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

Authors: Zhilong Li; Shujun Jiang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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


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## Body

## 1. Introduction

In 2007, Huang and Zhang [1] introduced cone metric spaces, being unaware that they already existed under the name K-metric and K-normed spaces that were introduced and used in the middle of the 20th century in [2–9]. In both cases, the set ℝ of real numbers was replaced by an ordered Banach space E. However, Huang and Zhang went further and defined the convergence via interior points of the cone by which the order in E 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 [10–18]). 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 θ-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 inθ-complete partial cone metric spaces over nonnormal cones, which improve the recent results of [22, 23].


## 2. Preliminaries

LetE be a topological vector space. A cone of E is a nonempty closed subset P of E such that ax+by∈P for each x,y∈P and each a,b≥0, and P∩(-P)={θ}, where θ is the zero element of E. A cone P of E determines a partial order ⪯ on E by x⪯y⇔y-x∈P for each x,y∈X. In this case, E is called an ordered topological vector space.A coneP of a topological vector space E is solid if int P≠Ø, where int P is the interior of P. For each x,y∈E with y-x∈ int P, we write x≪y. Let P be a solid cone of a topological vector space E. A sequence {un} of E weakly converges [22] to u∈E (denote un→wu) if, for each ϵ∈ int P, there exists a positive integer n0 such that u-ϵ≪un≪u+ϵ for all n≥n0.A subsetD of a topological vector space E is order-convex if [x,y]⊂D for each x,y∈D with x⪯y, where [x,y]={z∈E:x⪯z⪯y}. An ordered topological vector space E is order-convex if it has a base of neighborhoods of θ consisting of order-convex subsets. In this case, the cone P 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 N such that x,y∈E and θ⪯x⪯y implies that ∥x∥≤N∥y∥, and the minimal N is called a normal constant of P. Another equivalent condition is that
(1)inf⁡{∥x+y∥:x,y∈P,∥x∥=∥y∥=1}>0.
It is not hard to conclude from (1) that P is a nonnormal cone of a normed vector space (E,∥·∥) if and only if there exist sequences {un},{vn}⊂P such that
(2)un+vn→∥·∥θ⇏un→∥·∥θ,
which implies that the Sandwich theorem does not hold. However, the Sandwich theorem holds in the sense of weak convergence even if P is a nonnormal cone.Lemma 1 (Sandwich theorem).
LetP be a solid cone of a topological vector space E and {un},{vn},{wn}⊂E. If
(3)un⪯wn⪯vn,∀n,
and there exists some w∈E such that un→ww and vn→ww, then wn→ww.Proof.
Byun→ww and vn→ww, for each ϵ∈ int P, there exists some positive integer n0 such that, for all n≥n0,
(4)w-ϵ≪un,vn≪w+ϵ.
Thus, by (3) and (4), we have w-ϵ≪un⪯wn⪯vn⪯vn≪w+ϵ for all n≥n0; that is, wn→ww. 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.
LetP be a solid cone of a normed vector space (E,∥·∥). Then, for each sequence {un}⊂E, un→∥·∥u implies un→wu. Moreover, if P is normal, then un→wu implies un→∥·∥u.LetP be a cone of a normed vector space (E,∥·∥) and L:E→E. The mapping L is said to be a positive linear bounded mapping if L(P)⊂P, L(u+v)=Lu+Lv for each u,v∈E, and there exists some positive real number M>0 such that ∥L∥≤M. In the sequel, 𝔏 and I will denote the family of all positive linear bounded mappings and the identity mapping, respectively.Lemma 3.
LetP be a solid cone of a normed vector space (E,∥·∥), {Kn}⊂𝔏 and {un}⊂P. If un→wθ and sup⁡n⁡∥Kn∥<+∞, then Knun→wθ.Proof.
LetK~n=bKn, for all n, where
(5)b={1,sup⁡n∥Kn∥<1,1sup⁡n⁡∥Kn∥+1,sup⁡n∥Kn∥≥1.
