fixed point theorem, Mizoguchi-Takahashi 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 co-domain 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 Huang-Zhang is characterized by Al-Rawashdeh, Shatanawi and Khandaqji in [2]. Indeed (X, d) is a cone metric space if and only if (X, dE) is an E-metric space, where E is a normed ordered space, with Int(E+) * 0 ([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 tvs-cone 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 norm-valued cone metric spaces to tvs-valued 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 0. A nonempty subset P of E is called a convex cone if P + P £ P and IP £ P for l > 0. A convex cone P is said to be pointed (or proper) if P n (-P) = {0}; and P is a normal (or saturated) if E has a base of

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neighborhoods of zero consists of order-convex subsets. For a given cone P £ £, we define a partial ordering 4 with respect to P by x 4 y if and only if y -x e P; x < y will stand for x 4 y and x * y, while x « y stand for y - x e 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 0, P is a proper, closed and convex pointed cone in E with intP * 0 and 4 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 vector-valued function d : X x X ® E is said to be a tvs-cone metric, if the following conditions hold:

(C1) 0 4 d(x, y) for all x, y e X and d(x, y) = 0 if and only if x = y;

(C2) d(x, y) = d(y, x) for all x, y e X;

(C3) d(x, z) 4 d(x, y) + d(y, z) for all x, y, z e X.

The pair (X, d) is then called a tvs-cone metric space.

Remark 1.2. The concept of a cone metric space [1] (E is a real Banach space and d : X xX ® E satisfies (C1), (C2) and (C3)) is more general than that of a metric space, because each metric space is a cone metric space, where E = R and P = [0, (see [1, Example 1]). Clearly, a cone metric space in the sense of Huang and Zhang is a special case of tvs-cone metric spaces when (X, d) is tvs-cone metric space with respect to a normal cone P.

Definition 1.3. [7,13,14] Let (X, d) be a tvs-cone metric space, x e X and let {xn} be a sequence in X. Then

(i) {xn} tvs-cone converges to x whenever for every c e E with 0 « c there is a natural number n0 such that d(xn, x) « c, for all n > n0. We denote this by cone- limn®^

(ii) {xn} is a tvs-cone Cauchy sequence whenever for every c e E with 0 « c there is a natural number n0 such that d(xn, xm) « c, for all n, m > n0;

(iii) (X, d) is tvs-cone complete if every tvs-cone Cauchy sequence in X is tvs-cone convergent.

Let (X, d) be a tvs-cone 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].

(pi) If u 4 v and v « w, then u « w.

(p2) If u « v and v 4 w, then u « w.

(p3) If u « v and v « w, then u « w.

(p4) If 0 4 u « c for each c e intP, then u = 0.

(p5) If a 4 b + c, for each c e intP, then a 4 b.

(p6) If E is a tvs cone metric space with a cone P, and if a 4 la, where a e P and 0 < l <1, then a = 0.

(p7) If c e intP, an e E and an ® 0 in locally convex Hausdorff tvs E, then there exists an n0 such that, for all n >n0, we have an « c.

In [11], the concept of a set-valued contraction of Mizoguchi-Takahashi 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 tvs-cone 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

Let E be a locally convex Hausdorff tvs with its zero vector 0, P a proper, closed and convex pointed cone in E with intP * 0 and < be a partial ordering with respect to P. Let (X, d) be a tvs-cone metric space with a solid cone P and let A be a collection of nonempty subsets of X. According to [11], we denote

The following lemma will be used to prove Theorem 2.3.

Lemma 2.1. Let (X, d) be a tvs-cone metric space with a solid cone P in ordered locally convex space E, and let A be a collection of nonempty subsets of X. Then we have:

(1) For all p, q e E. If p < q, then s(q) c s(p).

(2) For all x e X and A e A . If 0 e s(x, A), then x e A.

(3) For all q e P and A, B e A and a e A. If q e s(A, B), then q e s(a, B).

(4) For all q e P and A, B e A . Then q e s(A, B) if and only if there exist a e A and b e B such that d(a, b) < q.

Remark 2.2. Let (X, d) be a tvs-cone metric space. If £ = R and P = [0, then (X, d) is a metric space. Moreover, for A, B e 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 e X.

Now let us prove the following main results of this article.

