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## Academic research paper on topic "Multivalued fixed point theorems in tvs-cone metric spaces"

﻿0 Fixed Point Theory and Applications

a SpringerOpen Journal

RESEARCH

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Multivalued fixed point theorems in tvs-cone metric spaces

Akbar Azam* and Nayyar Mehmood

Correspondence: akbarazam@yahoo.com Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan

Abstract

In this paper we extend the Kannan, Chatterjea and Zamfirescu theorems for multivalued mappings in a tvs-cone metric space without the assumption of normality on cones and generalize many results in literature. MSC: 47H10; 54H25

Keywords: tvs-cone metric space; non-normal cones; multivalued contraction; fixed points

1 Introduction

The notion of cone metric space was introduced by Huang and Zhang in . They replaced the set of real numbers by an ordered Banach space and defined a cone metric space. They extended Banach fixed point theorems for contractive type mappings. Many authors  studied the properties of cone metric spaces and generalized important fixed point results of complete metric spaces. The concept of cone metric space in the sense of Huang-Zhang was characterized by Al-Rawashdeh et al. in . 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+) = \$ (, Theorem 3.8).

Recently Beg et al.  introduced and studied topological vector space-valued cone metric spaces (tvs-cone metric spaces), which generalized the cone metric spaces .

Let (X, d) be a metric space. A mapping T: X ^ X is called a contraction if there exists X e [0,1) such that

d(Tx, Ty) < Xd(x,y)

for all x,y e X. A mapping T is called Kannan if there exists a e [0,2) such that

d(Tx, Ty) < a [d(x, Tx) + d(y, Ty)].

The main difference between contraction and Kannan mappings is that contractions are always continuous, whereas Kannan mappings are not necessarily continuous. Another type of contractive condition, due to Chatterjea , is based on an assumption analogous to Kannan mappings as follows: there exists a e [0, 2) such that

ft Springer

d(Tx, Ty) < a[d(x, Ty) + d(y, Tx)].

© 2013 Azam and Mehmood; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It is well known that the Banach contractions, Kannan mappings and Chatterjea mappings are independent in general. Zamfirescu  proved a remarkable fixed point theorem by combining the results of Banach, Kannan and Chatterjea. Afterwards, some authors investigated these results in many directions [29-34].

In the papers [35-39], the authors studied fixed point theorems for multivalued mappings in cone metric spaces. Seong and Jong  invented the generalized Hausdorff distance in a cone metric space and proved multivalued results in cone metric spaces. Shatanawi et al.  generalized it in the case of tvs-cone metric spaces. However, all these results presented in the literature [35-39] for the case of multivalued mappings in cone metric spaces are restricted to Banach contraction. In this paper, we have achieved the results for Kannan and Chatterjea contraction for multivalued mappings in tvs-cone metric spaces. We also extend the Zamfirescu theorem to multivalued mappings in tvs-cone metric spaces.

2 Preliminaries

Let E be a topological vector space with its zero vector Q. A nonempty subset P of E is called a convex cone if P + P c P and XP c P for X > 0. A convex cone P is said to be pointed (or proper) if P n (-P) = {Q}; and P is normal (or saturated) if E has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone P c E, we define a partial ordering ^ with respect to P by x ^ y if and only if y - x e P; x < y stands for x ^ y and x = y, while x ^ y stands for y - x e int P, where int P denotes the interior of P. The cone P is said to be solid if it has a nonempty interior. Now let us recall the following definitions and remarks.

Definition 2.1  Let X be a nonempty set, and let (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:

(CI) Q ^ d(x, y) for all x, y e X and d(x,y) = Q if and only if x = y;

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

(C3) d(x, z) ^ 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 2.1  The concept of cone metric space is more general than that of metric space, because each metric space is a cone metric space, and a cone metric space in the sense of Huang and Zhang is a special case of tvs-cone metric spaces when (X, d) is a tvs-cone metric space with respect to a normal cone P.

