Fixed point theorems for multivalued mappings in G-cone metric spaces

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.

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 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

$$H(Tx,Ty)\le \phi (d(x,y))(d(x,y))\mathit{\text{for all}}x,y(x\ne y)\in X,$$

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 non-normal 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 G-metric space which recovered the flaws of Dhage’s generalization [26, 27] of a metric space. Many researchers proved many fixed point results using a G-metric space [28, 29]. Anchalee Kaewcharoen and Attapol Kaewkhao [28] and Nedal et al. [30] proved fixed point results for multivalued maps in G-metric spaces. In 2009, Beg et al. [31] introduced the notion of G-cone metric space and generalized some results. Chi-Ming Cheng [32] proved Nadler-type 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 $P\ne \{\theta \}$,

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

• (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 $y-x\in P$.

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

$$\theta \preccurlyeq x\preccurlyeq y\text{implies}\parallel x\parallel \le K\parallel y\parallel .$$

The least positive number satisfying the above inequality is called the normal constant of P, while $x\ll y$ stands for $y-x\in intP$ (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 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 $x_n\to x$, $y_n\to y$ it need not follow that $d(x_n,y_n)\to d(x,y)$.

For the case of non-normal 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 non-normal.

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 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 $x,y\in X$.

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 G-cone metric space.

Proposition 2.1 [31]

Let X be a G-cone metric space, define $d_G:X\times X\to E$ by

$$d_G(x,y)=G(x,y,y)+G(y,x,x).$$

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 G-cone 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 G-cone 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 G-cone 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 G-cone 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 G-cone 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 G-cone 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 G-cone 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 G-cone metric spaces, respectively.

Let $(X,G)$ be a G-cone metric space. We define (see [33])

$$s(p)=\{q\in E:p\preccurlyeq q\}\text{for }q\in E,$$

and

$$s(a,B)=\bigcup _{b\in B}s(d_G(a,b))=\bigcup _{b\in B}\{x\in E:d_G(a,b)\preccurlyeq x\}\text{for }a\in X\text{ and }B\in N(X).$$

For $A,B\in B(X)$, we define

$$\begin{array}{c}\begin{array}{r}\hat{s}(A,B)=\bigcup _{a\in A,b\in B}s(d_G(a,b)),\end{array}\hfill \\ \begin{array}{r}s(a,B,C)=s(a,B)+\hat{s}(B,C)+s(a,C)=\{u+v+w:u\in s(a,B),v\in \hat{s}(B,C),w\in s(a,C)\},\end{array}\hfill \end{array}$$

and

$$s(A,B,C)=(\bigcap _{a\in A}s(a,B,C))\cap (\bigcap _{b\in B}s(b,A,C))\cap (\bigcap _{c\in C}s(c,A,B)).$$

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

• (i) Let $p,q\in E$. 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 G-metric 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 G-distance on $CB(X)$, i.e.,

$$\begin{array}{c}{H}_G(A,B,C)=max\{\underset{a\in A}{sup}G(a,B,C),\underset{b\in B}{sup}G(b,A,C),\underset{c\in C}{sup}G(c,A,B)\},\hfill \\ H_{d_G}(A,B)=max\{\underset{a\in A}{sup}{d}_G(a,B),\underset{b\in B}{sup}{d}_G(b,A)\},\hfill \end{array}$$

where

$$\begin{array}{c}G(x,B,C)=d_G(x,B)+d_G(B,C)+d_G(x,C),\hfill \\ d_G(x,B)=inf\{d_G(x,y),y\in B\},\hfill \\ d_G(A,B)=inf\{d_G(a,b),a\in A,b\in B\},\hfill \\ G(a,b,C)=inf\{G(a,b,c),c\in C\}.\hfill \end{array}$$

The above expressions show a relation between $H_G$ and $H_{d_G}$. Moreover, note that if $(X,G)$ is a G-cone metric space, $E=R$, and $P=[0,\mathrm{\infty })$, then $(X,G)$ is a G-metric 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 G-cone metric space. Then

• (a) $\hat{s}(\{a\},\{b\})=s(d_G(a,b))$ 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

$$\begin{array}{rcl}\hat{s}(\{a\},\{b\})& =& \bigcup _{a\in \{a\},b\in \{b\}}s(d_G(a,b))\\ =& s(d_G(a,b)).\end{array}$$

