We introduced a notion of topological vector space valued cone
metric space and obtained some common fixed point results. Our results generalize
some recent results in the literature.

1. Introduction

Huang and Zhang [1] generalized the notion of metric space by replacing the set of real numbers by ordered Banach space, deffined a cone metric space, and established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, several other authors [2–5] studied the existence of common fixed point of mappings satisfying a contractive type condition in normal cone metric spaces. Afterwards, Rezapour and Hamlbarani [6] studied fixed point theorems of contractive type mappings by omitting the assumption of normality in cone metric spaces (see also [7–14]). In this paper we obtain common fixed points for a pair of self-mappings satisfying a generalized contractive type condition without the assumption of normality in a class of topological vector space valued cone metric spaces which is bigger than that introduced by Huang and Zhang [1].

Let (E,τ) be always a topological vector space and P a subset of E. Then, P is called a cone whenever

P is closed, nonempty and P≠{0},

ax+by∈P for all x,y∈P and nonnegative real numbers a,b,

P∩(-P)={0}.

For a given cone P⊆E, we can define a partial ordering ≤ with respect to P by x≤y if and only if y-x∈P.x<y will stand for x≤y and x≠y, while x≪y will stand for y-x∈intP, where intP denotes the interior of P.

Definition 1.1.

Let X be a nonempty set. Suppose that the mapping d:X×X→E satisfies

0≤d(x,y) for all x,y∈X and d(x,y)=0 if and only if x=y,

d(x,y)=d(y,x) for all x,y∈X,

d(x,y)≤d(x,z)+d(z,y) for all x,y,z∈X.

Then d is called a cone metric on X and (X,d) is called a topological vector space valued cone metric space.

Note that Huang and Zhang [1] notion of cone metric space is a special case of our notion of topological vector space valued cone metric space.

Example 1.2.

Let X=[0,1],and letE be the set of all real valued functions on X which also have continuous derivatives onX,then Eis a vector space over ℝ under the following operations:
(f+g)(t)=f(t)+g(t),(αf)(t)=αf(t),
for all f,g∈E,α∈ℝ.Let τbe the strongest vector (locally convex) topology on E,then (X,τ) is a topological vector space which is not normable and is not even metrizable (see [15]). Define d:X×X→E as follows:
(d(x,y))(t)=|x-y|et,P={x∈E:x(t)⩾0∀t∈X}.
Then (X,d)is a topological vector space valued cone metric space.

Example 1.2 shows that this category of cone metric spaces is larger than that considered in [1–8] .

Definition 1.3.

Let (X,d) be a topological vector space valued cone metric space, and let x∈X and {xn}n≥1 be a sequence in X. Then

(i) {xn}n≥1 converges to x whenever for every c∈E with 0≪c there is a natural number N such that d(xn,x)≪c for all n≥N. We denote this by limn→∞xn=x or xn→x.

(ii) {xn}n≥1 is a Cauchy sequence whenever for every c∈E with 0≪c there is a natural number N such that d(xn,xm)≪c for all n,m≥N.

(iii) (X,d) is a complete topological vector space valued cone metric space if every Cauchy sequence is convergent.

2. Fixed Point

In this section, we shall give some results which generalize [6, Theorems 2.3, 2.6, 2.7, and 2.8] (and so [1, Theorems 1, 3, and 4]).

Theorem 2.1.

Let (X,d) be a complete topological vector space valued cone metric space and let the self-mappings S,T:X→X satisfy
d(Sx,Ty)≤kd(x,y)+l(d(x,Ty)+d(y,Sx)),
for all x,y∈X, where k,l∈[0,1) with k+2l<1. Then S and T have a unique common fixed point.

Proof.