Theorem 2.3. Let (X, d) be a tvs-cone complete metric space with a solid cone P, A = 0 be a collection of nonempty closed subsets of x and T : X ^ A be a multivalued map. If there exists a function ^ : P ® [0, 1) such that

lim y>(cn) < i (1)

for any decreasing sequence {cn} in P, and f

s(p) = {4 e £ : p ^ 4}

for p e E, and

s(a, B) = U s(d(a, b))

for a e X and B e A . For A, B e A, we denote

p(d(x, y)) d (x, y) e s(Tx, Ty)

for all x, y e X (x * y), then T has a fixed point in X.

Proof Let x0 e X, xi e Tx0 and assume x0 € Tx0. Then from (2), we have

y(d(x0, xi)) d (x0, xi) e s(Tx0, Txi).

Now by Lemma 2.1(3), we have ^>(d(x0, xi))d(x0, xi) e s(xi, Txi). By definition, we can take x2 e Tx1 such that ^(d(x0, x1))d(x0, x1) e s(d(x1, x2)). So, d(x1, x2) 4 ^(d(x0, x1))d(x0, x1).

Inductively, we can construct a sequence {xn} in X such that for all n e N , we have

d(xn, xn+1 ) 4 ^(d(xn— 1, xn)) d (xn— 1, Xn) , xn+1 e TXn.

If xn = xn+1 for some n e N , then T has a fixed point.

We may assume that xn ^ xn+1, for all n e N . From (3), {d(xn, xn+1)} s a decreasing sequence in P. Hence, from (1), there exists r e 0, 1) such that

Hm y(d(xn, xn+i)) = r.

Thus, for any 1 e (r, 1), there exists n0 e N such that for all n > n0, we have ^(d(xn, xn+1)) <1. Then we get, for n > n0,

d (xn f xn+1 ) 4 p{d{xn—1f xn)) d (xn—1f xn) ^ ^d(xn—1f xn) -< kn 710d(xn0, xn0 + l).

For m >n > no, we have

d(xn, xm) 4 d(xn, xn+1) + d(xn+1, xn+2~) + ' '' + d (xm—1/ xm)

xno, xno +

4 (kn—n0 + kn+1—n° + ••• + km—1—n0) d (xno, xno+1)

_№(1 —

d( xn0 , xn0 + 1).

Since A" —> 0 as n —> oo, we obtain that A" J d(xno,xno+i) 0 in the locally

convex space E, as n ® Now, according to (p7) and (p1), we can conclude that for every c e E with 0 « c there is a natural number n1 such that d (xn, xm) « c for all m, n > max {n0, n1} , so {xn} is a tvs-cone Cauchy sequence. Since (X, d) is tvs-cone complete, then {xn} is tvs-cone convergent in X and cone-limn®^ xn = x, that is, for every c e E with 0 « c, there is a natural number k such that d(xn, x) « c for all n > k.

We now show that x e Tx. Indeed, from (2) we have ^ (d (xn, x)) d (xn, x) e s (Txn, Tx) = s (xn+1, Tx) for n e N .By Lemma 2.1(3), there exists yn e Tx such that

y(d(xn, x)) d (xn, x) e s(d(xn+1, yn)).

Hence, d (xn+1, yn) 4 ^ (d (xn, x)) d (xn, x) 4 d (xn, x) . Moreover, for a given c e intP, we have

d(x, yn) 4 d(x, xn+1 ) + d (xn+l, yn) 4 d (x, xn+1) + d(xn, x)

<ii — + — = c, for n > k = k(c).

2 2 v '

Hence, according to Definition 1.3(i), we have that cone-limn®^ yn = x. As Tx is closed, then x e 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.

Example 2.4. Let X = [0, 1] and E = C[0, 1] be endowed with the strongest locally convex topology t(E, E*), and let P = {x e E :, x(t) > 0, t 0[1]}. Then the cone P is a normal cone with respect to the norm of the space E with a coefficient of normality K = 1. Also, P is a solid cone. Since t(E, E*) is stronger than a norm-topology, then the interior of P is nonempty with respect to t(E, E*). Thus, P is t(E, E*)-solid. This cone is nonnormal with respect to the topology t(E, E*). Indeed, if it were normal, then according to Theorem 2.1 of [14], the space (E, t (E, E )) would be normed, which is impossible as an infinite-dimensional space with the strongest locally convex topology cannot be metrizable (see [31]). Let us define the tvs-cone metric with d : X x X ® E, by d(x, y)(t):= |x - y|et. Let A be a family of nonempty closed subsets of X of the form A = {[0, x\,x e X}. Consider a mapping T : X A defined by T(x) = [0, §]. Let cp(c) = \ for all c e P. Obviously, the hypothesis (1) is satisfied. We now show that (2) is also a satisfied. Moreover, for x, y L X(x ^ y) we have,

(p(d(x, y)) d (x, y) e s(Tx, Ty) -o- —d(x, y) e s(Tx, Ty)

}') e ( H U I n ( f| IM* I

\aeTxbeTy ) \beTya eTx )

o (3a e Tx) (3b e Ty) id(x, y) e s(d(a, b)) d(a, b) —}').

