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Fixed point theorems for multivalued mappings in Gcone metric spaces
Journal of Inequalities and Applications volume 2013, Article number: 354 (2013)
Abstract
We extend the idea of Hausdorff distance function in Gcone metric spaces and obtain fixed points of multivalued mappings in Gcone metric spaces.
MSC:47H10, 54H25.
1 Introduction
The main revolution in the existence theory of many linear and nonlinear 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 let T:X\to {2}^{X} be a multivalued map such that Tx is a closed bounded subset of X for all x\in X. If there exists a function \phi :(0,\mathrm{\infty})\to [0,1) such that lim{sup}_{r\to {t}^{+}}\phi (r)<1 for all t\in [0,\mathrm{\infty}) and if
then T has a fixed point in X.
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 Rezapour and Hamlbarani [14] generalized a cone metric space with a nonnormal cone. Afterwards many researchers [15–24] have studied fixed point results in cone metric spaces. In [25] Mustafa et al. generalized the metric space and introduced the notion of Gmetric space which recovered the flaws of Dhage’s generalization [26, 27] of a metric space. Many researchers proved many fixed point results using a Gmetric space [28, 29]. Anchalee Kaewcharoen and Attapol Kaewkhao [28] and Nedal et al. [30] proved fixed point results for multivalued maps in Gmetric spaces. In 2009, Beg et al. [31] introduced the notion of Gcone metric space and generalized some results. ChiMing Cheng [32] proved Nadlertype results in tvs Gcone metric spaces.
In 2011 Cho and Bae [33] generalized a Mizoguchi Takahashitype theorem in a cone metric space. In the present paper, we introduce the notion of Hausdorff distance function on Gcone metric spaces and exploit it to study some fixed point results in Gcone 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\ne \{\theta \},

(b)
a,b\in R, a,b\ge 0, x,y\in P implies ax+by\in P, more generally, if a,b,c\in R, a,b,c\ge 0, x,y,z\in P\u27f9ax+by+cz\in P,

(c)
P\cap (P)=\{\theta \}.
Given a cone P\subset E, we define a partial ordering ≼ with respect to P by x\preccurlyeq y if and only if yx\in P.
A cone P is called normal if there is a number K>0 such that for all x,y\in E
The least positive number satisfying the above inequality is called the normal constant of P, while x\ll y stands for yx\in intP (interior of P), while x\prec y means x\preccurlyeq y and x\ne 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 nonnormal solid cones, cannot be provided by reducing to metric spaces, because in this case neither of the conditions of Lemmas 14 in [13] hold. Further, the vector cone metric is not continuous in a general case, i.e., from {x}_{n}\to x, {y}_{n}\to y it need not follow that d({x}_{n},{y}_{n})\to d(x,y).
For the case of nonnormal cones, we have the following properties.
(PT1) If u\preccurlyeq v and v\ll w, then u\ll w.
(PT2) If u\ll v and v\preccurlyeq w, then u\ll w.
(PT3) If u\ll v and v\ll w, then u\ll w.
(PT4) If \theta \preccurlyeq u\ll c for each c\in intP, then u=\theta.
(PT5) If a\preccurlyeq b+c for each c\in intP, then a\preccurlyeq b.
(PT6) If E is a real Banach space with a cone P, and if a\preccurlyeq \lambda a, where a\in P and 0\le \lambda <1, then a=\theta.
(PT7) If c\in intP, {a}_{n}\in \mathbb{E} and {a}_{n}\to \theta, then there exists an {n}_{0} such that, for all n>{n}_{0}, we have {a}_{n}\ll c.
In the following we shall always assume that the cone P is solid and nonnormal.
Definition 2.1 [31]
Let X be a nonempty set. Suppose that a mapping G:X\times X\times X\to E satisfies:
(G1) G(x,y,z)=\theta if x=y=z,
(G2) \theta \prec G(x,x,y), whenever x\ne y, for all x,y\in X,
(G3) G(x,x,y)\preccurlyeq G(x,y,z), whenever y\ne z,
(G4) G(x,y,z)=G(x,z,y)=G(y,x,z)=\cdots (symmetric in all three variables),
(G5) G(x,y,z)\preccurlyeq G(x,a,a)+G(a,y,z) for all x,y,z,a\in X.
Then G is called a generalized cone metric on X, and X is called a generalized cone metric space or, more specifically, a Gcone metric space.
The concept of a Gcone metric space is more general than that of Gmetric spaces and cone metric spaces (see [31]).
Definition 2.2 [31]
A Gcone metric space X is symmetric if G(x,y,y)=G(y,x,x) for all x,y\in X.
Example 2.1 [31]
Let (X,d) be a cone metric space. Define G:X\times X\times X\to E by G(x,y,z)=d(x,y)+d(y,z)+d(z,x). Then (X,G) is a Gcone metric space.
Proposition 2.1 [31]
Let X be a Gcone metric space, define {d}_{G}:X\times X\to E by
Then (X,{d}_{G}) is a cone metric space.
It can be noted that G(x,y,y)\preccurlyeq \frac{2}{3}{d}_{G}(x,y). If X is a symmetric Gcone metric space, then {d}_{G}(x,y)=2G(x,y,y) for all x,y\in X.
Definition 2.3 [31]
Let X be a Gcone metric space and let \{{x}_{n}\} be a sequence in X.
We say that \{{x}_{n}\} is:

(a)
a Cauchy sequence if for every c\in E with \theta \ll c, there is N such that for all n,m,l>N, G({x}_{n},{x}_{m},{x}_{l})\ll c.