Definition 2.2  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 if for every c e E with Q ^ c there is a natural number n0 such that d(xn,x) ^ c for all n > n0. We denote this by cone-limn^TOxn = x;

(ii) {xn} is a tvs-cone Cauchy sequence if for every c e E with Q ^ 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.

Remark 2.2  The results concerning fixed points and other results, in the case of cone spaces with non-normal solid cones, cannot be provided by reducing to metric spaces, because in this case neither of the conditions of Lemmas 1-4 in  hold. Further, the vector cone metric is not continuous in the general case, i.e., from xn ^ x, yn ^ y it need not follow that d(xn,yn) ^ d(x,y).

Let (X, d) be a tvs-cone metric space. The following properties will be used very often (for more details, see [16, 39]). (PT1) If u 4 v and v « w, then u « w. (PT2) If u « v and v 4 w, then u « w. (PT3) If u « v and v « w, then u « w. (PT4) If Q 4 u « c for each c e int P, then u = Q. (PT5) If a 4 b + c for each c e int P, then a 4 b.

(PT6) If E isa tvs-cone metric space with a cone P, and if a 4 Xa, where a e P and 0 < X < 1, then a = Q.

(PT7) If c e int P, an e E and an ^ Q in locally convex Hausdorff tvs E, then there exists an n0 such that, for all n > n0 ,we have an « c.

3 Main result

In the sequel, E denotes a locally convex Hausdorff topological vector space with its zero vector Q, 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.

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. Let T: X ^ A be a multivalued map. For x e X, A e A, define

Wx(A) = {d(x, a): a e A}.

Thus, for x, y e X,

Wx(Ty) = {d(x, u): u e Ty}.

Definition 3.1  Let (X, d) be a cone metric space with a solid cone P. A set-valued mapping F : X ^ 2E is called bounded from below if for all x e X there exists z(x) e E such that

Fx - z(x) c P.

Definition 3.2  Let (X, d) be a cone metric space with a solid cone P. The cone P is complete if for every bounded above nonempty subset A of E, sup A exists in E. Equiva-lently, the cone P is complete if for every bounded below nonempty subset A of E, inf A exists in E.

Definition 3.3 Let (X, d) be a tvs-cone metric space with a solid cone P. The multivalued mapping T: X ^ A is said to have lower bound property (l.b. property) on X if, for any x e X, the multivalued mapping Fx: X ^ 2E, defined by

Fx(y) = Wx (Ty),

is bounded from below. That is, for x,y e X, there exists an element lx(Ty) e E such that

An lx(Ty) is called lower bound of T associated with (x,y). By Lxy(T) we denote the set of all lower bounds T associated with (x,y). Moreover, UxyeXLxy(T) is denoted by LX(T).

Definition 3.4 Let (X, d) be a tvs-cone metric space with a solid cone P. The multivalued mapping T: X ^ A is said to have greatest lower bound property (g.l.b. property) on X if the greatest lower bound of Wx(Ty) exists in E for all x,y e X. We denote by d(x, Ty) the greatest lower bound of Wx(Ty). That is,

d(x, Ty) = inf{d(x, u): u e Ty}.

According to , let us denote

s(p) = {q e E: p ^ q} for q e E

Let us recall the following lemma, which will be used to prove our main Theorem 3.1.

Lemma 3.1  Let (X, d) be a tvs-cone metric space with a solid cone P in ordered locally convex space E. Then we have:

(i) Let p, q e E. If p ^ q, then s(q) c s(p).

(ii) Let x e X and A e A. If 0 e s(x, A), then x e A.

(iii) Let q e P and let A, B e A and a e A. If q e s(A, B), then q e s(a, B).

(iv) 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 3.1  Let (X, d) be a tvs-cone metric space. If E = R and P = [0, +c), 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 result which is a Kannan-type multivalued theorem in tvs-cone metric spaces.

