# Fixed point theorems for multivalued mappings in G-cone metric spacesAcademic research paper on "Mathematics"

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## Academic research paper on topic "Fixed point theorems for multivalued mappings in G-cone metric spaces"

﻿Azam and Mehmood Journal of Inequalities and Applications 2013, 2013:354 Journal of Inequalities and Applications

http://www.journalofinequalitiesandapplications.eom/content/2013/1/354 a SpringerOpen Journal

RESEARCH Open Access

Fixed point theorems for multivalued mappings in G-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

We extend the idea of Hausdorff distance function in G-cone metric spaces and obtain fixed points of multivalued mappings in G-cone metric spaces. MSC: 47H10; 54H25

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

1 Introduction

The main revolution in the existence theory of many linear and non-linear operators happened after the Banach contraction principle. After this principle many researchers put their efforts into studying the existence and solutions for nonlinear equations (algebraic, differential and integral), a system of linear (nonlinear) equations and convergence of many computational methods [1]. Banach contraction gave us many important theories like variational inequalities, optimization theory and many computational theories [1, 2]. Due to wide spreading importance of Banach contraction, many authors generalized it in several directions [3-9]. Nadler [10] was first to present it in a multivalued case, and then many authors extended Nadler's multivalued contraction. One of the real generalizations of Nadler's theorem was given by Mizoguchi and Takahashi in the following way.

Theorem 1.1 [11] Let (X, d) be a complete metric space, and letT: X ^ 2X be a multivalued map such that Tx is a closed bounded subset of Xfor all x e X. If there exists a function y : (0, x) ^ [0,1) such that lim supr^t+ y (r) < 1for all t e [0, x) and if

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

ft Spri

ringer

Suzuki [12] proved that Mizoguchi and Takahashi's theorem is a real generalization of Nadler's theorem. Recently Huang and Zhang [13] introduced a cone metric space with a normal cone with a constant K, which is generalization of a metric space. After that Reza-pour and Hamlbarani [14] generalized a cone metric space with a non-normal cone. Afterwards many researchers [15-24] have studied fixed point results in cone metric spaces. In [25] Mustafa etal. generalized the metric space and introduced the notion of G-metric space which recovered the flaws of Dhage's generalization [26,27] of a metric space. Many

© 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 originalwork is properly cited.

researchers proved many fixed point results using a G-metric space [28, 29]. Anchalee Kaewcharoen and Attapol Kaewkhao [28] andNedal et al. [30] proved fixedpoint results for multivalued maps in G-metric spaces. In 2009, Beg etal. [31] introduced the notion of G-cone metric space and generalized some results. Chi-Ming Cheng [32] proved Nadlertype results in tvs G-cone metric spaces.

In 2011 Cho and Bae [33] generalized a Mizoguchi Takahashi-type theorem in a cone metric space. In the present paper, we introduce the notion of Hausdorff distance function on G-cone metric spaces and exploit it to study some fixed point results in G-cone metric spaces. Our result generalizes many results in literature.

2 Preliminaries

Let E be a real Banach space. A subset P of E is called a cone if and only if:

(a) P is closed, nonempty and P = [0},

(b) a, b e R, a, b > 0, x,y e P implies ax + by e P, more generally, if a, b, c e R, a, b, c > 0, x, y, z e P =\$■ ax + by + cz e P,

(c) P n (-P) = [0}.

Given a cone P c E, we define a partial ordering 4 with respect to P by x 4 y if and only if y - x e P.

A cone P is called normal if there is a number K >0 such that for all x, y e E

0 4 x 4 y implies \\x\\<K||y||.

The least positive number satisfying the above inequality is called the normal constant of P, while x « y stands for y - x e int P (interior of P), while x -< y means x 4 y and x = y.

Rezapour [14] proved that there are no normal cones with normal constants K <1 and for each k > 1, there are cones with normal constants K > 1.

Remark 2.1 [34] 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 [13] hold. Further, the vector cone metric is not continuous in a general case, i.e., from xn ^ x, yn ^ y it need not follow that d(xn,yn) ^ d(x,y). For the case of non-normal cones, we have the following properties. (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 0 4 u « c for each c e int P, then u = 0. (PT5) If a 4 b + c for each c e int P, then a 4 b.