It is clear that ∥K~n∥≤b∥Kn∥<1 for all n, and hence, for all n, the inverse of I-K~n exists (denoted by (I-K~n)-1). It follows from {Kn}⊂𝔏 that for all n(I-K~n)-1∈𝔏 for all n, and then I-K~n∈𝔏 for all n. By Lemma 2 and un→wθ, for each ϵ∈ intP, there exists some positive integer n0 such that θ⪯un≪bϵ for all n≥n0. Note that I-K~n∈𝔏 for all n implies that K~nu⪯u for all n and each u∈P; then, θ⪯Knun=K~n(un/b)⪯(un/b)≪ϵ for all n≥n0; that is, Knun→wθ. The proof is completed.LetX be a nonempty set and let P be a cone of a topological vector space E. A partial cone metric on X is a mapping p:X×X→P such that, for each x,y,z∈X,(p1)
p(x,y)=p(x,x)=p(y,y)⇔x=y;(p2)
p(x,y)=p(y,x);(p3)
p(x,x)⪯p(x,y);(p4)
p(x,y)⪯p(x,z)+p(z,y)-p(z,z).The pair(X,p) is called a partial cone metric space over P. A partial cone metric p on X over a solid cone P generates a topology τp on X which has a base of the family of open p-balls {Bp(x,ϵ):x∈X,θ≪ϵ}, where Bp(x,ϵ)={y∈X:p(x,y)≪p(x,x)+ϵ} for each x∈X and each ϵ∈ int P.Let(X,p) be a partial cone metric space over a solid cone P of a topological vector space E. A sequence {xn} of X converges to x∈X (denoted by xn→τpx) if p(xn,x)→wp(x,x). A sequence {xn} of X is θ-Cauchy, if p(xn,xm)→wθ. The partial cone metric space (X,p) is θ-complete, if each θ-Cauchy sequence {xn} of X converges to a point x∈X such that p(x,x)=θ. Every complete partial cone metric space (X,p) is θ-complete, but the converse may not be true (see [23]).


## 3. Common Fixed Point Theorems

Let(X,p) be a partial cone metric space. The mappings T,S:X→X are called contractions restricted with variable positive linear bounded mappings if there exist Li:X×X→𝔏(i=1,2,3,4) such that
(6)p(Tx,Sy)⪯L1(x,y)p(x,y)+L2(x,y)p(x,Tx)+L3(x,y)p(y,Sy)+L4(x,y)[p(x,Sy)+p(y,Tx)],kkkkkkkkkkkkkkkkkkkkkl∀x,y∈X.
In particular, if (6) holds with Li(x,y)≡Ai(i=1,2,3,4) and Ai∈𝔏(i=1,2,3,4), then T and S 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,ℕ will denote the set of all nonnegative integer numbers.Theorem 4.
Let(X,p) be a θ-complete partial cone metric space over a solid cone P of a normed vector space (E,∥·∥), and let T,S:X→X be contractions restricted with variable positive linear bounded mappings. If
(7)ρ(L3(x,y)+L4(x,y))<1,ρ(L2(x,y)+L4(x,y))<1,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkk∀x,y∈X,l1l2<1 and l3<+∞, where ρ(·) denotes the spectral radius of linear bounded mappings,
(8)l1=sup⁡x,y∈X∥K1(x,y)∥,l2=sup⁡x,y∈X∥K2(x,y)∥,l3=sup⁡x,y∈X∥K3(x,y)∥,(9)K1(x,y)=L~1(x,y)[L1(x,y)+L2(x,y)+L4(x,y)],K2(x,y)=L~2(x,y)[L1(x,y)+L3(x,y)+L4(x,y)],K3(x,y)=L~2(x,y)[I+L3(x,y)+L4(x,y)],kkkkkkkkkkkkkkklkkki∀x,y∈X,
where L~1(x,y) and L~2(x,y) denote the inverses of I-L3(x,y)-L4(x,y) and I-L2(x,y)-L4(x,y), respectively. Then, T and S have a common fixed point in X. Moreover, if
(10)ρ(L1(x,y)+L2(x,y)+L3(x,y)+2L4(x,y))<1,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikk∀x,y∈X,
then T and S have a unique common fixed point x*∈X such that, for each x0∈X, xn→τpx*, where xn is defined by
(11)xn+1={Txn,nisanevennumber,Sxn,nisanoddnumber.Proof.