Wx(Ty) - lx(Ty) c P.

s(a,B) = ^J s(d(a, b)) = ^J {x e E: d(a, b) ^ x} for a e X and B e A.

beB beB

For A, B e A, we denote

Theorem 3.1 Let (X, d) be a complete tvs-cone metric space with a cone P, let A = \$ be a collection of nonempty closed subsets ofX, and let T : X ^ A be a multivalued mapping

having l.b. property. If for x, y e X there exist lx(Tx), ty(Ty) e LX (T) and k e [0,2) such that

k[ix(Tx) + ly(Ty)] e s(Tx, Ty), then T has a fixed point in X.

Proof Let x0 be an arbitrary point in X and xi e Tx0. By assumptions,

k[40(Tx0) + £X1(Tx1)] e s(Tx0, Tx{). If k = 0, then

Q e s(Tx0, Tx1). Thus, by Lemma 3.1(iii),

Q e s(x1, Tx1) = ^J ^d(x1, x)).

Thus, there exists some x2 e Tx1 such that

Q e s(d(x^x2)). Thus

d(x1, x2) = Q.

Hence x1 e Tx1. Now assume that k = 0. Then, by Lemma 3.1(iii), we obtain

k[4„(Tx0) + lx1(Tx1)] e s(x1, Tx1). Thus, there exists some x2 e Tx1 such that

k[tx0(Tx0) + ix1(Tx1)] e ^d(x1,x2^. Thus

d(x1,x2) 4 k[tx0(Tx0) + 4*1 (Tx1)].

Wx0 (Tx0) - ix0 (Tx0) c P, Wx1 (Tx1) - tx,(Txi) c P. It yields, tx0(Tx0) 4 d(x0,x1), tx1(Tx1) 4 d(x1,x2), thus d(x1,x2) 4 k[d(x0,x1) + d(x1,x2)\.

Again, by Lemma 3.1(iii), we obtain

k\_lx1(Txi) + 42(Tx2)] e s(x2, Tx2). Thus, we can choose x3 e Tx2 such that k[lx1(Txi) + £X2(Tx2)] e s(d(x2,x3)). Then

d(x2,x3) " k[¿x1(Txi) + £x2(Tx2^. Again, using the fact

Wx,(Txi)-lXi(Txi) c P, i = 1,2, we have

d(x2,x3) " k[d(x1,x2) + d(x2,x3)]. By mathematical induction, we construct a sequence {xn} in X such that

d(xn,xn+1) " k[d(xn-1,xn) + d(xn,xn+0], xn+1 e Txn for n = 0,1,2,3,____

It follows that

d(xn, xn+1) " Z T d(xn-1, xn) 1 - k

n(dxn_1,xn)

< n2d(xn-2,xn-3) -<----< nnd(x0,x1),

where n = 1-k. Now, for m > n, this gives

d(xn, xm) "" d(x0, x1). 1 - n

Since nn ^ 0 as n ^c, this gives us 1-nd(x0,x1) ^ 0 in the locally convex space E as n ^c. Now, according to (PT7) and (PT1), 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 > n1, so {xn} is a tvs-cone Cauchy sequence. As (X,d) is tvs-cone complete, {xn} is tvs-cone convergent in X and cone-limn^c xn = v. Hence, for every c e E with 0 ^ c, there is a natural number k1 such that

d(xn, v) ^ c, --d(xn,xn+1) ^ c for all n > k1. (3.1)

1-k n 1-k We now show that v e Tv. Consider

k[iXn (Txn) + 4(Tv)] e s(Txn, Tv).

As xn+1 e Txn, therefore, it follows that

k[txn(Txn) + tv(Txv)] e s(xn+1, Tv).

So there exists un e Tv such that

k[txn(Txn) + tv(Txv)] e ^d(xn+1, un)),

which implies that

d(xn+1, un) 4 k[txn (Txn) + tv(Txv)]

4 k[d(xn,xn+1) + d(v, un)].

Therefore, by (3.1),

d(v, un) 4 d(xn+1, v) + d(xn+1, un)

4 d(xn+1, v) + k[d(xn,xn+1) + d(v, un)]

4 -—-d(xn+1, v) + -—Td(xn,xn+1) 1 - k 1 - k

2 d(xn+1, v) 2k d(xn, xn+1)

4 I- 2 + 2

^ c for all n > k1 = k1(c). Hence, cone-limn^TO un = v. Since Tv is closed, so v e Tv. □

In the following we provide a Chatterjea-type multivalued theorem in a tvs-cone metric space.