(PT6) If E is a real Banach space with a cone P, and if a 4 Xa, where a e P and 0 < X <1, then a = 0.

(PT7) If c e intP, an e E and an ^ 0, then there exists an n0 such that, for all n > n0,we have an « c.

In the following we shall always assume that the cone P is solid and non-normal.

Definition 2.1 [31] Let X be a nonempty set. Suppose that a mapping G: X x X x X ^ E satisfies: (Gl) G(x, y, z) = Q if x = y = z, (G2) Q < G(x, x, y), whenever x = y, for all x, y e X, (G3) G(x,x,y) ^ G(x,y,z), whenevery = z,

(G4) G(x,y, z) = G(x, z,y) = G(y,x, z) = ••• (symmetric in all three variables), (G5) G(x, y, z) ^ G(x, a, a) + G(a,y, z) for all x, y, z, a e X.

Then G is called a generalized cone metric on X, and X is called a generalized cone metric space or, more specifically, a G-cone metric space.

The concept of a G-cone metric space is more general than that of G-metric spaces and cone metric spaces (see [31]).

Definition 2.2 [31] A G-cone metric space X is symmetric if G(x,y,y) = G(y,x,x) for all x, y e X.

Example 2.1 [31] Let (X, d) be a cone metric space. Define G : X x X x X ^ E by G(x, y, z) = d(x, y) + d(y, z) + d(z, x). Then (X, G) is a G-cone metric space.

Proposition 2.1 [31] LetX be a G-cone metric space, define dG : X x X ^ Eby

dG(x, y) = G(x, y, y) + G(y, x, x). Then (X, dG) is a cone metric space.

It can be noted that G(x,y,y) ^ §dG(x,y). If X is a symmetric G-cone metric space, then dG(x,y) = 2G(x,y,y) for all x,y e X.

Definition 2.3 [31] Let X be a G-cone metric space and let {xn} be a sequence in X. We say that {xn} is:

(a) a Cauchy sequence if for every c e E with Q ^ c, there is N such that for all n, m, l > N, G(xn,xm,xi) ^ c.

(b) a convergent sequence if for every c in E with Q ^ c, there is N such that for all m, n > N, G(xm, xn, x) ^ c for some fixed x in X. Here x is called the limit of a sequence {xn} and is denoted by limn^roxn = x or xn ^ x as n ^ro.

A G-cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.

Proposition 2.2 [31] LetX be a G-cone metric space, then the following are equivalent.

(i) {xn} converges to x.

(ii) G(xn,xn,x) ^ Q as n ^ro.

(iii) G(xn,x,x) ^ Q as n ^ro.

(iv) G(xm,xn,x) ^ Q as m, n ^ro.

Lemma 2.1 [31] Let {xn} be a sequence in a G-cone metric space X. If {xn} converges to x e X, then G(xm, xn, x) ^ Q asm, n ^ro.

Lemma 2.2 [31] Let {xn} be a sequence in a G-cone metric space X and x e X. If {xn} converges to x e X, then {xn} is a Cauchy sequence.

Lemma 2.3 [31] Let {xn} be a sequence in a G-cone metric space X. If {xn} is a Cauchy sequence in X, then G(xm,xn,xi) ^ Q, as m, n, l ^ x.

3 Main result

Denote by N(X), B(X) and CB(X) the set of nonempty, bounded, sequentially closed bounded subsets of G-cone metric spaces, respectively. Let (X, G) be a G-cone metric space. We define (see [33])

s(p) = {q e E: p 4 q} for q e E,

s(a,B) = U s(dG(a, b)) = U{x e E: dG(a, b) 4 x} for a e X and B e N(X).

beB beB

For A, B e B(X), we define s(A,B)= (J s(dG(a, b)),

aeA,beB

s(a,B, C) = s(a,B) + s(B, C) + s(a, C)= { u + v + w: u e s(a,B), v e s(B, C), w e s(a, C^,

s(A,B,C) = (p|s(a,B,Cm n (p|s(b,A,Cm n (p|s(c,A,B)\

\eA ' ^beB ' \eC '

Lemma 3.1 Let (X, G) be a G-cone metric space, let P be a cone in a Banach space E.

(i) Let p, q e E. If p 4 q, then s(q) c s(p).