For eachx,y∈X, by (7), the inverses of I-L3(x,y)-L4(x,y) and I-L2(x,y)-L4(x,y) exist. Then, it is clear that L~1 and L~2 are meaningful, and so K1,K2,K3 are well defined. Moreover, by Neumann’s formula,
(12)L~1(x,y)=∑i=0∞[L3(x,y)+L4(x,y)]i,L~2(x,y)=∑i=0∞[L2(x,y)+L4(x,y)]i,kkkkkkkkkkkkkkhhkkkk∀x,y∈X,
which together with Li:X×X→𝔏(i=2,3,4) implies that L~i:X×X→𝔏(i=1,2), and hence Ki:X×X→𝔏(i=1,2,3). By (6), (11), (p4), and L4:X×X→𝔏,
(13)p(x2k+1,x2k+2)=p(Tx2k,Sx2k+1)⪯L1(x2k,x2k+1)p(x2k,x2k+1)+L2(x2k,x2k+1)p(x2k,x2k+1)+L3(x2k,x2k+1)p(x2k+1,x2k+2)+L4(x2k,x2k+1)[p(x2k,x2k+2)kkkkkkkkkkkkkikk+p(x2k+1,x2k+1)]⪯L1(x2k,x2k+1)p(x2k,x2k+1)+L2(x2k,x2k+1)p(x2k,x2k+1)+L3(x2k,x2k+1)p(x2k+1,x2k+2)+L4(x2k,x2k+1)[p(x2k,x2k+1)kkkkkkkkkkkkkkik+p(x2k+1,x2k+2)],kkkkkkkkkkkkkkkkikkkkkkkk∀k∈ℕ,
and so
(14)[I-L3(x2k,x2k+1)-L4(x2k,x2k+1)]p(x2k+1,x2k+2)⪯[L1(x2k,x2k+1)+L2(x2k,x2k+1)kk+L4(x2k,x2k+1)]p(x2k,x2k+1),∀k∈ℕ.
Act the above inequality with L~1(x2k,x2k+1); then, by L~1:X×X→𝔏,
(15)p(x2k+1,x2k+2)⪯K1(x2k,x2k+1)p(x2k,x2k+1),∀k∈ℕ.
Similarly, by (6), (p3), (p4), and L4:X×X→𝔏,
(16)p(x2k+2,x2k+3)=p(x2k+3,x2k+2)=p(Tx2k+2,Sx2k+1)⪯L1(x2k+2,x2k+1)p(x2k+2,x2k+1)+L2(x2k+2,x2k+1)p(x2k+2,x2k+3)+L3(x2k+2,x2k+1)p(x2k+1,x2k+2)+L4(x2k+2,x2k+1)[p(x2k+2,x2k+2)kkkkkkkkkkkkkkk+p(x2k+1,x2k+3)]⪯L1(x2k+2,x2k+1)p(x2k+2,x2k+1)+L2(x2k+2,x2k+1)p(x2k+2,x2k+3)+L3(x2k+2,x2k+1)p(x2k+1,x2k+2)+L4(x2k+2,x2k+1)[p(x2k+2,x2k+1)kkkkkkkkkkkkkkk+p(x2k+2,x2k+3)],kkkkkkkkkkkkkkkkkkkkkkkk∀k∈ℕ,
and so
(17)[I-L2(x2k+2,x2k+1)-L4(x2k+2,x2k+1)]p(x2k+2,x2k+3)⪯[L1(x2k+2,x2k+1)+L3(x2k+2,x2k+1)kk+L4(x2k+2,x2k+1)]p(x2k+2,x2k+1),∀k∈ℕ.