For x0∈X and n≥0, define x2n+1=Sx2n and x2n+2=Tx2n+1. Then,
d(x2n+1,x2n+2)=d(Sx2n,Tx2n+1)⩽kd(x2n,x2n+1)+l[d(x2n,Tx2n+1)+d(x2n+1,Sx2n)]=kd(x2n,x2n+1)+l[d(x2n,Tx2n+1)]⩽kd(x2n,x2n+1)+l[d(x2n,x2n+1)+d(x2n+1,x2n+2)]=[k+l]d(x2n,x2n+1)+ld(x2n+1,x2n+2).
It implies that d(x2n+1,x2n+2)⩽[(k+l)/(1-l)]d(x2n,x2n+1). Similarly,
d(x2n+2,x2n+3)=d(Sx2n+2,Tx2n+1)⩽kd(x2n+2,x2n+1)+l[d(x2n+2,Tx2n+1)+d(x2n+1,Sx2n+2)]⩽kd(x2n+2,x2n+1)+l[d(x2n+2,x2n+3)+d(x2n+1,x2n+2)]=[k+l]d(x2n+1,x2n+2)+ld(x2n+2,x2n+3).
Hence, d(x2n+2,x2n+3)≤[(k+l)/(1-l)]d(x2n+1,x2n+2). Thus,
d(xn,xn+1)⩽λnd(x0,x1),
for all n≥0, where λ=((k+l)/(1-l))<1. Now, forn>m we have
d(xn,xm)⩽d(xn,xn-1)+d(xn-1,xn-2)+⋯+d(xm+1,xm)⩽(λn-1+λn-2+⋯+λm)d(x0,x1)⩽λm1-λd(x0,x1).
Let 0≪c. Take a symmetric neighborhood V of 0 such that c+V⊆intP. Also, choose a natural number N1 such that (λm/(1-λ))d(x1,x0)∈V, for all m≥N1. Then, (λm/(1-λ))d(x1,x0)≪c, for all m≥N1. Thus,
d(xn,xm)≤λm1-λd(x1,x0)≪c,
for all n>m. Therefore, {xn}n≥1 is a Cauchy sequence in (X,d). Since X is complete, there exists u∈X such that xn→u. Choose a natural number N2 such that d(xn,u)≪[c(1-l)/2(1+l)] for all n⩾N2. Thus,
d(u,Tu)⩽d(u,x2n+1)+d(x2n+1,Tu)=d(u,x2n+1)+d(Sx2n,Tu)⩽d(u,x2n+1)+kd(u,x2n)+l[d(u,Sx2n)+d(x2n,Tu)]⩽d(u,x2n+1)+kd(u,x2n)+l[d(u,x2n+1)+d(x2n,u)+d(u,Tu)]=(1+l)d(u,x2n+1)+(k+l)d(u,x2n)+ld(u,Tu).
So,
d(u,Tu)⩽[1+l1-l]d(u,x2n+1)+[k+l1-l]d(u,x2n)⩽[1+l1-l]d(u,x2n+1)+[1+l1-l]d(u,x2n)=c2+c2=c,
for all n≥N2. Therefore, d(u,Tu)≪c/i for all i⩾1. Hence, (c/i)-d(u,Tu)∈P for alli⩾1. Since P is closed, -d(u,Tu)∈P and so d(u,Tu)=0. Hence, u is a fixed point of T. Similarly, we can show that u=Su. Now, we show that S and T have a unique fixed point. For this, assume that there exists another point u* in X such that u*=Tu*=Su*. Then,
d(u,u*)=d(Su,Tu*)⩽kd(u,u*)+l[d(u,Tu*)+d(u*,Su)]⩽kd(u,u*)+l[d(u,u*)+d(u*,u)]⩽(k+2l)d(u,u*).
Since k+2l<1,d(u,u*)=0 and so u=u*.

The following corollary generalizes [6, Theorems 2.3, 2.7, and 2.8] (and so [1, Theorems 1 and 4]).

Corollary 2.2.

Let (X,d) be a complete topological vector space valued cone metric space and let the self-mapping T:X→X satisfy d(Tx,Ty)⩽ad(x,y)+bd(x,Ty)+cd(y,Tx) for all x,y∈X, where a,b,c∈[0,1) with a+b+c<1. Then T has a unique fixed point.

Proof.

The symmetric property of d and the above inequality imply that
d(Tx,Ty)⩽ad(x,y)+b+c2[d(x,Ty)+d(y,Tx)].
By substituting S=Ta=k and (b+c)/2=l in Theorem 2.1, we obtain the required result.

Theorem 2.3.

Let (X,d) be a complete topological vector space valued cone metric space and let the self-mappings S,T:X→X satisfy
d(Sx,Ty)≤kd(x,y)+l(d(x,Sx)+d(y,Ty)),
for all x,y∈X, where k,l∈[0,1) with k+2l<1. Then S and T have a unique common fixed point.

Proof.

