Research Article | Open Access

# Common Fixed Point Theorems of Contractions in Partial Cone Metric Spaces over Nonnormal Cones

**Academic Editor:**Sehie Park

#### Abstract

We prove some common fixed point theorems of contractions restricted with variable positive linear bounded mappings in -complete partial cone metric spaces over nonnormal cones and present some examples to support the usability of our results.

#### 1. Introduction

In 2007, Huang and Zhang [1] introduced cone metric spaces, being unaware that they already existed under the name -metric and -normed spaces that were introduced and used in the middle of the 20th century in [2â€“9]. In both cases, the set of real numbers was replaced by an ordered Banach space . However, Huang and Zhang went further and defined the convergence via interior points of the cone by which the order in is defined. This approach allows the investigation of cone spaces in the case that the cone is not necessarily normal. Since then, there were many references concerned with fixed point results and common fixed point results in cone metric spaces over a nonnormal cone (see [10â€“18]). In 2012, based on the definition of cone metric spaces and partial metric spaces introduced by Matthews [19], Sonmez [20, 21] defined a partial cone metric space and proved some fixed point theorems of contractions restricted with constants in complete partial cone metric spaces over normal cones. Recently, without using the normality of the cone, Malhotra et al. [22] and Jiang and Li [23] extended the results of [20, 21] to -complete partial cone metric spaces. In addition, the contractions considered in [23] are not necessarily restricted with constants but restricted with positive linear bounded mappings.

In this paper, we prove some common fixed point theorems of contractions restricted with variable positive linear bounded mappings in *-*complete partial cone metric spaces over nonnormal cones, which improve the recent results of [22, 23].

#### 2. Preliminaries

Let be a topological vector space. A cone of is a nonempty closed subset of such that for each and each , and , where is the zero element of . A cone of determines a partial order on by for each . In this case, is called an ordered topological vector space.

A cone of a topological vector space is solid if intâ€‰, where intâ€‰ is the interior of . For each with intâ€‰, we write . Let be a solid cone of a topological vector space . A sequence of weakly converges [22] to (denote ) if, for each intâ€‰, there exists a positive integer such that for all .

A subset of a topological vector space is order-convex if for each with , where . An ordered topological vector space is order-convex if it has a base of neighborhoods of consisting of order-convex subsets. In this case, the cone is said to be normal. In the case of a normed vector space, this condition means that the unit ball is order-convex, which is equivalent to the condition that there is some positive number such that and implies that , and the minimal is called a normal constant of . Another equivalent condition is that It is not hard to conclude from (1) that is a nonnormal cone of a normed vector space if and only if there exist sequences such that which implies that the Sandwich theorem does not hold. However, the Sandwich theorem holds in the sense of weak convergence even if is a nonnormal cone.

Lemma 1 (Sandwich theorem). *Let be a solid cone of a topological vector space and . If
**
and there exists some such that and , then .*

*Proof. *By and , for each intâ€‰, there exists some positive integer such that, for all ,
Thus, by (3) and (4), we have for all ; that is, . The proof is completed.

The following lemma is needed in further arguments, which directly follows from Lemma 1 and Remark 1 of [23].

Lemma 2. *Let be a solid cone of a normed vector space . Then, for each sequence , implies . Moreover, if is normal, then implies .*

Let be a cone of a normed vector space and . The mapping is said to be a positive linear bounded mapping if , for each , and there exists some positive real number such that . In the sequel, and will denote the family of all positive linear bounded mappings and the identity mapping, respectively.

Lemma 3. *Let be a solid cone of a normed vector space , and . If and , then .*

*Proof. *Let , for all , where
It is clear that for all , and hence, for all , the inverse of exists (denoted by ). It follows from that for all for all , and then for all . By Lemma 2 and , for each int, there exists some positive integer such that for all . Note that for all implies that for all and each ; then, for all ; that is, . The proof is completed.

Let be a nonempty set and let be a cone of a topological vector space . A partial cone metric on is a mapping such that, for each ,(p1);(p2);(p3);(p4).

The pair is called a partial cone metric space over . A partial cone metric on over a solid cone generates a topology on which has a base of the family of open -balls , where for each and each intâ€‰.

Let be a partial cone metric space over a solid cone of a topological vector space . A sequence of converges to (denoted by ) if . A sequence of is -Cauchy, if . The partial cone metric space is -complete, if each -Cauchy sequence of converges to a point such that . Every complete partial cone metric space is -complete, but the converse may not be true (see [23]).

#### 3. Common Fixed Point Theorems

Let be a partial cone metric space. The mappings are called contractions restricted with variable positive linear bounded mappings if there exist such that In particular, if (6) holds with and , then and are called contractions restricted with positive linear bounded mappings.

We first present a common fixed point theorem of contractions restricted with variable positive linear bounded mappings in a partial cone metric space over a nonnormal cone. In the sequel, will denote the set of all nonnegative integer numbers.