Act the above inequality with L~2(2k+2,x2k+1); then, by L~2:X×X→𝔏,
(18)p(x2k+2,x2k+3)⪯K2(x2k+2,x2k+1)p(x2k+1,x2k+2),kkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkk∀k∈ℕ.
Moreover, by (15), (18), and K1,K2:X×X→𝔏,
(19)p(x2k+1,x2k+2)⪯K1(x2k,x2k+1)kkkkkkkkkkkk×K2(x2k,x2k-1)⋯K1(x0,x1)p(x0,x1),p(x2k+2,x2k+3)⪯K2(x2k+2,x2k+1)K1(x2k,x2k+1)kkkkkkkkkk×K2(x2k,x2k-1)⋯K1(x0,x1)p(x0,x1),kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkklkkkkkkk∀k∈ℕ.
In the following, we will prove that
(20)p(xn,xm)→wθ.
For all m>n, we have four cases: (i) m=2p+1,n=2q+1; (ii) m=2p+1,n=2q; (iii) m=2p,n=2q+1; and (iv) m=2p,n=2q, where p and q are two nonnegative integers such that p>q. 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
(21)θ⪯p(xn,xm)=p(x2q+1,x2p+1)⪯p(x2q+1,x2q+2)+p(x2q+2,x2q+3)+⋯+p(x2p-1,x2p)+p(x2p,x2p+1)⪯pK1K2(x0,x1)=K1(x2q,x2q+1)×K2(x2q,x2q-1)⋯K1(x0,x1)p(x0,x1)+K2(x2q+2,x2q+1)K1(x2q,x2q+1)×K2(x2q,x2q-1)⋯K1(x0,x1)p(x0,x1)+⋯+K1(x2p-2,x2p-1)×K2(x2p-2,x2p-3)⋯K1(x0,x1)p(x0,x1)+K2(x2p,x2p-1)×K1(x2p-2,x2p-1)⋯K1(x0,x1)p(x0,x1),kkkkkkkkkkkkkkkkkkkkkkkkkkkikk∀p>q.
By l1l2<1,
(22)∥pK1K2(x0,x1)∥≤(l1q+1l2q+l1q+1l2q+1+⋯+l1p+1l2p+l1pl2p)∥p(x0,x1)∥=(l1∑i=qp(l1l2)i+∑i=q+1p(l1l2)i)∥p(x0,x1)∥≤(l1+l1l2)(l1l2)q∥p(x0,x1)∥1-l1l2,∀p>q,
which implies that pK1K2(x0,x1)→∥·∥θ, and hence pK1K2(x0,x1)→wθ by Lemma 2. Thus, by (21) and Lemma 1, p(xn,xm)→wθ; that is, (20) holds. It is proved that {xn} is a θ-Cauchy sequence in (X,p), and so by the θ-completeness of (X,p), there exists x*∈X such that xn→τpx* and p(x*,x*)=θ; that is,
(23)p(xn,x*)→wθ.
For all k∈ℕ, by (6) and (p4),
(24)p(Tx*,x*)⪯p(Tx*,x2k)+p(x2k,x*)=p(Tx*,Sx2k-1)+p(x2k,x*)⪯L1(x*,x2k-1)p(x*,x2k-1)+L2(x*,x2k-1)p(x*,Tx*)+L3(x*,x2k-1)p(x2k-1,x2k)+L4(x*,x2k-1)×[p(x*,x2k)+p(x2k-1,Tx*)]+p(x2k,x*)⪯L1(x*,x2k-1)p(x*,x2k-1)+L2(x*,x2k-1)p(x*,Tx*)+L3(x*,x2k-1)×[p(x2k-1,x*)+p(x*,x2k)]+L4(x*,x2k-1)×[p(x*,x2k)+p(x2k-1,x*)+p(x*,Tx*)]+p(x2k,x*),
and so
(25)[I-L2(x*,x2k-1)-L4(x*,x2k-1)]p(Tx*,x*)⪯[L1(x*,x2k-1)+L3(x*,x2k-1)i+L4(x*,x2k-1)]p(x*,x2k-1)+[I+L3(x*,x2k-1)+L4(x*,x2k-1)]p(x2k,x*).