For x0∈X and n≥0, define x2n+1=Sx2n and x2n+2=Tx2n+1. Then,
d(x2n+1,x2n+2)=d(Sx2n,Tx2n+1)⩽kd(x2n,x2n+1)+l[d(x2n,Sx2n)+d(x2n+1,Tx2n+1)]=kd(x2n,x2n+1)+l[d(x2n,x2n+1)+d(x2n+1,x2n+2)]=[k+l]d(x2n,x2n+1)+ld(x2n+1,x2n+2).
It implies that d(x2n+1,x2n+2)⩽[(k+l)/(1-l)]d(x2n,x2n+1). Similarly,
d(x2n+2,x2n+3)=d(Sx2n+2,Tx2n+1)⩽kd(x2n+2,x2n+1)+l[d(x2n+2,Sx2n+2)+d(x2n+1,Tx2n+1)]=kd(x2n+2,x2n+1)+l[d(x2n+2,x2n+3)+d(x2n+1,x2n+2)]=[k+l]d(x2n+1,x2n+2)+ld(x2n+2,x2n+3).
Hence, d(x2n+2,x2n+3)≤[(k+l)/(1-l)]d(x2n+1,x2n+2). Thus,
d(xn,xn+1)⩽λnd(x0,x1),
for all n≥0, where λ=((k+l)/(1-l))<1. Now, forn>m we have
d(xn,xm)⩽d(xn,xn-1)+d(xn-1,xn-2)+⋯+d(xm+1,xm)⩽(λn-1+λn-2+⋯+λm)d(x0,x1)⩽λm1-λd(x0,x1).
Let 0≪c. Take a symmetric neighborhood V of 0 such that c+V⊆intP. Also, choose a natural number N1 such that (λm/(1-λ))d(x1,x0)∈V, for all m≥N1. Then, (λm/(1-λ))d(x1,x0)≪c, for all m≥N1. Thus,
d(xn,xm)≤λm1-λd(x1,x0)≪c,
for all n>m. Therefore, {xn}n≥1 is a Cauchy sequence in (X,d). Since X is complete, there exists u∈X such that xn→u. Choose a natural number N2 such that d(xn,u)≪[c(1-l)/2(1+l)] for all n⩾N2. Thus,
d(u,Tu)⩽d(u,x2n+1)+d(x2n+1,Tu)=d(u,x2n+1)+d(Sx2n,Tu)⩽d(u,x2n+1)+kd(u,x2n)+l[d(u,Tu)+d(x2n,Sx2n)]⩽d(u,x2n+1)+kd(u,x2n)+l[d(u,x2n+1)+d(x2n,u)+d(u,Tu)]=(1+l)d(u,x2n+1)+(k+l)d(u,x2n)+ld(u,Tu).
So,
d(u,Tu)⩽[1+l1-l]d(u,x2n+1)+[k+l1-l]d(u,x2n)⩽[1+l1-l]d(u,x2n+1)+[1+l1-l]d(u,x2n)≪c2+c2=c,
for all n≥N2. Therefore, d(u,Tu)≪c/i for all i⩾1. Hence, (c/i)-d(u,Tu)∈P for all i⩾1. Since P is closed, -d(u,Tu)∈P and so d(u,Tu)=0. Hence, u is a fixed point of T. Similarly, we can show that u=Su. Now, we show that S and T have a unique fixed point. For this, assume that there exists another point u* in X such that u*=Tu*=Su*. Then,
d(u,u*)=d(Su,Tu*)⩽kd(u,u*)+l[d(u,u*)+d(u*,u)]=kd(u,u*).
Since k<1, d(u,u*)=0 and so u=u*.

The following corollary generalizes [6, Theorem 2.6] (and so [1, Theorem 3]).

Corollary 2.4.

Let (X,d) be a complete topological vector space valued cone metric space and let the self-mapping T:X→X satisfy d(Tx,Ty)⩽ad(x,y)+bd(x,Tx)+cd(y,Ty) for all x,y∈X, where a,b,c∈[0,1) with a+b+c<1. Then T has a unique fixed point.

Proof is similar to the proof of Corollary 2.2.

Example 2.5.

Let (X,d)be a topological vector space valued cone metric space of Example 1.2. Define S,T:X→Xas follows:
S(t)=T(t)={t3ifx≠1,16ifx=1.
Then,
|Sx-Ty|et≤k|x-y|et+l[|x-Sx|et+|y-Ty|et],
if k=1/6,l=5/18.Hence all conditions of Theorem 2.3 are satisfied.

Acknowledgments

The present version of the paper owes much to the precise and kind remarks of the learned referees.

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