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

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

Remark 3.1 Recently, Kaewcharoen and Kaewkhao [28] (see also [30]) introduced the following concepts. Let X be a G-metric space and let CB(X) be the family of all nonempty closed bounded subsets of X. Let HG(-, •, •) be the Hausdorff G-distance on CB(X), i.e.,

Hg(A, B, C) = max J sup G(a, B, C), sup G(b, A, C), sup G(c, A, B)},

'■aeA beB ceC '

HdG(A,B) = max] supdG(a,B), supdG(b, A)

'■aeA beB '

G(x,B, C) = dG(x,B) + dG(B, C) + dG(x, C), dG(x,B) = inf{dG(x,y),y e B,

dG(A,B) = inf{dG(a, b), a e A, b e B}, G(a, b, C) = inf{ G(a, b, c), c e C}.

The above expressions show a relation between HG and HdG. Moreover, note that if (X, G) is a G-cone metric space, E = R, and P = [0, c), then (X, G) is a G-metric space. Also, for A,B, C e CB(X), HG(A,B, C) = inf s(A,B, C).

Remark 3.2 Let (X, G) be a G-cone metric space. Then

(a) s([a}, {b}) = s(dG(a, b)) for a, b e X.

(b) If x e s(a, B, B) then x e 2s(dG (a, b)).

Proof (a) By definition

S([a}, {b}) = (J s(dG(a, b))

ae[a},be[b}

= s(dG(a, b)).

(b) Now let

x e s(a, B, B), then x e s(a, B, B) = s(a, B) + s(B, B) + s(a, B) ^ x e 2s(a, B)+s(B, B) ^ x e 2s(dG(a, b)) + s(0).

Let x = y + z for y e 2s(dG(a, b)) and z e s(0). Then by definition 0 4 z and 2dG(a, b) 4 y, which implies 0 + 2dG(a, b) 4 y + z = x. Hence 2dG(a, b) 4 x,so x e 2s(dG(a, b)). □

In the following theorem, we use the generalized Hausdorff distance on G-cone metric spaces to find fixed points of a multivalued mapping.

Remark 3.3 If (X, G) is a G-metric space, then (X, dG) is a metric space, where

dG(x, y) = G(x, y, y) + G(y, x, x).

It is noticed in [35] that in the symmetric case ((X, G) is symmetric), many fixed point theorems on G-metric spaces are particular cases of existing fixed point theorems in metric spaces. In these deductions, the fact G(Tx, Ty, Ty) + G(Ty, Tx, Tx) = 2G(Tx, Ty, Ty) = dG(Tx, Ty) is exploited for a single-valued mapping T on X. Whereas in the case of multivalued mapping T: X ^ 2X on a G-cone metric space,

s(Tx, Ty, Ty)= ( p| s(a, Ty, Ty)) n i p| s(b, Tx, Ty) J n i p| s(b, Tx, Ty)

\eTy ' \eTy

= i p| s(a, Ty, Ty)\ n i p| s(b, Tx, Ty)

aeTx ' beTy

= I p| 2s(a, Ty) I n I p| s(b, Tx) + s(Tx, Ty) + s(b, Ty) I

^aeTx ' \eTy '

= s(Ty, Tx, Tx).

Therefore,

( p| s(a, Ty) ] n i p| s(b, Tx)) = s(Tx, Ty, Ty) + s(Ty, Tx, Tx)

\eTx ' ^beTy '

and even in a symmetric case, we cannot follow a similar technique to deduce G-cone metric multivalued fixed point results from similar results of metric spaces.

In a non-symmetric case, the authors [35] deduce some G-metric fixed point theorems from similar results of metric spaces by using the fact that if (X, G) is a G-metric on X, then

S(x,y) =max{G(x,y,y), G(y,x,x)}

is a metric on X. Whereas, in the case of a G-cone metric space, the expression max{G(x,y,y), G(y,x,x)} is meaningless as G(x,y,y), G(y,x,x) are vectors, not essentially comparable, and we cannot find maximum of these elements. That is, (X, S) may not be a cone metric space if (X, G) is a G-cone metric space. In the explanation of this fact, we refer to Example 3.1 below, from [31]. Hence multivalued fixed point results on G-cone metric spaces cannot be deduced from similar fixed point theorems on metric spaces.