Act the above inequality with L~2(x*,x2k-1); then, by L~2:X×X→𝔏,
(26)θ⪯p(Tx*,x*)⪯K2,2k-1p(x*,x2k-1)+K3,2k-1p(x2k,x*),∀k∈ℕ,
where K2,2k-1=K2(x*,x2k-1) and K3,2k-1=K3(x*,x2k-1). It is clear that {K2,2k-1},{K3,2k-1}⊂𝔏 and sup⁡k⁡∥K3,2k-1∥<+∞,sup⁡k⁡∥K3,2k-1∥<+∞ by l1l2<1 and l3<+∞. Then, it follows from Lemma 3 and (23) that
(27)K2,2k-1p(x*,x2k-1)+K3,2k-1p(x2k,x*)→wθ,
which together with Lemma 1 and (26) implies that p(Tx*,x*)=θ. Therefore, Tx*=x* by (p1) and (p3). Similarly, we can show that Sx*=x*. Hence, x* is a common fixed point of T and S.
Now, we show the uniqueness of fixed point. Letx and x* be two common fixed points of T and S. Then, by (6), (p3), and Li:X×X→𝔏(i=2,3),
(28)p(x*,x)=p(Tx*,Sx)⪯L1(x*,x)p(x*,x)+L2(x*,x)p(x*,Tx*)+L3(x*,x)p(x,Sx)+L4(x*,x)[p(x*,Sx)+p(x,Tx*)]=[L1(x*,x)+2L4(x*,x)]p(x*,x)+L2(x*,x)p(x*,x*)+L3(x*,x)p(x,x)⪯[L1(x*,x)+L2(x*,x)i+L3(x*,x)+2L4(x*,x)]p(x*,x),
and so
(29)[I-L1(x*,x)-L2(x*,x)-L3(x*,x)l-2L4(x*,x)]p(x*,x)⪯θ.
It follows from (9) that the inverse of I-L1(x*,x)-L2(x*,x)-L3(x*,x)-2L4(x*,x) exists (denoted by [I-L1(x*,x)-L2(x*,x)-L3(x*,x)-2L4(x*,x)]-1), and [I-L1(x*,x)-L2(x*,x)-L3(x*,x)-2L4(x*,x)]-1∈𝔏 by Neumann’s formula. Act (29) with [I-L1(x*,x)-L2(x*,x)-L3(x*,x)-2L4(x*,x)]-1; then, p(x*,x)⪯θ, and hence x=x* by (p1) and (p3). The proof is completed.Remark 5.
Theorem 3 of [23] is a special case of Theorem 4 with T=S and Li(x,y)≡ciI(i=1,2,3,4), where ci(i=1,2,3,4) are nonnegative numbers such that c1+c2+c3+2c4<1.
Note that Theorem4 is still valid if Li(i=1,2,3,4) are replaced with nonnegative bounded real functions; then, we have the following corollary for which E is not necessarily confined to a normed vector space.Corollary 6.
Let(X,p) be a θ-complete partial cone metric space over a solid cone P of a topological vector space E and T,S:X→X. Assume that there exist four nonnegative bounded functions αi:X×X→[0,+∞)(i=1,2,3,4) such that
(30)p(Tx,Sy)⪯α1(x,y)p(x,y)+α2(x,y)p(x,Tx)+α3(x,y)p(y,Sy)+α4(x,y)[p(x,Sy)+p(y,Tx)],kkkkkkkkkkkkkkkkkkkki∀x,y∈X.