• (b) Now let

  $$\begin{array}{c}x\in s(a,B,B),\text{then}\hfill \\ x\in s(a,B,B)=s(a,B)+\hat{s}(B,B)+s(a,B)\hfill \\ \Rightarrow x\in 2s(a,B)+\hat{s}(B,B)\hfill \\ \Rightarrow x\in 2s(d_G(a,b))+s(\theta ).\hfill \end{array}$$

Let $x=y+z$ for $y\in 2s(d_G(a,b))$ and $z\in s(\theta )$. Then by definition $\theta \preccurlyeq z$ and $2d_G(a,b)\preccurlyeq y$, which implies $\theta +2d_G(a,b)\preccurlyeq y+z=x$. Hence $2d_G(a,b)\preccurlyeq x$, so $x\in 2s(d_G(a,b))$. □

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 $(X,d_G)$ is a metric space, where

$$d_G(x,y)=G(x,y,y)+G(y,x,x).$$

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 $G(Tx,Ty,Ty)+G(Ty,Tx,Tx)=2G(Tx,Ty,Ty)=d_G(Tx,Ty)$ is exploited for a single-valued mapping T on X. Whereas in the case of multivalued mapping $T:X\to 2^X$ on a G-cone metric space,

$$\begin{array}{rcl}s(Tx,Ty,Ty)& =& (\bigcap _{a\in Tx}s(a,Ty,Ty))\cap (\bigcap _{b\in Ty}s(b,Tx,Ty))\cap (\bigcap _{b\in Ty}s(b,Tx,Ty))\\ =& (\bigcap _{a\in Tx}s(a,Ty,Ty))\cap (\bigcap _{b\in Ty}s(b,Tx,Ty))\\ =& (\bigcap _{a\in Tx}2s(a,Ty))\cap (\bigcap _{b\in Ty}s(b,Tx)+\hat{s}(Tx,Ty)+s(b,Ty))\\ \ne & s(Ty,Tx,Tx).\end{array}$$

Therefore,

$$(\bigcap _{a\in Tx}s(a,Ty))\cap (\bigcap _{b\in Ty}s(b,Tx))\ne s(Tx,Ty,Ty)+s(Ty,Tx,Tx)$$

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

$$\delta (x,y)=max\{G(x,y,y),G(y,x,x)\}$$

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, $(X,\delta )$ 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 $X=\{a,b\}$, $E=R^3$,

$$P=\{(x,y,z)\in E:x,y,z\ge 0\}.$$

Define $G:X\times X\times X\to E$ by

$$\begin{array}{c}G(a,a,a)=(0,0,0)=G(b,b,b),\hfill \\ G(a,b,b)=(0,1,1)=G(b,a,b)=G(b,b,a),\hfill \\ G(b,a,a)=(0,1,0)=G(a,b,a)=G(a,a,b).\hfill \end{array}$$

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 $T:X⟶CB(X)$ be a multivalued mapping. If there exists a function $\phi :P\to [0,1)$ such that

$$lim\underset{n\to \mathrm{\infty }}{sup}\phi (r_n)<1$$

(a)

for any decreasing sequence $\{r_n\}$ in P, and if

$$\phi (G(x,y,z))G(x,y,z)\in s(Tx,Ty,Tz)$$

(1)

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

$$\phi (G(x_0,x_1,x_1))G(x_0,x_1,x_1)\in s(T{x}_0,T{x}_1,T{x}_1).$$

Thus, by Lemma 3.1(iii), we get

$$\phi (G(x_0,x_1,x_1))G(x_0,x_1,x_1)\in s(x_1,T{x}_1,T{x}_1).$$

By Remark 3.2, we can take $x_2\in T{x}_1$ such that

$$\phi (G(x_0,x_1,x_1))G(x_0,x_1,x_1)\in 2s(d_G(x_1,x_2)).$$

Thus,

$$2d_G(x_1,x_2)\preccurlyeq \phi (G(x_0,x_1,x_1))G(x_0,x_1,x_1).$$

Again, by (1), we have

$$\phi (G(x_1,x_2,x_2))G(x_1,x_2,x_2)\in s(T{x}_1,T{x}_2,T{x}_2),$$

and by Lemma 3.1(iii)