Now, taking a = i.r, and b = 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 = CR[0,1] with a norm ||w|| = ||w||„ + ||w'|U, u e E and let P = {w e E : u(t) > 0, t e 0[1]}. It is well known that this cone is solid but it is not normal. Now consider the space E = CR[0,1] 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).

Let X = {a, b, c} and define a tvs-cone metric d : X x X ® P by

d(a, b)(t) := 2 + 3t, d (b, c)(t) := 5 - 3t, d(a, c)(t) := 3, d(x, y) = d (y, x) and d(x, x) = 0 for x, y e X.

Then (X, d) is a complete tvs-cone 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 <p(c) = for all c e P. It is clear that the hypothesis (1) is satisfied. So let us prove that (2) is also satisfied, that is jd(x, y) e s(Tx, Ty), for all,*, ye X [x * y). Now, we have the following:

1 °\d(a, b) e s(Ta, Tb) = s({a, b}, {a, c}) 3«i e Ta,3b! e Tb such that d(a\, b\) =4 \d(a, b) Take ai = bi = a;

2°\d(a, c) e s(Ta,Tc) = s({a, b}, {a, b, c}) 3a2 e Ta,3b2 e Tc such that d(a2, b2) 4 \d(a, b). Take a2 = b2 = a;

3°\d{a, b) e s{Tb, Tc) = s({a, c], {a, b, c}) -o- 3a3 e Tb,lb3 e Tcsuch that d{a3,b3) 4 \d{a,b). Take a3 = b3 = 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 e B • X}, for a e X,

• H (A, B) = max {sup {D (a, B): a e A} , sup {D (b, A): b e B}}.

It is clear that H is a metric on CB (X), which is called the Hausdorff-Pompeu metric induced by d. A set-valued mapping T: X ® CB(X) is said to be a multi-valued contraction mapping, if there exists a fixed real number l, 0 < l <1 such that,

H(Tx, Ty) < kd(x, y),

for all x, y e X. A point x e X is called a fixed point of T, if x e Tx. Then as a consequence of Theorem 2.3 and in particular by taking E = R , P = [0, A = CB(X) (c) = l, for all c e P , we obtain the following corollary.

Corollary 2.6. (Nadler [30]) Let (X, d) be a compete metric space and let T: X ® CB(X) be a multi-valued contraction mapping. Then T has a fixed point. Also, according to Remark 2.2, we obtain the following corollary. Corollary 2.7. (Mizoguchi-Takahashi [29]) Let (X, d) be a complete metric space and let T : X ® 2X be a multi-valued mapping such that, Tx is a closed bounded subset of X, for all x e X. If there exists a function ^ : [0, ® [0, 1) such that,

lim <p(r) < 1 /or all £ e [0, and if

H(Tx, Ty) < q>(d(x, y))d(x, y), for all x, y e X(x ^ y), then T has a fixed point in X.

Acknowledgments

The authors would like to thank the referee and the editor for their valuable comments and suggestions. Vesna Cojbasic Rajic and Stojan Radenovic are thankfulto the Ministry of Science and TechnologicalDevelopment of Serbia

Author details

department of Mathematics, Hashemite University Zarqa, Jordan 2University of Belgrade, Faculty of Economics, Kamenicka 6, 11000 Beograd, Serbia 3University of Belgrade, Faculty of MechanicalEngineering, Kraljice Marije 16, 11 120 Beograd, Serbia 4Department of MathematicalSciences, UAEU, 17551 Al-Ain, United Arab Emirates

Authors' contributions

Allthe authors contributed equally. Allauthors read and approved the finalmanuscript. Competing interests

The authors declare that they have no competing interests.

Received: 24 February 2012 Accepted: 25 June 2012 Published: 25 June 2012

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doi:10.1186/1687-1812-2012-106

Cite this article as: Shatanawi et al.: Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces. Fixed Point Theory and Applications 2012 2012:106.