If
(31)α1(x,y)+α2(x,y)+α3(x,y)+2α4(x,y)<1,kkkkkkkkkkkkkkkkkkkkkkkkikkkkk∀x,y∈X,m1m2<1 and m3<+∞, where
(32)m1=sup⁡x,y∈Xα1(x,y)+α2(x,y)+α4(x,y)1-α3(x,y)-α4(x,y),m2=sup⁡x,y∈Xα1(x,y)+α3(x,y)+α4(x,y)1-α2(x,y)-α4(x,y),m3=sup⁡x,y∈X1+α3(x,y)+α4(x,y)1-α2(x,y)-α4(x,y).
Then, T and S have a unique common fixed point x*∈X such that, for each x0∈X, xn→τpx*, where xn is defined by (11).Corollary 7.
Let(X,p) be a θ-complete partial cone metric space over a solid cone P of a normed vector space (E,∥·∥), and let T,S:X→X be contractions restricted with positive linear bounded mappings. If
(33)∥A1+A2+A4∥+∥A3+A4∥<1,∥A1+A3+A4∥+∥A2+A4∥<1,
then T and S have a unique common fixed point x*∈X such that, for each x0∈X, xn→τpx*, where xn is defined by (11).Proof.
LetLi(x,y)≡Ai(i=1,2,3,4). It is easy to check that (6) holds with Li(x,y)≡Ai(i=1,2,3,4), Li,Ki:X×X→𝔏(i=1,2,3,4), where K1(x,y)≡(I-A3-A4)-1(A1+A2+A4), K2(x,y)≡(I-A2-A4)-1(A1+A3+A4), and K3(x,y)≡(I-A2-A4)-1(I+A3+A4). By (33) and Neumann’s formula,
(34)∥K1(x,y)∥≤∥(I-A3-A4)-1∥∥A1+A2+A4∥≤∥A1+A2+A4∥1-∥A3+A4∥<1,∥K2(x,y)∥≤∥(I-A2-A4)-1∥∥A1+A3+A4∥≤∥A1+A3+A4∥1-∥A2+A4∥<1,∥K3(x,y)∥≤∥(I-A2-A4)-1∥∥I+A3+A4∥≤21-∥A2+A4∥<+∞,
for each x,y∈X; that is, l1l2<1 and l3<+∞. Note that both (7) and (10) hold with Li(x,y)≡Ai(i=1,2,3,4) by (33); then, the conclusion directly follows from Theorem 4. The proof is completed.Note that (33) hold naturally if ∥A1∥+∥A2∥+∥A3∥+2∥A4∥<1. In this case, Corollary 7 holds true.The following common fixed point theorem improves Theorem 2 of [23].Theorem 8.
Let(X,p) be a θ-complete partial cone metric space over a solid cone P of a normed vector space (E,∥·∥) and T,S:X→X. Assume that there exists A∈𝔏 such that
(35)p(Tx,Sy)⪯Ap(x,y),∀x,y∈X.
If ρ(A)<1, then T and S have a unique common fixed point x*∈X such that, for each x0∈X, there exists some positive integer n0 such that xn→τpx*, where xn is defined by
(36)xn+1={Tn0xn,nisanevennumber,Sn0xn,nisanoddnumber.Proof.
Byρ(A)<1 and Gelfand’s formula, there exists 0<β<1 such that lim⁡n→∞⁡∥An∥n=ρ(A)≤β, which implies that there exists a positive integer n0 such that
(37)∥An∥≤βn,∀n≥n0.
By (35), (36), and (p2),
(38)p(x2k+1,x2k+2)=p(Tn0x2k,Sn0x2k+1)⪯An0p(x2k,x2k+1),p(x2k+2,x2k+3)=p(Tn0x2k+2,Sn0x2k+1)⪯An0p(x2k+1,x2k+2),∀k∈ℕ
and so
(39)p(xn,xn+1)⪯An0p(xn-1,xn),∀n∈ℕ,
which together with A∈𝔏 implies that
(40)p(xn,xn+1)⪯Ann0p(x0,x1),∀n∈ℕ.