Theorem 3.2 Let (X, d) be a complete tvs-cone metric space with a cone P, let A = \$ be a collection of nonempty closed subsets ofX, and let T: X ^ A be a multivalued mapping. If for x, y e X, there exist tx(Ty), ty(Tx) e LX (T) and k e [0,2) such that

k[tx(Ty) + ty(Tx)] e s(Tx, Ty),

then T has a fixed point in X.

Proof If k = 0, there is nothing to prove. Let k =0 and x0 be an arbitrary point in X, choose x1 e Tx0. Consider by assumption

k[tx0(Tx1) + tx1(Tx0)] e s(Tx0, Tx{).

This implies that

k[txQ(Tx1) + tx1 (Tx0^ e s(x, Tx1) for all x e Tx0.

As x1 e Tx0, so we have

k[tx„(Tx1) + tx1(Tx0)] e s(x1, Tx1). Then we can find x2 e Tx1 such that

k[tx„(Tx1) + tx1(Tx0)] e s(d(x1,x2)). Thus

d(x1,x2) 4 k[tx0(Tx1)+tx1(Tx0)]. Since

Wxa (Tx 1) - tx0 (Tx 1) c P, Wx1 (Tx0)- tx1 (Tx0) C P,

therefore

d(x1,x2) 4 k[d(x0,x2) + d(x1,x1)] 4 k[d(x0,x1) + d(x1,x2)]

4Tkrd(x0, x1). 1 - k

By the same argument we can choose x3 e Tx2 such that

k[tx1(Tx2) + tx2(Tx1)] e s(d(x2,x3)). Then

d(x2,x3) 4 k[tx1 (Tx2) + tx2(Tx 1)]. Again, using the fact

Wx1 (Tx2) - tx1 (Tx2) C P, Wx2 (Tx 1) - tx2 (Tx 1) c P, we have

d(x2,x3) 4 k[d(x1,x3) + d(x2,x2)] 4 k[d(x1,x2) + d(x2,x3)]

4 7krd(x1,x2). 1 - k

By mathematical induction we construct a sequence {xn} in X such that

d(xn,xn+1) 4 k , d(xn-1,xn) for n = 0,1,2,3,.... 1 - k

If n = 1-k, then n <1, and we have

d(xn,xn+1) " nd(xn-1,xn) < n2d(xn-2,xn-3) -< ••• -< nnd(x0,x1). Now, for m > n,we have

d(xn,xm) "" d(x0,x1). 1 - n

Since nn ^ 0 as n ^c, this gives us d(x0,x1) ^ 0 in the locally convex space E as n ^c. Now, according to (PT7) and (PT1), 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 > n1. So {xn} is a tvs-cone Cauchy sequence. Since (X, d) is tvs-cone complete, therefore {xn} is tvs-cone convergent in X and cone-limn^c xn = v. That is, for every c e E with 0 ^ c, there exists a number k2 such that

2(1 + ,k) d(xn+1, v) ^ c, -2k-d(xn, v) ^ c for all n > k2. (3.2)

1 - k 1 - k

We now show that v e Tv. For this consider

k[iXn (Tv) + lv(Txn)] e s(Txn, Tv). It follows that

k\lxn (Tv) +tv(Txn)] e s(xn+1, Tv), since xn+1 e Txn. So there exists un e Tv such that

k[txn(Tv) +tv(Txn)] e ^d(xn+1,Un)). Thus,

d(xn+1, Un) " k[txn (Tv) + tv(Txn)].

Using the fact

^Wxn (Tv) - txn (Tv) c P, Wv(Txn) - tv(Txn) c P,

we obtain

d(xn+1, un) " k[d(xn, un) + d(v,xn+^].