Theorem 4. *Let be a -complete partial cone metric space over a solid cone of a normed vector space , and let be contractions restricted with variable positive linear bounded mappings. If
** and , where denotes the spectral radius of linear bounded mappings,
**
where and denote the inverses of and , respectively. Then, and have a common fixed point in . Moreover, if
**
then and have a unique common fixed point such that, for each , , where is defined by
*

*Proof. *For each , by (7), the inverses of and exist. Then, it is clear that and are meaningful, and so are well defined. Moreover, by Neumannâ€™s formula,
which together with implies that , and hence . By (6), (11), (p4), and ,
and so
Act the above inequality with ; then, by ,
Similarly, by (6), (p3), (p4), and ,
and so
Act the above inequality with ; then, by ,
Moreover, by (15), (18), and ,
In the following, we will prove that
For all , we have four cases: (i) ; (ii) ; (iii) ; and (iv) , where and are two nonnegative integers such that . We only show that (20) holds for case (i); the proofs of the other three cases are similar.

It follows from (p4) and (19) that
By ,
which implies that , and hence by Lemma 2. Thus, by (21) and Lemma 1, ; that is, (20) holds. It is proved that is a -Cauchy sequence in , and so by the -completeness of , there exists such that and ; that is,
For all , by (6) and (p4),
and so
Act the above inequality with ; then, by ,
where and . It is clear that and by and . Then, it follows from Lemma 3 and (23) that
which together with Lemma 1 and (26) implies that . Therefore, by (p1) and (p3). Similarly, we can show that . Hence, is a common fixed point of and .

Now, we show the uniqueness of fixed point. Let and be two common fixed points of and . Then, by (6), (p3), and ,
and so
It follows from (9) that the inverse of exists (denoted by ), and by Neumannâ€™s formula. Act (29) with ; then, , and hence by (p1) and (p3). The proof is completed.

*Remark 5. *Theorem 3 of [23] is a special case of Theorem 4 with and , where are nonnegative numbers such that .

Note that Theorem 4 is still valid if are replaced with nonnegative bounded real functions; then, we have the following corollary for which is not necessarily confined to a normed vector space.

Corollary 6. *Let be a -complete partial cone metric space over a solid cone of a topological vector space and . Assume that there exist four nonnegative bounded functions such that
**
If
** and , where
**
Then, and have a unique common fixed point such that, for each , , where is defined by (11).*

Corollary 7. *Let be a -complete partial cone metric space over a solid cone of a normed vector space , and let be contractions restricted with positive linear bounded mappings. If
**
then and have a unique common fixed point such that, for each , , where is defined by (11).*

*Proof. *Let . It is easy to check that (6) holds with , , where , , and . By (33) and Neumannâ€™s formula,
for each ; that is, and . Note that both (7) and (10) hold with by (33); then, the conclusion directly follows from Theorem 4. The proof is completed.

Note that (33) hold naturally if . In this case, Corollary 7 holds true.

The following common fixed point theorem improves Theorem 2 of [23].

Theorem 8. *Let be a -complete partial cone metric space over a solid cone of a normed vector space and . Assume that there exists such that
**
If , then and have a unique common fixed point such that, for each , there exists some positive integer such that , where is defined by
*

*Proof. *By and Gelfandâ€™s formula, there exists such that , which implies that there exists a positive integer such that
By (35), (36), and (p2),
and so
which together with implies that
Thus, by (p4),
It follows from (37) that
which implies , and hence by Lemma 2. Therefore, by Lemma 1 and (41), we get is a -Cauchy sequence in . Then, by analogy with the proof of Theorem 4, by , and Lemma 3, we can prove that there exists some with such that , and is the unique common fixed point of and . For this , we have and ; that is, and are fixed points of and , respectively. It follows from (35) and that , and hence by (p1) and (p3). This shows that is a common fixed point of and . Note that is the unique common fixed point of and ; then, ; that is, is a common fixed point of and . Moreover, it is easy to show that is the unique common fixed point of and by and . The proof is completed.

*Example 9. *Let with the norm , and , which is nonnormal solid cone [24]. Define a mapping by
It follows from Example 2 of [22] that is a partial cone metric space. Let for each and , and for each . Clearly, is the unique common fixed point of and .

By the definitions of , , , and ,
that is, (35) is satisfied. It is clear that for each , and hence . Note that , and then
which implies that . Therefore, by Gelfandâ€™s formula, since , and hence and have a unique common fixed point by Theorem 8.

Finally, we present a fixed point theorem of contractions restricted with positive linear bounded mappings, which generalizes Theorem 3.1 of [22].

Theorem 10. *Let be a -complete partial cone metric space over a solid cone of a normed vector space and . Assume that there exist such that
**
If and
**
then and have a unique common fixed point such that, for each , , where for all .*

*Proof. *Let . Then, by (47), and so the inverse of exists (denoted by ). It follows from Neumannâ€™s formula that and
Let . Then, by and . Moreover, by (47) and (48),
By (46), (p4), and ,
Similarly, take and in (46), and we get
It follows from (50) and (51) that
Act the above inequality with ; then, by ,
and so, by ,
By (p4),
It follows from (49) that