(b)
a convergent sequence if for every c in E with \theta \ll c, there is N such that for all m,n>N, G({x}_{m},{x}_{n},x)\ll c for some fixed x in X. Here x is called the limit of a sequence \{{x}_{n}\} and is denoted by {lim}_{n\to \mathrm{\infty}}{x}_{n}=x or {x}_{n}\to x as n\to \mathrm{\infty}.
A Gcone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
Proposition 2.2 [31]
Let X be a Gcone metric space, then the following are equivalent.

(i)
\{{x}_{n}\} converges to x.

(ii)
G({x}_{n},{x}_{n},x)\to \theta as n\to \mathrm{\infty}.

(iii)
G({x}_{n},x,x)\to \theta as n\to \mathrm{\infty}.

(iv)
G({x}_{m},{x}_{n},x)\to \theta as m,n\to \mathrm{\infty}.
Lemma 2.1 [31]
Let \{{x}_{n}\} be a sequence in a Gcone metric space X. If \{{x}_{n}\} converges to x\in X, then G({x}_{m},{x}_{n},x)\to \theta as m,n\to \mathrm{\infty}.
Lemma 2.2 [31]
Let \{{x}_{n}\} be a sequence in a Gcone metric space X and x\in X. If \{{x}_{n}\} converges to x\in X, then \{{x}_{n}\} is a Cauchy sequence.
Lemma 2.3 [31]
Let \{{x}_{n}\} be a sequence in a Gcone metric space X. If \{{x}_{n}\} is a Cauchy sequence in X, then G({x}_{m},{x}_{n},{x}_{l})\to \theta, as m,n,l\to \mathrm{\infty}.
3 Main result
Denote by N(X), B(X) and CB(X) the set of nonempty, bounded, sequentially closed bounded subsets of Gcone metric spaces, respectively.
Let (X,G) be a Gcone metric space. We define (see [33])
and
For A,B\in B(X), we define
and
Lemma 3.1 Let (X,G) be a Gcone metric space, let P be a cone in a Banach space E.

(i)
Let p,q\in E. If p\preccurlyeq q, then s(q)\subset s(p).

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

(iii)
Let q\in P and let A,B,C\in B(X) and a\in A. If q\in s(A,B,C), then q\in s(a,B,C).
Remark 3.1 Recently, Kaewcharoen and Kaewkhao [28] (see also [30]) introduced the following concepts. Let X be a Gmetric space and let CB(X) be the family of all nonempty closed bounded subsets of X. Let {H}_{G}(\cdot ,\cdot ,\cdot ) be the Hausdorff Gdistance on CB(X), i.e.,
where
The above expressions show a relation between {H}_{G} and {H}_{{d}_{G}}. Moreover, note that if (X,G) is a Gcone metric space, E=R, and P=[0,\mathrm{\infty}), then (X,G) is a Gmetric space. Also, for A,B,C\in CB(X), {H}_{G}(A,B,C)=infs(A,B,C).
Remark 3.2 Let (X,G) be a Gcone metric space. Then

(a)
\stackrel{\u02c6}{s}(\{a\},\{b\})=s({d}_{G}(a,b)) for a,b\in X.