Example 3.1 [31] Let X = {a, b}, E = R3,

P = {(x,y,z) e E: x,y,z > 0}.

Define G : X x X x X ^ E by

G(a, a, a) = (0,0,0) = G(b, b, b),

G(a, b, b) = (0,1,1) = G(b, a, b) = G(b, b, a),

G(b, a, a) = (0,1,0) = G(a, b, a) = G(a, a, b).

Note that S(a, b) = max{G(a, a, b), G(a, b, b)} = max{(1,0,0), (0,1,1)} has no meaning as discussed above.

Theorem 3.1 Let (X, G) be a complete cone metric space, and let T : X —^ CB(X) be a multivalued mapping. If there exists a function y : P ^ [0,1) such that

lim sup y(rn) < 1 (a)

for any decreasing sequence {rn} in P, and if

y (G(x, y, z)) G(x, y, z) e s( Tx, Ty, Tz) (1)

for all x, y, z e X, then T has a fixed point in X.

Proof Let xo be an arbitrary point in X and xi e Txo. From (1), we have G(xo,x1,xO)G(xo,x1,x1) e s(Txo, Tx1, Tx1).

Thus, by Lemma 3.1(iii), we get

G(xo,x1,x^)G(xo,x1,x1) e s(x1, Tx1, Tx1).

By Remark 3.2, we can take x2 e Tx1 such that G(xo,x1,x^)G(xo,x1,x1) e 2s(dG(x1,x2)).

2dG(x1,x2) ^ G(xo,x1,x^)G(xo,x1,x1).

Again, by (1), we have

G(x1,x2,x2))G(x1,x2,x2) e s(Tx1, Tx2, Tx2),

and by Lemma 3.1(iii)

G(x1,x2,x2))G(x1,x2,x2) e s(x2, Tx2, Tx2).

By Remark 3.2, we can take x3 e Tx2 such that G(x1,x2,x2^G(x1,x2,x2) e 2^dG(x2,x3)).

2dG(x2, x3) ^ G(x1, x2, x2^ G(x1, x2, x2).

It implies that

2dG(x2,x3) ^ G(x1,x2,x2^G(x1,x2,x2)

^ G(x1,x2,x2))G(x1,x2,x2) + G(x1,x2,x2))G(x2,x1,x1)

^ G(x1, x2, x2^ [G(x1, x2, x2) + G(x2, x1, x1)]

= G(x1, x2, x2^ dG(x1, x2)

^ dG(x2,x3) ^ 2^(G(x1,x2,x2))dG(x1,x2).

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

dG(xn,xn+1) ^ -(p(G(xn-1,xn,xn))dG(xn-1,xn), xn+1 e Txn, for n = 1,2,3____ (2)

Assume that xn+1 = xn for all n e N. From (2) the sequence {dG(xn,x„+i)}„eN is a decreasing sequence in P. So, there exists l e (0,1) such that

lim sup dG(xn,xn+1)) = l.

Thus, there exists n0 e N such that for all n > n0, p(dG(xn,xn+1)) -< l0 for some l0 e (l,1). Choose n0 = 1, then we have

dG(xn, xn+l) ^ 2v{dG(xn-1, xn))dG(xn-1,xn) -< l0dG(xn-1,xn)

-< (l0)ndG(x0,x1) for all n > 1.

Moreover, for m > n > 1, we have that dG(xn, xm) ^ ( o), dG(xo, x1).

1 - io

According to (PT1) and (PT7), it follows that {xn} is a Cauchy sequence in X. By the completeness of X, there exists v e X suchthat xn ^ v. Assume k1 e N suchthat dG(xn, v) ^ § for all n > k1.

We now show that v e Tv. So, for xn, v e X and by using (2), we have

G(xn, v, v))G(xn, v, v) e s(Txn, Tv, Tv).

By Lemma 3.1 (iii) we have

G(xn, v, v))G(xn, v, v) e s(xn+1, Tv, Tv).

Thus there exists un e Tv such that

G(xn, v, v))G(xn, v, v) e 2s(dG(xn+1, un)).