Thus, by (p4),
(41)p(xn,xm)⪯∑i=nm-1p(xi,yi+1)⪯∑i=nm-1Ain0p(x0,x1),∀m>n.
It follows from (37) that
(42)∥∑i=nm-1Ain0p(x0,x1)∥≤∥p(x0,x1)∥∑i=nm-1∥An0∥i≤∥p(x0,x1)∥∑i=nm-1βin0≤βnn0∥p(x0,x1)∥1-βn0,∀m>n,
which implies ∑i=nm-1Ain0p(x0,x1)→∥·∥θ, and hence ∑i=nm-1Ain0p(x0,x1)→wθ by Lemma 2. Therefore, by Lemma 1 and (41), we get p(xn,xm)→wθ;thatis,{xn} is a θ-Cauchy sequence in (X,p). Then, by analogy with the proof of Theorem 4, by A∈𝔏, ρ(A)<1 and Lemma 3, we can prove that there exists some x*∈X with p(x*,x*)=θ such that p(xn,x*)→wθ, and x* is the unique common fixed point of Tn0 and Sn0. For this x*, we have Tn0(Tx*)=T(Tn0x*)=Tx* and Sn0(Sx*)=S(Sn0x*)=Sx*; that is, Tx* and Sx* are fixed points of Tn0 and Sn0, respectively. It follows from (35) and p(x*,x*)=θ that p(Tx*,Sx*)⪯Lp(x*,x*)=θ, and hence Tx*=Sx* by (p1) and (p3). This shows that Tx* is a common fixed point of Tn0 and Sn0. Note that x* is the unique common fixed point of Tn0 and Sn0; then, Tx*=Sx*=x*; that is, x* is a common fixed point of T and S. Moreover, it is easy to show that x* is the unique common fixed point of T and S by A∈𝔏 and ρ(A)<1. The proof is completed.Example 9.
LetE=Cℝ1[0,1] with the norm ∥u∥=∥u∥∞+∥u′∥∞, and X=P={u∈E:u(t)≥0,t∈[0,1]}, which is nonnormal solid cone [24]. Define a mapping p:X×X→P by
(43)p(x,y)={x,x=y,x+y,otherwise.
It follows from Example 2 of [22] that (X,p) is a partial cone metric space. Let (Ax)(t)=∫0tx(s)ds for each x∈X and t∈[0,1], Tx=Ax/2 and Sx=Ax/3 for each x∈X. Clearly, θ is the unique common fixed point of T and S.
By the definitions ofp, T, S, and A,
(44)p(Tx,Sy)={θ=Ap(x,y),x=y=θ,5Ax6⪯Ax=Ap(x,y),x=y≠θ,Ax2⪯5Ax2=Ap(x,y),x≠y,y=3x2,Ax2+Ay3⪯Ax+Ay=Ap(x,y),x≠y,y≠3x2;
that is, (35) is satisfied. It is clear that (Anx)(t)≤(tn/n!)∥x∥∞ for each t∈[0,1], and hence ∥Anx∥∞≤(1/n!)∥x∥∞. Note that (Anx)′(t)=(An-1x)(t), and then
(45)∥Anx∥=∥Anx∥∞+∥(Anx)′∥∞≤(1n!+1(n-1)!)∥x∥∞≤(1n!+1(n-1)!)∥x∥,
which implies that ∥An∥≤(1n/n!)+(1n/(n-1)!). Therefore, by Gelfand’s formula, ρ(A)=lim⁡n→∞⁡∥An∥n=0 since lim⁡n→∞⁡(1/n!n)=0, and hence T and S 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(X,p) be a θ-complete partial cone metric space over a solid cone P of a normed vector space (E,∥·∥) and T:X→X. Assume that there exist Ai∈𝔏(i=1,2,3,4,5) such that
(46)p(Tx,Ty)⪯A1p(x,y)+A2p(x,Tx)+A3p(y,Ty)+A4p(x,Ty)+A5p(y,Tx),∀x,y∈X.