(b)
If x\in s(a,B,B) then x\in 2s({d}_{G}(a,b)).
Proof (a) By definition

(b)
Now let
\begin{array}{c}x\in s(a,B,B),\phantom{\rule{1em}{0ex}}\text{then}\hfill \\ x\in s(a,B,B)=s(a,B)+\stackrel{\u02c6}{s}(B,B)+s(a,B)\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}x\in 2s(a,B)+\stackrel{\u02c6}{s}(B,B)\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}x\in 2s({d}_{G}(a,b))+s(\theta ).\hfill \end{array}
Let x=y+z for y\in 2s({d}_{G}(a,b)) and z\in s(\theta ). Then by definition \theta \preccurlyeq z and 2{d}_{G}(a,b)\preccurlyeq y, which implies \theta +2{d}_{G}(a,b)\preccurlyeq y+z=x. Hence 2{d}_{G}(a,b)\preccurlyeq x, so x\in 2s({d}_{G}(a,b)). □
In the following theorem, we use the generalized Hausdorff distance on Gcone metric spaces to find fixed points of a multivalued mapping.
Remark 3.3 If (X,G) is a Gmetric space, then (X,{d}_{G}) is a metric space, where
It is noticed in [35] that in the symmetric case ((X,G) is symmetric), many fixed point theorems on Gmetric 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)={d}_{G}(Tx,Ty) is exploited for a singlevalued mapping T on X. Whereas in the case of multivalued mapping T:X\to {2}^{X} on a Gcone metric space,
Therefore,
and even in a symmetric case, we cannot follow a similar technique to deduce Gcone metric multivalued fixed point results from similar results of metric spaces.
In a nonsymmetric case, the authors [35] deduce some Gmetric fixed point theorems from similar results of metric spaces by using the fact that if (X,G) is a Gmetric on X, then
is a metric on X. Whereas, in the case of a Gcone 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,\delta ) may not be a cone metric space if (X,G) is a Gcone metric space. In the explanation of this fact, we refer to Example 3.1 below, from [31]. Hence multivalued fixed point results on Gcone metric spaces cannot be deduced from similar fixed point theorems on metric spaces.
Example 3.1 [31]
Let X=\{a,b\}, E={R}^{3},
Define G:X\times X\times X\to E by
Note that \delta (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\u27f6CB(X) be a multivalued mapping. If there exists a function \phi :P\to [0,1) such that
for any decreasing sequence \{{r}_{n}\} in P, and if
for all x,y,z\in X, then T has a fixed point in X.
Proof Let {x}_{0} be an arbitrary point in X and {x}_{1}\in T{x}_{0}. From (1), we have
Thus, by Lemma 3.1(iii), we get
By Remark 3.2, we can take {x}_{2}\in T{x}_{1} such that
Thus,
Again, by (1), we have
and by Lemma 3.1(iii)
By Remark 3.2, we can take {x}_{3}\in T{x}_{2} such that
Thus,
It implies that
By induction we can construct a sequence \{{x}_{n}\} in X such that
Assume that {x}_{n+1}\ne {x}_{n} for all n\in N. From (2) the sequence {\{{d}_{G}({x}_{n},{x}_{n+1})\}}_{n\in N} is a decreasing sequence in P. So, there exists l\in (0,1) such that
Thus, there exists {n}_{0}\in N such that for all n\ge {n}_{0}, \phi ({d}_{G}({x}_{n},{x}_{n+1}))\prec {l}_{0} for some {l}_{0}\in (l,1). Choose {n}_{0}=1, then we have
Moreover, for m>n\ge 1, we have that
According to (PT1) and (PT7), it follows that \{{x}_{n}\} is a Cauchy sequence in X. By the completeness of X, there exists v\in X such that {x}_{n}\to v. Assume {k}_{1}\in N such that {d}_{G}({x}_{n},v)\ll \frac{c}{2} for all n\ge {k}_{1}.
We now show that v\in Tv. So, for {x}_{n},v\in X and by using (2), we have
By Lemma 3.1(iii) we have
Thus there exists {u}_{n}\in Tv such that
It implies that
So
Now consider
Therefore {lim}_{n\to \mathrm{\infty}}{u}_{n}=v. Since Tv is closed, so v\in Tv. □
The next corollary is Nadler’s multivalued contraction theorem in a Gcone metric space.
Corollary 3.1 Let (X,G) be a complete Gcone metric space, and let T:X\u27f6CB(X) be a multivalued mapping. If there exists a constant k\in [0,1) such that
for all x,y,z\in 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 Gmetric space, and let T:X\u27f6CB(X) be a multivalued mapping. If there exists a function \phi :[0,+\mathrm{\infty})\to [0,1) such that
for any t\ge 0, and if
for all x,y,z\in X, then T has a fixed point in X.
Corollary 3.3 [30]
Let (X,G) be a complete Gmetric space, and let T:X\u27f6CB(X) be a multivalued mapping. If there exists a constant k\in [0,1) such that
for all x,y,z\in X, then T has a fixed point in X.
In the following we formulate an illustrative example regarding our main theorem.
Example 3.2 Let X=[0,1], E=C[0,1] be endowed with the strongly locally convex topology \tau (E,{E}^{\ast}), and let P=\{x\in E:0\le x(t),\phantom{\rule{0.25em}{0ex}}t\in [0,1]\}. Then the cone is \tau (E,{E}^{\ast})solid, and nonnormal with respect to the topology \tau (E,{E}^{\ast}). Define G:X\times X\times X\to E by
Then G is a Gcone metric on X.
Consider a mapping T:X\to CB(X) defined by
Let \phi (t)=\frac{1}{5} for all t\in 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
and
Now
For s(x,Ty)=0=s(y,Tz), we have
and
Thus
Now
Hence,
All the assumptions of Theorem 3.1 also hold for other possible values of s(x,Ty) and s(y,Tz) to obtain 0\in T0.
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Azam, A., Mehmood, N. Fixed point theorems for multivalued mappings in Gcone metric spaces. J Inequal Appl 2013, 354 (2013). https://doi.org/10.1186/1029242X2013354
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DOI: https://doi.org/10.1186/1029242X2013354