$$\phi (G(x_1,x_2,x_2))G(x_1,x_2,x_2)\in s(x_2,T{x}_2,T{x}_2).$$

By Remark 3.2, we can take $x_3\in T{x}_2$ such that

$$\phi (G(x_1,x_2,x_2))G(x_1,x_2,x_2)\in 2s(d_G(x_2,x_3)).$$

Thus,

$$2d_G(x_2,x_3)\preccurlyeq \phi (G(x_1,x_2,x_2))G(x_1,x_2,x_2).$$

It implies that

$$\begin{array}{rl}2d_G(x_2,x_3)& \preccurlyeq \phi (G(x_1,x_2,x_2))G(x_1,x_2,x_2)\\ \preccurlyeq \phi (G(x_1,x_2,x_2))G(x_1,x_2,x_2)+\phi (G(x_1,x_2,x_2))G(x_2,x_1,x_1)\\ \preccurlyeq \phi (G(x_1,x_2,x_2))[G(x_1,x_2,x_2)+G(x_2,x_1,x_1)]\\ =\phi (G(x_1,x_2,x_2))d_G(x_1,x_2)\\ \Rightarrow d_G(x_2,x_3)\preccurlyeq \frac{1}{2}\phi (G(x_1,x_2,x_2))d_G(x_1,x_2).\end{array}$$

By induction we can construct a sequence $\{x_n\}$ in X such that

$$d_G(x_n,x_{n+1})\preccurlyeq \frac{1}{2}\phi (G(x_{n-1},x_n,x_n))d_G(x_{n-1},x_n),x_{n+1}\in T{x}_n,\text{ for }n=1,2,3\dots .$$

(2)

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

$$lim\underset{n\to \mathrm{\infty }}{sup}\phi (d_G(x_n,x_{n+1}))=l.$$

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

$$\begin{array}{rcl}{d}_G(x_n,x_{n+1})& \preccurlyeq & \frac{1}{2}\phi (d_G(x_{n-1},x_n))d_G(x_{n-1},x_n)\\ \prec & l_0{d}_G(x_{n-1},x_n)\\ \prec & {(l_0)}^n{d}_G(x_0,x_1)\text{for all }n\ge 1.\end{array}$$

Moreover, for $m>n\ge 1$, we have that

$$d_G(x_n,x_m)\preccurlyeq \frac{{(l_0)}^n}{1-l_0}{d}_G(x_0,x_1).$$

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

$$\phi (G(x_n,v,v))G(x_n,v,v)\in s(T{x}_n,Tv,Tv).$$

By Lemma 3.1(iii) we have

$$\phi (G(x_n,v,v))G(x_n,v,v)\in s(x_{n+1},Tv,Tv).$$

Thus there exists $u_n\in Tv$ such that

$$\phi (G(x_n,v,v))G(x_n,v,v)\in 2s(d_G(x_{n+1},u_n)).$$

It implies that

$$\begin{array}{c}2d_G(x_{n+1},u_n)\preccurlyeq \phi (G(x_n,v,v))G(x_n,v,v),\hfill \\ \begin{array}{rl}{d}_G(x_{n+1},u_n)& \preccurlyeq \frac{1}{2}\phi (G(x_n,v,v))G(x_n,v,v)\\ \preccurlyeq \phi (G(x_n,v,v))[G(x_n,v,v)+G(x_n,x_n,v)]\\ =\phi (G(x_n,v,v))d_G(x_n,v).\end{array}\hfill \end{array}$$

So

$$d_G(x_{n+1},u_n)\preccurlyeq \phi (G(x_n,v,v))d_G(x_n,v).$$

(3)

Now consider

$$\begin{array}{c}\begin{array}{rl}{d}_G(v,u_n)& \preccurlyeq d_G(x_{n+1},v)+d_G(x_{n+1},u_n)\\ \preccurlyeq d_G(x_{n+1},v)+\phi (G(x_n,v,v))d_G(x_n,v)\text{by using (3)}\\ \prec d_G(x_{n+1},v)+d_G(x_n,v),\end{array}\hfill \\ d_G(v,u_n)\ll \frac{c}{2}+\frac{c}{2}=c,\text{for all }n\ge k_1.\hfill \end{array}$$

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 G-cone metric space.