If ∥A2+A4∥<1 and
(47)∥A2+A3+A4+A5∥+∥2A1+A2+A3+A4+A5∥<2,
then T and S have a unique common fixed point x*∈X such that, for each x0∈X, xn→τpx*, where xn+1=Txn for all n∈ℕ.Proof.
LetB=(A2+A3+A4+A5)/2. Then, ∥B∥<1 by (47), and so the inverse of B exists (denoted by (I-B)-1). It follows from Neumann’s formula that (I-B)-1∈𝔏 and
(48)∥(I-B)-1∥≤11-∥B∥.
Let K=(I-B)-1(A1+B). Then, K∈𝔏 by (I-B)-1∈𝔏 and Ai∈𝔏(i=1,2,3,4,5). Moreover, by (47) and (48),
(49)∥K∥≤∥(I-B)-1∥∥A1+B∥≤∥A1+B∥1-∥B∥<1.
By (46), (p4), and A4∈𝔏,
(50)p(xn,xn+1)=p(Txn-1,Txn)⪯A1p(xn-1,xn)+A2p(xn-1,xn)+A3p(xn,xn+1)+A4p(xn-1,xn+1)+A5p(xn,xn)⪯A1p(xn-1,xn)+A2p(xn-1,xn)+A3p(xn,xn+1)+A4[p(xn-1,xn)+p(xn,xn+1)-p(xn,xn)]+A5p(xn,xn),∀n∈ℕ.
Similarly, take x=xn and y=xn-1 in (46), and we get
(51)p(xn+1,xn)=p(Txn,Txn-1)⪯A1p(xn-1,xn)+A2p(xn,xn+1)+A3p(xn-1,xn)+A4p(xn,xn)+A5p(xn-1,xn+1)⪯A1p(xn-1,xn)+A2p(xn,xn+1)+A3p(xn-1,xn)+A4p(xn,xn)+A5[p(xn-1,xn)+p(xn,xn+1)kkkkk-p(xn,xn)],∀n∈ℕ.
It follows from (50) and (51) that
(52)(I-B)p(xn,xn+1)⪯(A1+B)p(xn-1,xn),∀n∈ℕ.
Act the above inequality with (I-B)-1; then, by (I-B)-1∈𝔏,
(53)p(xn,xn+1)⪯Kp(xn-1,xn),∀n∈ℕ,
and so, by K∈𝔏,
(54)p(xn,xn+1)⪯Knp(x0,x1),∀n∈ℕ.
By (p4),
(55)p(xn,xm)⪯∑i=nm-1p(xi,xi+1)⪯∑i=nm-1Kip(x0,x1),∀m>n.
It follows from (49) that
(56)∥∑i=nm-1Kip(x0,x1)∥≤∥K∥n∥p(x0,x1)∥1-∥K∥,
which implies ∑i=nm-1Kip(x0,x1)→∥·∥θ, and hence ∑i=nm-1Kip(x0,x1)→wθ by Lemma 2. Therefore, by Lemma 1 and (55), p(xn,xm)→wθ; that is, {xn} is a θ-Cauchy sequence in (X,p). By analogy with the proof of Theorem 4, by Ai∈𝔏(i=1,2,3,4,5), ∥A2+A4∥<1, and Lemma 3, we can prove that there exists some x*∈X with p(x*,x*)=θ such that p(xn,x*)→wθ and x* is a fixed point of T. Note that (47) implies ∥A1+A2+A3+A4+A5∥<1; then, similar to the proof of Theorem 4, we can show x* is the unique fixed point of T. The proof is completed.Remark 11.
It is easy to check that all the conditions of Theorem10 are satisfied if ∑i=15∥Ai∥‍<1. Therefore, Theorem 10 is valid with ∑i=1n∥Ai∥<1, and hence Theorem 3.1 of [22] is a special case of Theorem 10 with Ai=ciI(i=1,2,3,4,5), where ci(i=1,2,3,4,5) are four nonnegative real numbers such that ∑i=15ci<1.



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*Source: 653841-2014-04-10.xml*