It implies that

2dG(xn+1, un) ^ G(xn, v, v))G(xn, v, v), dG (xn+1, Un) ^ 2 y>(G(xn, v, v))G(xn, v, v)

^ G(xn, v, v)) [G(xn, v, v) + G(xn,xn, v)] = G(xn, v, v)) dG (xn, v).

dG(xn+1, Un) ^ G(xn, v, v)) dG (xn, v). (3)

Now consider

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

4 dG(xn+i, v) + y (G(xn, v, v))dG(xn, v) by using (3)

-< dG (xn+1, v) + dG(xn, v), cc

dG(v, un) ^ - + - = c, for all n > k1.

Therefore limn^x un = v. Since Tv is closed, so v e Tv. □

The next corollary is Nadler's multivalued contraction theorem in a G-cone metric space.

Corollary 3.1 Let (X, G) be a complete G-cone metric space, and letT: X —► CB(X) be a multivalued mapping. If there exists a constant k e [0,1) such that

kG(x,y, z) e s(Tx, Ty, Tz)

for all x, y, z e X, then T has a fixed point in X.

By Remark 3.1, we have the following results of [30].

Corollary 3.2 [30] Let (X, G) be a complete G-metric space, and letT: X —► CB(X) be a multivalued mapping. If there exists a function y : [0, +x) ^ [0,1) such that

lim sup y(r)< 1

for any t > 0, and if

HG(Tx, Ty, Tz) < y(G(x,y,z))G(x,y,z) for all x, y, z e X, then T has a fixed point in X.

Corollary 3.3 [30] Let (X, G) be a complete G-metric space, and letT: X —► CB(X) be a multivalued mapping. If there exists a constant k e [0,1) such that

HG(Tx, Ty, Tz) < kG(x,y,z)

for all x, y, z e X, then T has a fixed point in X.

In the following we formulate an illustrative example regarding our main theorem.

Example3.2 Let X = [0,1], E = C[0,1] be endowed with the strongly locally convex topology t(E,E*), and let P = {x e E: 0 < x(t), t e [0,1]}. Then the cone is t(E,E*)-solid, and non-normal with respect to the topology t(E,E*). Define G: X x X x X ^ E by

G(x,y,z)(t) = Max{\x -y|, \y-z|, \x -z\\et.

Then G is a G-cone metric on X.

Consider a mapping T: X ^ CB(X) defined by

Tx = 0, — x 10

Let ^(t) = 5 for all t e P. The contractive condition of the main theorem is trivial for the case when x = y = z = 0. Suppose, without any loss of generality, that all x, y and z are nonzero and x < y < z. Then

G(x, y, z) = |x - z|et,

dG(x, y) = 2|x - y|et

s(x, Ty) =

s(y, Tz) =

0 if x < 10,

|x -10 |et if x > 10,

0 ify < l0, |y-10|et ify >10.

For s(x, Ty) = 0= s(y, Tz), we have s(x, Ty, Tz) = s(0),

p| s(y, Tx, Tz) = s( 2

p| s(z, Tx, Ty) = s(2

yx 10 -10

zxy 10 - 10-10

s(Tx, Ty, Tz) = (s(0)) n ^ 2

yx 10-10

If s(Tx, Ty, Tz) = si 2

zxy 10 - 10-10

e?\ n W 2

zxy 10-10 - 10

ef I, then

zxy 10-10 - 10

et < 2

zx 10 - 10

et, for t e [0,1]

= - |z - x|et = - Ma^j |x -y|, |y - z|, |x - z^ef

= - G(x, y, z);

1 t -\z - x\e = 5

1 G(x, y, z).

Max{\x -y\, \y-z\, \x -z\\ef

Hence,

-G(x,y,z) e s(Tx, Ty, Tz). 5

All the assumptions of Theorem 3.1 also hold for other possible values of s(x, Ty) and s(y, Tz) to obtain 0 e T0.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Both authors read and approved the finalmanuscript. Acknowledgements

We are very gratefulto the editor and anonymous referees for their valuable and constructive comments that helped us very much in improving the paper.

Received: 11 December 2012 Accepted: 15 July 2013 Published: 29 July 2013 References

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doi:10.1186/1029-242X-2013-354

Cite this article as: Azam and Mehmood: Fixed point theorems for multivalued mappings in G-cone metric spaces.

Journal of Inequalities and Applications 2013 2013:354.

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