Corollary 3.1 Let $(X,G)$ be a complete G-cone metric space, and let $T:X⟶CB(X)$ be a multivalued mapping. If there exists a constant $k\in [0,1)$ such that

$$kG(x,y,z)\in s(Tx,Ty,Tz)$$

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 G-metric space, and let $T:X⟶CB(X)$ be a multivalued mapping. If there exists a function $\phi :[0,+\mathrm{\infty })\to [0,1)$ such that

$$lim\underset{r\to t^{+}}{sup}\phi (r)<1$$

for any $t\ge 0$, and if

$$H_G(Tx,Ty,Tz)\le \phi (G(x,y,z))G(x,y,z)$$

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 G-metric space, and let $T:X⟶CB(X)$ be a multivalued mapping. If there exists a constant $k\in [0,1)$ such that

$$H_G(Tx,Ty,Tz)\le kG(x,y,z)$$

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 $P=\{x\in E:0\le x(t),t\in [0,1]\}$. Then the cone is $\tau (E,E^{\ast })$-solid, and non-normal with respect to the topology $\tau (E,E^{\ast })$. Define $G:X\times X\times X\to E$ by

$$G(x,y,z)(t)=Max\{|x-y|,|y-z|,|x-z|\}{e}^t.$$

Then G is a G-cone metric on X.

Consider a mapping $T:X\to CB(X)$ defined by

$$Tx=[0,\frac{1}{10}x].$$

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

$$G(x,y,z)=|x-z|e^t,$$

and

$$d_G(x,y)=2|x-y|e^t.$$

Now

$$\begin{array}{c}s(x,Ty)=\{\begin{array}{cc}0\hfill & \text{if }x\le \frac{y}{10},\hfill \\ |x-\frac{y}{10}|e^t\hfill & \text{if }x>\frac{y}{10},\hfill \end{array}\hfill \\ s(y,Tz)=\{\begin{array}{cc}0\hfill & \text{if }y\le \frac{z}{10},\hfill \\ |y-\frac{z}{10}|e^t\hfill & \text{if }y>\frac{z}{10}.\hfill \end{array}\hfill \end{array}$$

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

$$\begin{array}{c}s(x,Ty,Tz)=s(0),\hfill \\ \bigcap _{y\in Ty}s(y,Tx,Tz)=s\left(2\right|\frac{y}{10}-\frac{x}{10}\left|e^t\right),\hfill \end{array}$$

and

$$\bigcap _{z\in Tz}s(z,Tx,Ty)=s\left(2\right|\frac{z}{10}-\frac{x}{10}-\frac{y}{10}\left|e^t\right).$$

Thus

$$s(Tx,Ty,Tz)=(s(0))\cap \left(s\left(2\right|\frac{y}{10}-\frac{x}{10}\left|e^t\right)\right)\cap \left(s\left(2\right|\frac{z}{10}-\frac{x}{10}-\frac{y}{10}\left|e^t\right)\right).$$

Now

$$\begin{array}{c}\begin{array}{r}\text{If }s(Tx,Ty,Tz)=s\left(2\right|\frac{z}{10}-\frac{x}{10}-\frac{y}{10}\left|e^t\right),\text{then }\end{array}\hfill \\ \begin{array}{rl}2|\frac{z}{10}-\frac{x}{10}-\frac{y}{10}|e^t& \le 2|\frac{z}{10}-\frac{x}{10}|e^t,\text{for }t\in [0,1]\\ =\frac{1}{5}|z-x|e^t=\frac{1}{5}Max\{|x-y|,|y-z|,|x-z|\}{e}^t\\ =\frac{1}{5}G(x,y,z);\end{array}\hfill \\ \begin{array}{r}\text{If }s(Tx,Ty,Tz)=s\left(2\right|\frac{y}{10}-\frac{x}{10}\left|e^t\right),\text{then }\end{array}\hfill \\ \begin{array}{rl}2|\frac{y}{10}-\frac{x}{10}|e^t& \le 2|\frac{z}{10}-\frac{x}{10}|e^t,\text{for }t\in [0,1]\\ =\frac{1}{5}|z-x|e^t=\frac{1}{5}Max\{|x-y|,|y-z|,|x-z|\}{e}^t\\ =\frac{1}{5}G(x,y,z).\end{array}\hfill \end{array}$$

Hence,

$$\frac{1}{5}G(x,y,z)\in s(Tx,Ty,Tz).$$

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