Now using (3.2) we have

d(v, un) " d(xn+1, v) + d(xn+1, un)

" d(xn+1, v) + k[d(xn, un) + d(v,xn+1)]

4 d(xn+1, v) + k[d(xn, un) + d(v,xn+1)] 4 (1 + k)d(xn+1, v) + kd(xn, un) 4 (1 + k)d(xn+1, v) + k[d(xn, v) + d(v, un)]

2(1 + k) d(xn+1, v) 2k d(xn, v) 4 1-k 2 + 1-k 2

« 2 + 2 ^ c for all n > k2.

Therefore cone-limn^TO un = v. Since Tv is closed, so v e Tv. □

In the following we establish a Zamfirescu-type result in a tvs-cone metric space.

Theorem 3.3 Let (X, d) be a complete tvs-cone metric space with a cone P, let A = \$ be a collection of nonempty closed subsets ofX, and let T : X ^ A be a multivalued mapping having g.l.b. property on (X, d). If for x, y e X any one of the following is satisfied:

(i) k(s(d(x,y))) e s(Tx, Ty) for 0 < k <1;

(ii) k[d(x, Tx) + d(y, Ty)] e s(Tx, Ty) for 0 < k < ^

(iii) k[d(x, Ty) + d(y, Tx)] e s(Tx, Ty) for 0 < k < ^ then T has a fixed point in X.

Proof (i): A special case of  when y(d(x,y)) = k <1.

(ii): As d(x, Tx), d(y, Ty) e LX(T).

Put lx(Tx) = d(x, Tx), ly(Ty) = d(y, Ty), then for all x,y e X k[lx(Tx) + ly(Ty)] e s(Tx, Ty).

Now, by Theorem 3.1, T has a fixed point in X.

(iii): Put lx(Ty) = d(x, Ty), ly(Tx) = d(y, Tx), then for all x,y e X

k[lx(Ty) + ly(Tx)] e s(Tx, Ty). Now, by Theorem 3.2, T has a fixed point in X. □

Example 3.1 Let X = [0,1] andletE be the set of all real-valued functions on X which also have continuous derivatives on X. Then E is a vector space over R under the following operations:

(x + y)(t) = x(t) + y(t), (ax)(t) = ax(t)

for all x,y e E, a e R. Let t be the strongest vector (locally convex) topology on E. Then (X, t) is a topological vector space which is not normable and is not even metrizable. Define d: X x X ^ E as follows:

(d(x,y))(t) = |x -y|et, P = {x e E: Q 4 x}, where 0(t) = 0forall t e X.

Then (X, d) is a tvs-valued cone metric space. Let T: X ^ A be such that Tx =

Wx(Ty) = j d(x, u): u e

Denote d(x, Ty) by the greatest lower bound of Wx(Ty). Then 0 ifx < 30,

d(x, Ty)(t) = - y

(x - з°)et if x > —

= (29L\

d(x, Tx)(t)= ( et.

Moreover, if 0xy e E such that

0xy(t) = ^ ef,

s(Tx, Ty) = {f e E: 0xy " f}, d(x, Tx) + d(y, Ty) _ 1 ( 29x + 29y

, + —- e

3 \ 30 30

,x - — + y - — \ef 3\ 30 7 30/

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

> i(^x + V

3 30 30

> 30(x+y)et

> lx - y\pt = 0 (t)

> 30 e = xy(t).

It follows that

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

for all x, y e X and, similarly,

d(x, y)

e s(Tx, Ty)

e s(Tx, Ty).

Hence T satisfies all the conditions of Theorem 3.3 to obtain a fixed point of T.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Both authors contributed equally and significantly in writing this paper and approved the finalmanuscript.

Acknowledgements

The authors would like to thank the editor and the referees for their valuable comments and suggestions which

improved greatly the quality of this paper.

Reeeived: 5 April 2013 Aeeepted: 28 June 2013 Published: 12 July 2013

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doi:10.1186/1687-1812-2013-184

Cite this article as: Azam and Mehmood: Multivalued fixed point theorems in tvs-cone metric spaces. Fixed Point Theory and Applications 2013 2013:184.

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